VOL I ANAIS DO SIMPÓSIO LATINO-AMERICANO DE FÍSICA ...

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VOL I

ANAISDO SIMPÓSIO

LATINO-AMERICANODE FÍSICA DOS

SISTEMAS AMORFOS

EDITOR: ENRIQUE V. ANDA

CENTRO LATINO-AMERICANO DE FÍSICA — CLAFUNIVERSIDADE FEDERAL FLUMINENSE — UFF

ANAIS DO SIMPÓSIOLATWO-AMERICANO DE BSKA

DOS SISTEMAS AMORFOS

NITERÓI, 27 OE FEVEREIRO A 2 DE MARÇO DE 1984

EDITADO POR

ENRIQUE V. ANDAUNIVERSIDADE FEDERAL FLUMINENSENITERÓI - RJ

V O L U M E I

PUBLICAÇÃO DO CENTRO LATINO-AMERICANO DE FÍSICARIO DE JANEIRO - BRASIL

MAIO - 1985

SIMPÓSIO LATINO-AMERICANO DE FlSICA DOS SISTEMAS AMORFOS

NITERÓI - BRASIL, 27 DE FEVEREIRO A 2 DE MARÇO DE 1984

Organizadores

- Centro Latino-Americano de Física (CLAF)- Universidade Federal Fluminense (UFF)

PatrocinadoresCentro Latino-Americano de Física (CLAF)Universidade Federal Fluminense (UFF)Conselho Nacional de Desenvolvimento CientíficoTecnológico (CNPq)Fundação de Amparo â Pesquisa do Estado de SãoPaulo (FAPESP)Comissão de Aperfeiçoamento de Pessoal de NívelSuperior (CAPES)Sociedade Brasileira de Física (SBF)

Comissão Organizadora Nacional- Antonio Ferreira da Silva (INPE)- Francisco César de Sá Barreto (UFMG)- Frank Missel (USP)- Ivon Fittipaldi (UFPE)- Mario Baibich (UFRS)- Reiko Sato-Turtelli (UNICAMP)- Enrique V. Anda (UFF)- Mucio Amado Continentino (UFF)

Comissão Organizadora Local- Elisa Saitovich (CBPF)- Maria Augusta Davidovich (PUC)- Roberto Nicolsky (UFRJ)- Álvaro Ferraz Filho (UFF)- Fernando A. Oliveira (UFF)- Luiz Carlos S. do Nascimento (UFF)- Norberto Majlis (UFF)- Paulo Murilo C. de Oliveira (UFF)- Sergio S. Makler (UFF)- Silvia Selzer (UFF)

SUMÁRIO

V O L U M E I

INTRODUÇÃO i

CONFERÊNCIAS:

THEORY OF GIASS 1N. Rivier

PHASE TRANSITIONS IN GLASSY MATERIALS 92M. F. Thoipe

METALLIC GLASSES: ST; JCTURAL MODELS 110E. Nassif

AMORPHOUS MfcTAL? - FABRICATION AND CHARACTERIZATION . 128F. P. Missell

STRUCTURAL RELAXATION LOW TEMPERATURE PROPERTIES ... 144F. de la Ciuz

AMORPHOUS SUIERCONDUCTORS 161F. P. Missell

ON THE SUPERCONDUCTIVITY OF VANADIUM BASED ALLOYS ... 176F. Brouers ?nd J. Van der Rest

DENSIDADE DE ESTADOS ELETRÔNICOS EM VIDROS METÁLICOS. 186S. F. Pessoa

REAL SPACE RENORMALIZAfMN TECHNIQUES FOR DISORDEREDSYSTEMS 201

E. V. Anda

RESEARCH ON HYJ;ROGENATED AMORPHOUS SILICON AND RELATEDALLOYS FOR PHOT»')VOLTAIC APPLICATIONS 221

I. E. Chamboui'iyron

A GENERALIZED SLATER-PAUL]N3 CONSTRUCTION FOR MAGNETICAMORPHOUS ALLOTS 232A.P. Malozemoff, A.R. Wil/iams and V.L.Moruzzi

AMORPHOUS MAGrfETJfM '. 245H. R. RechenbfTg

REENTRANT FERROMAÒNETISM 258M. A. Continentino

INTRODUÇÃO

A realização do Simpósio Latino-Americano de Fí-sica dos Sistesas Aaorfos surgiu como conseqüência do interesse Manifestado por pesquisadores latino-americanos nessa área (eu crescente desenvolvimento na América Latina),eque já se constitui nua campo onde operam alguns gruposconsolidados (que trabalham ativamente) e outros que estãoem formação.

Os materiais anorfos apresentam uma rica fenomenologia ainda sem explicação satisfatória e possuem grandes implicações tecnológicas, o que representa um estímulopara cientistas e tecnõlogos. Frente a essa realidade, ecom o auspício do Centro Latino*Americano de Física, o Grupo de Física do Estado Sólido da Universidade Federal Fluainense (UFF) pensou na possibilidade de reunir os pesqiúsadores da América Latina engajados na área para facilitaro intercâmbio e a colaboração mais articulada e estreita entre os aesaos, além de permitir a realização de una avaliação das perspectivas do tema, tanto nos seus aspectos basi^cos como aplicados, dando um ênfase especial na integraçãoteõrico-experimental.

A oportunidade da realização do Simpósio nos pareceu particularmente apropriada como uma forma de promover o trabalho científico num momento político e econômicodifícil pelo qual atravessa a região, com manifestações ne_gativas nas atividades de pesquisa de nossos países, ativi^dades essas fundamentais para a resolução de muitos dosproblemas que a realidade latino-americana apresenta.

ii

O Simpósio teve lugar nas instalações do Institu

to de Química da Universidade Federal Fluminense, na cida

de de Niterói, durante o período de 27 de fevereiro a 2 de

•arco de 1984, e contou COB a participação de 144 pesquisa

dores. Entre as atividades do Simpósio, destacamos sessões

de painéis com apresentação de 56 trabalhos e a realização

de 9 palestras convidadas e 4 cursos sobre diferentes a£

pectos da área dos Sistcaas Anorfos Desordenados. Foram or

ganizados também dois grupos de trabalho sobre Semiconduto

res Amorfos e Metais Amorfos, o que permitiu fazer una ava

liação sobre os recursos humanos e de infraestrutura dispo

níveis na América Latina nessas duas áreas.

Nesta publicação incluem-se os cursos, as pales

trás e os trabalhos apresentados durante o Simpósio, cujos

textos foram enviados ao Comitê Organizador.

0 Simpósio foi realizado graças ao apoio finance^

ro do programa conjunto UNESCO/CLAF, do Conselho Nacional de

Desenvolvimento Científico e Tecnológico (CNPq), do UNIBAN

CO, da Universidade Federal Fluminense (hospedeira do even

to), da FAPESP, e de instituições latino-americanas que.co

brindo despesas de passagens, permitiram aumentar a part_i

cipação de colegas de outros países.

Gostaríamos de agradecer especialmente o apoio

fin .,.ceiro e institucional prestado pela Sociedade Brasilei^

ra de Física, bem como aos professores e funcionários do

Centro de Estudos Gerais e dos Institutos de Física e Qu£

nica da UFF, cujo trabalho e solidariedade tornaram possjf

vel a realização do Simpósio, colocando ã nossa disposição

a infraestrutura dessas unidades universitárias.

A secretaria administrativa esteve a cargo do

Vera Rosenthal e Humberto Teixeira, que muito contribui ram

para o êxito do evento.

THEORY OF GLASS

N. RttierCenter for Nonlinear SUdies

Los Alamos National LaboratoryLos Alamos, NM 87545, USA

Institute for Theori'ical PhysicsUniversity of California

Santa Barbara, CA 03106, USA

and (permanent address)

Blackett LaboraoryImperial College

London SW7 2BZ, Great Britain

OUTLINE

I Structure, from a topoáyg, ». VM. vpointD Gauge invarianceIII Tunneling modes

IV Supercooled liquid and glass transitionV Statistical crystallography

I - STRUCTURE OF GLASS FROM A TOPOLOGICAL VIEWPOINT

1.1 Introduction

Glass spans nearly 60 centuries of human activity, from early glazed objects

(4000 BC) and glassy beads which a p p e a l in Egypt between 3200 and 2500

yean BC (depending on the encyclopedia), to metallic glasses, discovered and

developed in the (19)sixties and seventies. Yet, from a condensed matter

physicist's poiBt of view, it b an ill-understood material, fall of fascinating and

surprising physical properties.

Tk* Methodology of condensed matter physics consists in identifying, and relat-

ing to taen other, the physical properties, structure, and constitutive dements of

a daw of materials. Glasses hare several physical properties which are both

universal and specific to disordered condensed matter (cf section 1.2), but

emphasis on ', and demonstration of (see eg. *) their universality have only

recently been made. To relate these properties to the structural constituents

requires an understanding of the structure of glass, which, at Erst sight, xtnsists

almost exclusively of negative statements: no metric geometry, trivial space

group, no generative symmetry, no Bloch states, no single ground state, no

Unique best structure, etc.

By contrast, perfect crystalline materials have easily discernible structure (space

group) and constituents (atoms, electrons, holes, unit cell). However, rare are the

physical properties of perfect crystals which depend specifically on their space

group. But imperfect crystals have properties directly and crucially affected by

extended structural constituents, or defects (vortices, dislocations, flux lines),

whose definition, label and existence (structural stability) is granted by the struc-

ture of the material3. We shall see that the physical properties of glasses are also

governed by a single, extended constituent, the odd line or 2T-disclinatR>n \

which is the only structural element surviving the absence of generative homo-

geneity and the triviality of the space group.

The main purpose of this lecture is to identify those universal properties and

structural constituents, both at the lowest level of specificity. The level of

understanding of glass is similar to that of crystalline solid state physics before

1910-301, Le. before quantum statistics, Bloch theorem, electron bands and dislo-

cations were introduced. General concepts are required, howrrer ovetsimplifi:..*

rather than the solution of a particular problem from fist principles. It wiD tun

out that even such a simplified description yields non-trivial results.

1.2

We shall list those properties which occur so widely in glasses that they can be

regarded as universal, and so rarely or never in crystals that they can be viewed

as specif c to disordered condensed matter.

a) Tunneling modes

At low temperatures, the glass is an elastic solid with overall homogeneity,

capable of supporting phonons of wavelength long enough to average over any

«homogeneity. Glass can ring, as Mozart, and probably others found out long

ago. Surprisingly, this is not all. At low temperatures, the specific heat is linear in

temperature, and this contribution dominates the T* phonon contribution below

IK. The thermal conductivity is lower than that of the corresponding crystalline

material (eg. quartz) and goes as 7*. The specific heat betrays the presence of

additional localized elementary excitations (MO per 10* atoms) , which can

absorb phonons and thereby reduce the thermal conductivity K It was suggested

by Anderson, Halperin and Varroa, and independently by W.A. Phillips, that

these additional excitations were tunneling modes between potential minima, or

valleys, distributed at random in configuration space *. These valleys are few

and far away m configuration space, so that tunneling occurs between pain,

thetuby forming two-level systems. The main evidence for tunneling, and for *

Inite nnmber of levels, comes from the fact that the ultrasonic absorption can he

•afurufeé'. Abo, echo spectroscopy can be performed, exactly as for an assembly

of spins 1/2 7.

The concept of tunneling modes has been entirely successful both m explaining

existing experiments, and in suggesting new ones. Tunneling modes occur m

Detalbc and covakut glasses •. But, what is H that mores, let alone tunnel ? How

are we to label the valleys in configuration space and determine their distances ?

(The height of the saddle points between valleys sfaouM be of the order of the

glass transition temperature). Answers in lecture ID.

b) Viscosity, relaxation rates, entropy and the Kauzmann paradox

Traditionally, glass transition b defined to coincide with the change in the

thermal expansion coefficient from a value characteristic of a liquid to that of a

solid. This corresponds to a viscosity of the order of 10u poises. The transition is

smooth and its temperature Tf depends on the cooling rate and on the thermal

history of the system. The viscosity (or any inverse transport relaxation rate)

increases smoothly and rapidly with decreasing temperature, and follows the

empirical Vogel-Fulcher formula11W

over n wide range of temperatures. Equation (1.1) is used by the National Bureau

of Standards for calibrating reference glasses *. The various names under which it

it known in different fields (WLF, Cohen-Turnbull, Tammann, Doolittle, etc., for

glasses, polymers, oik, etc.), should support its claim to universality, even

though, for some glasses, inite viscosities have been measured below T0, and the

viscosity seer» to cross over from (1.1) to Arrhenins behaviour as the tempera-

ture B lowered *. Thus, roughly, glass behaves like a supercooled Said above Tm,

and an elastic, random solid below T0. T, B well above room temperaUre for

window glass (a-SiOs), whereas T.=\V \K for glyeerol. Slow modes in the

sapertoolcsl fluid become frozen (qoencbed) below 7*#.

A measure of the entropy of a glass is obtained by integrating under the specifc

heat corre, with a known value for the entropy of the liquid as initial condition-

Its value depends on the cooling rate, but, if extrapolated to zero cooing rate,

the entropy becomes negligibly small at and below the same Enite temperature

Tf, at which the extrapolated viscosity (1.1) diverges '*. This vanishing entropy

at a finite temperature is called Kaiizmaan's paradox, even though it was prob-

ably known to Simon. A good discussion of the meaning of entropy of glasses as

measured by this method, can be found in a recent paper by Jaeckle ".

Unlike ÇWP5, whose viscosity is proportional to the atomic diffusion rate and

increases with increasing temperature, viscous or supercooled fluids have fluidity

1/iy - ability to yield to a shearing stress - proportional to the mobility of n

diffusing object or "defect", i.e. to its diffusion rate />, by Einstein's relation. One

would expect D to be activated in condensed matter, and 1/n to decrease with

decreasing temperature as in eq. (1.1), but with 7*,=0 I2. The other problem »:

which "defect" ? Answers in IV.

c) Energy gap

Window glass is transparent, and amorphous semiconductors have a gap.

Despite the absence of long-range order and of Bragg reflection, the existence of a

gap was shown by Weaire a to be a consequence of the fixed valence in amor-

phous semiconductors, i.e. to the uniformity in vertex coordination z, or regular-

ity of the graph describing their structure (cf section 1.3). Furthermore, for a

simple but realistic model, Weaire and Thorpe M have shown that the energy

spectrum consists of degenerate states and of the spectrum of eigenvalues of the

connectivity, or adjacency matrix (a matrix describing the topology of the

graph). The gap is filled by localized defects (eg. dangling bonds) and modified by

many-body effects (polarons, excitons). The subject is treated in M.H. Gphen's

lectures.

d) Hall effect

The Hall effect is doubly anomalous in amorphous semiconductors: "...two ord*

era of magnitude less than expected..., almost invariably it has the wrong

sign" lb. The magnitude of the effect suggests interference of the electron wave

packet, due to the fact that the space in which it propagates is not simply con»

netted. The cores of line "defects" puncture the space available to the charge car*

riers.

Structure

See references 16 and 17 for details.

Glasses belong to either of two classes; covalent glasses (like window glasses or

â-Si) or random packings (like metallic glasses). In both cases, their structure

can be described by a regular graph.

ID covalent glasses (Pl.l), vertices and edges of the graph are atoms and bonds,

respectively, (non-planar) faces are rings, and cells do not have a direct physical

interpretation. The atoms have fixed valency, so that the graph has fixed vertex

coordination z ( z = 4 for Si), and is said to be regular. Dangling bonds are per-

mitted. This graph is also called a continuouê random network (CRN).

In metallic glasses, only vertices have direct physical interpretation (atoms), and

neighbourhood (edges) must be defined precisely. This is done by

Voronoi construction, a partition of space whereby any point belongs to the ter-

ritory of the nearest atom. Every atom is thus ascribed a (convex) polyhedron or

cell, and the packing is a space-filling assembly of such cells. The assembly of

Voronoi polyhedra, with their (planar) faces, (straight) edges, and vertices, form a

graph, called Voronoi froth, which is regular (z—4 in 3D and r = 3 in 2D)

because any vertex with higher coordination can be split into several regular ver-

tices by an infinitesimal deformation of the packing (through transformation Tl

of section 1.5). They are not topologically or structurally stable, and occur with

negligible probability. Thus the Four Corners boundary between Colorado, Utah,

Arizona and New-Mexico, is a cartographer's conspiracy and is not geographically

stable. Similarly, in 3D, only edges where 3 faces or cells meet are stucturally

stable. Structural stability is what Lewis 2 I calls "random avoidance of the

niceties of adjustement", and the structural elements (vertices, edges, etc.) are

simply territorial boundaries of a random packing. By contrast, CRN (covalent

glasses) have vertex coordination given by chemical, not territorial considerations,

and more than 3 (non-planar) faces can meet on & single edge, even though their

P|.J. <3o»tintto«» rsndon} network, exbibiiicg homogeneity, non-coliirtearity and odd line» (thethread thronsh odd riugs exclti»ive!y)

FI2. Mndcracking

vertices are still - in SiO2 or a-Si - all tetrarcoordinated. The crystalline analo-

gues to the Voronoi construction are the Wigner-Seitz cells, or, in reciprocal

space, the Brillouin zones.

Let us return to the original packing. Each atom corresponds to a Voronoi

polyhedron or cell. Two atoms are neighbours, and will be joined by an edge, if

their Voronoi polyhedra share a face. Thus, two graphs, related by duality,

describe the same packing, the original atomic packing, and the Voronoi froth.

Only the Voronoi froth b regular. The packing has high, fluctuating vertex coor-

dination ( < z * > « 13.4, increasing with anisotropy of the Voronoi polyhedra,

and decreasing if the polyhedra have different volumes 18, as happens in real

metallic glasses made of at least two different elements). The self-dual condition

is 2 = z * = 6 in 3D and 4 in 2D (cubic and square lattices, respectively).

The essential feature is tLe considerable variety of shapes of Voronoi polyhedra.

Matzke w has identified 100 different kinds in soap bubble froth, out of which 20

occur frequently. Thus, there is no single unit cell in glasses, not even smooth

deformations thereof. The 20 Voronoi polyhedra are all topologically different,

and the structure of one metallic glass is a member of a statistical ensemble (of

most probable distributions - see lecture V). Moreover, glasses belong to a

different class of packings than the aperiodic tilings of Robinson and Penrose,

which have only a few elementary geometric tiles 20. The situation has been sum-

marized in 1043 by the botanist F.T.Lewis: "The average 14-hedral shape

observed in massed bodies of diverse surface tension may be due to random

avoidance of the niceties of adjustment. Failure to arrange the bodies so that 5 or

6 meet at a mathematical point, or form intersection where 4 meet along a

mathematical line is sufficient to account for promiscuous, vnoritntotcd

10

pçlfkeárm having an average of 14 facets" 21. This failure, associated with struc-

tural stability, gives rise to a substantial variety of cell shapes, and to statistical

equilibrium.

Voronoi froths and continuous random networks are topologicalrjr equivalent:

they have the same, low vertex coordination z=4 , but CRN's have bent edges,

nonplanar faces , nonconvex cells and two faces sharing more than one common

edge, whereas Voronoi froths have staight edges, planar faces, convex cells, but

unequal edge lengths. The ring statistics is accordingly different: Voronoi froths

have a majority of 5-bond rings, CRN's 7-bond ringsn .

The Voronoi partition of space can be extended to packings of different atoms (as

befits real metallic glasses) and unequal cells. This is not as elementary as it

sounds: Consider 3 atoms in 2D; they form the vertices of a triangle, whose per-

pendicular bissectors are concurrent, by an elementary theorem of geometry. If

the atoms are unequal, which perpendicular division retains concurrence, and

thus space-filling ? There is only one known answer, the radical «rú, a straight

line (a plane in 3D) wich is the locus of points with equal tangents to two circles

(spheres) representing the atoms. It is obvious that radical axes are concurrent,

loaf so that they are straight lines, and the proof of this last statement** (an

elegant exercise in inversion geometry) strongly suggests that the radical froth is

the only generalization of the Voronoi froth to unequal atoms. It has been applied

to metallic glasses by Gellatly and Finney 2*.

The Voronoi construction has been used to describe not only glasses, but also

ecological and geographical problems, soap bubbles froths, metallurgical aggre-

gate», convective cells, etc.17.

11

1.4 The need for topology «ad its consequences

Manifestly, amorphous structures have neither global length scale (ruler), nor

nor unique global reference frame orientation (compass). In fact, amorphous or

non-crystalline materials are characterized, at first sight, by what they are or

have not: No generative symmetry (translation or rotation), trivial space- or

point groups, MO unique ground state see tunneling modes), no Bloch

theorem,..., so that non-negative concepts are required, which are structurally

stable under those transformations wr;;h keep tje structure unchanged.

At a glance, one distinguishes the three mzn structural features, not only of

glasses, but of all large, space-filling random structures (metallurgical aggregates,

undifferentiated biological tissues, geological jointings, etc.):

i) Non-collinearity of local reference frames (or variety of cell shapes)

ii) Overall, but non-generative homogeneity

iii) Odd lines.

These three features are characteristic of the amorphous state, and can be taken

as its stuctural definition. If either (i) or (iii) are missing, one may still have a

random structure, but without topologieal disorder.

Homogeneity implies that atoms are distinct, but not physically distinguishable.

Glasses share this property with crystals. However, unlike in crystals, this homo-

geneity is not a generative symmetry, but only the fact that any objective (physi-

cal) statement (one which does not contain T , "this", "here",...) about one par-

ticular atom can equally well be made about any other, even though their local

environment (tetrapods attached to the Si atoms in a-Si or vitreous silica, Voro-

noi cells in metallic glasses) are manifestly different and non-collinear. (See Pl.l).

r» MtNã. I

f\%\. Topologicai invariuice apd transformation, iüusliated by Deni» We»irt (based on a realevent). ©Dcnii Weair«r 1982

13

A CRN construction can be continued ai nt/tntt«m. This is the kind of homo-

geneity experienced by getting lost in & forest, with trees taking the part of

atoms. This homogeneity of glasses manifests itself at a microscopic level, to the

surprise of early X-ray crystallographers: "...one of the most interesting

discoveries made in the comparatively early history of X-ray analysis was the fact

that silk and even paper are more crystalline than glass." (Dame Kathleen Lons-

dale, quoted in ref.[25], p.26). Thus the scale of homogeneity in glass is 3 order of

magnitude smaller than the smallest mkrocrystallites (10 versus 1000 atomic dis-

tances).

The automorphisms probing homogeneity are permutations of the atoms (or the

trees) and their surroundings, effected, for example, by local rotations of the

tetrapods. The physical properties must be unchanged under these local transfor-

mations, and homogeneity of glasses is a genuine, gauge symmetry. We shall dis-

cuss gauge invariance in detail in lecture II. At the structural level, the most ele-

mentary such automorphisms are automorphisms of the graph, that is permuta-

tions of the vertices which preserve adjacency. At a dynamical level, local rota-

tions must also be included.

In the absence of a global metric and reference frame orientation, the only invari-

ant under automorphisms probing the homogeneity of amorphous structure is

their connectivity or the spatial relationship imparted by, eg., Voronoi construc-

tion. (We shall see in lecture V that some transformations can switch neighbours

(Tl), but still leave tile statistical properties of the structure invariant). Thus,

the relevant geometry for amorphous condensed matter is topology (rubber

geometry), which replaces metric geometry of claasical crystallography. By the

•ame token, the 230 metric space groups are replaced by homotopy groups which

14

describe connectivity, k. states and Bloch theorem give way to topological sectors

atd to a theorem, also formulated by Bloch for superconductors n, stating that

lhe free energy is a periodic function of the flux triggering the gauge transforma-

tion;. The period corresponds to a large, or non-trivial gauge transformation,

which brings the system from one topological sector to another. The best illustra-

tion of topological invariance is given in a cartoon by Denis Weaire (Fig.})-

Consequences of this enforced retreat from metric to topology are that

a) One cannot distinguish structural constituents by their sizes alone.

b) The existence, and the definition of a structural constituent depends on its

structural stability, so that it cannot be made to disappear by small, continuous

deformations.

e) Its only distinctive feature fa its shape. Hence randomness and topological

disorder imp/f that cells have many different shapes, like the various soap bub-

bles identified by Matzke w . Glasses cannot be made of only one type of cell, as a

result of "random avoidance of the niceties of adjustement" 21. The problem of

describing the structure is a statistical one, and the methods of statistical

mechanics yield the average features (its "equation of state", which is called

l/ewis's law in the case of cellular stuctures, see refs.[27],[57]f or lecture V) of the

ideal random, space-filling structure (the class of most probable members of an

m$mN« of structures). This equation of state is a correlation between sizes and

shape» of the constituting cells, and implies a medium-range correlation (one

should not say order) in the amorphous structure.

1.6 Elementary structural transformations

There are three, locai, elementary transformations of a random structure

under small, continuous deformations 17>28.

i) Tl , or neighbour-switching. (Fig.2).

ii) T2, or cell disappearance. (Fig.3).

if) Mitosis, or cell divbion. (Fig.4).

All of which occur in 2D and in 3D. In addition, in 3D, one has

iii) Face disappearance.

All these processes (or their inverses) occur in froths, foams or emulsions, in rock

or mud crackings, and in convection cells. Voronoi froths have a conserved

number of cells, and do not accomodate (ii) or (ii'). Covalent networks (CRN)

only have Tl transformations, which is there a local valence or bend exchange.

(The term "valence alternation" is now used to describe a particular class of

non-local transformations, accompanied by charge transfer and change in vertex

coordination, which control the electronic and optical properties of amorphous

chalcogenides n). The T l or neighbour-switching process is also that which res-

tricts topological stability to 3- (in 2D) and 4-coordinated vertices (in 3D), only.

Note that it conserves the vertex coordination of the froth, but not that of the

dual graph. In this respect, it affects adjacency, but we shall see in lecture V that

the statistical properties of the structure remain invariant.

There has been several recent attempts to construct random networks from by

applying successive, random T l transformations on an initially regular (hexago-

nal or diamond) lattice in 2D " " o r 3 D " .

I-is. 2 1 lt-;ikin.ir\ i.K.i| KMi

TI process

incnl of ivlk

T2 process

X. Vanishing of ;i cell.

mitosis

-I (VII division.

l-'tj!. *. A s ? p;iir nf ivIK. m.ikmir up ;i

These transformations change the number of edges of the faces involved. In 3D,

this corresponds to the introduction of topologiral disclinations (rotation disloca-

tions). Hexagons tile the plane (or the floor of a kitchen). Removing or adding

one edge to one hexagon can be done by making a cut in the tiling, removing or

adding a wedge (segment) of material and reglueing. One recovers a perfect hex-

agonal tiling apart from the cell at the end of the cut which is now pentagonal or

heptagonal, and the fact that the tiling is now warped. It has positive, or nega-

tive curvature. The non-hexagonal cell is a disclination, and imparts curvature,

through the Gauss-Bonnet theorem of geometry. 12 pentagonal cells transform

your kitchen floor into a football, an elementary consequence of Euler's theorem

to be discussed in the next Section.

This geometrical picture no longer holds completely in 3D, where there is no

Gauss-Bonnet theorem. In relaxed networks or froths, faces have on average 5.1

edges (Voronoi froth 18) or ^ 7 edges (CRN a ) so that there is no unique regu-

lar, flat packing. More to the point, disclinations exist, and are labelled in amor-

pbous materials whose space group is trivial, by the fundamental group of rota-

tions 3'4 nx{SOD). In Z>=2, 7|(502) = #> an<* disclinations, labelled by sign and

intensity, exist, whereas KX(SOD) = Z2 for all D>S, and the only topologically

stable line "defects" are odd lines (see below). Nevertheless, even in 3D, curvature

associated with the number of edges per face, remains an useful concept to

describe the local strain.

A dipole of neighbouring 5- and 7-sided faces form a dislocation (Fig.5). (Again,

this is only a local concept in 3D, without topological stability). The dislocation

has the following physical properties in random networks:

IS

1) It can glide (by u^ng T! rans'ormatioos). This determines the elastic amá

plastic properties of commercial foanis tt. It also constitutes a very efficient «ay

of dissipating shear energy. This property finds an airusing illustration in the

structure of daisies, y>>aecones or pineapples (phyUotaxb) M .

2) It can climb. A mitosis vor a T l tracsformaticn) creates a pair of dislocations,

and subsequent mitoses of the larger cells, corresponding to a glide of the two

dislocations away from each other, tearing an additional layer of cells inbetwecn

(Fig.6). Again, as far as elastic energy is concerned, this is a very efficient method

of solving the problem of copii.3 with addition»* material (growth of biological

tissues M or of metallurgical grains s&). Hillerts mechanism is slightly more com-

plicated as it involves creation of 3-sided celb (in 2D) only (a combination of T l

and T2 transformations, exclusively).

ii1 Ire lion ;i::^ «li-. iM;t!:on »rf .1 iji-l«K.ition pair by SUCCCIMVC cell divnior».

Morral and Ashby 3e uave given a very clear rrpresentation of these transforma-

tions and defects in 3D ordered foams or grains packings. In particular, the dislo-

cation core forms one single odd line (cf. fig. Ha of rcf. (36J), which shows thai

topological concepts applicable io 2D cannot always be transposed verbatim to

3D. The glide of the dislocation occurs through succesive Tl transformations, as

m JD. It is abo emphasised that a l tkcsc ar* bat

3) Dbiocataos» scree» the stiam d«e to discMsmtiosa (cwratmre). For example,

stnKtwaBy, a radnbria (mietoseopie mame aainal) h a huge footbaB, and ases

dishxatioas to screea the steam from the 12 dwctiaatioa repaired by topology

(football » sphere). The same «rn ramg of (topolopcaBy stable) disdiaatioas by

distocatioas (which are aot topofopeaBy stable ia passes) oetar ia aa elastic

4) Disfecatioas Jiable disdiaatioas to more. (For example, a Tl toaasformatioa

aezt to a disdmatioB creates a disfecatioa aad mores the disdiaatioa).

AD these properties emphasize the efmtmicd role of dislocations, which is partic-

ularly efecthre m raadom stractvres, where dislocations are not topologieaDy

stable, aad caa be regarded as local strain lactvatioas falfiUiag a specific, physi-

cal pvrpose.

Although these traasfonnatioas have beea iatrodaced here from a purely mechaa-

khl point of view, one can show (lectare V) that they abo leave the statistical

properties of the raadom structure (average shapes and their correlations) invari-

ant, so that they caa occar independently of each other, anywhere in space or

time, without affecting the statistical equilibrium of the structure. They play the

same part in statistical crystallography that meroremr$Mlitf plays in statistical

thermodynamics.

20

1.6 Topological conservation laws in random structures

Topologically stable constituents of disordered condensed matter are those

which retain their identity (and their existence) under small, continuous deforma-

tions of the system (including its motion). For example, a dislocation in a crystal

is topologically stable, an arbitrary elastic distortion, usually not. Topological

stability can be expressed as a conservation law or a continuity equation. In ran»

dom networks or froths, there are two conservation laws:

1) Euler's relation

F-E + r = X (2Z>)

- C + F - E + V = O (ZD) (1.2)

for a space-filiing graph with V vertices, E edges, F faces and C cells (including

the cell at oo). The Euler-Poincare characteristics x is a topological invariant of

the manifold (space) containing the network or the froth. It is an integer of order

1 ( « V,E,F in systems with a large number of elements). Euler's relation is

easily proven by induction, adding elements to a given network. In particular,

one can verify that the left-hand sidM of equations (1.2) are invariant under the

elementary transformations of section 1.5. Indeed, their physical properties sets

the time scale for topological evolution, which is much longer than that of non-

topological deformations (the latter, typically of order a/t, where a is the aver-

age spacing between atoms and e, the speed of sound).

2) Odd lines

The second conservation law is even simpler, and applies to 3D structures: The

presence of five- or seven-sided faces in a structure is symptomatic of its non-

crystalünity, being incompatible with simultaneous rotation and translation sym*

metrics. Odd-membered rings (faces with an odd number of edges) are not found

in isolation, but are threaded through by uninterrupted lines, which form closed

loops or terminate on the surface of the material 4 (cf. the ribbons of Pl.l)

Thus, odd faces are not isolated elements, but linear objects which are topologi-

cally stable. They are characterized by oddness, rather than intensity. However,

because these objects are lines, they remain difficult to eliminate despite their

modulo 2 algebra.

There are two, similar proofs of this conservation law. Both rely on the construc-

tion of an arbitrary, closed surface S, homeomorphic to a sphere for simplicity,

which intersects the random network at its vertices. S contains therefore some

vertices, edges and faces of the network, which triangulate S (this can always be

arranged by small deformation of $). The assertion is proven if, for any S, there

is an even Dumber of odd faces on S (providing an exit for any odd line entering

S).

i) Ascribe a weight Jt—{ 1) to every edge e on S, and an index

* / = I I Jf ~ ± 1 (even/odd face), to every face / on S. Then,

JJ 4> = f | JeJ = l, (Je enters twice in the product because every edge

ft 5 t ' li

belongs to 2 fact» on S). q.e.d. 37.

ii) Let «,/,- denote the number of edges, i-sided faces on S. The incidence rela-

tion between edges and faces, 2e = £]»'/,, implies that ][] í /,- = even, and,

because í /,• and / , have the same parity for i odd, VJ /,- = even. q.p.d. *.

The second proof suggests a generalization of the odd line conservation law to 2D

surfaces of arbitrary EuJer-Poincare characteristics and networks of constant ver-

tex coordination 38, and also to the structure of polyhedral networks (a model of

amorphous packings with tetrahedra and octohedra of atoms as elements, pro-

posed by Ninomiya M to describe medium-range "order" in amorphous packings).

For some amorphous materials, like polymer glasses, it may be too difficult or

complicated to construct a random network describing the structure. The sim-

plest, and least specific description of any glass is as an elastic continuum, with a

trivial space group (ie. without any infinitesimal translations! and rotational sym-

metries). (A crystal can also be represented as an elastic continuum, but with

either full (VoUerra continuum), or discrete (one of the 230 space groups) sym-

metry). Odd line is the only stable structural constituent surviving the transition

from discrete, cellular network, to continuum. (Cells,..., ie. simplices have disap-

peared). It takes then the form of 2ir-disclination. This can be shown very sim-

ply 40: Consider a configuration at a point, and take it for a walk in space, while

maintaining its orientation relative to the local reference frames (parallel tran-

sport) (Fig.7). Upon returning to the starting point, the configuration, which

determines the physical properties (strain, etc.) of the system, must either be

restored to its original orientation, or any mismatch must be physically

irrelevant. The latter situation applies to crystals, where possible mismatches are

elements of the space group, which label the Burgers vectors of dislocations and

disclinations. In glasses, the space group is trivial and the configuration must be

restored to its original orientation. Nevertheless, there are still two possible

transformations of the local configuration upon circumnavigation: a rotation by

4ÍT is homotopic (continuously deformable) to the identity, but rotation by 2ir

entangles connections of the local configuration with the rest of the system. Thus,

one remains with one single structurally stable constituent in continuous amor-

phous, condensed matter, the 2ff-disclination which has the same algebra as odd

lines in networks. Both have cores puncturing space (there is no way of filling an

odd ring because the relationship between any two vertices depends on the path

chosen around the ring - zero is an even number), and both are sources of non-

collinearity (see PI.l). They are therefore the same specific and universal consti-

tuents of glasses which we were seeking. They are also the only ones: transla-

tions are homotopically trivial, so that dislocations of any kind are not structur-

ally stable. Neither are point defects.

In summary:

Group of all possible local transformations (excitations) G = T3 A S 0 3

Space group H — 1 A 1 (trivial)

Topologically stable constituents (labelled by non-trivial homotopy groups of

G/H 3:

Odd lines - *i{SO3) = Z2

Non-singular textures - JT3(S03) = Z (yet to be identified)

Odd lines are therefore what count as configurations in the glass (eg. in the resi-

dual entropy *''"), and also as slow modes in a viscous liquid or in the glass

above To. Indeed, as long as the time scale for topological transformations is

longer than that for non-topological deformations (~a /c ) , the flow of topologi-

cally stable objects is a slow- or hydrodynamic mode, and any time-independant

conservation law has a time-dependant correspondant. Such conservation laws are

identities expressing topological stability, called Bianchi identities. [In elec-

tromapetism, the first couple of Maxwell equations include a time-independant

(div J9 = 0) and a time-dependant (0(D + curl E = 0) conservation laws for

24

the density B of topological objects. In elasticity, the topological objects are dis-

clinations in general (dislocations in crystals, where disclinations cost a prohibi-

tively high strain energy, and are excluded by an ad hoc hypothesis (distant

parallelism)), and inclusion of time-dependance constitutes the standard,

phenomenological generalization of elasticity to rheology and visco-elasticity 43.]

Thus, Bianchi identity plays two parts. It grants the object its topological stabil-

ity. But also, it is a topological conservation law which constraints the motion of

the object and of the fluid containing it.

Note that disclinations also appear in Kleman and Sadoc 4* description of the

structure of glasses, as perfect crystals in an "ideal" (in the sense of Plato),

curved space, which are projected into our usual, Euclidean space (the cave).

Overall homogeneity is guaranteed by construction in the curved space, as is the

best local packing (tetraledral or icosahedral, and therefore incompatible with

Euclidean space-filling requirements), and disclinations lines appear as a result of

the projection ib. Whereas this approach and our own agree in 2D, the disclina-

tions of Kleman and Sadoc conserve a sign and intensity in 3D, and are

prevented to cut across each other by topological obstruction (they are labelled

by & non-abelian group). Consequently, glass above To and supercooled liquid

cannot be described by the same curved space model (where Sow would be akin

to solving the Rubik cube), unless the tetrahedron SiO4 (in covalent glasses), or

the icosahedral cluster (in metallic glasses) order parameters, vanish at To.

J.C.Phillips has proposed a cluster model for chalcogenide and oxide glasses

glasses **. The size of the cluster (radius & 30 A) fits with homogeneity discussed

in section 1.4, but its surface represents a wall-like defect, which is not

2S

topologically stable (?o('903) = 1). It can be healed into line defects, as demon-

strated explicitely in the case of metallic clusters by Sadoc and Mosseri *7>4&, and

also in actual crystalline phases like Wp Cu*>Mg, Mna, Up 48. This lack of topo-

logical stability does not mean, of course, that local strains are not concentrated

on internai surfaces in glasses, and several experimental properties are best

explained by Phillips's model. Our topological point of view analyzes glasses at a

lower level of sophistication. Even then, the odd lines, which are the only

ingredients at that level, are not physically irrelevant but can account for non-

trivial, specific and universal properties of glasses.

n - GAUGE INVARIANCE

2.1 Discrete gauge in variance in spin glasses

Gauge «variance in disordered condensed matter was first mentioned in a

paper by Toulouse4fl dealing with spin glass on a lattice, described by the

Ed wards-Anderson Hanultonian,

where S ,=± 1 (Ising spins on lattice points i). and tbe coupling between nearest

neighbour spins is Jjj—± .!. according to some probability distribution given

a priori. Hamiltonian (2.1), and thus the physics of the system, are invariant

under tbe lotai transformation

S{' ^TtS, , J;i> ^TtJij7j (2.2)

parametrized by r,—±1 {Z* gauge transformation). This is a local transforma-

tion, involving variables which live on vertices (dynamical variables) and on edges

(connections); it is therefore a ga»ge transformation. Transformation (2.2) is

exact. It is also not very useful, since it involves two variables of physically

different status: Tho spins 5,- arc dynamical variables, allowed to reach thermo-

dynamic equilibrium within a canonical ensemble (Boltzmann distribution). By

contrast, the couplings J l ; are fixed (quenched) when sample under investigation

was made. They arc only random in the ensemble of different, bu! physically

equivalent, realizations of similar spin glasses. It is tbe free energy of every par-

ticular spin glass realization which is averaged over the coupling». In short-, half

of the gauge transformation J—*J' is not physically realizable on a pive.n sample.

Invariants under transformation (2.2) include not only the Hamiltonian, but abo

the face (or plaquette) index • / = f l ^ • an<l therefore the frustration (or odd-«7

ness in the language of lecture I) tig <fr = - 1 . (It b easy to show that frustration

forms dose loops or lines terminating on the surface of the material, using, eg.,

the proof (i) of the theorem of section 1.6). Frustration is a geometrical property,

independant of the "matter" field 5,, so that geometry (and essential, topologkal

disorder) is preserved under gauge transformation. Moreover, the partition func-

tion for a given sample, £[{ J}], and thus its physics, b independant of all the

details of the distribution of couplings {J} apart from those which are gauge

invariant **:

Z[{J'}) = Z[{J}} (2.3)

In other words, the tiling (warping, curvature, oddness,...) {$} completely charac-

terizes the geometry of the system, and is gauge invariant. As a corollary, the

unfnistrated or Mattis model (J,y=J, Jy so that all $ig • = 1 ), has the same

statistical mechanics as a ferromagnet.

[ It B amusing to see what happens to gauge invariance within the replica formal*

ism **. For simplicity, consider the infinite range model for N spins M, with gaus-

sian distribution of couplings. Then,

> = £ exp[/J*/V(2iV) Y,Y.S?S?SfSf),

which is invariant under two types of transformations: Either a local spin flip

S*' = TjSf, in all replicas, or a global rotation of each replica independently,

S°' = R ( l a ) Sa. The Edwards-Anderson order parameter ? „ £ = <5<o rS/> is

not invariant under the latVer transformation, which expresses the fact that the

symmetry between replicas is broken (<â ; ?>^0). The system is trapped in one

of tbe many valleys (replicas) in configuration space. 2nd hydrodynamic modes

«re associated with this broken symmetry S3&1. ]

Transformation (2.2) cannot meaningfully be generalized to continuous (XY or

Heisenberg) spins, because the couplings J,y are essentially real numbers. We

shall see that, by going from a lattice to the continuum, and from a microscopic

(spins) to a semi-macroscopic description of the matter field, a full exploitation of

gauge invariance in disordered condensed matter (Yang-Mills theory) becomes

possible.

This simple example (2.2) of a spin glass on a lattice emphasize? the main charac-

teristics of gauge invariance in disordered systems:

a) Gauge invariance is an exact symmetry of / /

b) It preserves essential geometrical ingredients (frustration or odd lines)

c) It enables us to recover as much generativo homogeneity as is compatible with

(b), for example, by treating J as dynamical variables, and including (b) as con-

straints (source terms)50: From eq (2.3), Z -> £ Z[{J}\ 6[frustration).

An integral representation of the delta function restores the full gauge invariance,

at the price of a more complicated effective Hamiitonian.

2.2 Gauge invariance at a semi-macroscopic scale as a genuine sym-

metry

It was argued in section 1.4 that the non-gener.ilive homogeneity of glasses,

29

associated with randomness (non-collinearity of local reference frames), is a

genuine, 50(3) gauge symmetry. Indeed, as Jaynes has aptly put it in a

different, but relevant context (the Bertrand "paradox9 of probability theory),

"Every circumstance left unspecified in the statement of a problem (here, the

frames' orientations) defines an invaríance property which the solution must have

if there is to be any definite solution at all. The transformation group, which

expresses these invariances mathematically (here, the gauge group of local rota-

tions), imposes definite restrictions on the form of the solution, and in many cases

fully determines it" 81. Let us now be more specific.

The seminal 1078 paper of Dzyaloshinskii and Volovik (DV) M begins with the

statement that the temptation to use the concept of local exchange invaríance to

describe the spin glass state, is difficult to resist. Most readers did, I believe,

agree immediately with this preamble, albeit for their own, different reasons, even

though they did not accept at face value the specific modol of DV.

Here were my own reasons: Elementary excitations are deviations of the spins,

SiOt tetrapods,... from the orientation of the local reference frame, and their

(exchange, twist,...) energy is given by comparing deviations at different points,

each with a different frame orientation. The ordinary derivative dS, which meas-

ures these deviations in uniform magnets, must be replaced by a covariant

derivative DS, with D = d + iA, where A is the connection or gauge field,

which defines parallelism in the frames at two different points. The gauge field

provides the answer to the technical problem of connecting points with different

frames. The next step is to show that the particular orientation of the frame at a

given point has no physical importance (gauge invaríance).

30

Hertz51 argues by analogy with ferro- and antiferromagnets: The exchange

energy is given, typically, by | (#-«$)# |2t where ^ is, for example, the angle of a

XY spin, and <? is the wave vector of the magnetic modulation, 3 = 0 in fer-

romagnets, $5^0 in antiferromagnets (Ginzburg-Landau free energy). In spin

glasses, (${2) is a random variable with a given distribution; it is the gauge field

or connection.

In fact, gauge invariance should occur whenever one can define noncollinear, local

reference frames. Connection between neighbouring frames is defined arcwise, by

requiring that two overlapping neighbourhoods have local frames fitting together

without any rotation. However, a finite circumnavigation docs not necessarily

restore the frame to its original orientation; it only does so in the absence of cur-

vature, or of disclinations (Fig.7).

Fig.7. a) Example of parallel transport and non-unkity of the local reference frame in the pres-eaee of curvature. Tbe frame ia A has been rotated upon circumnavigation. The cone, tangent tothe trajectory, can be flattened on a plane to define parallel transport.

b) la a crystal, rotation of tbe reference frame mast be an element of the space group.

31

Consequently, the orientations of the local reference frames cannot be defined

uniquely everywhere in the presence of disclinations. It is locally arbitrary, but

the physical properties of the system are independant of this arbitrariness, ie.

invariant under a local rotation of the reference frame (given a connection

between neighbouring frames), which is precisely a SO3 gauge invariance.

Local frames occur in continuum elasticity theory. Glass can be regarded, on a

semi-macroscopic scale (whenever all relevant length scales are longer than the

interatomic distance a), as an isotropic, elastic continuum, with frozen-in internal

stresses 40>57>58. In metallic glasses, the stresses are due to the fact that Euclidean

space cannot be filled by atomic configurations miniming locally the energy (or

maximizing packing: tetrahedra, icosahedra) 44>4*. In covalent glasses, there is an

entropy barrier preventing crystallization (and inducing non-coUinearity of tbe

SiO4 tetrapods, and stresses), when cooled from the melt. Thus, stresses are asso-

ciated with Don-collinearity of the frames, which are in turn related to disclina-

tions or curvature 40>S7^. Continuum elasticity is, accordingly, a gauge

theory w.**.57.58.*9. Unfortunately, almost no three-dimensional crystals have any

disclination (because their strain energy is prohibitively high), so that, histori-

cally, the full gauge invariance of tbe theory was lost by the introduction of an

ad hoe, distant parallelism (zero curvature, global reference orientation)

hypothesis, justifiable in crystals, but not in glasses. This hypothesis breaks most

of the gauge invariance of the theory at the onset. In glasses, dislocations screen

tbe strain energy of disclinations (odd lines), so that they can axist on energy, as

weft as on topologccal grounds *°.

Comtet has found an explicit relation between integrability condition (connec-

tion) for frames on a minimal surface and the Euler-Lagrange equations of a class

32

of two-daaeasioaal 5Ü, gauge field theories n . Gauge iavariaace of the field

theory correspoads to rotatioa of the frames in the tangent plane of the surface.

Tab relation caa probably be generalized to three-dimensional, Yang-Milb {SOJ

field theory and the frames of a three-dimeasioaal hypenurface, thereby estab-

hshmg explieitery the correspoadaace betweea aoa-coBiaear frames aad SOt

gauge mvariaace at the core of thb lecture. As a bonus, the correspoadaace of

Comtet requires the exbtaace of aa underlying curved hypersurface, which may

tun out to be aa explieK reaHiatioa of the ideal, curved space of Klemaa aad

Sadoe".

Gauge theory b therefore the proper method to go from a discrete lattice to the

continuum, aad in particular, to generalize tike coupling Jtj to all points betweea

i aad / (which b easy to do if J>0, but not so if / < 0 ) . So, roughly, we have

some matter field, coupled through a covariant derivative to a gauge field

representing the coupling between spins. Specific questions remain:

i) What b the matter field ?

ii) What b the gauge field, which replaces {/} in the continuum ?

in) How should the disorder be quenched ?

The answer to (i) was an important idea of DVW . The matter field #(! )

represents, not one single «pin, but a group of spins within aa elementary cube of

the discrete lattice centered at 1 (semi-macroscopic representation). Because frus-

tration induces non-cottinearity of the spins, 4 b no longer a vector of ghrea

length (with values on a sphere (Hebenberg spins) or a circle (XY spins)), but a

hedgehog, or hirsute object, whose manifold of states b that of the fuD rotation

group S09 (Hebenberg) or SOt (XY). Similarly, in glasses, •<*) b an object br-

ing in the fall rotation group SOit rather than a single tetrapod (< SOrfT, where

7* b the tetrahedral group), for eovalent glasses, 01 a single kosahedroa

(c SOJI, where / b the kosahedral group), for metalbc glasses. Thb b because

the field • must be defined everywhere in a continuous space, and not only at the

centre of the tetrapod or ieosahedroa.

In fact, 4(i) must be chosen so that tt yields the correct "defects* (vortices, frus-

tration, odd fines) **. Hebeuberg spas would not have given any frustration

(*t(Sj) — I). Similarly, single tetrapod», tetrahedra or ieosahedra would have

given rise to line defects which would have been topotogkalry entangled (since

their lespective *, are aon-abeiian •*), thereby preventing the material to flow, so

that the Mine model of the structure of glass could cot describe the dynamics of

Ms supercooled or viscous liquid, even though both states have the same struc-

ture. (At aay rate, there b no generative tetrahedral or kosahedral symmetry in

the glass, hence no reason to restrict the manifold of • to a coset of SOS]

Hydrodynamics (irrotational fluids have for field a scalar potential 4(7), with

velocity 7 = v ^ , to be replaced by 7 itself when the field b rotational (with «3=

curl 7 as vortex density)), dectromagnetbm without, or with magnetic monopoks

[A replaced by 5= carl Ã as the field), and continuara elasticity without, or

with dbclinations **, provide classical examples of the overriding influence of the

defects on the selection of the proper matter field. Cf. abo our introductory

remarks (section 1.1).

Consequently, the matter field can be expressed in terms of a rotation operator

•(*) . X(*) 0(7) (2.4)

(Stueckelberg decomposition). Whereas in spin glasses, the amplitude X(7) can

34

be regarded as the order parameter M which vanishes at the spin glass transition,

in glasses, it represents the size of the tetrapods and can be taken as a constant,

X,, everywhere except at the core of the odd lines where it vanishes. The charac-

teristic length associated with fluctuations in X is #»« , the interatomic spacing

in the network or the size of a plaquette. \jíQ corresponds to the chemical pro-

perties of the constituting atoms, and is taken for granted at all temperatures

relevant for the glassy, liquid or solid states. This approximation is sometimes

referred to as the nonlinear <r model.

(ii) We postpone discussion of the precise meaning of the gauge field AJfi) until

next lecture (eq.3.2), apart from remarking that it should appear naturally and

automatically in the free energy density through the covariant derivative of the

matter field, and that it is directly related to the non-collinearity of the tetra-

pods. In fact, introduction of a gauge field A^H) is necessary to make the

derivative covariant (eq. 2.6 below). As it is its only physical purpose, one can

assume that the covariant derivative also provides the only coupling between

gauge and matter fields (minimal coupling). The length scale of the gauge field, /,

(penetration depth in superconductors) is the range of non-collinearity. Thus (see

Pl.l), t»&&a, and glasses are type II gauge materials (in the superconductivity

terminology). They exhibit vortices rather than full Meissner effect.

(iii) From our experience with discrete lattices (end of section 2.1), it seems evi-

dent that gauge fields should be treated as dynamical variables (and full gauge

invariance or homogeneity maintained) as much as possible. Only the source of

gauge fields (the frustration or odd lines), ie. the essential disorder, need be

quenched. This remark is in contrast with the assumptions of DV and of Hertz,

who both felt that all disorder terms should be quenched. Hertz M quenches the

35

gauge field Â(2), and DV5 5 give it a mass term in the free energy. Both pro-

cedures break gauge invariance. Ours is gauge invariant. Furthermore, the

sources of gauge field - the (geometrical) odd lines, quenched below T0 in glasses,

and at all temperatures in spin glasses - can be treated simply as punctures of the

space E into which matter and gauge fields are put, exactly like flux lines in

extreme type II superconductors. Boundary conditions on the puncture are free.

This allows gauge and matter field (with amplitude X=c«f) to take up

configurations (eg. rotated by 2* around the puncture), which would have been

forbidden if the space had been simply-connected. Thus, we replace » simply-

connected, Euclidean space by a punctured one (E). This complicates slightly the

geometry but simplifies enormously the algebra, notably by letting

X = | $ | = est everywhere in E, and by restoring full gauge invariance to the

system within E.

For example, consider a superconducting ring E. The free energy within E is

gauge invariant, the magnitude of the matter field (superconducting order param-

eter) is uniform within E, and the new configurations associated with multiple-

connectivity are those of quantized fluxoid. The degeneracy between these

configurations is only lifted if the (electromagnetic) free energy outside £ (chiefly

inside the ring's hole) is included.

In summary, sources of gauge field are puncture», with free boundary conditions

(except that every configuration must be single-valued, and rotate by a multiple

of 2* about every puncture). The free energy is fully gauge invariant in punc-

tured space E, where the magnitude of the matter field can be taken as constant.

The multiple-connectivity of E implies that there are several possible ground

states (valleys), all degenerate because they are related to each other by gauge

36

transformations (see lecture III), instead of the unique ground state in simply-

connected space, which is one of the cornerstones of classical solid state physics.

2.3 Gaage invariant model free energy for glass

Without further ado, we can now write down the model free energy for

glass87.

F = / « * / ( * )E

Âp\ (2.5)

where E is the Euclidean space, punctured by odd lines, with free boundary con*

ditions.

/(?) is invariant under gauge transformations, the /oco/ rotations

Â ; = üÀpü-1 + - í / ^ c r 1 ) (2.8)

under which the derivative is covariant,

37

Fpy is gauge covamnt,

F¥J = ÜF^ÍT1 (2.7)

(unlike its abelian counterpart, the electromagnetic induction S — curl Ã, which

is gauge invariant), and is therefore only related to the non-collinearity density

Fpy, a physical observable which must be gauge invariant (eq.3.2).

Free energy (2.5) was written down as early as 1954 by Yang and Mills M in the

completely different context of elementary particles. This suggests that there is

little arbitrariness in the selection of a gauge invariant free energy.

Gauge invariance imposes the introduction of a new field, the gauge field. On the

other hand, it severely restricts the possible free energy densities Consequently,

eq. (2.5) has very little arbitrariness: Only,

1. Minimal coupling between gauge (À^ and matter (0) fields (solely through the

covariant derivative)

2. One energy (density) scale t = (i\jg)2 = l/{g2l4) = kB TJl*

3. One length scale (the penetration depth) / = l / f v ^ X , )

4. The distance ç between punctures (l»ç»^ a), and their configuration

(semi-dilute)

have or will be chosen on physical grounds.

Odd lines, and gauge invariance will enable us to construct explicitely the many

valleys and two-level systems responsible for the low-temperature properties of

glasses

38

m - TUNNELING MODES

8.1 Many potential valleys in configuration space

The anomalous properties of glasses at low temperatures, briefly reviewed in

section 1.2.a, and in detail in refs.[2] and (5], are properly described by the con*

cept of tunneling modes ' , which are the elementary excitations of systems with

several deep potential minima in configuration space. In this lecture, I shall use

the model free energy (2.5), ie. odd lines and gauge invariance, to locate precisely

the many valleys in configuration space, to label them, and to calculate the tun-

neling rate between different valleys.

It is elementary to show that a distribution JV(Atf) of two-level systems (2LS)

split by an energy A, (proportional to the tunneling rate), yields a specific heat

increasing linearly with temperature,

(assembly of Fermi-Dirac oscillators), where kB is Boltzmann's constant, as long

as the density of two-level systems remains finite as A,-+0. The coefficient

C/Tfal^Wêif^K'2 is typically an order of magnitude smaller than that of a

good metal (fvidr* W^K'* for Cu), suggesting ÍO^-IO"6 2LS / atom active at#

« IK, with a size of «40.4, since resonance (T2) measurements indicate that

the 2LS are roughly independant excitations 2-105.

Three remarks can be made at the onset:

39

1) In tunneling modes, the higher and broader the potential barrier, the smaller

the tunneling rate and the level splitting. In glasses, splittings down to

£&{10~2K)kg are observed (no departure from linearity, or time dependance of the

specific heat down to these temperatures), suggesting high (^kBT0) and broad

potential barriers, that is a lot of atoms moving very little. The size of 2LS indi-

cates that, as far as quantum coherence goes, they are semi-macroscopic: consid-

erably larger than a single electron or alpha particle, but still smaller than a

SQUID or Schroedinder's cat.

2) The classical potential barrier may even be infinite (and will be so for our con-

tinuum model free energy (2.5)), but, as long as W is integrable, the tunneling

rate remains finite. Then, the 2LS have no classical equivalent. This may explain

why the explicit nature of tunneling modes has remained elusive for 10 yean.

3) Saturability 5, and coherence 7 of the 2LS indicate that the elementary excita-

tions within each potential we'll are thermally inaccessible below &1K. They lie

at much higher energies than the splitting A0 between ground states of different

wells. Thus, only the ground state within each well is relevant to the physics ol

glass at low temperatures. The situation is reminiscent of the tight-binding, oi

LCAO model of electronic band structure of metals and molecules, where only

one orbital per atom is relevant to the physical or chemical properties in a lim-

ited energy range, from which core levels, and higher atomic excited states, are

inaccessible.

These remarks are sketched in Fig.8, which is also an adequate summary of the

conclusions of this lecture.

40

CONR SPACE

UO 2 3 D)

!27T>

_mm CONF. SPACE(ID/OOO LINE)

Fig.8. Tbe many valleys in configuration space, giving rise to tunneling modes.

a) Old picture (ref.|6j).

b) New picture (this lecture).

Note that the lO^dimensional conflgoration space has been redwed to ID per odd line; the vat-

leys (topological sectors) can be labelled ( | 0 > , |2f>), and their distance (or tbe energy splitting

A,) calculated.

Dotted Unes: excited states in each sector (irrelevant at low T)

Foil line»: 2LS

Only a rough topography of tbe valleys is required to obtain groand state and elementary excita-

tions of a glass.

41

The specific heat is approximative)? linear in the range Q.Q25K-IK. This sug-

gests a constant density of states for 2LS, N(A#) = N(0), in thb energy range.

In fact, Lasjauias ef «I9* hare suggested that N(At) has a gap below 16 mK.kB,

thereby giving a limit for the slowest tunneling rate observable experimentally. It

must be emphasized that several problems of detail remain with the tunneling

mode concept (only approximate linearity of C, 7 s contribution to C

significantly above that expected from Debye phonons, response of the system to

short thermal pulses, etc " A 1 * ) , bat the major theoretical challenge is univer-

sality: "...no plausible argument has been presented yet why all amorphous sub-

stances have approximately the same density of states of tunehng defects which,

and that b probably even more puzzling, scatter the phonons with almost equal

strength" a (US, not HM spelling). The experimental situation will abo be

clarified, as new and direct methods of investigation and comparison are

developed (effects of high pressure w , or, as in epoxy resins, of the size of the net*

work constituents n ) .

3.2 Classical, ground state configurations

According to Fig.8, it b sufficient to identify and label the ground state or

metastable configuration of every valley. These ground state configurations

{À0t(l} are classical solutions of the Euler-Lagrange (EL) equations obtained by

minimizing the free energy (2.5) in punctured space E (the punctures being put in

• priori - quenched at random below T0), with free boundary conditions on the

puncture. Free energy, boundary conditions, and EL equations, are all gauge

invariant. The EL equations look frighteningly complicated 37, but we shall not

need An explicit solution.

42

The matter field ft is obviously parametrized by rotations, and one anticipates

two ground state configurations per puncture, one corresponding to a rotation of

Ò upon circumnavigation around the puncture by 0 or a multiple of 4ir, the other

to rotation by an odd multiple of 2*. (The latter is the new configuration per-

mitted by the multiple-connectivity of £). All other configurations can be con-

tinuously deformed into these two. (Recall our discussion of section 1.6, and the

fact that the rotation group is not simply connected (jr,(S0s) = Z2), necessary to

justify quantization of the electron's spin in half integral multiples of h/2x). We

must show that the gauge field Âf is aho parametrized by rotations. Then, we

will have proven that there are two ground state configurations per puncture,

parametrized by rotations.

To do so we construct a linear combination of matter and gauge fields which is

gauge invariant, so that a covariant gauge transformation (rotation) of the

matter field ft induces a contravariant rotation of the gauge field. This procedure

is familiar in superconductivity and electromagnetism, where the phase 9 of the

order parameter ^(7) = p(?)e2p [#(?)] forms with the vector potential X[t), the

gauge invariant combination <5(a?) =

A' = A +

0' =0-{hc

(This constitutes also the simplest derivation of the Higgs mechanism (massive

vector boson) in minimally coupled, abelian gauge theories).

The construction proceeds in a few steps: 37>68

43

i) A given configuration of the matter in £ can be written in terms of its orienta-

tion at some point ?#, Ò# = Ô(?#), and a rotation operator W(t) c SOS), which

is independant of the path from % to ?, apart from a winding number (rotation

by 2*n, n=O,l) around every puncture. Then, tl{t) = W(2)ilt W~l(&), and the

"phase gradient" (the phase orientation density x* =s (dflW)W~1) are path-

independant.

ii) One then goes to a rotated frame, defined by Ã = WTlkWt for any operator

A. The new covariant derivative Õjb. = ÚT^D^W, involves the phase £„ « d

the gauge field Â^, in a linear combination C^, (Stueckelberg decomposition)

= gÀ,(t) + xfi) - \Cfi)\' (3.1)

which is gauge invariant, as is any operator Ã in the rotated frame. (Under gauge

transformation ÜÇ2), W'{?) — C/(7)^(aT), and Â' — Ü\ÍTl impües Ã' = Ã ) .

This includes F^,

which, being gauge invariant, measures the physical density of non-collinearity.

Similarly, the free energy density (2.5) can be written in term of the gauge invari-

ant C,.,

— • F F t SSS I ITI lasol

iii) The combined gauge field C^ is accordingly gauge invariant, and therefore

unique. The rotation WÇZ), which parametrizes the matter field Xf, or 0, also

44

parametrizes the gauge field À0. Because there are only two possible

configurations per puncture of the matter field, up to continuous deformations,

corresponding to rotation by 0 or 2JT, there are also two configurations of the

gauge field, and therefore, two ground state configurations per puncture {tl^p},

solutions of the EL equations for the free energy F in punctured space E. q.e.d.

Invariance of the free energy under a non-trivial gauge transformation (rotation

by 2x), implies that F is periodic in he fiux triggering the gauge transformation,

as long as the source of non-colliacarity vanishes in E (cf. superconducting ring,

where F is periodic in the applied magnetic flux, as long as the applied magnetic

induction vanishes inside the superconductor). This generalizes Bloch's theorem

for superconductors **, to any minimally coupled gauge theory.

Similarly, the restriction on the matter field (which must be uniform for a given

configuration, ie. returned to the same orientation after circumnavigation),

translated into a restriction on the gauge field, corresponds to fluxoid quantiza-

tion in, superconductors. It is a direct consequence in minimally coupled gauge

theories, of the non-triviality of some gauge transformations.

3.S Tunneling states, two-level systems

We have identified and labelled two ground state configurations per puncture

(odd line), which will serve as the "atomic orbitais* of our LCAO. These two

"tunneling states" | 0 > and | 2?> are characterized by rotations of 0 and 2JT

upon circumnavigation about the puncture. They are degenerate in energy (by

gauge invariance of F). But neither classical configuration is gauge invariant,

45

and can qualify as the true ground state of the glass, which must be gauge invari-

ant. (There is no reason to assume that gauge invariance should be broken in

glasses, us we have argued in lecture II). Tunneling, however slow, must take

place to restore gauge invariance. The true ground state and elementary excita-

tion, per odd line, are the gauge invariant combinations,

l ± > = ^ l | O > ± \2x>] (3.4)

which are the 2LS, split by (fc/2x) times the tunneling rate.

3.4 Tunneling rate

We shall assume that the frozen configuration of punctures is semi-dilute, that

is, intermediate between dilute (alphabet soup), and melt (dense spaghetti bun-

dles) (The appellation is borrowed from polymer physics and may not correspond

to gastronomic states). In this case, only one length describes the frozen, isotropk

configuration, the distance ç between non-adjacent segments of odd lines (which

may equally well belong to the same or to different odd lines). Thus, to transform

configuration | 0 > into |2*> about a given odd line, it is sufficient for a quan-

tum flux of rotation of length ç to tunnel through a distance ç only. Further away

lies the region influenced by other odd lines.

Justification for a semi-dilute distribution relies either on the dynamics of odd

lines when the glass is cooled from the melt (lecture IV), or on maximizing the

entropy (most probable distribution) &7. Qualitatively, the most probable distri-

bution of odd lines is that which maximizes the density of low-energy states, per

unit volume and energy interval. This suggests the semi-dilute distribution, in

46

which the loops are concentrated enough for the odd lines to intertwine and the

nearest non-adjacent segment to belong to another loop (maximum namber of

2LS per unit volume), but not concentrated enough that their cores overlap

(maximum number of 2LS per energy interval). This picture has not yet been

confirmed experimentally.

The energy splitting A# is given by the transition amplitude,

- 1 ^ - = <012r> = / [DÜ] [DÂJ cjp {-{2x/h)Jdr L (r) ) (3.5)

for all paths in imaginary time r connecting | 0 > and |2x>, where L(T) is the

Lagrangian, and the energy has been adjusted to vanish at the initial and final

configurations. The leading contribution to A# is given by the classical paths

from 10> to | 2JT> minimizing the exponent (action) in (3.5). One faces, at once,

two major problems: 1) To obtain a Lagrangian from the free energy (2.5), ie.

dynamics from thermodynamics. 2) There are no continuous, classical paths from

|0> to 12x>, since the configurations are not homotopic. A trick must be used:

For a semi-dilute distribution of odd lines / » ç » a , the free energy (2.5) or

(3.3) can be linearized (the free energy is dominated by terms with the highest

power in /, that is the linear terms in C^ in Fpin because the length scale of dC

is now ç rather than / as for a single odd line or in the dilute limit (>/) . The

linearized free energy is similar to that of an extreme type II superconductor,

with penetration depth / and constant magnitude of the order parameter. A# will

therefore be given by the tunneling rate of a flux quantum across a superconduct-

ing ring of radius f, height ç and radius of puncture a. This rate can then be

evaluated by opening a thin slit (width a) across the cylinder. This trick pro-

vides at a stroke 1) a Lagrangian, 2) a classical path.

4?

One has, in fact, the geometry of a Josepkson transmission hne, along which the

magnetic lux 4 satbfes a sine-Gordon equation n , whose Lagrangian b well-

known

where X/ b the Josephson penetration depth and «, the warefront Telocity. For a

thin slit of width c, \j^l, and c"«<:\/«7(*fj

A typical superconducting ring has height L, radhts & and L > ç » X j , so that a

tall, thin solitoa propagates along a long slit. The cosine term dominates the

dynamics, and the tunneling amplitude goes as expf- cat Lç\ which is utterly

negligible, so that superconducting magnets can hold their magnetic field for long

enough to be useful.

By contrast, in glass, the rotation flux quantum is a very fat X/^/ , short (height

f) object, which has to more very little (distance ç«l). The [cos* - l]/\j term

b negligible, and jir L{i) «s the energy of a 2D vortex. The tunneling ampli-

tude is much larger **,

A, = ÎI. exp \-D (Ç/B) lnfr/«) ] (3.7)

as long as ç is not macroscopic (semi-dilute limit). Here D =

Typically, ç/« « 40, T.(*sdO0K, (kc/2wkB 7 » » 10 , yielding an exponent

Abo, a dbtribution of distances Ç,

(f/a)l exp [-

48

corresponds to a flat and broad density of 2LS energies /V(A,)«(t&T*)"1 for

At<kBTó. Numbers obtained from (3.7) are therefore reasonnable. Note that

the density of 2LS N(Aa), is inversely proportionnal to the only energy scale of

the problem kg To (see eq. (2.5)). Thus, in glass, tunneling does occur, and yields

the dominant excitations below some low temperature proportionnal to

1/y/kgT,. This temperature may be anomalously low if the elastic energy kB T,

is large, as is the case in a-Ge, or if the distribution of odd lines is not semi-

dilute. On the other hand, no odd lines, and no tunneling modes are expected in

glasses which can be described by a bichromatic model containing even rings

only 71, or its continuous equivalent, a pure gauge model. a-GaAs may be a sys-

tem without odd member rings, as long as wrong bonds {As-As or Ga-Go) are

excluded by electronegativity.

The density of 2LS is inversely proportionnal to the only energy scale of the

problem kB To. (Cohen and Grest also obtain this relation in their free volume +

percolation model72). This should be directly observable by comparing viscosity

and low T properties of the same system under pressure. Measurements by Bar-

tell and Hunklinger M in vitreous Si02 show that the low temperature ultrasonic

attenuation (ie. the density of 2LS) surprisingly increases with increasing pres-

sure. This is consistent with the anomalous pressure dependance of the viscosity

(which decreases as p increases) observed in GeO2 (and Si0 2 is expected to follow

the same behaviour 73, to the delight of geologists, worried about the viscosity of

the earth's mantle) through Tt decreasing with pressure (although why this is so,

is not clear microscopically). On the other hand, neutron irradiated silica

apparently follows the opposite, less unexpected, trend: Increase of the mass den-

sity upon irradiation is accompanied by a decrease in the density of 2LS, N(A0),

ie. an increase of the thermal conductivity and a decrease of the specific heat u.

49

However, irradiated and fused silica have strikingly different structural charac-

teristics, even if they are both amorphous. (The intensity of the 600 cm ~* (sharp]

Raman line of fused silica increaes by an order of magnitude upon irradiation 76).

8*S Can one see what it fa t ha t tunnels ?

A superconducting ring can be prepared in one of its free energy minima.

r>ecause an external magnetic flux is readily available irom a magnet. Tunneling

between energy minima is then observed as Josephson effect. In glass, by con-

trast, rotation flux does not correspond to a commercially available knob. How-

ever, there exists a method to prepare the system in one of its classical ground

states, which takes advantage of the coherence of tunneling between two levels,

the phonon or electric (when the 2LS have a dipole moment) echoes7, the 2LS

counterparts of spin (5=5=1/2) - echo spectroscopy in magnetic resonance.

In order to find out precisely what it is that tunnels, one would like to tune a

structural probe (pulsed neutron beam) to the coherently oscillating tunneling

modes, so that neutrons take successive snapshots of the 2LS when they are in

the classical state | 0 > , say. Constructive interference between successive

snapshots can be arranged by splitting the neutron beam, scattering beam 1 on

the sample at time t—0, when the 2LS are in the classical state

| 0 > = (l/ \ /2)[ |+ > + | -> ] i then delaying the scattered beam, while beam 2

is first delayed, then scattered by the s; .nple at time t~2t0 when the 2LS are

again in state | 0 > . The two beams are combined at some later time t°>2t0, and

superposition of neutron diffraction of the same 2LS in the same, classical state

| 0 > , should show constructive interference78. (Fig.Q). Varying the delay time

50

detunes neutron and 2LS, and leads to destructive interference between scattered

beams. Varying the time of scattering of the first beam (taken above to be f =0),

will find the 2LS outside the classical state, within the barrier, and the construc-

tive interference of the scattered beams destroyed. (Even though neutrons are

quantum probes, which could couple to 2LS through the off-diagonal operators ax

or o$, as well as through the diagonal ones, i or <rx, the 2LS involve too many

atoms in a disordered configuration, to be detected by neutrons as anything else

than density fluctuations (through operator i). In this case, echo (coherence) is

observed only for the classical states 10> and 12?> (eq.3.8)). Combination of

constructive and destructive interferences should fingerprint what it is that tun-

nels.

Phonon echoes are obtained as follows: One applies to the 2LS, described by the

Hamiltonian

( | ± > are eigenvectors of <rz), two high amplitude , A,, electrical or acoustic

pulses of duration r (SET pulse) and 2r (REVERSE pulse), respectively. The fre-

quency A = w(A/2ff) of the pulses should be ^ A < ( ( « A 1 ) , for resonance.

Hi(t) = — Ai(<

The SET pulse induces a phase shift

S = ^ ( A - A J 2 * A,2 T (2ir/A) « A, r (2TT/A) = JT/4

The evolution of the 2LS is evaluated in a rotating frame, where the effective

Hamiltonian is time-independant. The time-table of the tuned echo-neutron pulse

51

system is sketched in Fig.9. A r/4 pulse prepares the 2LS in classical state jO>

at f=0. A JT/2 puke at t=t, reverses the 2LS state, from a | + > + b | - > to

« | - > + 6 | + >- The echo is observed at f =2* ,+ 2r^2^, when the 2LS is again

in the classical state | 0 > . The two pulsed neutron beams (*) measure the state

of the system at i = 0 and 2t,+ 2r.

n. nn . nSET REVERSE ECHO

T <- t0 - 2r - t, —

Fig.9. Time-table of the tuned 2LS echo-neutron system

A distribution of 2LS {A,,} stills shows coherence (echo), as long as the reverse

pulse is short compared to the time of free propagation Ar«A t f i < ) , and the 2LS

are initially in a classical state:

| + / ->(a i f=0)-> C\-/+> + t,-deptndant term {at f= / # +2r) (3.8)

Thus,

| + > ± \->(at t—O)-+±C(\+>± j - > ) + incoherent terms {at t—to+1r)

A,where C=-isin(2S)sin0 and 0 — tan"1 •-• •. Maximum coherence occurs at

resonance A « Í A P , where 2S=0—ic/2.

In this lecture, we have exhibited explicitely the many potential valleys in a

52

continuous model of glass. The situation is summarized in Fig.8. Some progress

has recently been made towards repeating the same analysis for discrete random

networks, and obtaining again two classical configurations per odd lines 7I.

53

IV - SUPERCOOLED LIQUID AND GLASS TRANSITION

4.1 Introduction

This lecture will be concerned with the regime T>T9 above the glass transi-

tion temperature, where the structure can be deformed under shear, and glass

behaves like a supercooled, highly viscous liquid instead of a structurally invari-

ant, elastic solid below Tf. Viscosity, or any inverse structural relaxation rate,

follows the empirical Vogel-Fulcher relation (1.1), at least approximatively, as

described in detail in section 1.2b.

Above To, odd lines can move as the structure is deformed. However, they obey

a topologkal conservation law (section 1.6), so that they can expand or shrink,

but remain uninterrupted, assuming as before that bonds are broken and recon-

structed over a much shorter time scale than that associated with the fluidity.

Odd lines are the slow - or hydrodynamic modes of the system. Viscosity, or any

structural relaxation rate, is associated with the diffusion of some "defect", in the

crude but adequate model of Glarum, and Phillips, Barlow and Lamb 78. The

"defect" must be of sufficient generality to account for the universality of the

relaxation process in glass, and of sufficient stability to retain its identity and

avoid disintegrating during (slow) diffusion. Odd lines, or 2^-discJinations, as the

only structurally stable ingredients of any glass, clearly fill the bill on both

counts. Free volume would not, on its own, be stable enough.

The glass transition at T0 corresponds to the freezing or "condensation" of the

odd lines. We know already (section 2.2), that glass transition does not

54

correspond to a vanishing amplitude of some order parameter (unlike the transi-

tion in spin glasses), nor to the breaking of some (gauge) symmetry. A mean field

calculation of the density of free, or mobile odd lines will yield a condensation

similar to the Kosterlitz-Thouless transition, but in 3D, which may be frustrated

close to 7*# by a crossover to Arrhenius behaviour (see section 4.5).

Ideally, one would like to start from the free energy (2.5), and allowing the punc-

tures to move freely, like vortices in hydrodynamics and flux lines in supercon-

ductivity. Unfortunately, to the best of my knowledge, no method has been found

as yet to isolate the "vortices" as elementary excitations of a field theory in the

continuum, similar to the Villain method on lattices ". [ In electromagnetism,

thb would mean partitioning the integral over all gauge field configurations A (7)

into a discrete sum over configurations of different, quantized flux £ A{2) & ].

We must therefore calculate the energy E and entropy 5 of an assembly of free

(mobile) odd lines of density pj. Minimization of the free energy

F\pf] = E[pf]-TS[pf] , 6F/6pf =0, yields the density of free odd lines in

thermodynamic equilibrium Pj(T). A semi-dilute distribution of odd lines, has

their dimensionless density related to their shortest mutual distance ç by

P = (a/çf (4.1)

This is a pedestrian treatment of the model free energy (2.5) at high tempera-

tures when the punctures are mobile, but it is the best we can do. The argument

was first sketched by Anderson 1, and worked out in ref.[86].

This method has the free odd lines as the only (slow) dynamical variables or

degrees of freedom, through their density pj. The other (frozen) odd lines do not

contribute to the dynamics and thermodynamics as a first approximation. They

cannot be displaced or adjusted to lower the free energy of the sytem. At T4, all

odd lines are frozen pf{TJ=O (eq.4.5). Just above 7*#, there are so few free odd

lines that they form a dilute distribution, even though the configuration of all

odd lines, frozen and free, b semi-dilute at all temperatures, as discussed in sec-

tion 3.4.

All the results of this lecture have also been obtained from continuum elasticity

theory, allowing for (2JT-) disclinations, which are, as we have seen in section 1.6,

the only topologically stable, line constituents in a medium with trivia] space

group. Dislocations are not topologically stable, and screen the strain energy of

disclinations, which are then allowed to exist in 3D glasses on energy grounds

(their strain energy is unacceptably high in crystals, where it increase» linearly

with the distance between two disclinations). Most of the continuum elasticity

results on glass are due to Duffy, and have been published extensively 40-80'57.

4.2 Energy of an assembly of odd lines

We start with the fields and geometry of free energy (2.5) or (3.3). Assuming

(to be confirmed a posteriori) a semi-dilute distribution of free odd lines at

T»T0, ç«l (where / is the length scale of the gauge field, which measures

the range of non-collinearity), we can linearize the theory: The quartic terms in

the free energy density (2.5) or (3.3) are negligible when compared to the qua-

dratic terms involving spatial derivatives , because, in terms of rcscaled gauge

fields <?„ - /.

idC « //f » C

56

The free energy density (3.3) reduces to

(4.2)

which is that of eiectromagnetism of current loops (apart from the fact (taken

care of in the trace) that C is a tensor rather than a vector). In particular, the

Euler-Lagrange equations become simply

curl curl C =

where are the odd loops, sources of gauge field C. Consequently, the energy E

of the loops takes the Ampere form,

dTdT K* ~ < S/J -rr-fr + E W i: = « E

I ' 1t) I ' . - ' / I í i -1

E — F\-Apf In pj + Bpf] (4.3)

for all, semi-dilute loop geometries 57. Here, it is convenient to refer sums or

integrals to an underlying random network of edge length a, with F faces, L odd

loops making up a total of K segments of of length a (or threading through K

odd faces), core energy ecort, so that pj =s= KJF — (a/f)2. r;- is the distance

from segment / to the nearest, non-adjacent one. In eq. (4.3), the parameters

take the values

A = i i / 3 , B ^ etm + £ <ln(r í/í)>

The leading term in (4.3) is proportionnal to the only energy scale c, and only B

depends (weakly) on the microscopic structure of the glass through the core

structure. This is, of course, the result of the semi-dilute distribution of free odd

57

loops: A semi-dilute distribution has been assumed twice in the derivation of the

energy (4.3). First in the linearization of the field theory (2.5-3.3), and second in

the evaluation of the Ampere integral (4.3). When the distribution of free odd

lines becomes dilute (when T approaches To as we shall see in section 4.5), the

logarithmic contribution in (4.3) vanishes, and the energy of each loop adds up,

E — B' fij (dilute).

Elasticity theory yields the same result (4.3) 4°.80-*7. Outside the cores of odd

lines, the strains and connections are small and the theory can be linearized.

Dislocations (which are not topologically stable) screen the strain energy of decli-

nation down to the Ampere form. They also enable disclinations to move, as dis-

cussed in section 1.5.

4.3 Entropy of an assembly of free odd loops

The calculation of the previous section, despite starting from a free energy

density, kept the odd lines frozen in order to calculate their energy. The entropy

of free odd lines, which can move, expand or shrink, cannot be evaluated by the

same method, but must be calculated directly.

Let us calculate the number of configurations fi of K odd faces (in an underlying

random network with a total of F faces), making up an arbitrary number L of

closed loops. The first loop threads through ni odd faces, the second through n2,

etc. Lm is the maximum possible number of loop, n0 the minimum number of

odd faces threaded through by a single loop, thus X=Lm/K=l/n^ — 0(1). Let

C(n)/n be the number of configurations of a single n-stcp loop with arbitrary

58

starting point. Typically,

C(n)/n

with fl&3 in 3D, but actual numbers are of little relevance to the final result for

the entropy (4.4). The number of configurations is given by

The binomial factor is responsible for the entropy of mixing between loops, and

C(n)/n, for the configurational entropy. In mixtures of loops of arbitrary length,

the entropy of mixing of the smaller loops ( ^ \aK\ *=» K ln/T) is expected to

dominate the configurational entropy ^ K In(z-l).

One introduces the generating function

in terms of which ft is expressed as

Whenever l « L m « F , the most probable distribution has overwhelmingly

more configurations than all the others, as is well-known in statistical mechanics,

thus,

since Lm <<(1/2)F, and the entropy is obtained in terms of pj^

5 = kB\nÇl = kBFi~Xpf lapf + Ypf) + O(\nK) (4.4)

It is dominated, as anticipated, by the binomial coefficient ie. by the entropy of

mixing {X=Lm/K). The second term in (4.4), l^YV*- Yit has both mixing

(Y^X-XlnX) and confipurational {Y2~ln{z-l)-Xln(n^) origins.

4.4 Density of free odd Unes in thermodynamie equilibrium

Although glass is not strictly in thermodynamic equilibrium, and the observed

"glass transition" is essentially kinetic in character (the fluid-solid crossover is

smooth, and occurs at a temperature above T, which depends on the cooling rate

and on the thermal history of the sample), the time scale for topological

modification and for change in the physical properties is much longer than that

relevant to the propagation of elementary excitations (eg. phonons) about a

metastable state. Thus, slow modes (the density of free odd lines) can be

obtained from a thermodynamic derivation, and describe the physical behaviour

of the system, even if its linear response is so sluggish as to be inaccessible experi-

mentally very close to To.

The density of free odd lines at a temperature T, Pf{T), is that minimizing the

free energy F — E ~ TS, with E(pf) and S(pf) given by eq. (4.3) and (4.4),

respectively. One obtains,

, / ( D « » e - í ' í r - T - ) (4.5)

which has the Vogel-Fulcher form (1.1) because both energy and entropy have

the same functional dependance, pf\npf. Here kBT0=A/X^ c is the only

energy scale of the problem, as expected (section 3.4). 6=exp(yr/X)-l and

60

kBc={B-AY/X)/X is a positive number because et#re>«, (B>A).

The density of free odd lines vanishes at Tt, where all odd lines have been frozen

by their mutual interaction. Similarly, their entropy abo vanishes exponentially

at Tt (Kauzmann paradox),

Below T#, the odd lines arc frozen, and their entropy (which could then be calcu-

lated from the free energy density /(7) (3.3)) is not accessible thermodynamicaUy

(eg. from the specific heat). This condensation of odd lines at 7*# is a genuine

phase transition, the extrapolation to zero cooling rate of the kinematic 'glass

transition". However, the assumption of a semi-dilate distribution of odd loops,

responsible for the pflnpf dependance of the energy (4.3), is not valid close to

r#, and the pj[T) crosses over from Fulcher to Arrhenius behaviour, as we shall

discuss in secton 4.5.

The viscosity, or any structural relaxation time, then follows the Vogel-Fulcher

law (1.1), since its inverse, fluidity, is due to the diffusion of some topologically

stable, universal "defect" in glass 78>12'1, here, the odd lines. By dimensional

analysis, the length of free odd line per unit volume is pj a~2, the average dis-

tance between any site in the glass and the nearest odd line is pj1^*, and the

average time 7 for the odd line to diffuse to that site in order to relax it, is pro-

portionnal to pjla2, so that the fluidity is itself proportionnal to the density of

free odd lines in thermal equilibrium

61

4.6 Crossover to Arrbenfos

The Vogd-Fulcber condensation of free odd lines is only a mean field resvlt,

valid as long as the distribution of free odd lines remains scmi-dihrte (cf. deriva-

tion of the energy (4.3)). This is not valid dose to T#l where pj is very amafl.

How close will be discussed in this section.

The semi-dilate regime corresponds to

where the crossover temperature T, b given by

and eq (4.5). In this r^me. the Tree energy can be linearized, one has a similar

situation to electrodynamics of ~urrent loops, &nd E a

However, when

T.<T<T., p , <{a/!f, ç > í

the distribution of <«\i Hu<*s is dilute, the energy is the sum of individual loop

contributions E a pf. A, ic. lhe r Té" in eq.(4.5) vaol«b, and the density of free

odd lines crosses ovrr to Arrhenius behaviour, pj is the larger than it would have

been in tb^ semi-dilate regime at the same temperature (4.5), and the viscosity

accordingly falls below the Vogel-Fulcher curve.

Whether the crossover to Arrhenius is observable in a given glass, depends on the

non-collinearity length /= l / ( / i (X , i ) . If / is larg*, 7,«* T,, and n{T.) is large,

the crossover is not observable in an experiment lasting a finite time. On the

other hand, the crossover is obervable if / is small. The status of several glasses

is reviewed in Cohen and Grest9.

Although the behaviour of a glass above To can be described by the motion of

odd lines, which show a tendency to freeze in a Vogel-Fulcher fashion, the

existence of a thermodynamic phase transition is offset by crossover to dilute dis-

tribution of odd lines, and to non-linear elastic behaviour. Even within the semi-

dilute regime (4.3), the precise nature of the condensation (its universality class)

is not known, and cannot be obtained from our rough mean field calculation.

For completeness, and without discussion, let me mention different approaches to

the glass transition:

i) All kinematics (no transition)

ü) Statistical mechanics o* long polymer molecules fô. This approximate calcula-

tion leads to a phase transition of a very particular type (the continuous relation

between energy and entropy cannot persist below a finite temperature), but a

counter-example for networks of coordination 2D has been given by Gujrati and

Goldstein M .

iii) Free volume + percolation theory 72>0.

iv) Floppy (underconstrained) and rigid (overconstrained) regions (first discussed

by J.C.Phillips 4t) + percolation theory M.

The experimental situation is reviewed in ref.[85].

The present approach is by far the most economical, both in the number of

parameters and of specific ingredients. We have shown in these four lectures that

the combination of odd lines (as essential structural constituents), and gauge

63

invariance (describing the non generative homogeneity of glass) can explain and

control directly and specifically the universal properties of glasses and super-

cooled liquids.

64

V - STATISTICAL CRYSTALLOGRAPHY

5.1 And now for something completely different

In this lecture, I shall return to the structure, not only of glasses, but of ran-

dom, space-filling, cellular structures in general, which will be referred to as ran-

dom tissues, froths, or mosaics in the case of a 2D pattern. Glasses are

represented in this class of structures by the Voronoi froths of amorphous pack-

ings and metallic glasses. But such structures abound in nature and in inrolun-

tary art (Fig. 10). In 3D, metallurgical aggregates, foams, soap bubble froth,

undifferentiated biological tissues, lead-shot packings (Bernal model for the struc-

ture of a liquid); in 2D, geological jointings (cracked lava flows like the Giant's

Causeway), mud crackings (P1.2), photographic emulsions, etc., all are random

froths.

Fig.10. Wiodow áedm of an archaic, but common variety in England. "The Bell", Narborough,Leici.: Gentlemen lavatory, left window.

65

To a first approximation, all these structures are indistinguishable, apart from an

obvious scaling factor. This structural similarity is exhibited dramatically in

Fig.l of Dormer's book 87I in Figs.2-7 of Weaire's essay w , and in a recent review

of 2D random patterns 17. One notices also (as befits random structures - section

1.3) a considerable variety of cell shapes. It is improbable that specific physical,

or biological forces should be responsible for such identical but variegated archi-

tectural style, so that one looks again for universality: An ideal random space*

filling structure, determined solely by inescapable, mathematical constraints and

the fact that the structure is the most probable one. It is in statistical equili-

brium, in that any topological rearrangement of the cells leaves its "arbitrariness"

invariant, the arbitrariness being measured precisely by the entropy or informa-

tion contained in the structure. One sees immediately the analogy of this pro-

gramme wih statistical mechanics, and with the Maximum Entropy formalism of

probability theory Wi9°. If such an ideal structure exists, it is the (most probable)

representative of an ensemble of structures. It is not unique. Accordingly, criteria

for ideality will be relations between averaged, measurable properties of the

structure, like the ideal gas law in thermodynamics, rather than geometrical data

like unit cells or Bragg spots. One is looking for the statistical analogue of the

simple cubic structure in crystallography, or of the ideal gas law in thermo-

dynamics, from which departures can be measured to identify the forces which

may differentiate between structures at a second order of approximation.

[ A few words of apology: To the addict, random patterns are a cause of many

delightfully wasted hours. By others, it is regarded as an amusing, and rather

trivial hobby, harmless, but without artistic pretensions or cosmic significance. In

this respect, it suffers from its very universality: a subject which, spans beer froth,

crazy paving, cucumber skin and ideal partition of Ireland may not be

66

immediately associated with deep mathematics or ethereal beauty. It is rather

vulgar (Fig. 10). It is also littered with empty experiments (eg. measuring Euler's

theorem, as quoted in ref.[87|), false statements repeated blindly over centuries

(eg. that in 3D, cells are 12 -, or, later on, 14-sided on average lfl>17), and irredu-

cible positions stated in dramatic language: Contrast "There are aspects of tissue

geometry so obvious that they can hardly escape the attention of any person who

seriously considers the question at all. The appreciation that cells are polyhedral

figures came with the very first histological report ever published" (ref.{87), p.7-

8), with a referee's report from Nature "...the paper, which deals with cells as if

they were polyhedra, which they are not...". Incidentally, Dormer is right, but the

paper was rejected. The subject started (in the 17th century) as useless,

because w the standard activity at the time consisted in packing cannon balls,

and random packing of these is unlikely to make one the Ruler of the Queen's

Navy, but (in the 1940's), botanists have taken the trouble to compare plant tis-

sues with the structures of soap bubble» froths, and lead-shot packings, in order

to find out which physical force (surface tension or bard-sphere repulsion) was

responsible for the structure of biological tissues. Without success as all three

types of structures are roughly identicall0. ]

There are also several beautiful examples of man-made and natural foams in

ref.[104]. M.F.Asbby's conclusions (on the mechanical properties of cellular solids)

are very similar to those of this lecture: Importance of the structure in determin-

ing the mechanical properties of cellular solids, which are roughly independant of

the chemical or physical properties of their constituting materials, and universal-

ity of the structures, which are well described by a few parameters: cell aniso-

tropy, open or closed cells, and relative density.

6?

6.2 The random tissue or froth

Random froth labels the class of structures with which we shall be concerned.

It is a maximally random, space-filling, cellular structure. Maximal randomness

(or "random avoidance of the niceties of adjustment" 21) means that all vertices,

edges and faces are structurally stable (their connectivity is unchanged under a

small deformation), ie., in 3D, 4 edges, 6 faces and 4 cells meeting at every ver-

tex, every edge shared by 3 faces and cells, besides the general property that

every face separates 2 cells. In 2D, vertices have 3 incident edges and faces,

besides every edge separating 2 faces. Exceptional vertices with more than 4

edges are not structurally stable in 3D: They can be split into 2 normal vertices

by infinitesimal transformation (half of T l of Fig.2). Probability of their

occurrence in a random packing is negligible. The same holds for exceptional

edges with more than 3 faces. CRN are excluded from this restricted class of

random structures, because their edges may share more than 3 (non-planar) faces,

even if their vertices still have 4 edges (albeit for chemical rather than topological

reason). Consequently, the 3D relation (eq.(5.2) below) between average numbers

of faces per cell and of edges per face, should be modified for CRN. Indeed, it has

already been remarked that <n>£& 7 in CRN 22.

The structural stability of random froths was discussed in lecture I, with Voronoi

froth as paradigm. They exhibit all the vocabulary (dislocations, disclinations)

and the grammar (elementary transformations) of section 1.5.

There are two topological random variables for the 3D froth, the number / of

faces per cell, and the number n of edges per face. For the 2D froth, n is the

only topological random variable. They are not independant: Euler's relation

6S

(1.2), and the valence relations between incident edges and vertices, etc., dis-

cussed above, yield immediately tbe followL •£ topological identities for the ran-

dom froth

< n > = 6 (2D) (5.1)

^ (5-2)

which relate the expectation values of the topological random variables. Eq.(5.2)

is an identity, which holds both for any individual cell in the froth, and for the

froth as a whole. Indeed, there are two statistical problems in 3D, one at the

froth level, described by the random variable / , and the other at the cell level,

where n can still fluctuate (with < " > f t a determined by eq.(5.2)). The two prob-

lems are related by (5.2). All this is summarized in the considerable variety of cell

shapes observed lfl.

Equations (5.1) and (5.2) are the only topological constraints on the statistical

structure of the froth.

In 2D, the expectation value of the topological random variable < n > , is fixed

by eq. (5.1). In 3D, it is found empirically that most random froths have

< / >:=» 14, but it is, emphatically, not an exact result, or even a limit, despite

repeated statements in the literature to the countrary (cf. reis.[19] and [17]). À>

random froth with isotropic cells of equal volumes has < / >=13.40, correspond-

ing to the impossible feat (in Euclidean space) of packing 5.1 equal, regular

tetrahedra in the dual graph (cf. section 1.3). Fluctuations in the volumes of the

celb reduce < / > , whereas fluctuations in the angles (anisotropy of the cells)

increase < / > l8, as observed ««.«.«.«.«. (The high values of < / > observed for

lead-shot packings w , are probably due to uniaxial compression causing cell

b9

aaisotropy, as D.E.G.Williams has aptly remarked). For a Voronoi froth with

centres at random (Poisson distribution), Meijering81 obtains </>p=15.54

exactly. The cells appear indeed highly anisotropic.

We now turn to the correlation between cell shapes (ie. between the topological

variables in neighbouring cells). One knows from experience that large cells tend

to have small neighbours, and vice versa. The precise formulation of this result is

due to Aboav (empirically)*4, and was made plausible by Weaire *5.

6.3 Topological correlations, Aboar-Weaire law and mkroreversibility

In this section, we shall investigate shape correlations between neighbouring

cells of the random froth. We shall use a derivation of Aboav-Weaire's relation

due to Blanc and Mocellin M, mainly because it investigates the random froth

under the elementary structural transformations of section 1.5. In other, fashion-

able words, it follows the froth under structural renormalization. As a bonus, we

shall obtain the same recursion relation (5.3) for shape correlation under all ele-

mentary transformations, Tl, T2, mitosis and their inverses. This implies, first,

microrevcrribility: all these transformations can occur independently of each

other in space or time, without affecting the statistical equilibrium of the struc-

ture. Second, êtatiêtieal equilibrium itself: a random froth is already very close to,

or at a fixed point under structural transformations, with both expectation value

(hi 2D, through (5.1)) and correlation of its topological variable invariant under

these transformations.

Let us start with the froth in 2D. The random variable is n, its expectation value

70

is fixed by eq.(5.1), and the topological correlations are given by m(n), the aver-

age shape (number of sides) of the nearest neighbours of a n-sided cell. For short,

label each cell by the number of its sides. Consider transformation T2 first

(Fig.3): triangular cell d, neighbour to cell a, disappears. Among cell fl's neigh-

bours, cells b and e lose one side in the process (as does cell a). The other cells

(«) remain unaffected. Thus

a m(a)=6+ e+d+t -+ (a-1) m{a-l)—{b-l)+ (c-l)-f t

Since d=3, one has the recursion relation,

a m{a) = {a-1) m(o-l) + 5 (5.3)

Similarly, for a Tl transformation (Fig.2): a - m ' = a - l , 6f =6-1, e ' = e + l ,

rf'=rf+l, «'=«, hence,

a m(a) = (tf-1) m(a-l) + < 6 ' > - 1

But cell b' is no longer nearest neighbour to a (whereas b was, prior to the

transformation), so that, assuming no correlation beyond nearest neighbours, we

have <b' > = 6 (eq.5.1), and obtain the same recursion relation (5.3) under T l as

under T2. Under the same assumption, one obtain the r'th iterate of (5.3) under

mitosis where cell a splits into cells r and a-r+ 4,

a m(a) = r m(r) + 5(a-r)

Since the same recursion relation holds under Tl , T2, mitosis and their inverses,

all these transformations can occur independently in space and time (roicrorever-

sibility), without affecting statistical equilibrium.

7 !

The recursion relation (5.3) is readily solved:

() 5+n

(5.4)

which is the Aboav 84-Weaire °5 law. It is well obeyed experimentally 17. (It was

originally a purely empirical relation M, like the other equation of (the statistical

equilibrium) state, Lewis's law discussed in the next section). A sum rule due to

Weaire,

< n m(n)> = < n 2 > = / i 2 + < n > 2 (5.5)

relates the parameter B to the variance //2 of the distribution of {n}.

In 3D, there are aow two random variables n and / , and there is no relation

involving only / and the average number of faces of cells neighbouring cell / , as

far as I am aware. However, one obtains the same results a» in 2D (Aboav rela-

tion, unique recursion relation, and microreversibility) if one considers what hap-

pens to a n-sided face of a f-faceted cell, and its neighbours under structural

transformations Tl, T2 and mitosis *7, ie. the statistical description of the froth

at the cell level. Consider a given, f-faceted cell. Denote as before, by w»/(n), the

average number of sides of the neighbouring faces to face n on / . Transforma-

tion Tl conserves / , but T2 reduces it: / ' = / - J (T2). Also, there is now a corre-

lation between a' and b' f, despite the fact that they are DO longer nearest

peighbours (cf, Tl), because they belong to the same, finite cell where eq.(5.2)

must hold.

the dust settles, one obtain the 3D Aboav relation °7

^= 5/ - 11 - C{f-\-n) (5.6)

72

where C is a parameter of the froth, independant of / . There has been, so far, no

experimental verification of eq.(5.6). Weaire's sum rale (5.5) (for a given cell) and

eq.(5.2), inserted into (5.0), yield an interesting relation between the variance of

the distribution of sides on a f-sided cell, /»2,/=<[«-<» > / J 2 > / ? *&d f

Calculating bounds for /i2>/ for small cells (/ =4,5), one obtains - 1 < (7<2.

5.4 Ideal random froth, most probable distribution and Lewis's law

Let us define statistical equilibrium and universality of random structures.

We begin with the remark that ail random, space-filling structures enumerated in

section 5.1 are, roughly, identical. They are therefore unlikely to depend on the

^particular physical, biological or chemical properties of their constituting materi-

als, except for their single length scales. Indeed, the very fact that random struc-

tures occur (at least in 2D where there is uo conflict (frustration) between local

packing and global ordering), suggests that specific, short-ranged, directional

forces are less important in -framing the structure1 than the inescapable,

mathematical and .-universal constraints ($1-2, 5.8-0)' pertaining 'to the space

which the cells are filling. (Shortoanged foroes in 2D gite rise to triangular pack-

ings or honeycombs, possibly with a few dislocations). f

The miracle is that these two mathematical constraints are sufficient to frame in

a preei$e, ob$ervable fashion the structures generated under their sole or overrid-. f

ring influence. This is due to randomness and to the fact, well-known in statistical

thermodynamics, that the most probable distribution of cells is overwhelmingly

more probable than any other (cf. comments preceding eq.(4.4)).

The complete statistical problem involves two random variables (in 2D), the

number n of sides per cell - the topological variable, and the area .A of a cell -

the metric variable. To obtain an equation of state, it is sufficient to concentrate

on the topological variable n only. A full calculation of the most probable distri-

bution p(n,A) will be published elsewhere97. Correlation between shapes of

neighbouring cells, discussed in the last section, is an automatic consequence of

statistical equilibrium, and not an additional constraint.

One looks for the most probable distribution, {/>„}, of shapes of the cells in the

structure, where p. is the probability of finding a n-sided cell.

[ p. = / dA p{n,A) , Ãm = f iA p(n,A) A \

It is that distribution which maximizes the entropy or information 8fl

(5.7)

subject to constraints corresponding to our prior knowledge of the system. For

00 random space-filling structures, the constraints are

H = 1 (normalization)

».Ã» = A*IF (spacefilling) (5.8)

$ > . i i = Ô (topology) (5.1)

in 2D. In 3D, (5.8) and (5.1) are replaced by

? = V,/C (3D space-filling) (5.9)

7-1

(3D topology) (S.2)

respectively. Here .4m is the average area of a it -sided cell, and A,, the total area

available to the F cells io the 2D mosaic. Similarly, mvtatis mutandis, in 3D.

This is all. The problem, as formulated, b entirely mathematical. The constraints

are also all mathematical. Physics (or biology,...) is absent at this level, so that

the resulting structures arc universal. The only subjective step is in the coding of

the structure by the sole , topological parameter n, and the requirement that it b

in statistical equilibrium.

We do not need to evaluate the entropy, or the most probable distribution {pn};

the "equation of state" can be obtained by the following argument of Lissowski

and myself v (fully confirmed by a full calculation *7<s7). The constraints are a

linear systems of equations between p%, so that the smaller the dimensionality of

the space of constraints (<3), the larger that of the space of possible solutions

f=(P3tP4,P^f ' ' ' ). and the more probable one such solution will be. The most

probable distribution is obtained by reducing as much as possible the dimen-

sionality of the space of constraints, by making them linearly dependant,

K = (. 1,/F)X[n-(6-l/X)l » {AjF)\Mn 6) + 1} (5.10)

The averaged area of a n sided cell, Ant is linearly related to the number n of its

sides. [The intercept n# = (6-1/X)}. This relation was suggested empirically in

1928 by Lewis "a, on the basis of observations on cucumber epidermis, human

annion and the pigmented epithelium of the retina. Most undifferentiated biolog-

ical tissues obey i t M . It is abo obeyed by Voronoi froth generated from Poisson-

distributed centres M, albeit with a smaller intercept (n#£&0 instead of « 1 2 in

biological mosaics), but not, usually, by metallurgical grain aggregates, as will be

discussed below. It is a relation bet wees averaged, observable parameters of the

random froth, ie. the equation of state of statistical crystallography describing an

ideal froth (in the same sense as "ideal gas" in thermodynamics).

The effective elimination of one constraint, leading to lewis's law (5.10),

increases the entropy further. This is because, as in statistical thermodynamics,

the most probable entropy 5(X,) (5.7) is a convex function of its variables, the

right-hand sides of the constraint equations (S.I) and (5.8), written generally

y,f»cai=Xi, and imposed by Lagrange multipliers X,- n ,

(specific heats are positive). Let i=s be the constraint we wish to eliminate,

imposed by Lagrange multiplier X,. As a function of A,, 5 is maximum when

dS/dX, = 0. But dS/dX, = -X#, and X, = 0 states precisely that constraint s

b no longer operative, in our case because it b no longer independant of the oth-

ers57.

This derivation of Lewis'9 law goes therefore one step beyond the standard appli-

cations of the Maximum Entropy formalbm n>M, in which the functional form of

the constraints b known a priori. Here, we have taken advantage of the liberty

to tijust the functional form of the constraints in order to maximize further the

entropy.

There exbts, at least in principle, an alternative to Lewis's law in decreasing the

76

dimensionality of constraint space: It relates linearly the space-filling constraint

to the normalization, instead of the topology constraint, and yields ,4B=cst,

independant of n, that is no correlation between cell shapes and sizes. Even

though this solution also maximizes further the entropy, it does not occur in

natural froths or mosaics, but for no mathemetical reason as far as I know. This

remarks shows that further increase of the entropy corresponds to a choice

between discrete alternatives.

The parameter X in (5.10) is the undetermined Lagrange multiplier imposing the

linearly dependant constraints (5.1) and (5.8). (The other multiplier is eliminated

because of the linear dependance of the constraints). It is related to the slope

and intercept of Lewis's relation, and is therefore an important descriptive

parameter of the structure. Moreover, Lagrange multipliers have a habit in ther-

modynamics and in mechanics of possessing a physical meaning of their own.

They are not merely arbitrary mathematical factors. What is, therefore, the

meaning of X ? We shall see in next section that it measures the ageing of the

structure.

Lewis's law is not obeyed by 2D metallurgical grain aggregates, whether experi-

mentally, or in computer simulations by the EXXON group 10°, where it is the

gram's radius (or its perimeter) Wn, which is proportional to n, rather than its

area as in Lewis's law. According to the methodology of Maximum Entropy for-

malism, this fact demonstrates the existence of new constraint, besides the

mathematical ones. Clearly, in the computer simulations of grain growth and

statistics by the EXXON group, and as demonstrated experimentally 1 0 \ it is the

energy associated with grain boundary length which is the driving mechanism for

grain growth and statistical equilibrium. [ The EXXON group model grain

statistics and dynamics by a (/=64ôo-state) ferromagnetic Potts model (a gen-

eralization of the Ising model to more than 2 states), on which they carry Monte-

Carlo simulations. Every grain is characterized by a different orientation of the

Potts spin, ie. by a different value of /. All the energy is carried by the inter-

faces, where the "spins" are not "parallel", so that the energy is proportional to

the perimeter (ie. /?„) of the cell ]. The additional, physical constraint is simply

the energy

E P . S . = E (5.H)

This provides an alternative to Lewis's law in reducing by 1 the dimensionality of

the constraint space, namely *7

Rn=a(n-n,') (5.12)

as observed experimentally and in the simulation. One can show that at

(ie. with the Lagrange multiplier imposing the energy constraint (5.11) ôo), the

maximal entropy using alternative (5.12) is larger than that using Lewis's alter-

native (5.10) ®7. This example emphasizes the diagnostic power of the Maximum

Entropy formalism. A relevant physical constraint has been uncovered by notic-

ing the discrepancy between observed and ideal equations of state.

The most drastic reduction (by 2) of the dimensionality of constraint space, and

the largest entropy, would occur in a polycrystalline aggregate where all con-

straints are linearly dependant, ie. WH ~ An ~ n, or its equivalent in 3D. Such

scaling beween perimeter and area of cells is clearly impossible in 2D, but would

be possible in principle in 3D, if the cell boundaries were fractal (dendritic). But

then, reduction in the energy would no loager be the driving force for grain

growth and statistical equilibrium.

78

It is elementary to generalize Lewis's law to 3D random froths 1S

V> = (VJC)\\f-«f >-l/X)] (5.13)

but this relation has not yet been confirmed experimentally.

5.6 Evolution of a random froth. Von Neumann's law

Many random cellular structures evolve (slowly) in time. In soap bubble

froths, gas diffuses across the interface between bubbles. Biological tissues

undergo growth and cell divisions. In metallurgical aggregates, large grains grow

at the expense of small ones, in a controlled process called sintering, etc. The

time scale for this evolution is much longer than thbt associated with mechanical

response of the structure, so that it can be assumed that the random froth, once

in statistical equilibrium, remains in statistical equilibrium at all times. Conse-

quently, Lewis's law (5.10-13) is expected to hold throughout the evolution of the

structure. The parameters Ao, F, and especially the hitherto undetermined

Lagrange multiplier X, have their own, specific evolution.

The simplest case is the evolution of a 2D froth with constant total area Ao and

constant number of cells F. It refers to time intervals not long enough for bub-

bles to disappear (no T2 process) in a soap bubble froth, or for a cell to divide in

a biological tissue. Differentiating Lewis's law with respect to time, one obtains

the rate of growth of an averaged n-sided cell,

This result was actually derived by von Neumann 102, for 2D soap bubble froths

79

exclusively. (Actually, von Neumann's result for soap bubbles is slightly stronger

than eq.(5.14), as it involves the rate of growth of every individual bubble rather

than its average for all n-sided cells as in (5.14)). Von Neumann's derivation

relied on the physical mechanism of evolution of a soap bubble froth and on

three assumptions specific to this particular sytem, even though the final result

(5.14) is topological: A pentagonal cell loses area at the same rate as a heptagonal

cell gains it, and at half the rate of an octogonal cell, etc., regardless of their

geometrical shapes or areas. Our derivation 103 is completely general and topolog-

ical, so that von Neumann's law (5.14) is applicable to any evolving mosaic, and

generalizable to 3D random froths, since Lewis's law, and statistical equilibrium,

can also be generalized to 3D.

[ Von Neumann's derivation of (5.14) required a) a specific physical mechanism

for the evolution of a soap bubble froth: diffusion of the (incompressible) gas

across the interface between two bubbles, at a rate proportional to the pressure

difference between the bubbles, ie. to the radius of curvature of their interface,

and b) three, crucial assumptions: i) incompressible gas, ii) interfaces meeting at

120' on vertices, and iii) 2D system. Yet, the result is topological. ]

For a soap bubble froth, (A^/F)\ — 7 = (2x/3)&r > 0, where a is the surface

tension of the interfacial liquid, and 6, its diffusivity 102. 7 is therefore a constant

of the froth, invariant in time, at least as long as the interfacial liquid does not

change appreciably in thickness. This provides the physical interpretation of the

Lagrange multiplier X:

X measures therefore the ageing of the - -ructure (or, simply, the time). This is so

80

even if F, A4 and 7 evolve with time (mitosis, T2 process, etc.),

dt (5.15)

but then, the cells which do not evolve in time are no longer hexagonal as in

(S.14), but n |-sided, with n t given by

ft 1 d{AJF)

Computer-generated Voronoi froths with Poisson centres m are "as-quenched",

younger structures (X=l/6 f Lewis intercept »f l=0) than biological tissues 27>M,

which are "aged" (X=l/4, n,=2). Ageing affects also the appearance of the

structure, where cells become more isotropic and diflerentiated in sizes as time, ie.

X or n0 increases.

As Lewis's law, and statistical equilibrium can be extended to 3D froths, von

Neumann's law can also be generalized to 3D structures t03. In particular, over

short periods when C and < / > are constant in time, one has

with < / > given by eq.(5.2). The Lagrange multiplier in Lewis's law still meas-

ures ageing of the structure.

We end with a direct check of statistical equilibrium. From Lewis (5.10), and

<n>=6(5 .1 ) ,

* dt ~ it

81

From the space-filling constraint (5.8),

Si ~ ^p-rfT + ^ A- I T

Since Aa > 0 for all n, one obtains

that is, a stationnary distribution or statistical equilibrium.

6.7 Concrasiona

We have seen that statistical crystallography, a method to describe and clas-

sify the structures of .amorphous materials, can usefully be set up, along the

guidelines of established statistical thermodynamics M. It actually goes one step

beyond statistical thermodynamics, as one can take advantage of the arbitrari-

ness in the functional dependance of one of the constraints to maximize further

the entropy. The constraints (ie. the statistical ensemble) are easily selected:

They arc purely mathematical, pertaining to the topology of the space which the

sructure is filling. There may be additional constraints, which are then physical

or biological. In the absence of these additional, specific constraints, the random

structure is called ideal, by analogy with ideal gas in thermodynamics. Its equa-

tion of state (Lewis's law) is experimentally observable, as is the ideal gas law.

Deviations from Lewis's law betray the presence and structural relevance of

specific (physical, biological,...) forces, which are obviously worth identifying,

exactly as departures from the ideal gas law yield information, through the virial

expansion, on interatomic forces and, ultimately, on other phases of matter. As in

82

statistical thermodynamics, statistical equilibrium, and Lewis's law, are conse-

quences of the balance between entropy (most probable distribution), and the

lowest level of organization (space-filling, territorial partition, encoded by the

constraints).

Acknowledgements

Denis Weaire has kindly let me use hb cartoon (Fig.l) to illustrate topological

equivalence. I am very grateful to him also for all the arguments we have had

over the years about many points raised in these lectures. Figs.2-6 have been

taken from ref.[17], Fig.7 from ref.[S7].

Dorothy Duffy, Andrei Lissowski and Helena GUchrist have made substantial con-

tributions to the material presented in these lectures. Discussions with D.Bovet,

AComtet, N.R.DaSilva, F.Dowell, J.-M.Dubois, C.Eilbeck, H.L.Frbch,

J.P.Gaspard, M.Goldstein, O.J.Greene, H.Guentherodt, J.Jaeckle, R.Kemer,

M.Kleman, J.Malbouisson, R.Mosseri, T.Ninomiya, J.F.Sadoc, J.Sethna, J.Spalek,

G.Toulouse, M.F.Thorpe and D.E.G.WiUiams are gratefully acknowledged.

I am grateful to the CNLS and CMS, Los Alamos National Laboratory, and to

the organizers of the Latin-American Symposium on the Physics of Amorphous

Systems, Niterói 1084, for the opportunity to present and prepare these lectures.

Financial support by the British Council, CNPq, SERC and CNLS-CMS is ack-

nowledged with thanks.

83

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92

Phase Transitions in Glassy Materials

M.F. THORPE

Department of Physics and AstronomyMichigan State UniversityEast Lansing, HI 48824

U.S.A.

We examine the mechanical properties of covalent random networks

with high and low mean coordination. It is shown that networks with

high mean coordination (amorphous solids) have elastic constants de-

termined by the covalent forces whereas networks with low mean coor-

dination (polymeric glasses) have elastic constants determined by the

longer range molecular forces.

These ideas can be made rigorous by considering the number of

continuous deformations (i.e. zero frequency modes) allowed within

the network. In the transition from one kind of network to another,

rigidity percolates through the system. This leads to a picture in

which polymeric glasses have large floppy or spongy regions with a

few rigid inclusions. On the other hand in amorphous solids, the

rigid regions have percolated to form a rigid solid with a few floppy

or spongy inclusions.

1. Introduction

In this lecture we will summarize some ideas concerning the

nature of random networks. Although these ideas have not been firm-

ly established either theoretically or experimentally, there is

strong circumstantial evidence that they are correct. We present

these ideas in this paper. Much of this work has been published pre-

viously [1] although some comments relating to elasticity are new.

It is quite reasonable to consider a network consisting of long

polymer chains with a few cross links [2] to be quite different in

terms of its rigidity from a random network describing an amorphous

solid [3] like Si. The former can be deformed easily whereas the

latter is rigid. Ideas somewhat along these lines have led Phillips

[4-6] to postulate the notion of "overconstrained" and "undercon-

strained" glasses. His arguments attempt to delate the entropy and

strain at the glass forming temperature T with the average cuordi-

nation. While we have found his general ldoas to be insightful and

physically applealing, two objections must be raised.

The first Is that any discussion of entropy and strain at T is

bound to be imprecise as the processes taking place are complex and

poorly understood. However the important physical insights of

Phillips caj» be incorporated into arguments that are rigorous if the

following viewpoint is adopted.. Given a network structure at some

reasonably low temperature (< T_ where T_ is the Debye temperature

and T_. < T ), let us enumerate the number of wavs M in which theD g o

network can be continuously deformed with no cost in energy. This

is equivalent to asking for the number of zero frequency modes and

is a well posed mathematical problem. In this paper we argue that

94

for low scan coordination <r> the network is a polymeric glass in

which the rigid regions are isolated as sketched in fig. 1. As the

•can coordination <r> increases, these rigid regions increase in vol-

ume until <r> - r when a percolation transition takes place to a

rigid network or amorphous solid also sketched in fig. 1. Although

this «ay seem at first sight to be just another simple variant of the

percolation problem, albeit with a rather unusual quantity (rigidity)

doing the percolating, in fact the situation is rather more subtle

and complex. This is because the floppy regions contain both rigid

and floppy modes. Indeed the rigidity, associated with finite fre-

quency modes, and the floppiness, associated with zero frequency

modes, have many similarities with extended and localized modes

found in Anderson localization [7]; the finite frequency modes being

extended and the zero frequency modes localized. However, we prefer

to focus on the percolation aspects of the problem as it is the per-

colation of rigidity that drives the transition. In the next sec-

tion we develop a mean field theory which leads .to the transition

taking place at

r p-2.4 (1)

as previously obtained by Dohler et. al. [8].

The second objection is that whilst these ideas lead inevitably

to the concept of floppy and rigid regions and rigidity percolation,

they do not lead inevitably to the formation of molecular clusters

14-6]. What we are saying is that, ideas based on constraints can-

not be used as an agrument to support.the existence of molecular

clusters. On the other hand, the existence of such clusters is not

inconsistent with these ideas either.

95

The layout of this paper is as follows. In Che next section ve

focus attention on the number of zero frequency modes and rederive

the result r =2.4. We note that chis sane result can also be de-P

rived from a mean field theory of the cross linking of polymer chains.

In §3 we argue that this is really a percolation problem. The count-

ing of linearly independent constraints is crucial and not always

easy. In section four we set up a simple soluble model, that con-

tains more some of the features of interest. We show that a third

order phase transition takes place in this model. We comment on the

related problem of spin wave stiffness and infer the behaviour of the

elastic constants in networks.

2. Zero frequency modes

Imagine that a particular network with N atoms has been con-

structed with n atoms having r bonds

rSmall vibrations about this (equilibrium structure are described by a

1/2 £ ^ ( - ^ i , ) 2 + 1/2 Z e <A6 ) 2 , (3)

where Ar.. is the change in the nearest neighbour bond length be-

tween atoms <ij> and A9 . is the change in the bond angle between

two adjacer* bonds <ij> and <jk>. The a.., 6 are force constants

whose precise value will not concern us (an alternative form for the

angular part of eq.(3) such as that used by Keating [9] would not

affect the argument). These forces are known to be the most impor-

tant in covalent molecules and solids [10] and all the comments a-

part from those at the end of section four assume that the network

is described by the potential (3). We seek to calculate the number

96

M of vibrational nodes with zero frequency. These correspond to

ways In which the network can be continuously defomed at no cost in

energy.

In general the number of Bodes with zero frequency M is giveno

by the difference between the number of degrees of freedom 3N and the

number of linearly independent constraints N

M - 3N - N . (4)o c

These constraints are just the number of eigenvectors of the dynami-

cal matrix formed from eq.(3) that correspond to non-zero frequencies.

Put another way N is the rank of the dynamical matrix [11]. The

number of zero frequency modes N could be ascertained directly by

diagonalizing the dynamical matrix or by determining its rank. An-

other way is to estimate the number of constraints . These are;

(A) one per bond that is associated with the first term in

eq.(3) and

(B) for a 2 coordinated atom there is a single angular constraint

associated with the second term in eq.(3). Adding each additional

bond gives two more constraints because the angles with two existing

bonds must be specified. As we are only concerned with continuous

deformations, other discrete possibilities are ignored. This gives a

total of 2r - 3 linearly independent constraints around an atom with

r 2 bonds [8]. This has previously been overestimated [4,8] to be

r(r - 1)11 which gives 1, 3, 6 for r » 2, 3, 4 rather than the

Similar arguments have been used before. In the context of randomnetworks these go back to Weaire and Thorpe [12]; as applied to thevibrational problem see Weaire and Alben [13] and Sen and Thorpe [14]and particularly Thorpe and Galeener [15] where it was shown thatM - 3N - N, for nearest neighbour central forces only in networksWith r • 2, 3 and 4 where N. is the number of nearest neighbour bonds.

97

correct values 1, 2, 5.

Using the constraints associated with both terms in the potential

(3) we have

Mo « 3 j > r - £ n r [ r / 2 + <2r - 3 ) ] ,r r

which setting Mn = 0 yields

rp Z r £ r <>r r

Note that in applying eq. (6), dangling bonds (r = 1) are excluded.

They take no part in the connectivity of the network and can be pro-

gressively removed until no more exist. They are relevant in reduc-

ing the average co-ordination of the remaining network (e.g. H in

amorphous Si, see for example ref. 16).

For <r> > r , the network, is rigid and we refer to it as an amor-

phous solid; whereas for <r> < r , the network is not rigid and can be

macroscopically deformed. This is referred to as a polymeric glass.

Because r is not too much greater than 2, it is instructive to re-

examine an isolated polymer chains with N' atoms where,

M = N + 1.5n , (7)o a e

and n are the number of ends (two/chain). We can imagine a (math-

ematical) cross-linking in which 3 and A co-ordinated sites are form-

ed by fusing together pairs of atoms as indicated in fig. 2. In order

not to have any free ends, the number of 3 co-ordinated atoms n_ = n .

Each time a 3 co-ordinated atom is made, the total number of atoms is

reduced by one (costing 3 constraints to coalesce the atoms) and 2 ad-

ditional angular constraints are required. The situation is similar

when a 4 co-ordinated atom is made, except that 3 rather than 2 addi-

tional angular constraints are required. It there are N atoms when

this process is complete

98

• " Na -»3 ~ V (8)

and

i- 2(N - n, - n.)]/H, (9)

and M Q given in eq.(8) has been reduced to

Mô " Na + 1*5n3 " ° + 2)n3 " ° * 3) V

At this point it is convenient to introduce the quantity f,

which is the fraction of zero frequency modes in the system

f - MJ/3K (11)

and from eqs.(9)-(ll) we find that

f - (12 - 5<r>)/6, (12)

which is shown as the dashed line in fig. 3. The quantity f goes to

zero at <r> • r - 2.4 as before. Of course f cannot be negative and

so vithin this simple mean field scheme we have f = 0 for <r> > r .

This particular way of achieving the final result is somewhat arbi-

trary and other schemes could be used. Using polymer language,

<r> « 2.4 implies that 40Z of all atoms are cross linked if the cross

linkages are all of the type 3 shown in fig. 2 whereas only 20Z are

cross linked if they are all of the type marked 4.

3. Rigidity percolation

The ideas of the previous section argue that covalent networks

can be divided into two classes. Those of type I for which f > o

(polymeric glasses) and those of type II for which f • 0 (amorphous

solids). It is clear that by definition f cannot be negative. If,

in attempt to count the number of constraints, f should appear to be-

come negative, then the constraints are not all linearly Independent

constraints that reduce the number of zero frequency modes.

For a single long polymer chain, there is one bond and one angle

99

constraint per site so that f " i for <r> « 2. Note* that dangling

bonds, if they ever existed, are removed by progressive elimination

as mentioned in the previous section. For <r> close to 2, the ini-

tial slope df/d<r> * - ^ as <r> •» 2 is also given correctly be mean

field theory. However as <r> increases, mean field theory become in-

creasingly unreliable as <r> approaches r . This is because, mean

field theory overestimates the number of linearly independent con-

straints.

In fig. 3 we show in the solid curve the conjectured behaviour

of f against <r>. As <r> is increased from 2, the small rigid re-

gions grow in size until they percolate [17] at r . The quantity f

is not zero at r because snail floppy inclusions exist in the perco-

lated rigid region. The whole solid is rigid for <r> > r . The be-

haviour for <r> - r is harder to conjecture and for this we rely

heavily on the simple soluble model described in section four. This

would suggest that f and its first two derivatives are continuous at

r but the third derivative is discontinuous,P 3

d fr discontinuous at <r> • r . (13)

d<r>3 p

One of the main deficiencies of the simple model in S4 is that

it is completely random, whereas the network situation is much less

so especially when chemical ordering is taken into account (i.e. 2

co-ordinated atoms avoiding each other, etc.). It is harder to say

where the tail terminates and f may well be zero for <r> > 3. This

tall is somewhat akin to a Lifshitz tail [18] depending upon how

random the configurations are above r .

4. A simple soluble nodel

In order to understand better how the effects that we have been

100

discussing occur, consider the following simple model system. Mass

points m are placed at the sites of a (crystalline) lattice and their

motion is described by the Lagrangian

L -SiEaK:.)2 - !j£ a(z. - z.)2 Pii, (14)

j> J J

where z. is the displacement from equilibrium of the mass at site i in

a direction orthogonal to the dimensions that define the lattice geom-

etry. The sum over <ij> is over nearest neighbour pairs. The quanti-

ties p., = 1, 0 with probability p, 1 - p so that we have a bond per-

colation problem [17]. Note that every site is always occupied by a

•ass point. An example would be a square lattice defined in the xy

plane with particle motion in the z direction. Again we ask for the

number of zero frequency modes M . It is easy to see that there is

exactly one such Coldstone mode associated with each isolated cluster

[19-21] so that

Mo = W P ) > + N(1 " P)Z' (15)

where <Y (p)> is the average number of distinct connected clusters per

bond for bond percolation, N is the number of bonds and N(l - p) isb

the number of isolated sites. Therefore we have

* - z/2 <YC(P)> + (1 - P)Z (16)

and che problem of finding f is reduced to that of knowing <y (p)> for

this simple model. An identical result to eq.(16) would be obtained

for the fraction of zero frequency spin wave modes in a bond dilute

*Heisenberg ferromagnet .

In order to link up with our previous notation,, we will define

Another interpretation for f is that it is the ratio of the entropyat zero temperature to that at infinite temperature for a dilute bondspin h Ising model [22]. Each cluster behaves like a spin with twoorientations at zero temperature.

101

<r> • zp. (17)

The quantity <Y (p)> is not known analytically for any real lat-

tice although excellent numerical simulations exist [20J and its over-

all behaviour and critical behaviour is well understood. <y (p)>

plays the role of a free energy [19-21J for the percolation problem

and so f defined in eq.(16) also behaves like a free energy analogue

although of course there is no temperature defined in this problem.

In general f and its first two derivatives are continuous at p so

we may regard the transition at p as being third order [23] (in the

Ehrenfest sense) with

3 3d f/d<r> discontinuous at r . (18)

P

Put another way, a "specific heat" that can be formed from <Y (p)> by

taking two derivatives is continuous, but its derivative is discontinu-

ous. The specific heat has critical exponents [20],

o • a1 • - 0.60 ± 0.1 in two dimensions,(19)

a » a1 » - 0.58 ± 0.11 in three dimensions,

which correspond to a cusp.

In order to have an analytic expression for f for this model we

have examined it on a Bethe lattice. Using the formalism of Fisher

and Essam [24] we find that for p < Pc(i.e. <r> < r )

f - 1 - <r>/2, (20)where zp - r • z/(z - 1). The form (20) is universal for all z.

c pFor p > p , (i.e. <r> > r ) the result depends on z. For z * 3,

f - [(3 - <r>)/<r>]2 (<r> - l)/2 (21)

and for z • 4,

f • [(4 - <r>)/<r»]4 { <r> - 4 + [<r>(16 - 3<r>)]>i } /2. (22)

These results are shown in fig. 4, where the values of r are marked

102

by crosses as a discontinuity in the third derivative is not apparent

to the eye!

For a real lattice vith rings etc. like the square net, it is

possible to count the small clusters and show that for <r> < rP

f - 1 - <r>/2 + (nz/2s)(<r>/z)s + 0(<r>s"1), (23)

where the first two terms are universal and the leading corrections

depend on the size of the smallest rings which have s bonds with n of

them through each edge (2 for the square net). For <r> < z, we have

f - [(2 - <r>)/<r>]Z, (24)

which is obtained by isolating a single site by removing the z bonds

around it. For the square net the value of <Y (p)> at the critical

point is known (25] and from eq.(24) leads to f - 0.16 at r . We have

used this and the results (23) and (24) to sketch the result for the

square net shown in fig. 4.

Adding the next bond may either reduce the number of zero fre-

quency modes by 1 or 0. For small <r>, it will almost always be 1 but

as the density of bonds gets larger it may be 0 if the two sites, as-

sociated with the bond, already belong to a connected cluster. This

means that

- h <_ df/d<r> £ 0, (25)

where the lower value (— h) is attained for small <r> and the slope

gradually goes to 0 as <r> •* z.

The important aspect of this simple model is that there is a

phase transition of the percolation kind 117]; that this phase transi-

tion is third order; and that the behaviour of f with <r> is rather

universal. It is interesting to note that the result for the linear

chair, eq.(20) for <r> <l, is linear all the way like the mean field

103

result of §2 whereas the Bethe lattice results show more curvature

as z increases. The value of f at r lies between 0.16 and 0.33 forP

the two-dimensional networks shown in fig. 4.

It is clear that it would be almost impossible c- determine the

high order derivative in f experimentally, even if some method of

measuring f were to be found. However, fortunately there is another

quantity that manifests the phase transition much more directly and

is easy to measure experimentally. The model described in this sec-

tion is isomorphous to the Heisenberg ferromagnet which has a dis-

persion relation for spin wave excitations

« - Ak2 (26)

where u is the frequency, k the wave vector and A is the spin wave

stiffness. The spin wave stiffness is zero for <r> < r as the sys-

tem is made up of finite islands that cannot communicate with each

other. For <r> > r , the spin wave stiffness increases until it ob-

tains its pure system value at <r> « 4.

By analogy we expect similar behaviour for the elastic constants

in the rigidity percolation problem, with the elastic constants beingzero for <r> < r and finite for <r> > r . This has been demonstrated

P P

in computer experiments on a triangular net with bonds removed at

random [26], The transition takes place at p = — which is well be-

fore the lattice falls apart which is at the percolation concentration

p - 0.3473 where p is the fraction of bonds present. This is be-

cause certain configurations that are effective in carrying current

are ineffective in transmitting rigidity information. We expect sim-

ilar effects in 3D random networks.

Of course in reality the elastic constants in the low coordina-

104

tion polymeric glasses will not be zero but will be determined by

the weaker force constants of longer range. These have values typi-

cal of molecular crystals. In the high coordination amorphous solids,

the elastic constants will be higher and determined by the covalent

forces [i.e., the a and @ terns in equation (3)].

5. Conclusions

We have presented a new framework within which to look at the

differences between random networks with high and low mean co-

ordination. The original ideas of Phillips 14-6] focused on the free

energy F » E - TS at the temperature T - T at which the network is

formed. This is an extraordinarily complex quantity involving both

the energy E and the entropy S. We have adopted a different and

simpler viewpoint in this paper. Given a network at some tempera-

ture, we do not worry about how it was formed or what is its energy

or entropy. We ask only if it can be continuously deformed. This

is a well posed mathematical problem.

We have shown that a mean field theory of rigidity percolation

leads to a linear dependence off on <r> and gives r « 2.4. The

network is divided into two kinds of regions that are designated

rigid and floppy as shown in fig. 1. For <r> < r , we have a poly-

meric glass whereas for <r> > r we have an amorphous solid.

We have shown that the key quantity if f, the fraction of zero

frequency modes. This can never be negative and avoids the mathe-

matically undefined notion of an "overconstrained network" {4]. In-

stead the apparent constraints become more and more linearly depend-

ent as r is approached from below. Indeed it is untrue [4] to say

2that strain builds up like R (where R is the radius) when hand built

105

models are constructed; instead the strain is held within the small

rings and does not propagate. Thus it is possible for both a poly-

mer with a few cross links and a tight random network like amorphous

Si to exist. The discussion then passes to the zero frequency modes.

Whether or not such infinite random networks do actually exist in

nature is beyond the scope of the arguments given in this paper.

We have examined a simple model for which f can be calculated

as a function of <r>. The model shows that there is indeed a phase

transition but that it is the third derivative of f that is dis-

continuous. If this result transfers to random networks, it means

that attempts to locate r by numerically diagonalising dynamical

matrices, or by finding their rank, are doomed. It will be necessary

to examine other quantities such as the elastic constants that are

expected to be more singular at r .

I should like to thank the O.N.R. for supporting this research.

I should also like to thank the organisers of the Latin American

Symposium on Amorphous Materials for a very pleasant week.

106

References

til M.F. Thorpe, J. Non-Crystalline Solids _57_, 355 (1983).

[2] P. Flory, Statistical Mechanics of Chain Molecules (Wiley,Mew York, 1969)-

[3] D.E. Polk, J. Non-Crystalline Solids 5 (1971) 365.

[4] J.C. Phillips, J. Non-Crystalline Solids 34 (1979) 153; 43(1981) 37.

[51 J.C. Phillips, Phys. Stat. Sol. (b) 101 (1980) 473.

{6] J.C. Phillips, Phys. Today 35 (1981) 27.

17) P.W. Anderson, Phys. Rev. 109 (1958) 1492.

18] G.H. flbhler, R. Dandoloff and H. Bilz, J. Non-CrystallineSolids 42 (1980) 87.

[9] P.N. Keating, Phys. Rev. 145 (1966) 637.

[10} K. Nakamoto, Infrared and Raman Spectra of Inorganic andCoordination Compounds (Wiley, New York, 1963).

[11] R.A. Frazer, W.J. Duncan and A.R. Collar, Elementary Matrices(Cambridge, England, 1938) p. 18.

[12] D. Weaire and M.F. Thorpe, Phys. Rev. B4 (1971) 2508.

[13] D. Weaire and R. Alben, Phys. Rev. Lett. 29 (1972) 1505.

[14] P. Sen and M.F. Thorpe, Phys. *ev. B15 (1???) ^ O .

[15] M.F. Thorpe and F.L. Galeener, Phys. Rev. B22 (1980) 3078.

[16] M.H. Brodsky, M. Cardona and J.J. Cuomo, Phys. Rev. B16(1977) 3556.

[17] J. Essam, in: Phase Transitions and Critical Phenomena, Vol.2, eds., C. Domb and M.S. Graen (Academic Press, New York,1974) p. 197.

[18] I.M. Lifshitz Adv. Phys. 13 (1969) 483.

[19] C M . Fortini and P.W. Kastelyn, Physica 57 (1977) 536.

[20] S. Kirkpatrick, Phys. Rev. Lett 36 (1976) 69.

[21] M.F. Thorpe and S. Kirkpatrick, J. Phys. A12 (1979) 1835.

[22] A.R. McGurn and M.F. Thorpe, J. Phys. C 12 (1979) 2363.

107

[23] A.B. Pippard, Classical Thermodynamics (Cambridge, England,1966).

124] M.F. Fisher and J.W. Essaj, J. Math. Phys. 2 (1961) 609.

[25] H.N.V. Temperley and E.H. Lieb., Proc. Roy. Soe. A322 (1977) 251.

[26] S. Feng and P.N. Sen, Phys. Rev. Lett. 52,, 216 (1984).

108

X Polymeric Gloss

Amorphous Solid

Fig.l — The rigid and floppy regions in networks of type I

(polymeric glass) and type II (amorphous solid).

Fig.2 - Isolated polymer chains can be cross linked to form

3 and 4 co-ordinated atoms by fusing pairs of atoms

together.

109

40

Fig.3 - The fraction of zero frequency nodes f versus mean

co-ordination <r>. Rigidity percolation occurs at r .

The dashed curve iV a result of a mean field calcula-

and the solid curve is a sketch of the conjectured be-

haviour with the transition taking place at the cross.

os

. 04 •

Fig.4 - The behaviour of f with <r> for the simple model of

§4. The dashed line is for the linear chain. The

transitions in the solid curves take place at the

crosses. The Bethe lattice results are exact and the

result for the square lattice is estimated (see text).

METALLIC GLASSES: STRUCTURAL MODELS

E. Massif, Depto. de Física, Facultad de Ingenierla, U.B.A.

Paseo Colon 850 (1063) BUENOS AIRES, ARGENTINA

INTRODUCTION: Many models have been proposed in the last

few years to describe the structure of metallic glasses.

These models can be regarded as microscopical descriptions

of the atomic arrangement in the system considered, in

which the fundamental unit is formed by a single atom or a

group of atoms. In order to check the validity of any of

these models, the ability of the proposed structure to re-

produce the main macroscopical features of the analized

system must be tested. From a structural point of view our

most important macroscopical parameters in the case of a

disordered system are basically two: the structure factor

S(Q) and any of the atomic distribution functions [RDF(r) ,

g(r) or G(r)]. To define therefore the efficiency of a

definite kind of structural model, it is necessary to know

accurately the experimental values of those functions, as

well as their partial components from which they can be ob-

tained as a linear combination.

As we shall see this has been possible only in a few

experimental investigations up to now, making difficult to

draw conclusions about the fit of the different known mo-

dels to the atomic structure of the studied system. The

aim of this talk is to give a summary of the attempts made

up to the present in order to describe by structural models

the atomic arrangement in metallic glasses, showing also

why the structure factors and atomic distribution functions

cannot be always experimentally determined with a reasonable

accuracy.

STRUCTURE FACTORS AND DISTRIBUTION FUNCTIONSj The Structure

factor for a one-component system can be defined as:

S(0) * Icoh(q)/Nf2(Q) (1)

Ill

where Q= |{J'-§0|=4ir senÔA is the absolute value of the sca-ttering vector, X is the radiation wavelenght, 20 the sca-ttering angle and ICoh(Q)/N the coherently scattered in-tensity per atom.

- < |A(Q)|2> - f2(Q) < X 2 exp{-i(5(rrrk)>>J *

(2)

where f(Q) is the atomic scattering factor, A(Q) is the am-plitude of the scattered radiation and rj the position ofthe atom j. (The brackets < > denote the statistical avera-ge) .

In the case of a two-component system, there are threedifferent definitions for the structure factor, followingthree different formalisms. Considering the definition ofAshcroft and Langreth (1) we have, for the total structurefactor of a binary alloy:

iíüLílíül S-LÍ» (3,N<f2(Q)> TJ I j <f2(Q)>

where <f2(Q)> = Cjff(Q) + c2f|(Q) being CJJ the atomic con-centrations of the atoms of the species i,j respectively andSjj(Q) the partial structure factors, which verify:

Mm S^(Q) » 0 ; 11m sft(Q) -Mm. S^Q) = 1r ^ ti»j ^^

In the Faber-Ziman formalism (2) the structure factor

can be written as follows: r

SFZ(Q)N

- «f2(Q)> -

22 CfCt f f(Q) fj(Q) . -

- i t J J sff(Q)• J <f(Q)>2 J

with <f(Q)> = Cif^QJ + CafziQ) and where {<f2(Q)>-< f(Q)> 2} *" Cîfi-fa)2 i s a term usually known as the "Laue monotonediffraction", and the partial structure factors verify:

112

"m S^(Q) - 1 Yi.j 1.J - 1,2

Comparing both formalisms one can easily state that(SAL(Q) -i)/(sFZ(Q) -i)<i and TJin SFZ(Q) = TJn SAL(Q) = 1 whichmeans, that the Faber-Ziman formalism is more sensitive tothe oscillations of the structure factor. The Aahcroft-Langreth formalism,1s, on the other hand, simpler, moresensible to concentration fluctuations and with the advan-tage that it does not diverge when <f> * 0 , which occurs inthe case of negative scattering lengths usual in some neu-tron diffraction experiments.

A third description of the structure factor is thatgiven by the Bhatia-Thornton formalism (3) in terms ofwhich the structure factor can be written as follows:

<f2(Q)>

<f(Q)>2 , ^ i j W ^ Q ) }SNM(Q) + SCC(Q)

<f2(Q)> NN <f2(Q)> "

ifi(Q)-f2(Q)>

where SNN(Q) , SCC(Q) are, respectively, the partial struc-ture factors of the correlations between number densityfluctuations and of the correlations between concentrationfluctuations; SNç(Q) is the partial structure factor of thecross correlations between concentration fluctuations anddensity fluctuations. They can be expressed in the form»

SNN(Q) " f <N*(Q) «(Q)> ; SCC(Q) » Ç <C*(Q) C(Q)>

SNC(Q) - Re <N*(Q) C(Q)>

being N(Q) and C(Q) the Fourier transform of the local de-viation in the total number density and In the concentra-tion, respectively. It can be shown that:

Mm SBT(Q) - 1 • 11m SNNCQ) - 11m SCC(Q) ; 11m SNC(Q) • 0

113

Comparing the Bhatia-Thornton formalism with those pre-viously seen, it can be stated that the total structure fac-tors are numerically identical in the BT and the AL forma-lisms, being however the interpretation of the structure interms of the partial structure factors completely different.The Bhatia-Thornton formalism is particularly useful to des-cribe the structure of substitutional alloys, in which theatomic size of the components is almost equal (R1âfR2) andcan be verified SNC«0 . Since the partial structure factorof the correlations between concentration fluctuations. Sec»is extremely sensitive to those fluctuations,these formalismis also particularly effective in those systems with a strongchemical short range order (CSRO). (In the Paber-Ziman for-malism, on the other hand, one has for substitutional alloys

sH-sH-sB) .Whereas the total structure factors are almost directly

available from the experimental data through Icoh(Q) (whichis indeed the measured difracted intensity of the radiation,subsequently corrected for sample absorption, polarization,incoherent or Compton scattering and finally normalized (4)),the distribution functions have a more complicated mathema-tical treatment from experimental data but their physicalmeaning remain much more evident than that of the structurefactors. The so called radial distribution function RDF(r)defined as:

RDF(r) * 4*r2p(r) (6)

(where p(r) is the local atomic density at a distance r

from a given atom), measures the number of atoms in a sphe-

rical shell of radius r and thickness unity, and verifies:

Mm RDF(r) * 47rr2p0

(with PQ being the mean atomic density of the glass). Byintegration of the RDF(r) between two consecutive minima,one can compute the coordination number, that means, thenumber of atoms located between distances R1 and Ru from

114

a given atom:

2 P » Í " RDF(r) dr (7)JR1

where Rl ,Ru define usually the minima just by a Maximumof the RDF(r) determining the p-th shell and therefore thep-th coordination number Zp . Although the RDF is quiteuseful, the pair probability function g(r) is more closelyrelated to the scattering pattern (as we shall see later)and is defined as follows:

g(r) - (8)

where Mm g(r) = 1 .Thr most frequently used distribution function is, ho-

wever, the so called pair correlation function (also calledthe reduced atomic distribution function) G(r), which canbe expressed, following the Faber-Ziman formalism, as:

6(r) = 4*rP0lg(r) -1} = 4wr U>(r) -/>„] (9)

verifying 11m 6(r) » 0 .This atomic distribution function has the advantages,

In comparison with those previously seen, that the densityfluctuations can be more clearly appreciated and that hisexpression as function of the structure factor is simpler,which can be shown in the following:

6(r) « 4rr»0íg(r) - 1] = \ Í"QÍS(Q) - 1] sen (Q.r) dQ (10)Jo

S(Q) • 1 • **/><> frtg(r) -1] s e nW' r) dr

H d r (11,

The total pair correlation function can be writtenalso as a function of the partial terms like:

115

(12b)

<f>2 ^ j t j z )GNN(r) + ~ ~ S — 6 c c ( r )

(fx-f2)GNC(r) (12c)(f2>

in the Faber~Ziman, Ashcroft-Langreth and Bhatia-Thornton

formalisms, respectively.

Two important features must be remarked:

i) the expressions of the total structure factor and the

total pair correlation function as function of the partial

contributions in the Ashcroft-Langreth formalism are asy-

mmetric, and

ii) the partial distribution functions ( gf.(r-) ,/t>;j(r) and

Gjt(r)) are the same in the Faber-Ziman and the Ashcroft-

Langreth formalisms. This is not true, however, for the

partial structure factors.

The GNN(r) partial pair correlation function in the

BT formalism represents the topological short range orde-

ring (TSRO), whereas Gcc(r) are related to the chemical

short range ordering (CSRO) and Gtyçfr) represents the size

effect, which is caused by different atomic volumes of the

components (see, for example, Wagner and Ruppersberg (5a)

or Chieux and Ruppersberg (5b) ) , and equals zero in the

case of substitutional alloys.

Unfortunately, the distribution functions, being one

dimensional, are not capable of discriminating between clo-

sely related models of the structure of metallic glasses,

such as the dense random packing of hard spheres or the

polyhedral model (that we shall see in the second part of

this talk), and more details about the structure must be

determined. For this reason, it is of the greatest impor-

116

tance to calculate the partial structure factors (whichmeans also the partial distribution functions), in orderto determine the local atomic arrangement in metallic gla-sses, In the case of binary alloys, three partial func-tions, SU(Q) , S12(Q) and S22(Q) (°* their correspondingSfw(Q) * SNC(Q) andSçç(Q) ) must be calculated. This can bedone quite readily, in principle, by measuring three inde-pendent total scattering functions S(Q) and solving forthe three unknowns S|j(Q) or S ^ Q ) , since S(Q) is theweighted sum of the three partial functions» The weigh-ting factors depend upon the atomic concentration c; andthe scattering factor fj of the element i in the alloy,as we can see following, for example, eg. (4):

(13)

with;

wfj c | C j

(Faber-Ziman formalism).

in matrix notation like:

with:

IT(Q)J

Equation (13) can be re-written

(14)

Sn(Q)

S22ÍQ)Si2(Q)

MQ)J«22 »Í2

W3W12

where [R(Q)J corresponds to the three partial structure

factors, (w(Q)] represents the weighting factors for di-

fferent scattering abilities, and [T(Q)1 are the total

structure factors observed experimentally for each indepen-

dent event. The solution for the partial structure factors

can be written as:

1R(Q)I - (w(Q)J-1 [T(Q)] = [V(Q)] tT(Q)] (15)

117

A unique solution is found for [R(Q)1 only if the de-terminant of [w] is different from zero. A measure of howgood the equations system is conditioned is given by |w|f, ,the normalized determinant of the system of three equationswhose value can be, at most 1. Unfortunately, in most ex-perimental cases, the normalized determinant for metallicglasses is very small, giving thus an indication of the di-fficulties to be expected in order to solve equation (15).That means, even very small experimental uncertainties leadto drastic uncertainties in the resulting partial func-tions, making therefore very difficult to decide whethera definite structural model describes the atomic arrange-ment of a glassy metal in a better way than others.

In order to separate the partial structures measu-ring the total structure factors by three different expe-riments, several methods are used. In principle they canbe divided into four kinds:

a) the three different radiation techniques using X-rays,neutrons and electrons (see, for example, Paasche (6));

b) the isotope-enrichment technique for neutron diffrac-tion in which the scattering power of the constituents arevaried by using different isotopes (see Sperl (7));

c) the polarized neutron technique which is applicable tomagnetic materials; d) the anomalous scattering techniquefor both X-rays and neutrons (see Waseda (8)) and of course;e) any assortment of the above techniques as, for example

a combination of X-rays with the isotope-enrichment tech-nique (Nold (9)) or a combination of X-rays with the pola-rlzed-neutron technique (see, for example, Sadoc and Dix-mier (10)). From all these attempts to separate the par-tial structures the most succesful was that performed bySperl (7) for the NÍ8iB19 metallic glass, using the isoto-pe enrichment technique for neutron diffraction, who had avalue of 0.51 for the determinant of ]wjn •

With the improvement of the experimental techniques,better data can be obtained because of the new develop-ments of more intense sources and efficient detectors.

118

Some of this techniques (energy dispersive X-ray diffrac-

tion (EDXD) or tine of flight measurements with pulsed neu-

trons) enable us to carry out measurements of the scatte-

red intensities up to Q-40A'1, which permits the evalua-

tion of high resolution Fourier transforms which can pro-

vide a more detailed description of nearest neighbour in-

teractions in binary metallic glasses. Measurements of

extended X-ray «absorption fine structure (EXAFS) and Moss-

bauer spectroscopy provide us a rather accurately informa-

tion about the local atomic structure. For a more detai-

led description of the separation methods and of these new

techniques see, for example, Waseda (8), Hafner (11) or

Egami (12).

STRUCTURAL MODELS: A comparison between the structure of

the amorphous state and that of the liquid state shows

(see, for example, Waseda (8)) that the general features

of the structure of the former are similar to those of

the latter, except for a shoulder on the second peak ob-

served in both S(Q) and g(r) in the amorphous diagrams.

From the similarity of the gross features of the S(Q)

and g(r) for metallic glasses, the fundamental configura-

tion of atoms should be considered as liquid like. On

the other hand, the ratio (r2/rj) between the positions of

the second peak (r2) and of the first peak (r\) in the

amorphous state (about 1.67), is rather similar to the va-

lues obtained for the c/a ratio in close packed hexagonal

structures (1,63) and for the (r$/r\) ratio in a fee

structure (1.73), falling rather far away from the values

of (r2/»*i) and {r$/r\) for the liquid state (about 1.86

and 2.70 respectively). This Implies that the short ran-

ge order of near neighbours In metallic glasses is affec-

ted more or less by the atomic arrangement of the crysta-

lline state.

From these considerations the two basic tendences

that exist in the formulation of structural models for me-

tallic glasses can be well understood: on one side (1) the

mlcrocrystalllne disorder models (also called stereocheml-

119

cally defined models), on the other side (ii) the topologi-cal disordered models.

i) Microscrystalline models

These models are based on the similitudes that have

been observed between the short range order in metallic

glasses and the corresponding crystalline phase at the sa-

me concentration or in the same concentration range. Hama-

da et al (13) proposed a structural model for the Fe-B me-

tallic glass, consisting on "crystalline-embryos" with a

bcc structure (similarly to crystalline Fe3B) surrounded

by a statistical atomic distribution in the boundary re-

gions between the embryos.

The total structure factor S(Q) calculated with

this model reproduces acceptably some structural features

seen with the experimental S(Q) like the reduction of the

shoulder on the right hand side of the second maximum of

the structure factor with increasing boron content.

A good agreement between the model and the measured

data was also observed in Aglf8Cu52 (14) and Fe75P25 (15)

metallic glasses, proposing in both cases a fee structure

for the crystalline embryos.

Kuhnast et al (16) presented a microcrystalline model

to explain the structure of amorphous NiçgB3l, , starting

from the crystalline phase NÍ3B with which he could success-

fully described the local ordering for r<6A . Although a

good agreement could be found in all these cases, efforts

to obtain agreement for other materials have been unsuc-

cessful. Cargill (17) has showed that in order to fit the

experimental structure factors with these models, too ma-

ny parameters (such as sizes and strain distributions of

the mlcrocrystals) must be varied.

We can now summarize the basic properties of the mi-

crocrystalline models, following Gaskell (18):

i) In a A. B alloy, one of the possible local configura-

tion of A and B atoms is preferred basically on energetic

grounds.

ii) This arrangement thus represents the dominant coordi-

nation polyhedron over a wide range of concentrations CA

120

and radius ratios IVR8 ' being the coordination number of

the B species (the smaller atoms) relatively independent

of these quantities.

iii) Differences in structure with concentration and cha-

racter of A and B atoms, and the distinction between amor-

phous and crystalline phases, are described by variations

in the way local structural units are interconnected,

iv) There is a potential local structure equivalence of

the crystalline and amorphous phases.

v) In this kind of models, the long range order dissapears

because of the random orientation of the microcrystals.

Historically, the validity of such "embryo-models" was

supported by some X-ray small angle scattering and trans-

mission electron microscopy measurements that showed the

existence of a number of inhomogeneities in the amorphous

state.

ii) Topological disorder models

These structural models (often called homogeneous di-

sordered models) can be separated in two main groups: a)

dense random packing of hard spheres (DRPHS); b) dense ran-

dom packing of coordination polyhedra (DRPCP).

a) DRPHS Modelst They are the natural extension of the

structural models originally employed in the description

of monoatomic liquids and are based on the general simila-

rities of S(Q) and g(r) between the liquid and amorphous

states. The starting point of all these models is the hard

sphere model first proposed by Bernal (19). The spheres

are dense in the sense that no "internal holes" great

enough to accomodate another sphere can be found, and

they are at random because only weak interactions are pre-

sent between spheres separated by four or five atomic dia-

meters. That means, there are no regions of long range or-

der as in the crystalline structures.

In Bernal1s model, the topology is described in terms

of polyhedra (also called "Bernal holes") in which the ver-

tices are defined by the sphere centers. Bernal found 5

different types of polyhedra: 1) tetrahedta, 2) octahedra,

121

3) archimedean trigonal prisms, 4) archimedean antiprisms,

5) tetragonal dodecahedra. The introduction of archimedean

polyhedra prevents from any long range order and from the

realization of the ideal structure precisely. This model

was later improved by J.L. Finney (20), who obtained the

radial distribution function for a dense random packing of

about 8000 hard spheres with much better resolution than

previously available.

Finney's model was used to fit the experimental va-

lues of the reduced radial distribution function G(r) ob-

tained by Cargill (17) for the NiygP^ metallic glass.

The results were rather satisfactory since we take into

account that only one parameter namely the atomic size

(R^jsRp) was varied.

Polk (21) proposed a model consisting of a Bernal

structure (DRPHS) which is primarily metallic, with the

metalloid (the smaller atoms) filling some of the larger

holes inherent in the random packing. Polk suggested

that it is this special relationship between the metal

and the metalloid which could lead to the stabilization

of the amorphous structure. Unfortunately, none of the

holes are as large as originally believed by Polk, not

allowing, therefore, their filling up by the metalloid

atoms. Polk (22) later generalized his view of the DRPHS

void-filling model to allow the metal atoms to occupy

random packing structures somewhat less dense than those

of Bernal and Finney, which should provide more larger

holes to accomodate the metalloid atoms.

Ichikawa (23) constructed a DRPHS model suggesting

some modifications of Bennett's criterion (24). Bennett's

algorithmus consisted on adding spheres to 'an initial equi-

lateral triangle formed by three spheres, enumerating all

possible sites for which an added sphere would be in hard

contact with three spheres already in the cluster, but

would not overlap with any of them and selecting among

them the nearest to the center of the cluster. In his mo-

del, Ichikawa introduced a measure of the tetrahedral per-

fection of the pocket formed by three spheres, defined as:

T = r^/ÍRj+Rj) (16)

where rf*x is the maximum distance between the centers of

spheres i and j in the distribution of three spheres, and

R} and Rj are their radius respectively. In this model

the assembly of hard spheres with diameter T is conside-

red, in which the tetrahedron of four spheres is first

constructed. Then the other spheres are arranged in a

growing process of tetrahedral clusters, under the condi-

tion that the desired sphere occupies only the position of

having been in contact with three spheres of surface within

the distance T - Thus, deviation from the condition T=1.0

allows the formation of a slightly deformed tetrahedron.

Ichikawa suggested that the model calculation with the pa-

rameter T=1.2 (which corresponds to a relatively rigid

packing of tetrahedrons) is more compatible with the expe-

rimental data for the amorphous state. He showed also that

the packing became obscure for T=2.0 , that is the funda-

mental tetrahedra are distorted, and in this case the spli-

tting of the second peak in both S(Q) and g(r) is no

longer observed, a fact that is in good agreement with the

experimental results for the liquid state. This means

that the amorphous structure requires something to be

added to the simple model structure of hard sphere liquids.

In fact more rigid configurations of atoms must be consi-

dered, this corresponding to the introduction of the para-

meter for tetrahedral perfection.

The main problem of this kind of model is that if

atomic arrangements in the amorphous metal-metalloid

alloys were truly like those of spheres in such dense

random packings, then the metalloid atoms would have the

same average surroundings as the noble or transition metal

atoms, a fact that, as showed by several experimental obser-

vations, is unlikely. Binary DRPHS models with smaller

spheres representing metalloid atoms rand with no metalloid-

metalloid nearest neighbours is probably a more realistic

structural model for the metal-metalloid amorphous alloys,

than simple dense random packing of equal size hard spheres

123

with metal and metalloid atoms occupying random sites. Such

a model was proposed by Sadoc etal (25) who used Bennett's

algorithm to construct hard sphere packings with two sizes

of spheres in which the smaller spheres, representing meta-

lloid atoms, were not allowed to occupy adjacent sites*

The introduction of small spheres requires the large sphe-

res to form a looser random packing to provide enough lar-

ge holes to accomodate the small spheres. This tendences

could be even observed in the liquid state, as suggested

by this author (26) for the Ni-B melt.

Cargill and Kirkpatrick (27) obtained a rather good

fitting of the experimental RDF(r) with a binary DRPHS

model using Bennett's global criterion for rare earth-

transition metal alloys, namely Tb3 3Pe6 7,Gd36Fe6it and

GdigCo62* Waseda et al (25,28) used a binary model based

on the previous works from Ichikawa (23) and Cargill and

Kirkpatrick (27), to describe the experimental behaviour

of the partial structure factors Sjj(Q) and pair distri-

bution functions g|j(r) for amorphous Fe-P and Cu-Zr.

An important improvement of binary DRPHS models was that

made by Connell (29) and von Heimendahl (30) who relaxed

the model through energy minimization of the assumed pair

potential. It is important to note that the relaxation

does not conserve the original topology, which means that

one has to make a topological analysis after the structu-

re was relaxed. Boudreaux (31) relaxed a DRPHS model using

a Lennard-Jones potential for the Fe-B system at different

concentrations finding a coordination number of 6.6 around

the boron atoms. This leads Finney (32) to propose a mo-

del based on local structural units (trigonal prisms) with

six iron atoms around a boron one, like in crystalline

Fe3B (cementite structure) with the rest of the boron

atoms distributed as a "statistical adhesive" between the

molecular units. Some investigators (see, for example,

Gaskell (18)) questioned this criterion, pointing out

that the model gives a poor information about that "statis-

tical cement". Another improvement of the DRPHS models

has been carried out through softening of the hard spheres,

allowing them to oscillate around their equilibrium sites

with a defined frequency which corresponds to a given tem-

perature. This leads to a broadening of the first coordi-

nation shell in tha pair correlation functions. With such

dense Random packing of about 5000 soft spheres, Blétry

(33) obtained rather satisfactory results to fit the very

accurately experimental results of the partial structures

determined by Sperl (7) with the NiaiB19 amorphous

alloy. Other calculations based on the same experimental

data were made by Beyer and Hoheisel (34) using molecular

dynamics and taking a Iennard-Jones potential for a sys-

tem under conditions similar to those of the liquid state.

Though this model made reasonable predictions for the va-

lues of the GNf_f|f(r) and GNJ_B(r) partial pair correla-

tion functions, the values of the experimental Gg_g(r)

show considerable differences with those predicted by the

model. Furthermore, the assumption that the glassy alloy

could be handled as a binary liquid alloy on thermal equi-

librium at high temperatures seeias to be unlikely,

b) DRPCP models; The DRPHS models seen before start from

the similarity of the pair correlation functions in the

liquid and in the amorphous states. While this remains

obviously true for the long and medium range structure,

the more recent diffraction experiments support the point

of view that the short range order in the amorphous phase

may be similar to the crystalline structure. Thus one

may begin with larger local units having the desired coor-

dination and a topology borrowed from the crystalline

structure. The model is built by a random stacking of tho-

se local units. In this senss some investigators (see,

for example, Gaskell (13)) supported this kind of models

because they allow a better description of the CSRO (che-

mical short range order) than the DRPHS models. The

opposite argument is that the danse random packing of

coordination polyhedra models doesn't take into account

satisfactorily the random elements , namely the disorder.

Two of the most important D?J?CP models are those proposed

by Takeuchi (35) and Wright (36) for monoatomic systems

125

based on geometrical considerations. As showed by Hoare

and Pal (37), the icosahedron is the I3-atoms cluster con-

figuration with the lowest energy. The icosahedron can be

separated in 20 almost regular tetrahedra. Considering all

atoms of the icosahedron as equivalent, the whole amorphous

structure can be idealized as an array of tetrahedral units.

The model is constructed in such c. way that each new tetra-

hedron is put above one of three faces of the previous te-

trahedron so that the vertex of the new tetrahedron comes

to the farthest distance from the origin. The average

coordination number obtained in this way for the first

shell is about 13.4 which agrees rather well with the va-

lues obtained experimentally for most of the metallic gla-

sses. Sperl (7) presented a very good agreeement between

the normalized atomic distances (rj/n)f»if2... , obtained

from his experimental partial pair correlation function

GNj_Nf(r) and the values taken from Wright's (36) and Ta-

keuchi's (35) models. Other DRPCP models considering lar-

ger fundamental units have been presented, such as that of

Kuhnast et al (16), who proposed a dense random packing of

icosahedral units to describe the structure of amorphous

Ni71B2g. Finally, another important DRPCP model, succes-

fully employed for the Pd-Si amorphous alloy, is that pre-

sented by Gaskell (38) who takes as fundamental units a tri-

gonal prism (with 6 Pd atoms around a Si atom) as well as a

trigonal prism capped with three half-octahedra (9 Pd atoms

around a Si atom) with a similar topology as the cementlte

structure (crystalline Fe3C). The model was relaxed using

Lennard -Jones potentials and supplementary conditions to

maintain the coordination after the relaxation procedure.

This structural model was also employed to reproduce the

above mentioned experimental data of Sperl (7) . The results

were not very satisfactory: a better agreement between the

model and the measured data for the pair correlation func-

tion would require a greater diameter of the metalloid

atoms. Furthermore, the first boron-boron atomic distance

calculated with this model is somewhat smaller than the one

obtained from the experimental curves.

126

The essential features of the homogeneous disordered

models (DRPHS and DRPCP) can be thus summarized as follows:

i) In a Aj_xBx amorphous alloy, the smaller atoms B, are

mainly surrounded by A atoms. This has been experimentaly

demonstrated for some glassy metals and it seems to be a

general phenomenon in these materials.

ii) The atomic arrangement in local coordination polyhedra

is essentially at random and dictated by geometrical (radius

ratio) and compositional factors alone. There is no prefe-

rence, energetic or otherwise, for a particular local sy-

mmetry or coordination number so that the average local

geometry becomes a function of the radius ratio and the

concentration of each species (as pointed out by Jansen

(39)).

CONCLUSIONS: We can summarize the former discussions saying

that three main descriptions can be found for the structure

of metallic glasses: a) Stereochemically defined models; b)

Dense random packing of hard or soft spheres; and c) Dense

random packing of polyhedral units.

There are obvious similarities between these three ty-

pes of models. In each case the structure consists of a

dense, space filling arrangement of local structural units.

The unit, however, can range from a single atom through a

group containing, say, five or seven atoms to a cluster of

several hundreds. The topologlcal disordered models, how-

ever, seem to describe the structure of glassy metals in a

better way than the microcrystalline models and to provide

a more realistic framework in which one can analize the me-

chenical, magnetic and electronic properties of these mate-

rials. In view of the increased Information contact it is

of the greatest importance that not only total but also

partial distribution functions might be accurately determi-

ned for these amorphous alloys and in this way there would

be a lot of experimental work to do.

127

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I . Ashcroft N. and Langreth D.. Phys. Rev. 159,500 (1967)2. , Faber T. and Ziman J.. Phil. Mag.. 11,153 (1965)3. Bhatia A. and Thornton D.. Phys. Rev., B2.3004 (1970)4. Wagner C . Journal of Non Cryst Solids, 31,1 (1978)5a. Wagner C and Ruppersberg H., Atomic Energy Review, 1.101 (1981)5b. Chieux P. and Ruppersberg H., Journ. de Physique, C8.41,145 (1980)6. Paasche F.. Doktorarbeit, Univers. of Stuttgart (1981)7. Spert W., Ooktorarbeit. Univ. of Stuttgart (1982)8. WasedaY., 'The structure of norxrysUiline materials". M e Graw-Hill (1980)9. Nold E., Doktorarbeit. Univ. of Stuttgart (1981)10. Sadoc J, and DixmierJ., Mat Sci.Eng. 23,187(1976)II. Hafner J., •'Glassy Metals I" ed. by H. Guntherodt and H. Beck, Springer Vertag (1981)12. Egami T., in "Glassy Metals I"13. Hamada et al. Proceedings of the RQM IV (4th. International Conference on rapidly quenched

metals) Sendai (Japan) 1981.23 (3)14. Wagner C , Light T.B., Haider N. and Lukens W., Journal of Appi. Phys., 39,3960 (1968)15. Waseda Y.. Okazaki H. and Masumoto T . , Sci. Rep. Res. Inst Tohoku Univers., 26a. 202 (1977)16. Kuhnast F., Machizaud F. and Ftachon J., Journ. de Physique, C8,41,250 (August 1980)17. Cargill Ml G., Solid State Advances in Research and Applications, Vol. 30,227 (1975) Academic

Press, New York18. Gaskell P., Proceedings of the RQM IV, Sendai (Japan) 1981, p. 24719. Bemal J.. Proc R. Soe. A280, 299 (1964)20. Finney J., Proc R. Soe., A319,479 (1970)21 . Polk D., Src Metall, 4,117(1970)22. Polk D., Acta Met, 20,485 (1972)23. Ichikawa T., Phys. Stat. Sol. (a). 29,293 (1975)24. Bennett C , J. Appl. Phys., 43,2727 (1972)25. Sadoc J., Dixmier J., Guinier A., Journal of Non-Cryst Solids, 12,46 (1973)26. Nassif E., Lamparter P., Sedelmeyer B., Steeb S., Z. fur-Naturforsch, 38a, 1098 (1983)27. Cargill DIG., Kirkpatrick S.. AtP Conf. Proc., 31,339 (1976)28. Waseda Y., Masumoto T., Tomizawa S., Can. Met. Qurt, 17.142 (1977)29. Connell G., Solid State Communic, 16,109 (1975)30. von Heimendahl K., H. Phys. F., 5, L141 (1975)31. Boudreaux D., Phys. Rev., B18,4039 (1978)32. Finney J., Wallace J., Journal of non-cryst. solids, 43,165 (1981)33. Blétry J.. Z. fur-Naturforsch. 33a. 327 (1977)34. Beyer O., Hoheisel C , Private Communication, Univers. Bochum (1983)35. TakeuchiS.,KobayashiS.,Phys.Stat.Sol.(a)65,315(1981)36. Wright J., Inst. Phys. Conf. Ser., number 30,251 (1977)37. HoareM.. Pat P., Adv. Phys., 20,161(1971)38. ' Gaskell P., Journal of non-crystalline solids, 32,207 (1979)39. Jansen H.f Boudreaux D. and Snljders H., Phys. Rev., B21, 2274 (1980)

128

AMORPHOUS HETALS - FABRICATION AND CHARACTERIZATION

Frank P. Hissell

Instituto de Física, Universidade de São Paulo,

C P . 20516, Sio Paulo, S.P.

We present a review of the principal methods used to

prepare metallic alloys in the amorphous phase, focussing on

two which have come into widespread use: melt-spinning and

sputtering. Methods of characterizing the amorphous state

«re discussed. Finally we discuss briefly a kinetic approach

to understanding the formation of metallic glasses.

•*••• •'•"•• • • • • • ' • • - • ' • • . I

Probably the earliest appearance of amorphous metals

was in the form of films of Ni-P and Co-P, obtained by elec-

troless deposition. Electroless plating or deposition is the

controlled autocataiytic deposition of a continuous film by

the interaction of a metal salt in solution with a chemical2

reducing agent . The Ni-P alloy possesses good corrosionresistance and hardness and was used as a replacement for

hard chromium. Thus, the initial interest in these materials

was exclusively for engineering purposes and only later did

their amorphous nature become an object of study.

Films of pure metals were prepared in the amorphous

state by Bucket and Hi lsch* Who evaporated metals in high

vacuum and deposited the vapor onto a substrate at He-temp-

eratures. The atoms lose their energy rapidly upon striking

the substrate and the formation of a periodic crystal is

avoided. In this was, the pure metals Bi and Ga can be ob-

tained in the amorphous phase, while other metals like Pb,

Sn, In, Tl, etc. can be forced into the amorphous phase by

adding 10-20% of another component . The effects of prepar-

ing metals in this manner are dramatic: crystalline Bi is

not a superconductor, while amorphous Bi possesses a super-

conducting transition temperature T "6.1 K. The low temp-

erature electrical resistivity also shows large variations

from the crystalline state. However, amorphous Bi cannot

129

be s t u d i e d at room t e m p e r a t u r e b e c a u s e , at a t e m p e r a t u r e of

about 20 K, it u n d e r g o e s an i r r e v e r s i b l e t r a n s i t i o n to the

crystal 1i ne state .

A s i g n i f i c a n t a d v a n c e in the s t u d y of a m o r p h o u s m e t a l s

was m a d e in 1959 when D u w e z , W i l i e n s and Klement d e v e l o p e d

a t e c h n i q u e for the " s p l a t " q u e n c h i n g of m a t e r i a l s . T h e i r

" g u n " t e c h n i q u e d e v e l o p e d c o o l i n g rates g r e a t e r than 10 K / s e c

by f o r c i n g a d r o p of liquid a l l o y to impinge v e r y r a p i d l y

o n t o a h i g h l y c o n d u c t i n g s u b s t r a t e . S o o n a f t e r w a r d , t h e s e

a u t h o r s p r o d u c e d a thin ( M O y ) f l a k e of Au_,.Si2l- (all c o m p o -

s i t i o n s are a t o m i c %) a l l o y w h o s e x - r a y d i f f r a c t i o n p a t t e r n

s h o w e d no s t r o n g peaks a s s o c i a t e d w i t h B r a g g s c a t t e r i n g . The

m a t e r i a l p r o v e d to be u n s t a b l e at room t e m p e r a t u r e , a n d , a f -

ter Ik hours d e c o m p o s e d into p h a s e s w i t h c o m p l e x c r y s t a l

s t r u c t u r e s . The w o r k of D u w e z e_£ ^J_. , h o w e v e r , g e n e r a t e d m u c h

interest b e c a u s e they p r o d u c e d an a m o r p h o u s metal w h i c h could

be e x a m i n e d at room t e m p e r a t u r e . T h e i r s was a l s o the f i r s t

use of x-ray d i f f r a c t i o n to p r o v i d e e v i d e n c e of the a b s e n c e

of long range c r y s t a l l i n e o r d e r in a rapidly q u e n c h e d m e t a l .

In 1 9 6 7 , D u w e z and Lin p r o d u c e d a t e r n a r y F e - P - C a l l o y n e a r

the e u t e c t i c c o m p o s i t i o n w h i c h p r o v e d to be f e r r o m a g n e t i c ,

s t a b l e , and t o t a l l y a m o r p h o u s . The p o s s i b i l i t y of p r o d u c i n g

c h e a p , F e - b a s e d , a m o r p h o u s m a g n e t s p r o v i d e d f u r t h e r impetus

for w o r k in this a r e a .

A n o t h e r s i m i l a r m e t h o d for o b t a i n i n g the rapid s o l i d i f i -Q

c a t i o n o f m e t a l s is t h e p i s t o n a n d a n v i l t e c h n i q u e . I n t h i s

c a s e , a d r o p o f m e t a l i s f o r c e d f r o m t h e b o t t o m o f a c r u c i b l e

i n s u c h a w a y a s t o t r i p a p h o t o c e l l a n d a c t i v a t e a p i s t o n

w h i c h q u e n c h e s t h e d r o p a g a i n s t a n a n v i l . T h e r e s u l t i s s i m i -

l a r t o t h a t o f t h e g u n t e c h n i q u e , a n d o n e o b t a i n s a s p l a t o f

a c e n t i m e t e r o r t w o in d i a m e t e r . A r a t h e r s o p h i s t i c a t e d v a r i -

a t i o n o f t h i s m e t h o d h a s r e c e n t l y b e e n u s e d t o p r o d u c e h i g hq

q u a l i t y s a m p l e s f o r U P S a n d X P S m e a s u r e m e n t s . In A r g e n t i n aa p i s t o n a n d a n v i l a p p a r a t u s is i n u s e a t U n i v e r s i d a d d e

B u e n o s Ai r e s .

T h e f o r m a t i o n o f c o n t i n u o u s m e t a l l i c f i l a m e n t s o f a m o r -

p h o u s a l l o y s p r o b a b l y h a s i t s o r i g i n i n t h e w o r k o f L a n g

w h o m a d e s o l d e r w i r e o n t h e p e r i p h e r y o f a r o t a t i n g d r u m .

130

M o r e r e c e n t l y , the m e l t - s p i n n i n g p r o c e s s of Pond e m p l o y e d

the inner s u r f a c e of a r o t a t i n g c o n c a v e d i s h to r a p i d l y cool

a j e t o f l i q u i d m e t a l . C o n t a c t w i t h the s u b s t r a t e w a s e n h a n -

ced by the c e n t r i f u g a l f o r c e . P o n d ' s i n v e n t i o n w a s i m p r o v e d12by P o n d a n d M a d d i n w h o u s e d the inner s u r f a c e of a r o t a t i n g

d r u m to cool t h e l i q u i d . T o a v o i d the s u p e r p o s i t i o n o f l a y e r s ,

the c r u c i b l e w a s t r a n s l a t e d p a r a l l e l to the a x i s of the d r u m ,

p r o d u c i n g a c o n t i n u o u s h e l i c a l r i b b o n . It w a s w i t h t h i s d e v i c e

that M a s u m o t o and M a d d i n p r o d u c e d a l e n g t h ('VjjO c m ) o f r i b -

bon o f a m o r p h o u s P d o o S * 2 0 'n " 7 1 «C o n t i n u o u s f a b r i c a t i o n o f m e t a l l i c f i l a m e n t s w a s a c c o m -

1 Lp l i s h e d by P o l k a n d Chen , u s i n g a C h e n - M i l l e r t w i n r o l l e r

d e v i c e . In t h i s c a s e , a c o n t i n u o u s a m o r p h o u s r i b b o n w a s

f o r m e d by q u e n c h i n g a j e t of l i q u i d metal b e t w e e n two r o t a t i n g

r o l l e r s . T h i s d e v i c e w a s u s e d to- s u r v e y the g l a s s f o r m i n g1 ka b i l i t y o f a n u m b e r of Fe a n d / o r Ni a l l o y s . T h ? s t e c h n i q u e

w a s a d e q u a t e f o r m a k i n g s c i e n t i f i c s a m p J e s , but it w a s s o m e -

w h a t d i f f i c u l t to c o n t r o l t h e p o s i t i o n of the j e t o f l i q u i d

m e t a l and to o b t a i n a d e q u a t e q u e n c h r a t e s . For this r e a s o n ,

B e d e l l and W e l l s l a g e r , at A l l i e d C h e m i c a l , t r i e d d i r e c t i n g

the l i q u i d m e t a l a g a i n s t the o u t e r s u r f a c e of a c o p p e r d i s c ,

r o u g h l y 10 cm in d i a m e t e r , r o t a t i n g w i t h a s u r f a c e v e l o c i t y

o f a b o u t 30 m / s e c . A l t h o u g h t h i s d e v i c e did n o t b e n e f i t f r o m

c e n t r i f u g a l f o r c e to m a i n t a i n g o o d c o n t a c t b e t w e e n t h e s u b -

s t r a t e a n d the m e l t , it w a s , in f a c t , s u c c e s s f u l a n d is t h e

b a s i s of the t e c h n i q u e n o w k n o w n as m e l t - s p i n n i n g . L a t e r ,

B e d e l l s h o w e d h o w the c o n t a c t b e t w e e n w h e e l and m e t a l c o u l d

be e n h a n c e d a n d the p o i n t of d e p a r t u r e of the r i b b o n c o n t r o l -

led.

In F i g . 1 we s h o w a s c h e m a t i c d i a g r a m of a m e l t - s p i n n e r

s u c h a s t h a t w h i c h m i g h t be u s e d to p r o d u c e s a m p l e s f o r r e -

s e a r c h p u r p o s e s . T h e d i s c is f r e q u e n t l y of c o p p e r (The a p p r o -

p r i a t e f i g u r e o f m e r i t f o r the c h i l l s u b s t r a t e is kdC , the

p r o d u c t of the t h e r m a l c o n d u c t i v i t y , d e n s i t y a n d s p e c i f i c1 8h e a t . S i n c e c o p p e r h a s a h i g h e r kdC p r o d u c t than o t h e r

m e t a l s , its u s e is j u s t i f i e d . ) and r o t a t e s w i t h a s u r f a c e

v e l o c i t y w h i c h Is c o n t i n u a l l y v a r i a b l e up to a b o u t 50 m / s e c .

131

The sample is heated in a quartz tube with an induction coil

to the melting point. H o w e v e r , the high surface tension of

the liquid metal prevents it from leaving the crucible. The

application of a pressure of Ar gas above the me!t forces it

out a small hole in the crucible onto the rotating disc, where

it forms a ribbon of the amorphous alloy.

The cooling rate of the metal during melt spinning is

easily estimated. For e x a m p l e , a F e . - N i . ^ .B, alloy melts at

a temperature of about 10009C and is cooled to approximately

room temperature. The ribbon maintains contact with the wheel

during an angle of about 109, which, for a surface velocity

of 30 m/sec, corresponds to a time of 10 sec. Thus the cool-

ing rate is d T / d t M O K/sec.ARGON

PRESSURE.

FUSEDSILICATUBE-

MOLTEN ALLOY

AMORPHOUSALLOYFILM

COIL

ROTATING DISC

Fig. 1. Schematic diagram of a m e l t - s p i n n e r .

A discussion of ribbon formation in a melt-spinner was1 a

g i v e n by l i e b e r m a n n a n d G r a h a m p in t e r m s of the B e r n o u l l i

e q u a t i o n . T h e s e a u t h o r s o b t a i n e d an e q u a t i o n f o r the r i b b o n

c r o s s - s e c t i o n a l a r e a A in t e r m s o f P, the a p p l i e d g a u g e p r e s -

s u r e of the e j e c t i o n g a s , d, t h e d e n s i t y of t h e l i q u i d m e t a l ,

D , t h e d i s c d i a m e t e r , S, the f r e q u e n c y of r o t a t i o n , a n d 9, the

l i q u i d j e t d i a m e t e r , t a k e n e q u a l to the o r i f i c e d i a m e t e r :

132

A »r

The above equation was shown to be in quantitative agreementwith experimental data for fretnN'li0B20 rib')ons* T n e u s e

Bernoulli equation determines the conditions under which acontinuous ribbon will be formed in the melt-spinner. It has

20also been used to discuss ribbon formation by Takayama and 01 .

A study of the thermal and energetic constraints on rib"18bon formation in a melt-spinner has been made by Kavesh f who

also obtained equations relating the ribbon dimensions to the

process parameters. In this case, the fundamental parameters

were taken to be the chill surface velocity V and the flux Q

of material incident on the chill surface. Kavesh considered

two different possible modes of ribbon formation:(1) a mode

in which the ribbon is formed by transport of momentum and/or

heat between the melt puddle and the chill surface, and (2)

the case where the ribbon is formed by wetting the substrate

and, therefore, the ribbon dimensions are determined by the

equilibrium of energy at the chill surface. In the first case,

Kavesh concludes that heat transport is dominant over momen-

tum transport and that the ribbon width w and thickness t are

given by: w»c"Q n/V 1" n and t-Q 1" n/c"V n, with n-0.75- In the

case where energy balance is the important mechanism for rib*

bon formation, the ribbon dimensions are given by wMi and

t M / V . Experimental data for Fe. nNi . nP,. B, were found to be In

agreement with the following equations: w»0.$253 Q ' /V '

and t-1,585 Q O è 1 7 / V 0 ' 8 3 . Thus it is difficult to decide which

mechanism Is dominant in ribbon formation. For the same mater-21iai, H M I m a n and Hiizinger also observed w and t to decrease

with increasing velocity as W W ' and tM/ , in substan-

tial agreement with Kavesh.

The melt-spinner has become a widely used technique for

preparing metallic glasses from the melt and is used by many

experimental groups. In Brazil, melt-spinners are In use at

Universidade de São Paulo and Universidade Federal do Rio

Grande do Sul, and, in Argentina, at Centro

Atômico Bariloche . Several other institutions are In the

133

p r o c e s s of i n s t a l l i n g this d e v i c e .

W h i l e c o n t i n u o u s r i b b o n s can be p r o d u c e d by the above

p r o c e s s , the w f d t h of the ribbon is limited to about 5 mm.

T o p r o d u c e w i d e r r i b b o n s it is n e c e s s a r y to e s t a b l i s h a rec-

t a n g u l a r melt p u d d l e on the chill s u r f a c e and this cannot be

p r o d u c e d by a c y l i n d r i c a l j e t . A l s o , r e c t a n g u l a r j e t s d e g e n *

o r a t e rapidly d u e to the low v i s c o s i t y and high s u r f a c e ten-

sion of the m e l t . T h e p r o b l e m of p r o d u c i n g w i d e r ribbons w a s22solved by M, C, N a r a s i m h a n w h o invented the p l a n a r flow

c a s t i n g p r o c e s s - A s c h e m a t i c view of the p l a n a r flow casting

p r o c e s s is shown in Fig. 2, In this c a s e , the m o l t e n metal is

Substrate

Fig. 2 S c h e m a t i c v i e w of the p l a n a r flow c a s t i n g p r o c e s s .

forced through a r e c t a n g u l a r o r i f i c e in the b o t t o m of a c r u -

cible in close p r o x i m i t y to the chill s u b s t r a t e - Flow is be-»

sically p r e s s u r e c o n t r o l l e d but is c o n s t r a i n e d by orifice

w i d t h * s u b s t r a t e s p e e d and o r i f i c e - s u b s t r a t e d i s t a n c e . This

p r o c e s s or s i m i l a r v a r i a t i o n s have been used to p r o d u c e

ribbons w i t h w i d t h s up to 10 cm. Commercial p r o d u c t i o n of Fe-

based wide a m o r p h o u s ribbons is c u r r e n t l y u n d e r w a y by the

A l l i e d C o r p , , V a c u u m s c h m e l z e GmbH, and Hitachi M e t a l s Ltd,

A n o t h e r t e c h n i q u e w h i c h has come into w i d e s p r e a d use for

the p r o d u c t i o n of a m o r p h o u s alloys is s p u t t e r i n g . This p r o c e s s23

w a s d i s c o v e r e d • by Sir W i l l i a m Robert Grove in 1852 when he

n o t i c e d that the w a l l s of the d i s c h a r g e tube that he was

S t u d y i n g b e c a m e coated w i t h metal from the e l e c t r o d e s . Later

F. Stark gave the first c o r r e c t e x p l a n a t i o n of the p h e n o m e n o n

«nd J. vJ. T h o m p s o n p r o d u c e d the name " s p l u t t e r i n g " to d e s c r i b e

134

the process. Sputtering was used commercially as early as

by the Western Electric Co. to deposit a thin conducting

metallic film onto wax phonograph records. The mechanism of

dc (direct current) sputtering was studied by Wehner e± al.

With the development of rf (radio frequency) sputtering ,

applications for this technique expanded greatly. Recent

Improvements In vacuum technology have turned sputtering into

a major industry.

The basic mechanism of sputtering is illustrated in

F|g. 3» where the diode sputtering configuration is shown.

An expendable target is attached to the cathode, while the

«node supports the substrates onto which the target material

I» to be deposited. The process is carried out in a vacuum

chamber, which is first evacuated to high vacuum and then

filled with a low pressure of argon gas. The cathode voltage

TARGET M*T£Bl*l

ÍUISTSATES

Fig. 3 Pi ode sputtering configuration.

Is adjusted to - 500-600 V and a plasma is created in the

vacuum chamber. The ionized Ar are attracted to the cathode,

and., upon collision with the target, the kinetic energy of

the; Ar ions is transferred to atoms of the target material.

The target material is then deposited onto the substrates.

Thç system functions in either a dc or rf mode when the tar-

get material is a metal. However, the rf excitation mode must

be used when the target material is a dielectric in order to

«void charge accumulation on the cathode, which would reduce

the sputtering rate,

135

O t h e r s p u t t e r i n g c o n f i g u r a t i o n s h a v e been used in o r d e r

to i n c r e a s e s p u t t e r i n g r a t e s . For e x a m p l e , the t r i o d e c o n f i g -

u r a t i o n has s i g n i f i c a n t rate a d v a n t a g e s o v e r the d i o d e m o d e

but at a c o n s i d e r a b l e i n c r e a s e in the c o m p l e x i t y o f the s y s -26tem . High s p u t t e r i n g rates are a l s o o b t a i n e d w i t h m a g n e t r o n

s p u t t e r i n g . In this c a s e , a m a g n e t i c field is a l i g n e d p e r p e n -

d i c u l a r to the e l e c t r i c e x c i t a t i o n f i e l d o f the s p u t t e r i n g

s y s t e m . T h e m a g n e t i c field t r a p s and c o n f i n e s the e l e c t r o n s

in the p l a s m a to a r e g i o n v e r y c l o s e to the target s u r f a c e ,

r e s u l t i n g in high ion c u r r e n t s and h i g h s p u t t e r i n g r a t e s . T h e

dc m a g n e t r o n m o d e is used to s p u t t e r m e t a l s , w h i l e the rf

m a g n e t r o n e x c i t a t i o n m o d e is used for d i e l e c t r i c s .

The s p u t t e r i n g t e c h n i q u e has a s e r i e s of a d v a n t a g e s w h e n

c o m p a r e d to o t h e r thin f> 1m d e p o s i t i o n p r o c e s s e s . For e x a m p l e

c h e m i c a l v a p o r d e p o s i t i o n (CVO) p r o c e s s e s can be u s e d to p r o -

d u c e a b r o a d s p e c t r u m of m a t e r i a l s . H o w e v e r , a m a j o r d i s a d -

v a n t a g e of the C V D p r o c e s s is that m a j o r c h a n g e s m u s t b e m a d e

in the a p p a r a t u s in o r d e r to p r e p a r e d i f f e r e n t m a t e r i a l s .

V a c u u m e v a p o r a t i o n u s i n g r e s i s t i v e or e l e c t r o n beam h e a t i n g

of the melt is a r e l a t i v e l y s i m p l e and e a s i l y c o n t r o l l e d m e -

thod. H o w e v e r , the d e p o s i t i o n rates vary g r e a t l y w h e n the v a r

ious c o m p o n e n t s of an a l l o y have d i f f e r e n t vapor p r e s s u r e s ,

and so it is d i f f i c u l t to d e p o s i t c e r t a i n a l l o y s . T h e s e d i s -

a d v a n t a g e s are not p r e s e n t in s p u t t e r i n g . The e q u i p m e n t is

v e r s a t i l e in that o n l y the target need be m o d i f i e d to p r o d u c e

a d i f f e r e n t a l l o y . F u r t h e r m o r e , target m o d i f i c a t i o n s m a y be

a c c o m p l i s h e d s i m p l y by c o v e r i n g part of the target w i t h a n o -7 7 9ft

ther m a t e r i a l ' . D e p o s i t i o n rates in the s p u t t e r i n g p r o -

cess are a l s o m u c h m o r e u n i f o r m than for e v a p o r a t i o n .

The e f f e c t i v e q u e n c h rate d u r i n g s p u t t e r i n g may be e s t i -

m a t e d from the o b s e r v a t i o n that the d e p o s i t e d atom will lose

its k i n e t i c e n e r g y in a time c o m p a r a b l e to a few ionic v i b r a -

t i o n s . T h u s , if the atom is in e q u i l i b r i u m w i t h the p l a s m a

at ^ 5 0 0 ? C , and loses its e n e r g y in a time 5 x 10 s e c , then1 U

the e f f e c t i v e q u e n c h rate will be d T / d t ^ 10 K / s e c . Thei n c r e a s e d q u e n c h rate d u r i n g s p u t t e r i n g c o m p a r e d w i t h p r o -

c e s s e s such as m e l t - s p i n n i n g has a p r a c t i c a l c o n s e q u e n c e :

136

amorphous alloys are formed over a much wider composition

range using sputtering. For example, for the 2r-Rh system»

the eutectic occurs around 25(atomic)$ Rh and melt-spinning

may be used to produce amorphous alloys containing 20-301 Rh.

On (he other hand, with sputtering, amorphous alloys

containing as much as 43% Rh may be obtained . The properties

of alloys made by sputtering are generally comparable to those

produced by melt-spinning .

Sputtering has been gsed to produce amorphous alloys at

Universidade de $ao Paulo and Universidade Estadual de Cam-

pinas in Brazil. In Argentina, a sputtering system is being

installed at Centro Atômico Bariloche.

Another technique with a very high effective quench rate

is ion implantation, in this case, beams of ions ar^ acceler-

ated by an electrostatic particle accelerator and are direc-

ted onto a substrate. Using accelerating voltages pf kOO kV,

it is possible to implant ions to a depth of around 3000A.

Thus it is possible to form thin films of amorphous alloys

on the surface of a substrate. A variation of this technique,

known as ion beam mixing, uses a beam of rare gas tons such

as Ar or Kr to mix atoms deposited in a series of very thin

(MQ0 8) layers. The resultant mixing produces an amorphous

alloy , In Brazil, ion implantation techniques are in use

at Universidade de Sao Paulo and Universidade Federal do Rio

Grande do Sul.

Finally, we mention that amorphous alloys have recently

been formed by solid state reactions. For example, an amor-

phous Zr-Ni phase has been produced by annealing vapor depo-32sited crystalline layers of elemental Zr and Ni . Thus the

already lengthy list of techniques for producing amorphous

metals fs constantly growing.

Characterization of the amorphous state begins with an

x-ray diffraction sc^n. The graph of x-ray intensity vs,

scattering angle shows no strong narrow peaks, such as those

which result from Bragg scattering. Instead, there are a num-

ber of weaker broad maxima, the most intense of which

137

g e n e r a l l y o c c u r s at the angle c o r r e s p o n d i n g to the most inten-

se Bragg p e a k . In Fig. k w e show x-ray intensity v s . angle

28 for three samples of ^ o n B ^ o * The uppermost trace c o r r e s -

ponds to a sample which is totally a m o r p h o u s , while the m i d d l e

trace belongs to a sample showing slight signs of crystal U n -

ity. In the lower t r a c e , the Bragg peaks are already becoming

e v i d e n t , for this sample is much more c r y s t a l l i n e than the

o t h e r two. The first indication of an a m o r p h o u s s a m p l e , t h e r e -

fore, is the absence of Bragg p e a k s .

Fig. k X-ray intensity v s . angle 28 for three samples ofF e 8 0 B 2 O '

For the u p p e r m o s t trace of Fig. k, one can use the

S c h e r r e r e q u a t i o n to make a simple e s t i m a t e of the m a x i m u m

dimension 0 of the region of the sample from which there is a

coherent s c a t t e r i n g c o n t r i b u t i o n to the broad x-ray maximum.

If B is the full width at half maximum of the intensity curve

and X is the w a v e l e n g t h of the radiation used, then

0-0.9 A/ B c o s 8 . For the upper trace of Fig. k, D ^ 2 o 8 . More

detailed structural information, such as the radial d i s t r i b u -

tion f u n c t i o n , mav be obtained from a detailed numerical

analysis of the x-ray diffraction v

A rapid way to test samples to determine if they are

amorphous is to make use of the 180? bend test. This test,

138

w h i c h w o r k s for m a n y F e - b a s e d a l l o y s , is b a s e d on the f a c t

that m a n y a m o r p h o u s a l l o y s a r e m o r e d u c t i l e than t h e i r

c r y s t a l l i n e c o u n t e r p a r t s . For e x a m p l e , the s a m p l e c o r r e s p o n d -

ing to the u p p e r m o s t t r a c e o f F i g . k w a s a b l e to s u s t a i n an

a c u t e 180? bend w i t h o u t b r e a k i n g . The m i d d l e s a m p l e a l m o s t

s u s t a i n e d the s h a r p b e n d , w h i l e the l o w e r s a m p l e b r o k e b e f o r e

It w a s bent t h r o u g h an a n g l e o f 9 0 ? . T h i s t e s t w o r k s for

r e l a t i v e l y d u c t i l e m a t e r i a l s , but m u s t be u s e d in c o n j u n c t i o n

w i t h x - r a y d i f f r a c t i o n m e a s u r e m e n t s .

A n o t h e r p r o p e r t y w h i c h is q u i t e d i f f e r e n t in the c r y s -

t a l l i n e and a m o r p h o u s s t a t e s is the e l e c t r i c a l r e s i s t i v i t y

and the t e m p e r a t u r e d e p e n d e n c e of the r e s i s t i v i t y . D u e to the

I n c r e a s e d a t o m i c d i s o r d e r in the a m o r p h o u s s t a t e , the e l e c -

t r o n i c m e a n f r e e p a t h is g r e a t l y r e d u c e d in t h e s e m a t e r i a l s .

T h u s , the e l e c t r i c a l r e s i s t i v i t y of m e t a l l i c g l a s s e s is h i g h

( 1 0 0 - 2 0 0 pfi-cm) and v e r y s i m i l a r in m a g n i t u d e to that of the

liquid s t a t e . S i n c e the t e m p e r a t u r e c o e f f i c i e n t of the r e s i s -

t i v i t y is a l s o v e r y small ( ( 1 / p ) d p / d T ^ 1 0 ~ 5 / K ) , as in the

liquid s t a t e , one m i g h t e x p e c t that the t h e o r y of liquid

m e t a l s w o u l d be u s e f u l in u n d e r s t a n d i n g r e s u l t s in m e t a l l i c

g l a s s e s . S u b s t a n t i a l p r o g r e s s has b e e n m a d e in t r e a t i n g the

teer.36

r e s i s t i v i t y o f liquid m e t a l s , and t h e s e r e s u l t s h a v e been

v e r y u s e f u l in i n t e r p r e t i n g d a t a in m e t a l l i c g l a s s e s '

One must e x e r c i s e a c e r t a i n c a u t i o n , h o w e v e r . Early e x p e r i -

m e n t a l w o r k s s o m e t i m e s a f f i r m e d that a n e g a t i v e t e m p e r a t u r e

c o e f f i c i e n t o f the e l e c t r i c a l r e s i s t i v i t y w a s in i t s e l f a n

i n d i c a t i o n of the a m o r p h o u s s t a t e . M o r e recent t h e o r e t i c a l

w o r k , h o w e v e r , has s h o w n that a m o r p h o u s m a t e r i a l s m a y h a v e

e i t h e r p o s i t i v e or n e g a t i v e t e m p e r a t u r e c o e f f i c i e n t s o f r e -

s i s t i v i t y , and t h e s e c o n c l u s i o n s h a v e been c o n f i r m e d by e x -

p e r i m e n t .

G l a s s e s a r e u n s t a b l e r e l a t i v e to the c r y s t a l l i n e s t a t e

and c r y s t a l l i z e w h e n h e a t e d b e y o n d the c r y s t a l l i z a t i o n t e m p -

e r a t u r e T • T is r o u g h l y one h a l f of the m e l t i n g t e m p e r a t u r ex x

T and is u s u a l l y s l i g h t l y h i g h e r than the g l a s s t e m p e r a t u r eT . T h e c r y s t a l l i z a t i o n p r o c e s s o c c u r s as an e x o t h e r m i c9

r e a c t i o n and is c l e a r l y v i s i b l e in the s p e c i f i c h e a t , in

139

c o n t r a s t t o t h e g l a s s t r a n s i t i o n , w h i c h is u s u a l l y d i f f i c u l t

t o o b s e r v e . T w o t e c h n i q u e s w h i c h a r e u s e d t o s t u d y t h e t h e r -

m a l b e h a v i o r o f g l a s s e s a r e d i f f e r e n t i a l t h e r m a l a n a l y s i s

( D T A ) a n d d i f f e r e n t i a l s c a n n i n g c a l o r i m e t r y ( D S C ) . D S C w a s

f i r s t a p p l i e d t o t h e s t u d y o f t h e g l a s s t r a n s i t i o n in t h e3 7a m o r p h o u s a l l o y A u _ _ G e 1 _ /;Si Q . by C h e n a n d T u r n b u l l '.

B o t h a l l o w T to b e e a s i l y d e t e r m i n e d . A s t u d y o f T a s aX X

f u n c t i o n of the s a m p l e h e a t i n g r a t e a l l o w s a d e t e r m i n a t i o n

of the a c t i v a t i o n e n e r g y for c r y s t a l l i z a t i o n .

V a r i o u s t h e o r i e s for the f o r m a t i o n o f m e t a l l i c g l a s s e s1,0

h a v e b e e n p r o p o s e d . B e n n e t t , P o l k , and T u r n b u l l d i s c u s s e d

the r o l e of c o m p o s i t i o n in m e t a l l i c g l a s s f o r m a t i o n , in p a r t ,

b a s e d on a s t r u c t u r a l m o d e l o f a m o r p h o u s a l l o y s w h i c h had

b e e n p r o p o s e d e a r l i e r by P o l k . N a g e l and T a u c c o n s i d e r e d

the r o l e o f the e l e c t r o n gas in g l a s s f o r m a t i o n a n d s t a b i l i t y

of a m o r p h o u s a l l o y s . T h e s e t h e o r i e s tend to e q u a t e s t a b i l i t y

w i t h e a s e of g l a s s f o r m a t i o n . On the o t h e r h a n d , in a k i n e t i c

a p p r o a c h to g l a s s f o r m a t i o n , o n e a t t e m p t s to d e t e r m i n e t h e

q u e n c h r a t e n e c e s s a r y to o b t a i n the a m o r p h o u s p h a s e by

c o o l i n g f r o m the l i q u i d p h a s e . T h u s o n e a t t e m p t s to c a l c u l a t e

a c r i t i c a l c o o l i n g r a t e R for the s y s t e m u n d e r c o n s i d e r a t i o n ,c hi

T h i s k i n d of a p p r o a c h has b e e n a d o p t e d by D a v i e s and by

U h l m a n n and o t h e r s . A r e v i e w of t h e s e t r e a t m e n t s h a s b e e n

g i v e n by D a v i e s . We d e s c r i b e b r i e f l y the k i n e t i c a p p r o a c h

w h i c h s e e m s to be the m o s t r e l e v a n t to t h e e x p e r i m e n t a l fact

that h i g h c o o l i n g r a t e s are n e c e s s a r y for the f o r m a t i o n of

m e t a l 1Ic g l a s s e s .

U h l m a n n a d o p t e d the c l a s s i c a l J o h n s o n - M e h i and A v r a m i

t r e a t m e n t of t r a n s f o r m a t i o n k i n e t i c s , r e l a t i n g the f r a c t i o n

of the t r a n s f o r m e d p h a s e to the h o m o g e n e o u s n u c l e a t i o n f r e -

q u e n c y , t h e g r o w t h r a t e and the t i m e . P h y s i c a l l y , the f o r -

m a t i o n of c r y s t a l l i n e s o l i d s from l i q u i d s t a k e s p l a c e by a

n u c l e a t i o n and g r o w t h p r o c e s s b e c a u s e the a t o m i c s t r u c t u r e

of a s o l i d , w i t h t r a n s 1 a t i o n a l and p o i n t s y m m e t r y , is i n c o m -

m e n s u r a t e w i t h the s t r u c t u r e of the l i q u i d . T h e i n t e r f a c e

b e t w e e n the c r y s t a l l i n e and l i q u i d p h a s e s c o r r e s p o n d s to

140

Lo..;,o time

Fig. 5 Time-temperature-transformation curve.

locally distorted atomic arrangements and the energy neces-

sary to produce the interface is positive. Thus, even below

the solidification temperature, where the energy of the

crystalline phase is lower than the liquid phase, it takes

a finite time for crystalline nuclei to reach a critical size

for growth into larger grains. Using accepted theories of

crystal growth and homogeneous nucleation, as well as esti-

mated values for the liquid viscosity between T and T ,

Uhlmann constructed time-temperature-transformation curves

for a detectable fraction of a crystal. The crystallization

time becomes shorter at first as the driving force due to

cooling below the solidification temperature is increased.

As the temperature is reduced, however, the diffusivity of

the liquid phase is reduced, which, in turn, slows down the

growth of crystalline nuclei, and the time necessary for

crystallization increases. Thus one obtains a curve such a^

that of Fig. 5 above. The critical cooling rate for glass

formation is then' approximately given by the cooling curve

necessary to avoid the nose of this time-temperature-trans^

formation curve: R « (T -T )/t , where T and t are respec-c m n n n n

tively the temperature and time corresponding to the nose of

the curve. As shown by Davies and Uhlmann, the kinetic

approach gives good agreement with experiment (depending upon

the accuracy of the viscosity data) for glasses ranging from

SiO, to P d 8 2 S ' i g to amorphous Ni. The kinetics of glass for-

141

mat!on by cooling from the melt is governed largely by the

reduced glass temperature T /T . Through this kinetic theory,

the formation of metallic glasses has been approximately

rationalized in terms of theories of nucleation and crystal

growth.

REFERENCES

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(1959).

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and H. Beck (Springer, New York, in press).

10. E. M. Lang, U. S. Patent 112,50* (Feb. 21, 1871).

11. R. B. Pond, U. S. Patent 2,825,108 (March *, 1958).

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2*75 (1969).

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16. See L. A. Davis, N. J. DeChristofaro, and C. H. Smith, in

Proc. Conf. Met. Glasses:Sci. and Tech., Budapest, 1980.

17- J. Bedell, U.S.Patent 3,862,658 (Jan. 28, 1975).

18. S. Kavesh, in Metal 1i c Glasses, ed. J. J. GiIman and H.J.

Leamy (Amer. Soc. Metals, Metals Park,OH, 1978)p. 36.

19. H. H. Llebermann and C. D. Graham,Jr.,IEEE Trars. Mag.

1-12, 921 (1976).

142

20. S. Takayama and T. Oi , J. Appl. Phys. J50.4962 (1979).

21. H. Hilman and H. R. Hilzinger, in Rapidly Quenched Metals

III, ed. B. Cantor (Metals S o c , London, 1978)vol . 1 ,p.3O.

22. H. C. Narasimhan, U.S.Patent 4,142,571 (Mar. 6, 1979).

23. W. R. Grove, Phil. Tran. Roy. Soc. J_42, 87 (1852).

2^. S. K. Wehner, Advances in Electronics and Electron Physics,

VII I (Academic, New York, 1955)p. 239-

25. G. S. Anderson, W. N. Hayer and G. K. Wehner, J. Appl.

Phys. JJ), 2991 (1962).

26. Sputter Deposition and Ion Beam Processes, publication

of the Education Committee of the American Vacuum Society.

27. J. J. Hanak, J. Vac. Sci. Technol . £, 172 (1970.

28. A. K. Ghosh and D. H. Douglass, Phys. Rev. Lett. 37' 32

(1976).

29. F. P. Missell, S. Frota-Pessôa, J. Wood, J. Tyler, and

J. E. Keem, Phys. Rev. J3£7, 1596 (1983).

30. F. P. Missell, R. Bergeron, J. E. Keem and S. R. Ovshin-

sky, Solid State Commun. ^ 7 , 177 (1983).

|1. See Metastabie Materials Formation by Ion Implantation,

eds. S. T. Picraux and W. J. Choyke(EIsevier, Mew York,

1982).

32. B. M. Clemmens, Buli. Amer. Phys. Soc. 2_9, 506 (1984).

33. B. D. Cullity, Elements of X-ray Diffraction, 2 ed.,

(Addison-Wesley, Reading,MA, 1978)p. 102.

J4. G. S. Cargilt, M l , in Solid State Physics, Vol. 30.eds.

H. Ehrenreich, F. Seitz, and D. Turnbuil (Academic, New

York, 1975) p. 227.

35. G. Busch and H. Guntherodt, in Solid State Physics,Vol.29,

eds. H. Ehrenreich, F. Seitz, and D. Turnbu11(Academíc,

New York, 1974) p. 235-

36. H. J. Guntherodt and H.U.Kunzi, in Metal Ii c Glasses, ed.

J. J. GiIman and H. J. Leamy (Amer. Soc. Metals, Metals

Park, OH, 1978)p. 247.

37. H. S. Chen and D. Turnbull, J. Appl. Pbys.J_0, 284 (1967).

38. H. E. Kissinger, J. Res. Natl. Bur. Stan. $]_, 217 (1956).

39. P. C. Boswell, J. Therm. Anal. )£> 3 5 3 (1980).

kO. C. H. Bennett, D. E. Polk, and D. Turnbull, Acta metal 1.

143

19, 1295 (1971).

41. S. R. Nagel and J. Tauc, Phys. Rev. Lett.3^, 380 (1975).

J. Tauc and S. R. Nagel, Comments Sol. St. Phys. 2»^9(1976)

42. H. A. Davies, Phys. and Chem. Glasses JJ, 159 (1976).

43. D. R. Uhlmann, J. Non-Cryst. Solids 7, 337 (1972).

Work supported by FAPESP, CNPq and FINEP (Brazil).

144

STRUCTURAL RELAXATION: LOW TEMPERATURE PROPERTIES

Francisco de la Cruz

Centro Atômico Báriioche

8'+00 - S.C. de Bariioche (R.N.), Argentina.

ABSTRACT

We discuss the changes in transport and superconduct-

ing properties of amorphous Zr70Cu30, induced by thermal

relaxation.

The experimentai results are used to investigate the

relation between the microscopic parameters and the

observed physical properties. It is shown that the density

of electronic states determines the shift in T as well as

c

the variation of the electrical resistivity.

It is necessary to assume strong hybridization between

a and d bands to understand the eiectrodynamic response of

the superconductor.

145

INTRODUCTION

In this lecture I will refer to the research made In the

Low Temperature Lab in Bariioche, during the last four

years, concerning the normal and superconducting properties

of metallic amorphous systems. The title of the talk is

misleading. I am not an expert in relaxation and you will

see that the heat treatment is only used to induce changes

in the physical properties of our samples, in order to

study the behaviour of the microscopic parameters of these

metals.

The materials investigated are lOum thick ribbons of

Zr 7 0Cu 3 0 and La 7 0Cu 3 0 alloys, obtained by melt spinning.

In most of this talk we will refer to the results obtained

from the Zr70Cu30 system.

Before we start to show and discuss the experimental

results I will remark some properties chat are common to

all the transition metals amorphous alloys:

a) High electrical resistivity: p » 200uftcm.

b) Non-validity of Matthiessen's rule when applied to the

temperature dependence of the electrical resistivity.

This result is known as Mooij's criterium.

c) The thermodynamic and transport properties of the

amorphous materials at low temperatures are characteriz-

ed by the presence of the low energy excitations, TLS.

d) Since the electron mean free path l, is of the order of

interatomic distances the heat is mainiy carried by

phonon3. As a consequence, amorphous systems are ideal

146

materials to investigate the phonon thermal conduction

in metals.

We will now focus our attention in some questions

related to the properties we have indicated:

i) Is the high electrical resistivity of these amorphous

metals due to the d-eiectron contribution in

transition metals?

ii) Which is the origin of the negative temperature

coefficient of P(T)?

iil) Assuming that in this amorphous metals it is possible

to define a Fermi wave vector, it is found that

kp«i=«l. Is the BCS-Gorkov theory adequate to describe

superconductivity in this extreme dirty limit?

iv) Another question related to the previous one is: are

the Gorkov equations valid when the d and s electrons

contribute to the transport properties and super-

conductivity?

v) Is there any dependence between the superconducting

critical temperature, T , and the density of TLS,

n(0)?

vi) Is the Matthiessen's rule valid when applied to the

phonon thermal conduction in amorphous metals?

THEORETICAL AND EXPERIMENTAL BACKGROUND

Following the BCS scheme, superconductivity arises

from the competition between an attractive phonon-electron

147

*i n t e r a c t i o n , c h a r a c t e r i z e d by a parameter X end a

r e p u l s i v e Coulomb i n t e r a c t i o n » u . As a r e s u l t the

c r i t i c a l temperature should be a function of these two

parameters

Tc = f(X M ) (1)

Due to the lack of tunneling data in amorphous metals

the electron-phonon parameter can be approached by X «

N(0)I2/92, where the symbols are those typically used in

the literature. The parameter y » 0-1 is usually accepted

for transition metals.

A review of the critical temperature behaviour of

amorphous metals can be found in ref.2. There it is

indicated that relation (1), with X and u as described

previously, is enough to understand most of the

experimental data. Nevertheless, I believe there are some

questions that have no definite answer. One of the

problems is the possibility that the TLS contributes to

T . If this is the case, T should also be a function ofc c

n(0) and expression (1) shouid then be generalized. It is3

also important to remark that if the TLS can modify the

* '+effective x , recent calculations indicate that disorder

could Increase v . This is an important result since it

indicates that T , H . and p could be correlated through

the degree of electron localization and would indicate that

in the extreme dirty limit, the critical temperature should

also be a function of the electron mean free path. From

the experimental point of view there are no answers co

148

these questions.

Measurements in the Zr Cu, system indicate that the

increase in x induces a rise in T together with a decrease

in p . The behaviour of T has being explained taking into

account the measured behaviour of N(0). The decrease in

9 10p has also been related ' to the increase in the density

of states due to contribution of the d-Zr band. In those

experiments it is difficult to separate the contribution of

TLS (if any) and/or, of localization. There is not enough

systematic investigation of a possible direct correlation

between T and n(0). In this lecture we will discuss some

results related to this topic.

Other superconducting parameter related to the

electronic properties of the material is the upper critical

field, H 2* Within the Gorkov theory and for the dirty

limit

H c 2 - Vk f N(0)pf(T). (2)

To obtain (2) it has being used the Ginzburg-Landau

coherence length in the dirty limit, given by Ç2(0,0 »ÇQ4,

with ç0 - 0.18 hvF/kTc , vF » k2S/6hy, p-1-(2/3)e2vFN(0)£,

and Y-(2/3)(wk)2N(0). Here 40 is the BCS coherence

lengths, S is the area of the Fermi sphere, I the electron

mean free path and y the coefficient of the electron heat

capacity. Ail these expressions have been obtained

assuming that kp* ~» 1. As was mentioned before this limit

is not adequate for the amorphous samples used in our

experiments.

149

Within the same approximation the superconducting

response to the presence of a low magnetic field Is

determined by the superconducting penetration depth

X(l,t) « ^(OMCo/Oi'Zfit). (3)

For T near Tc

where the London penetration depth x1(0) « 3h*l/2Yl/2/ekS.

It Is Interesting to recall that A. (0) Is only related to

the normal properties of the material and that expressions

(3) and (31) indicate a correction to *T(0) through the

square root of the ratio of two distances. Superconducti-

vity only appears through the definition of ÇQ.

Expressions (2) and (3') can be verified since ail physical

quantities that appear in them are experimentally

accesible. Although the verification of expressions (2)

and (31) is interesting from the point of view of the

effects induced by an extreme short i, we believe that

there is other related point that has to be considered when

studying transition metals. It was realized by Bergmann

that expression (2) should not be valid when applied to

metals that can be characterized by the presence of two

bands (d and s, in our case). Following the same arguments

we will see that it is difficult to Justify the validity of

expression (3'). The coherence length ç0 is strongly

associated to the interaction energy necessary to form the

Cooper pairs (kTc). The critical temperature in a d and s

150

band superconductor, is believed to be determined by the d

band density of states, N,. On the other hand in an

independent two band model the s electrons contribution

7 8should at least be competitive with the d electrons ' .

9 10Recent work ' gives the experimental results of the

Theresistivity of Zr Cu._ as a function of xo

concentration dependence has being explained , on the basis

of a two band model, where the contribution to p from d and

s electrons are found to be comparable. We believe that if

this is the correct explanation expression (31) should not

be applicable.

Until here we have referred to changes in the physical

properties of the amorphous material induced by changing

concentration. We have other available experimental

technique to change the behaviour of the material at

constant concentration. It has being shown in the last

years that thermal heat treatment modifies the normal

and superconducting properties of these materials. In the

case of Zr Cu, it has being suggested chat the supercon-

ducting critical temperature is determined by the

electronic density of states, in agreement with Varraa and

Dynes. This result has being obtained from the analysis

of the variation of Tc and N(0) with concentration . If

the analysis is correct the change in T Induced by thermal

relaxation should also be determined by a corresponding

change in N(0).

151

EXPERIMENTAL RESULTS AND DISCUSSION

We have measured the thermal conductivity of Zr70Cu30

amorphous ribbons, in the range of temperature between

O.i°K and 7°K. The results are shown in Fig.l.

Experimental details can be found in ref. 12. It is

clearly seen that the thermal conductivity below the

critical temperature of the alloy is monotonically

increased with annealing. Further annealing is not

possible because the sample starts to crystallize as

indicated by X-ray diffraction analysis and electrical

resistivity measurement . The critical temperature is

11 'decreased when annealing, as it is also indicated by the

12structure of the thermal conductivity plot in Fig.l.

The T2 dependence of the thermal conductivity at low

temperature is characteristic of phonon-TLS resonant

scattering. Since annealing does not change the

temperature dependence but increases the thermal

conductivity we conclude that annealing increases the

coefficient of the T2 dependence. That is to say, it

decreases the product of the number of scattering cento's,

n(0), times the square of the coupling matrix between the

phonon and TLS. Considering only thermal conductivity

measurements that is all we can say. In any event, these

measurements indicate that these mild heat treatments can

modify considerably the TLS behaviour. In Fig.2 we plot

Che thermal conductivity of the amorphous sample taken at

T-0.5K, normalized by the value of the as quenched one, as

152

a function of the critical temperature, also normalized by

the critical temperature of the as quenched sample. From

these results it is tempting to say that there is a

correlation between the TLS behaviour and the critical

temperature. We will see later that this is not necessari-

ly true and that the change in T can be explained without

involving the assistance of the TLS.

In Fig.3 we show the effect of annealing in the

critical temperature and electrical resistivity. In the

plot of T vs pj normalized by the respective values of the

npn annealing sample, we can clearly distinguish two

regions. First, the critical temperature decreases at

almost constant p, later there is a rapid decrease in p

without major changes in T . This indicates two thermally

Induced processes. To detect structure changes during

annealing we have investigated the X-ray diffraction

pattern. In the first region, we were not able to

distinguish any change within our experimental error, in

the second when p('4k)/ p^C+k)- 0.8 it was detected a weak

structure typical oi crystallization. A n the results we

discuss here, including the thermal conductivity

measurements, correspond to thermal heat treatment in the

first region.

As was mentioned in the Introduction, if there Is only

one microscopic parameter that determines T , the change In

the parameter that correaponoa to a given AT should be

independent of the method used to vary T .

153

From specific heat and H » measurements the change in

density of states as a function of concentration is known.

From these data we obtain that the change we should expect

from the AT induced by annealing is only a few percent,c

Since it ia very' difficult to achieve the necessary

precision by measuring specific heats we decided to use

the H 2 And p measurements, together with expression (2),

to determine the relation between N(0) and T when

5 18annealing. There is experimental evidence ' indicating

that this expression is valid when applied to splat cooled

samples. In this work we assume the validity of expression

(2) and we will discuss later some related experimental

results, obtained in our laboratory.

Figure % shows the results of N(0) obtained from H j

as a function of T c. The dotts correspond to the

variation of N(0) with T , induced by annealing and the

full line is an interpolation from the data obtained by

changing concentration. We see that the data obtained by

thermal heat treatment are quite similar to that obtained

from the change in concentration. As a consequence, the

correlation suggested by Fig. 2 is not more than spurious,

indicating that there is no direct relation between T and

n(0). It would be intereting to understand why the thermal

relaxation changes the electronic density of states as well

as that of the TLS.

154

Let us focus our attention on the behaviour of the

electrical resistance. We have found that the resistivity

increases when the sample is annealed. Since we will not

discuss the kinetics of the relaxation process and it is

found that T is strongly correlated with the hehaviour of

the resistivity, we plot the resistivity change as a

function of the variation of T , see Fig. 5. The increase

in resistivity found for these alloys seems co be

characteristic of amorphous transition rcetais and, to my

knowledge, there is no explanation for such behaviour.

It is interesting to remark that in the range of

9 10concentration ' we investigate T decreases with N(0) and

p increases when N(0) is diminished. Since we know the

experimentai relation between concentration and N(0) we can

determine & P / A K ( 0 ) in the range of concentration of

19interest. It is found that the Ap/iN(0) obtained from

Figs. '* and 5 is smaller by a factor between 1.3 and 2.2

when compared with that obtained from the change ino jo

concentration'' . The range in the slope values is due to

the difference between the experimentai values of refs. 9

and 10. Considering the difficulties in determining the

geometrical factor of amorphous ribbons, we think chat çhe

19similarity found between the Ap/AN(0) obtained from the

change in concentration and annealing experiments, is

strongly indicating that N(0) is also the fundamental

parameter determining the behaviour of p.

Let us now discuss the results obtained from the

155

penetration depth measurements. Details on the

experimentai technique used to measure x(t) can be found in

reference 20. We wiii not discuss here the temperature

dependence of A(t). We will refer only to the relations

between X(0), p and T , as given by expression (3). We

19 20have measured ' Ã ( 0 ) , T and p for different amorphous

alloys, the results are shown in table I. We see that

expression (3) is verified within a 10% error. Since the

error in the geocetricai factor is not less than 10X we

find che agreement surprising and good. These results are

important since until now we hav~ indicated that N(0) is

the main microscopic parameter determining the behaviour of

several properties of the Z r ^ C u ^ systems-

In a two band model the density of electronic states

should be maimy related to the d contribution. As we said

in the introduction, expression (3) seems to be

incompatible with a two band model since the correction due

to a finite mean free path is given by a ratio of two

lengths, one characteristic of the superconducting state,

ÇQ, che other, l, related to the transport properties in

the normal state. In a two band jnodei T is determined by

c J

the d density of states but che t that appears in (3)

should not be the one that determines the measured

electrical conductivity.

The experimental verification of expression (3) is

strongly suggesting that in thest; transition amorphous

metal there is a single type of carriers contributing to

156

Che chernodynamic and transport properties. These results

5 18are in agreement with those ' supporting the verification

of expression (2). We believe that the suggestion made byo

tenBosch and Bennemann concerning to hybridization of d

and 8 bands is of fundamental importance for a correct

understanding of transport properties in amorphous

transition metals.

We have not been able to complete the discussion

proposed at the introduction but I hope that future work

will serve to verify the ideas exposed previously and will

clarify the rest of the remarks made at the beginning of

this lecture.

ACKNOWLEDGEMENTS

I want to thank M.E. de la Cruz for her help during

the preparation of the manuscript. The results that have

been discussed here were obtained at the Low Temperature

Group in Bariloche. I gracefully acknowledge many

suggestions and discussions with the staff of che group.

Many points discussed in che lecture become clearer after

useful conversations with 0. Balseiro.

157

REFERENCES

1 H. Tuczauer, P. Esquinazi, M.E. de ia Cruz and F. de

ia Cruz, Rev. Sci. Instrum. 51, 5'+6 (1980).

2 W.L. Johnson, in Glassy Metais I, Vol. '+6 of Topics in

Appiied Physics, edited by H.J. Güntherodt and H. Beck

(Springer, N.Y. 1981).

3 R. Harris, L.J. Lewis and M.J. Zuckermann, J. Phys. F

13, 2323 (1983), S.V. Maieev, Sov.Phys. JETP, 57, l'+9

(1983).

% P.W. Anderson, K.A. Muttalib, and T.V. Raroakrishnan,

Phys. Rev. B 28, 117 (1983); Liam Coffey, K.A.

Muttaiib and K. Levin, Phys. Rev. Lett. 51, 783

5 F.P. Misseii, S. Frota-Pessoa, J. Wood, J. Tyier and

J.E. Keens, Phys. Rev. B 27, 1596 (1983); K. Samwer

and H.v. Lohneysen, Phys. Rev. B 26, 107 (1982).

6 G. Bergmann, Phys. Rev. B 7, '+850 (1973).

7 G.F. Weir and G.J. Morgan, J. Phys. F. U, 1833

(1981).

8 A. cen Bosch and K.H. Bennemann, J. Phys. F. 5, 1333

(1975).

9 D. Pavuna, J. of Non Cryst. Sot. - in print.

10 M.N. Baibich, W.B. Muir, Z. Aitounian and Tu Guo-Hua,

Phya. Rev. B 2_7, 619 (1983).

11 J. Guimpei and F. de ia Cruz, Solid State Commun. '*\,

1045 (1982).

158

12 P. Esquinazi, M.E. de ia Cruz, A. Ridner and F. de la

Cruz, Solid Stace Commun. j-^, 9'4l (1982).

13 H.J. Schink, S. Grondey and H.v. Lohneysen, in

Phonon Scattering in Condensed Matter, Ed. W.

Eisenmenger and S. Dõttinger (Springer Series in

Solid State Sciences, Vol. 51, 1984).

1'* J.C. Lasjaunias, A. Ravex, and 0. Bêthoux, in Phonon

Scattering in Condensed Matter, Ed. V. Eisenmenger and

S. D'dttlnger (Springer Series in Solid State

Sciences, Vol. 51, 198'4).

15 L. Civale, F. de ia Cruz and J. Luzuriaga, Solid State

Cornmun, Vb, 389 (1983).

16 P.H. Kes and C.C. Tsuei, Phya. Rev. B J29, 5126 (1983).

17 Ç.M. Varma and Dynes, in Superconductivity in d and f-

band Metals, edited by D.H. Douglass (Plenum, N.Y.,

1976).

18 M.G. Karkut and R.R. Hake, Phys. Rev. B 28, 1396

(1983).

19 F. de ia Cruz, M.E. de la Cruz, L. Civale and R. Arce,

to be published.

20 R. Arce, F. de la Cruz and J. Gulmpel, Solid State

Cotnmun. '+7, 885 (1983).

O

w.

159

' "2T ! K )

Fig.l - Thermal conductivity, k, as a function of tenr-erature,T, for amorphous Zr7,.Cu_c, for different heat treat-ments. See ref.12.

1 - a A

Õ.9 Tc/Tc,

Fig.2 - Thermal conductivitv nt 0.5k as a function of the

change in T_, induced by annealing..

160

OHi

oi

Fig.3 - Variation of the critical temperature as a function

of the chanpe in resistivity, induced by annealing.

L...27 / "

Fig.4 - Density of states, N(0), as a function of T^. Open

circles correspond to the value obtained by annea-

ling, the full curve is an interpolation from ref.5.

F \

us!

Fip.5 - Variation of the electrical resistance

as a function of the critical temera-

turc.

161

A M O R P H O U S S U P E R C O N D U C T O R S

Frank P. M i s s e l l

I n s t i t u t o de F f s i c a , U n i v e r s i d a d e de S ã o P a u l o ,

C P . 2 0 5 1 6 , S ã o P a u l o , S . P .

W e d e s c r i b e b r i e f l y the s t r o n g c o u p l i n g s u p e r c o n d u c t i -

v i t y o b s e r v e d in a m o r p h o u s a l l o y s b a s e d u p o n s i m p l e m e t a l s .

For t r a n s i t i o n m e t a l a l l o y s w e d i s c u s s the b e h a v i o r of the

s u p e r c o n d u c t i n g t r a n s i t i o n t e m p e r a t u r e T , the u p p e r c r i t i -

cal f i e l d H , and the c r i t i c a l c u r r e n t J . A s u r v e y of c u r -

rent p r o b l e m s is p r e s e n t e d .

T h e f i r s t a m o r p h o u s s u p e r c o n d u c t o r w a s p r e p a r e d ín 1951»

by B u c k e l and H M s c h . T h e y e v a p o r a t e d Bi in h i g h v a c u u m

and d e p o s i t e d the v a p o r o n t o a s u b s t r a t e at l i q u i d H e t e m p -

e r a t u r e . T h e e f f e c t o f p r o d u c i n g B; in this m a n n e r is d r a -

m a t i c : in t h e n o r m a l c r y s t a l l i n e p h a s e Bi is not a s u p e r -

c o n d u c t o r , w h i l e a m o r p h o u s Bi has a s u p e r c o n d u c t i n g t r a n s i -

tion t e m p e r a t u r e T = 6 . 1 K. P u r e Ga can a l s o be p r e p a r e d in

the a m o r p h o u s p h a s e , as can m e t a l s s u c h as P b , S n , T l , In,

e t c . by the a d d i t i o n of 1 0 - 2 0 % of a n o t h e r c o m p o n e n t . O n e

g e n e r a l l y f i n d s that T i n c r e a s e s a b o v e the c r y s t a l l i n ec 2v a l u e for t h e s e a m o r p h o u s a l l o y s . In a d d i t i o n . from t u n n e l -

ing e x p e r i m e n t s o n e f i n d s that the v a l u e of 2 A / k T is of the

o r d e r of k.5 for t h e s e m a t e r i a l s ( w h e r e 2A is the z e r o

t e m p e r a t u r e e n e r g y g a p ) . T h i s v a l u e c o n t r a s t s w i t h the v a l u e

of 3 * 5 2 , a p p r o p r i a t e for a w e a k - c o u p l i n g 8 C S s u p e r c o n d u c t o r ,

and s u g g e s t s that t h e s e s u p e r c o n d u c t o r s a r e s t r o n g - c o u p l i n g ,

that i s , t h e y h a v e a s t r o n g e 1 e c t r o n - p h o n o n c o u p l i n g . In

T a b l e I, w e p r e s e n t v a l u e s of T and 2 A / k T for s e v e r a l

a m o r p h o u s a l l o y s . A l s o s h o w n in t h i s t a b l e a r e v a l u e s of >.,

the e l e c t r o n - p h o n o n c o u p l i n g c o n s t a n t , as d e t e r m i n e d f r o m

t u n n e l i n g e x p e r i m e n t s . A s t r o n g e l e c t r o n - p h o n o n c o u p l i n g

l e a d s to an e n h a n c e m e n t of the e l e c t r o n i c d e n s i t y of s t a t e s

by a f a c t o r 1+X, w h e r e >.«2/oi2F(w)d(i}/u). The E l i a s h b e r g f u n c -

162

T a b l e I. P r o p e r t i e s o f a m o r p h o u s s u p e r c o n d u c t o r s b a s e d o n2

s i m p l e tneta 1 s .Al l o y

Bi

Ga

S n 9 0 C u 1 0P b 9 0 C u 1 0P b75 B 525l n 8 0 S b 2 0T 1 9 0 T e 1 0

T c(K)

6.1

8.4

6.76

6.5

6.9

5-6

4.2

2A/kTc

4.60

4.60

k.k6

4.754.98

U.XO

i. .6

2

1

X.2-2.

.94-2

1.84

2.0

2.76

1.69

1 .70

46

• 25

t i o n a 2 F ( c o ) i s g i v e n in t e r m s o f a t h e a v e r a g e e l e c t r o n -

p h o n o n m a t r i x e l e m e n t a n d F ( u > ) , t h e d e n s i t y o f p h o n o n s t a t e s .

In t h e n o r m a l s t a t e C X 2 F ( O J ) is p r o p o r t i o n a l t o t h e p r o b a b i -

l i t y t h a t a n e x c i t e d e l e c t r o n c a n e m i t a p h o r t o n w i t h f r e q u e n -

cy to. T h e f u n c t i o n O I 2 F ( Í J Ü ) c a n b e d e t e r m i n e d e x p e r i m e n t a l l y

b y i n v e r t i n g d a t a f r o m t u n n e l i n g e x p e r i m e n t s .

F i g . 1 T h e E ' l i a s h b e r g f u n c t i o n a z F i u ) f o r c r y s t a l l i n e a n d

d i s o r d e r e d P b a n d a m o r p h o u s P b n C u j ( , .

In F i g . 1 w e s h o w t h e E l i a s h b e r g f u n c t i o n f o r c r y s t a l -

l i n e P b , f i n e - g r a i n e d c r y s t a l l i n e P b , a n d a m o r p h o u s P b ^ / ^ C u ^

d e t e r m i n e d f r o m t u n n e l i n g e x p e r i m e n t s b y K n o r r a n d B a r t h .

F o r c r y s t a l l i n e P b , v.he p e a k s c o r r e s p o n d i n g t o t h e t r a n s -

v e r s e a n d l o n g i t u d i n a l p h o n o n s a r e c l e a r l y v i s i b l e , w h i l e

t h e c u r v e f o r a m o r p h o u s P b C u . s h o w s a "large c o n t r i b u t i o n

a t l o w f r e q u e n c i e s . E v i d e n t l y t h i s l a r g e i n c r e a s e at l o w

163

frequencies is responsible for the increased values of X in

the amorphous alloys. It i? possible to show experimentally

from Mossbauer measurements of the Debye-Waller factor that

this increase is due primarily to an enhanced electron-

phonon interaction. The Mossbauer effect furnishes

/F(w)du/w at zero temperature, a quantity which weights the

phonon density of states in the same manner as the integral

defining A. Bolz and P o b e M measured this quantity for both

crystalline and amorphous Sn, obtaining the same result to

within 10%. Thgs the large differences observed in X between

then crystalline and amorphous states must be related to an

increased electron-phonon interaction a 2.

To calculate the electron-phonon coupling constant, it

is necessary to have some idea of the electronic structure

of these amorphous metals. There is experimental evidence

thst the amorphous superconducting alloys may be treated as

free electron metals. Measurements of the Hall effect for

n ç w t r a n s i t i on metals have yielded, with few exceptions,

the free electron value of the Hall constant . Furthermore,

the measured values of the optical surface resistance for

amorphous Ga agree with the predictions of the free elec-gtron model to '-;ithin e x p e r i m e n t a l u n c e r t a i n t y . Similar

9

agreement is o b t a i n e d for BÍ . A calculation of the electron-

phonon interaction, based on the free e l e c t r o n m o d e ! and

neg l e c t i n g c o n s e r v a t i o n of m o m e n t u m , has been carried out

by Bergmann , He showed that the increased phase space for

electron and phonon interactions could e x p l a i n the increased

values of a 2 in the amorphous s t a t e . Thus, it appears that

the increased values of X and 2A/kT can be explained in

terms of the changes in a 2F(u>), w h i c h , in turn, can be

related to d i s o r d e r . Other c o n s e q u e n c e s of strong coupling,

such as a modified temperature d e p e n d e n c e of the energy gap

and critical field as compared with the BC3 theory, have2

been observed and are discussed by Bergmann ,The first s y s t e m a t i c study of amorphous transition

meta,l alloys was carried out by Collver and Hammond w h o

employed e l e c t r o n - b e a m e v a p o r a t i o n onto cryogenic s u b s t r a t e s

Io4

to o b t a i n a large n u m b e r of m e t a s t a b l e m a t e r i a l s b a s e d on

the ki and 5d transition m e t a l s . T h e s e data are n o t e w o r t h y

?n that the v a r i a t i o n of T w i t h e l e c t r o n / a t o m (e/a) ratio

is m a r k e d l y different than in c r y s t a l l i n e m a t e r i a l s . For

the a m o r p h o u s t r a n s i t i o n metal a l l o y s , T varies slowly w i t h

e/a and e x h i b i t s a broad m a x i m u m a r o u n d e / a ^ 6 . 5 . In c o n t r a s t ,

T in c r y s t a l l i n e a l l o y s s h o w s two sharp peaks near e/a v a l u e s

of k.S and 6-5 ( M a t t h i a s ' r u l e ) . T h e s e peaks have been

a t t r i b u t e d to structure in the d-band density of s t a t e s for

c r y s t a l l i n e m a t e r i a l s . T h i s c o n t r a s t is shown in Fig. 2 .

P r e v i o u s l y , Crow et a_l . had m a d e similar o b s e r v a t i o n s for

c e r t a i n t r a n s i t i o n metal a l l o y s and a t t r i b u t e d the i n c r e a s e d

T^ values in the a m o r p h o u s s t a t e to m o d i f i c a t i o n s in the

e l e c t r o n i c density of s t a t e s due to the p r e s e n c e of d i s o r -

d e r . If T is d e t e r m i n e d m a i n l y by the e l e c t r o n i c d e n s i t y

of states N ( E p ) aná if N ( E p ) p o s s e s s e s a single b r o a d m a x -

imum for a m o r p h o u s alloys b a s e d on the Ad t r a n s i t i o n m e t a l s ,

then the Collver and H a m m o n d result could be a r e f l e c t i o n

of a s t r u c t u r e l e s s d band for these a m o r p h o u s t r a n s i t i o n

m e t a ! a 1 loys.

20

16 -

12 -

2'I

1 I

-

-

i f •

A>

§•

h\4 -

3 4 S 6 7 8 9(V) (Zf) INb) (Mo) (Tc) (Rui (Rh)

i/m

F i g . 2 T h e v a r i a t i o n o f T f o r kà t r a n s i t i o n m e t a l s in t h e

c r y s t a l l i n e ( d a s h e d c u r v e ) and a m o r p h o u s ( s o l i d c u r v e )

s t a t e s , f r o m C o l l v e r a n d H a m m o n d .

165

Th e a m o r p h o u s n a t u r e of the films o f C o l l v e r and Ham ~

m o n d w a s d e d u c e d i n d i r e c t l y from the sharp drop in the e l e c -

trical r e s i s t a n c e w h e n the fi l m s w e r e h e a t e d to room t e m p e r -

a t u r e . In a few c a s e s , such as for al l o y s of M o - R u o r N b - Z r ,

the a m o r p h o u s p h a s e wa.s s t a b l e at room t e m p e r a t u r e and it

wa s p o s s i b l e to v e r i f y the a m o r p h o u s n a t u r e by m e a n s of

t r a n s m i s s i o n e l e c t r o n m i c r o s c o p y . The first m e t a l l i c g l a s s e s

e x h i b i t i n g s u p e r c o n d u c t i v i t y w e r e La-Au a l l o y s , P r e p a r e d by

splat q u e n c h i n g . Many o t h e r s u p e r c o n d u c t i n g t r a n s i t i o n

m e t a l a l l o y s h a v e s i n c e been p r e p a r e d w h i c h a re s t a b l e at

room t e m p e r a t u r e . The T v a l u e s of some of the s e a l l o y s are

given in Ta b l e II. We note that the h i g h e s t t r a n s i t i o n temp-

e r a t u r e is T = 9 . 0 0 K for M o 8 o P i O B 1 0 ' T h e s e a 1 1 o v s f a " i n t o

two b r o a d c a t e g o r i e s : e a r l y t r a n s i t i o n m e t a l s ( Z r , Ti , N b )

wit h late t r a n s i t i o n m e t a l s (Rh, Cu, Pd) or t r a n s i t i o n m e t a l s1 k 15wi t h a n o n - t r a n s í t i o n metal or a m e t a l l o i d . J o h n s o n '

has p r e s e n t e d reviews of the p r o p e r t i e s of these m a t e r i a l s .

T a b l e II, T v a l u e s for some a m o r p h o u s s u p e r c o n d u c t o r s

Al loy T XiSi

( M o 0 . 8 R u 0 . 2 ) 8 0 P 2 C 7 .31

*VioBio 9 - 0 0

( M o 0 . 8 R e 0 . 2 ) 8 0 P 1 0 B 1 0 8 ' 7 1

L a 8 0 A u 2 0 3 ' 5

L a 8 0 G a 2 0 3 - 8 4

7 5 5

Z r70P d30 2 .807 3

Nb,Ge 3-6

NbjSi 3-9

Mo 6 8 Si 3 2 7.4«°80N20 8 ' 3

T h e s u p e r c o n d u c t i v i t y of a m o r p h o u s t r a n s i t i o n metal

a l l o y s p r e s e n t s d i f f e r e n t f e a t u r e s than that of a l l o y s based

1Ó6

upon simple m e t a l s . For exa m p l e , results of tunneling exper-

iments furnish values of 2A/kT which are in good agree-

ment with the value obtained from BCS theory. Thus these

alloys are weak coupling superconductors with X < 1 , and we

expect that many of their superconducting properties will

be in agreement with the BCS or Ginzb u r g - L a n d a u - Abrikosov-

Gor'kov (GLAG) theory. The situation is more complex,

however.

Early m e a s u r e m e n t s of the upper critical field H (T)

for amorphous transition m e t a l s showed rather large values

for this quantity, consistent with a small coherence

length £ and large penetration depth \., characteristic of

materials possessing a short mean free p a t h . Furthermore,

the field slope at T , (dH _ / d T ) _ , is also very largec ci i

(20-l»0 kG/K) , cons i s t en t w i t h the r e s u l t of the extended

GLAG t h e o r y 1 7 ' 1 8 :

T = " 6 J & £ £ pN { E F ) ( 1 >c TT

where p »s the normal resistivity, N (E_) is the electron-

phonon dressed density of states at the Fermi level for one

spin direction, and 8 is an enhancement factor (of order 1)

for strong coupled superconductors. Since the amorphous

transition metal alloys have large normal resistivities and

large densities of states, it is reasonable that the field

slope is also large.

The temperature dependence of H „ was observed to be

linear over a substantial portion of the temperature range

below T '"15> 9 |.ater, Tenhover et al . made a carefulc -

comparison of H ?(T) to the GLAG theory for the amorphous

alloys M o 3 0 R e 7 ^ ( H o ^ R u ^ ^ B , v and (Mo0 s*u Q J ^? n,

produced by the hammer-and-anvi1 method. These authors found

the experimental values of H « to be larger than the pre-

dictions of the GLAG theory for low values of the reduced21temperature t»T/T . Carter et al. showed how these enhan-

c «—«——

ced values of H might arise from a fine-scale inhomogen-

efty In the sample.

167

They suggested that this inhomogeneity might result from the

fact that the quench rate in the experiments of Tenhover

et al . (dT/dtM 0 K/sec.)might not have been sufficient to

produce a thoroughly homogeneous sample. As evidence, they

noted that their sputtered films of and M oç 2G el»8

were in complete agreement with the GLAG theory. Recent22

measurements on melt-spun and sputtered samples of Zr.^Rh.

show good agreement of H ,(T) with the GLAG theory Jn both

cases. Thus it appears that the linear behavior observed

for H ~(T) down to low reduced temperatures, may not be

general behavior for all amorphous superconductors, but may

be restricted to certain sample compositions or to certain

sample preparation techniques. Further experiments are need"

ed in this area.

0

Fig. 3 The upper critical field H . vs temperature T for

sputtered ( Z r7^ R h24^ an(* "ît'sP" 0 (Zr _Rh )material,

Solid lines are predictions of the GLAG theory in the

absence of paramagnetic limiting. Data can be brought

into agreement with theory by considering spin-orbit

scattering and paramagnetic limiting.

The homogeneous nature of amorphous superconductors

becomes quite evident when we consider the critical current

density J The J values encountered are very low in thecas-quenched material. In Fig.

6 R u0 i»^80B10^'i0

we compare J for two samplescby the hammer-and-anv i I

16S

IO4

^. IO1

10

I ' I • I • I

"A.-nofphoo«"

T-I.6K

K

«T-4.2K

.• I • I M , 1O 1 2 3 4 5

10' _

!0* -

101 -

10»

r

-

T M

UI

;

ran9e totJei0)

, L ...

i | r f i i . j

Airwphoui withcryitallinc inclusions _

_ . T-2.57 K -

x :T - 4 . I B K :

•

! i I • • • ' •0 1 2 3 * 5

ll.T

F i g . í» C r i t i c a i c u r r e n t d e n s i t y v s . m a g n e t i c f i e l d f o r t w o

s a m p l e s o f ( M o 0 ^ Ü Q } 8 ( JS i , Q 3 1 fl ,

t e c h n i q u e . In o n e c a s e , the s a m p l e h a d b e e n p a r t i a l l y r e -

c r y s t a l H z e d by m e a n s o f a n n e a l i n g a n d the r e s u l t a n t c r y s -

t a l l i t e s s e r v e as f l u x - p i n n i n o s i t e s . A s a r e s u l t , J h a s3 c

i n c r e a s e d by a f a c t o r of 10 at 20 k G . A p r o m i s i n g m a t e r i a l

in t e r m s o f i is t h e a l l o y ( Z r Q 7 H f Q . ) , n V , . . In the a m o r -

p h o u s p h a s e , t h i s m a t e r i a l e x h i b i t s g r e a t d u c t i l i t y b u t Is

not s u p e r c o n d u c t i n g . A f t e r an a p p r o p r i a t e h e a t t r e a t m e n t ,

the b r i t t l e C - 1 5 p h a s e f o r m s w i t h a J in e x c e s s o f lOOkA/cir/

at 1 5 0 k G . P o s s i b l e a p p l i c a t i o n s o f a m o r p h o u s m a t e r i a l s h a v ebeen d i s c u s s e d e l s e w h e r e25

As in the case of H -,(í), the T behavior* of amorphousc2 c

transition metal alloys sti1 1 presents some d i f f i c u l t i e s .

The suggestion by Crow et a 1., that disorder-induced changes

ín the density of electronic states N ( E _ ; may be responsible

for the differences in T behavior between •rystalline and

amorphous stater., has been discussed extent vely by

Johnson ' in terms of the McMillan equal on . The Mc-

Millan equation gives T_ in terms of <w z> the mean square

phonon frequency, u , tê Coulomb pseudopotentfa 1, and >,

the e1ectron-phonon coupling constant a s :

169

r i.oi(i+\)

L" I^+TnõTr2xTTc = TTfT exp

McMillan showed that X,as defined previously, can be rewrit-

ten as A»N(E F)<I2>/M<a) 2>, where N ( E F ) = N

+ ( E F ) / O + X ) is the

bare density of electron states at the Fermi level, <\2> is

an average squared electronic matrix element, with the

average taken over all electron-phonon scattering processes,

and M is the Ionic mass.

Now in order to apply this equation to the problem of

amorphous transition-metal superconductors, Johnson employed

a tight-binding analysis of the electron-phonon coupling27constant X due to Varma and Dynes . These authors used a

simple model of the d band density of states, in which s

electrons and s-d hybridization were neglected. For the case

in which there is one orbital at the Fermi surface, Varma

and Dynes argued that the constant X is given by X=N(E )W(1+5),

where W is the width of the d band and S is the overlap in-

tegral of atomic d orbitais on neighboring sites. The factor

(ITS) takes into account the nonorthogonal ity of atomic d

functions on different sites. The mi nus(p1 us) sign is appro-

priate for E_ in the lower(upper) half of the d band des-

cribed by N ( £ _ ) . To the extent that the width V is a constant

for a given transition metal series, then X^N(E_) within

the upper and lower (antibonding and bonding) halves of the

d band. Thus, N(E_) is the principal parameter governing

superconductivity in these materials.

In order to compare the Varma-Dynes predictions with

experiment, Johnson obtained N(E_) from the low temperature

specific heat coefficient y and obtained X by inverting the

McMillan equation. The ratio X/N(E_) was compared for several

alloys and was found to be constant, as one might expect

from the Varma-Dynes calculation . In an attempt to study

the Varma-Dynes mode) for a greater number of alloys, Flasck

et a l / c plotted T c vs. N+(F.p) for (Mo, - x K % ) 8 o

p2 0 .

("VôVi-xV ("VeKVî^V and (Hoi-yVi-*sV

where T is a transition metal. N (E p) was determined from

170

Eq.(i), using measured values of p and dH _/dT. Although

Eq.(i) was derived for free electrons, a number of experi-28 +ments show that N (E_), derived from this equation and Y»

agree to within 5-10?. Thus, if N+(E ) is the most important

parameter in determining T , a graph of T vs. N (E_)

should have a positive slope. In fact, a correlation was

observed between T and N (E_), but this correlation depends

upon the metalloid involved in the alloy. For the alloys

involving Si, the slope of the curve T vs. H (E_) turns out

to be negative, i.e., larger values of N (Ep) result in

smaller T . Since the electronic structure of these alloys

is not well understood, this result does not necessarily

represent a contradiction of the Varma-Dynes model. It is

possible, for example, that hybridization effects might

invalidate some of the assumptions of the model.

In an attempt to study the Varma-Dynes model for a si-

tuation where its assumptions should be valid, alloys with a29simple electronic structure were considered . For example,

Zr-Cu alloys should satisfy the assumption of one orbital

at the Fermi level, since calculations performed using the

; I29

recursion method'' have shown that the Cu 3d states lie about

3-5 eV below E _ . Zr-rich alloys of Zr-fl.h also seem 7 to have

only Zr states near E_. In these two cases, the curve

T v s . N (E ) has a positive slope and thus larger values of

M (E-) result in larger values of T . Thus it appears that

the Varma-Dynes model might have a certain limited validity,

in cases where its assumptions are clearly satisfied. This

does not mean to say that all amorphous alloys based on

early/late transition metals satisfy the assumptions of the

model. Recent measurements of N (E_) in Zr-Pd and (slightly

contaminated)Zr-Ni alloys indicate that, in these m a t e r i a l s ,

N (E-) is not the dominant factor in determining T .

The question then arises as to whether the Collver-

Hammond curve does indeed result simply from a single broad

maximum in a featureless d band. It is known, for example,

that calculations of the d band for amorphous Co, simulated

by a dense random packing of hard spheres relaxed through a

171

lennard'Jones potential, show two broad peaks, one for the32

bonding and the other for the antibonding states . Thus a

simple rigid-band approach to this double-peaked band would

not explain the Col 1ver-Mammond data, in order to a n s w e r

this question, calculations are under way, using the recur-

sion methpu, of N(E_) for amorphous Mo.. Ru alloys . Since

e/a"6 for pure Mo and e/a^tJ for pure Ru, these c a l c u l a t i o n s

will furnish H(E-) «round the peak of the Col 1ver-Hammand

curve and allow a test of the arguments which have so often

been put forth to explain these d a t a 1 1 ' 1 2 ' ' **.

Recent studies of the density of states N (E p) In amor-

phous alloys have raised another problem related to the

experimental determination of this quantity. For e x a m p l e ,

Altounian and Strom-Olsen- have recently determined N (E_)

for melt-spun Zr-Ni alloys using E q . ( 1 ) . Their results are

in reasonable agreement with those of Kroeger et a l . who

studied melt-spun Zr-Ni alloys and determined N (E-) from

specific heat m e a s u r e m e n t s , On the other hand, the N (E_)

values obtained for sputtered Zr-Ni alloys are in agreement

with results from melt-spun material for Zr-rich c o m p o s i t i o n s ,

but there is some disagreement as the Zr fraction Is redu-

c«d . This disagreement may .esult from the different

quench rates obtained In m<?lt spinning and s p u t t e r i n g . T h u s ,

the melt-spun m a t e r i a l , produced at a lower cooling rate

( d T / d t M O K / s e c , ) , may show a slight chemical short range

ordering, which would tend to decrease N(E_) in these materials

and which would not be present in the sputtered m a t e r i a l s1 kbecause of the higher quench rate ( d T / d f U O K / s e c ) . Indeed

Krocger et a 1 .have seen evidence of chemical ordering in

their Zr-Ni #)loys. W h e t h c or not this is the e x p l a n a t i o n

for these discrepancies is difficult to decide since there

has been very little in the way of structural studies on

these materials. It would be Interesting to repeat these

measurements on melt-spun and sputtered samples w h i c h have

been thoroughly characterised by x-ray and neutron s c a t t e r -

ing experiments.

Another possible e x p l a n a t i o n for the different v a l u e *

172

of N + ( E p ) observed in these experiments lies in the use of

two different methods (specific heat and Eq. (?)) for deter-

mining N + ( E p ) . Recently, Ravex et al. determined N + ( E _ )

for sputtered Zr-Cu alloys using both specific heat measure"

msnts and Çq, (1) . The value obtained from Eq.(l) was ^ 0 %

smaller than that obtained directly from the low temperature

specific heat- This is the first and only report of a serious

disagreement between these two methods of measuring H (E p) .

I % must be teken seriously,however,sSnce Eq.(l)was derived

for free electrons.We note that the results on m e l f s p u n34 35Zr-Ni material generally agree among themselves, even

though in one case Eq.(1) was used to obtain N + ( E _ ) while in

pec i f131,36

the o t h e r çaçe t h i s q u a n t i t y was o b t a i n e d f rom the s p e c i f i c

h e a t . A s i m i l a r s i t u a t i o n h o l d s f o r s p u t t e r e d m a t e r i a l

F u r t h e r m o r e , many o t h e r examples e x i s t o f good agreement

between t besç two methods o f d e t e r m i n i n g N ( E , . ) . Nevertheless,

i t wou ld be u s e f u l t o have f u r t h e r t h e o r e t i c a l s t u d i e s o f

t he a p l i c a b i j i t y o f the GLAG t h e o r y to amorphous s y s t e m s .

F i n a l l y we men t i on the e f f e c t o f a n n e a l i n g on t he super38 ~"

c o n d u c t i n g p r o p e r t i e s . Drehman and Johnsor r s t u d i e d the e_f

f e e t on t h e r m a l r e l a x a t i o n on the s u p e r c o n d u c t i n g propert ies

o f an amorphous Zr -Rh a l l o y . These a u t h o r s obse rved a M 5 &

decrease i n T r , as w e l l as o t h e r changes upon a n n e a l i n g a t

t e m p e r a t u r e s w e l l be low the c r y s t a l i z a t i o n t e m p e r a t u r e . M o r e

r e c e n t l y Qaroche e t a i . • s t u d i e d the e f f e c t o f t h e r m a l r e l £

x a t l o n on amorphous Zr -Cu and no ted the changes w h i c h oc -

c u r r e d i n T-, p, O. , and A. These r e s u l t s y/ere c o n s i s t e n t l y

i n t e r p r e t e d |n terms o f the M c M i l l a n e q u a t i o n f o r T - . l n s p f -

t© o f t h i s , however , the T« r e d u c t i o n sugges ts a r e d u c t i o n

i n t he e f f e c t i v e e l e c t r o n - e l e c t r o n i n t e r a c t i o n . At t h e same

t i m e , a n n e a l i n g has been sliown t o reduce the low t e m p e r a t u -

re s p e c i f i c heat , a r e d u c t i o n wh ich has been a t t r i b u t e d to

a r e d u c t i o n |n the number o f t w o - l e v e l «.ystems ( T L S ) . The

TI.5 a re g e n e r a l l y accep ted as the e x p l a n a t i o n f o r the low

t e m p e r a t u r e the rma l and u l t r a s o n i c anoma l i es o b s e r v e d in

g l a s s e s . A model f o r TLS i n m e t a l l i c g l a s s e s has been d e -h]

v e l o p e d by B?nvi l le and Harr is , and models fo r the enhancement of T-by the TLS in m e t a l l i c g l a s s e s have been proposed . However,

173

recent experiment by Grondey et a l , * In amorphous Zr-Cg

Has thrown doubt on this e x p i r a t i o n of the annealing

Vior of the superconducting properties of amorphous alloy?.

These authors measured the low temperature heat capacity and

thermal conductivity, for several well characterized alloys,

as a function of annealing. Large change? in the thermal çqn.

d.uctivity were clearly observed, but the corresponding ch$n

ge$ in the specific heat were much smaller than expected.

Thus there appears to be some doubt whether the reduction

Of TLS upon annealing really causes the changes in the heat

capacity and thermal conductivity, or if it only occasionally

a,ççompanys these changes.

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38. A.J. Drehman and W.L. Johnson, Phys. Stat. Solidi (a)

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41. M. Banvilie and R. Harris, Phys. Rev. Lett. _44_, 1136

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Work Supported by FAPESP, CNPq and FINEP (Brazil)

176

ON THE SUPERCONDUCTIVITY OF VANADIUM BASED ALLOYS

F. Brouers

Freie U&iversitat Berlin, Inst i tut für Theoretische Physik

Arnimallee lkt D-1000 Berlin 33

and

Department of Physics, University of the West Indies

Mona, Kingston 7, Jamaica W. I .

and

J. Van der RestInst i tut de Physique, Université de Liege,Belgium

Abstract

We have computed the electron density of states of solid

solutions of vanadium based transition metal alloys V 5Í by using

the tight-binding recursion method for degenerate d-bands in order

to calculate the alloy superconducting transition temperature with

the McMillan formula. As observed experimentally for X on the left

hand side of V in the periodic table one obtains an increase of T

while for X on the right hand side of>V the cr i t ical temperature

decreases.

The detailed comparison with experiments indicate that

when the bandwidths of the two constituents are different, one can-

not neglect the variation of the electron-phonon interactions.

Another important conclusion is that for alloys which are

in the split-band limit like VAu, VPd and VPt, the agreement with

177

experimental data can be obtained only by assorting that these alloys

have a short-range order favouring clusters of pure vanadium.

1. Introduction

There i s currently an active interest in the effect of

disorder on the superconducting properties of materials. Superconduc-

tivity in alloys and amorphous metals and alloys has been the subject

of a great number of experimental and theoretical studies.

From these efforts, one expects to obtain not only a better

understanding of the basic nature of superconductivity in highly dis-

ordered metallic systems but also some important technological appli-

cations of new class of materials. This interest is related among

others to the search for superconductors which show minimum degrada-

tion of superconducting properties by neutron irradiation.

To analyze the variation of the critical superconducting

temperature with disorder, experimentalists use the McMillan formula

(McMillan 1968)

( ! )X -u* - 0.62Xu*

6. i s the Debye temperature and X is the Fermi surface averaged

enhancement of the electron mass which corresponds to a reduction

of the electron velocity due to electron-phonon scattering by a

factor (1 + A) McMillan formula (1) is an extension of the BCS

formula

Tc = 1.13 8D exp | £• | (2)

178

obtained by introducing an electron mass enhancement in the two-

particle propagator of the "vertex function", which has the effect

of replacing the exponent A~l by (1 + A)/A, by taking account of

the effective Coulomb repulsion u* between the electrons and by

considering the T-dependence of X.

Moreover, McMillan (1968) showed that X can be factorized

into electronic and primarily phonon dependent factors by writing

H(Er) < I2 >A = i (3)

M < o»2 >

where H(E_) i s the bare density of states at Fermi level, <I2> the

Fend surface average of the square of the electron-phonon matrix

elements and

« 2

/dw o2 (u)F(w)w~1

and average over the frequency distribution fmction multiplied by

a factor a(u) depending on the phonon induced electron-electron

scattering.

For transition metals, the calculated values of < Iz> and M< (o2>

show a maximum near the middle of the series.

It is not surprising that the electron-phonon interaction should be

strong near the center of the transition metal series. Assuming the

atomic orbitais to decay exponentially as 0(r)exp(-qQr), one has

< I2 > * q* W2 (5)

where w, is the d-bandwidth which peaks near the middle of the

transition series.

179

Although the variation in < I2> overcomes the variation of

M < w2 > , the parallel variation of < I2> and M <u>2 > with atomic

number has led to the conclusion that one can consider the ratio con-

stant and the effect of disorder on T has been mostly discussed by

considering only the variation of N(Ep)/pynes and Vanna 1976/.

However, the comparison of the superconductivity of crystalline and

amorphous phases of transition metals has shown deviations with respect

to this simple rule.

Recent comparative studies of crystalline and amorphous phase

of Nbjàe suggest that there is no strong correlation between N(E_) and

T in strong coupled A15-superconductors /Tsui 1980/. In particular,

the high T superconductivity of Nb.Ge does not appear to stem from

an unusually high N(E_), but rather from a relatively strong electron-

phonon coupling enhanced by some localized electron states. Hanke et

al. (1976), Cowan and Carbotte (1978).

These recent works show the interest of a systematic study

of the variation of T with N(Ep) and electron-phonon interaction in

disordered systems.

In the present paper we want to analyze the experimental

data of Daumer et al (1982) who have studied the superconductivity

of vanadium solid solution VQ X where X is a transition metal.

These authors have applied the McMillan formula assuming that A varies

only with N(Ep). The agreement is good for the alloys where specific

heat measurements are available. For the others these authors have

estimated N(E_) from the measured values of T .r C

180

We have made a systematic calculation of the density of

states at Fermi-level of these solid solutions as well as other for

which data are not yet available* We have used the technique of

calculating the S-fold degenerate d-band density of states froa a

continuous fraction expansion of the tight binding Green's function

with the help of the recursion method of Haydock, Heine and Kelly

(1980). We did not try to f i t the experimental data. We have chosen

the alloy and disorder parameters used by Van der Rest et al. (1975)

to calculate alloy energies of formation.

From these results, we can conclude that, when the bandwidths

of the two constituents are different, one cannot neglect the variation

of the electron-phonon interaction as i t was assumed by Da user

(19B2).

2. Method of Calculation

We have calculated the density of states using the recursion

method of Haydock, Heine and Kelly (1980).

For each alloy, we have calculated the local densities of

states on a V and on a X atom for the seven most probable configurations

of the first neighbors. The occupation of more distant atoms i s

chosen randomly. For a concentration of 90% of V, we have treated

exactly 96% of the first neighbors possible configurations. The

average density of states i s obtained by averaging over these partial

densities of states.

We have used the BCC structure and we have considered the

first and second neighbors hopping integrals and calculated the first

181

seven steps of the continuous fraction. This corresponds to clusters

of 1695 atoms. We have only considered the d-density of states. The

s-density of states i s supposed constant aid equal to the inverse of

the s-bandwidth.

The energy levels E. , the electron nushers N. as well as

the d-bandwidths of the pure metals (W.) are given in Table 1.

The hopping integrals between two V neighboring atoms are

taken to be (in eV)

dd o1 = - 1.05, dd TTj = 0.5, dd 6± = - 0.065

The hopping integrals between second neighbors are taken to be

(for a discussion of these values see Derenne et al (1982))

dd o2 = - 0.i»3, dd Tt2 ? 0.16, dd 62= - 0.016.

The hopping integrals between the X atoms are obtained by multiply-

X V

ing the V-hopping integrals by the ratio W./W.. The hopping in-

tegrals between a V-atotn and a X-atom are obtained by taking the

geometric average of V- and X- hopping integrals. When the s-density

of states i s added to the d-density of states, the f i l l ing of the

d-band i s fitted in order to obtain the pure vanadium N(E-) calculated

by Glotzel et al. (1979).

The energy levels and charge transfers have been calculated

self-consistently.

3. Numerical Results

The results of our calculations are summarized in Table 1.

182

The e and W parameters are the ones used by Van der Rest et al.

(1975). To calculate T we have used McMillan formula (1). lhec

Debye temperature is the average of the pure &.. T is the

critical temperature assuming that X depends only on N(Ep). The

proportionality constant i s chosen to f i t the experimental T of(2)pure vanadium. T is the calculated critical temperature assuming

that the electron-phonon interaction varies as the averaged bandwidth

following Eq. (S).

t . Conclusions

From the observation of Table 1 , one can conclude that

1) when X i s on the lef t hand side of the periodic table with

respect to V, T i s enhanced, when X is on the right hand

side i t decreases (Zr and Hf are between brackets because

they do not form 10at% solid solutions);

2) when the d-bandwidths are different, one cannot neglect

the variation of the electron-phonon interactions. It i s

interesting to note that such an effect appears already

for alloys of relatively small (10at%) X-concentration.

3) The calculations (Brouers and Van der Rest 1981) reveal

that the three alloys which exhibit a large discrepancy

with experimental data (AuV, PdV, PtV) are in the spl i t

d-band limit and have partial densities of states the

shape of which varies strongly with local environment.

Cohesive energy calculations indicate that the short-range

order favours the formation of pure V-clusters in the alloy.

183

In that case, the density of states H(E_) of Pt and Att

are reduced respectively to 1.11 and 1.12 and the respective

Tc to 0.<t and 0.3 °K which i s in agreement with experimental

data.

Our calculation show that more correlated experimental data (specific

heat and critical teaperature measurements) are needed for nore alloys

and for different concentrations i f one wants to have a better under-

standing of the effect of disorder on superconductivity of alloys.

184

TABLE 1

V

Cr

Ma

(Zr)

Mb

Mo

Pd

(Hf)

TaW

Re

Pt

A.U

Ed(eV)

0

-0.5

-1

0.6

-0.1

-0.8

-3.6

0.6

-0.1

-0.8

-1.5

-3.6

-5

Wd(eV)

7

6.5

6

7

9

9

6

8

11

11

10.5

7.5

1.5

N(Ep)(StateseV/at)

1.80

1.51

1.45

2.01»

1.75

1.15

1.25

2.00

1.72

1.50

1.17

1.22

1.32

T (Dc

5.3

2.6

2.1

7.6

4.2

2.3

0.8

7.2

4.1

2.6

0.6

0.7

1.2

T c( 2 )

5.3

2.5

1.9

7.6

5.9

3.1

0.7

7.5

6.3

4

1.1

0.7

0.9

5.3a)b)

2.818) - 2.166)

2.1ia)

».08a)" - 5.6C)

2.85a) - 3.1b)

0.37a)

-

3.59a) - 4.47C)

3.5«*a) - 4.08d)

0.76a)

0.37a)

0.30a)

a) Daumer et ai (1982)

b) Andres et ai (1969)

c) Corsan and Cook (1970)

d) Shivkov et ai (1975)

e) R. Kuentzler (1982)

185

References

- Andres, K., Bucker, E . , Maita, J .P. and Sherwood, R.C (1969)

Phys. Rev. 178_, 702.

- Brouers, F. and Van der Rest, J. (1984) t o be published.

- Corsan, J.M. and Cook, A.J. (1970) Phys. Stat . Sol. W,, 657.

- Cowan, N.B. and Carbotte, J.P. (1978) J. Phys. C 11^ L 265.

- Dauner, W., Kettschau, A. , Khan, H.R., Lviders, K., Raub, Ch.J.»

Rieseneier, H. and Roth, 6. (1982) Phys. Stat . S o l . ( b ) . 112_, K67.

- Derenne, M., Brouers, F. and Van der Rest, J. (1982) Phys. Rev.

B26, 1048.

- Glotzel , D. , Rainer, D. and Schrober, H.R. (1979) Z.Physik

B35, 317.

- Hanke, W., Hafher, J. and Bi lz , H. (1976) Phys.Rev.Lett. £7 , 1560.

- Haydock, R., Heine, V. and Kelly, M.J. (1980) Solid State Physics

35, (Acad. Press) .

- Shikov, A.A., Chemoplekov, N.A., Panova, G.K., Samoilov, B.N.

and Zhernov, A.P. (1975) JETP 4_2_, 927.

- Tsuei, C.C. (1980) in "Superconductivity in d- and f-band metals,

ed. by H.Suhl and M.B. Maple, Acad.Press p.233.

• Van der Rest, J . , Gautier, F. and Brouers, F. (1975) J. of Phys. F5_,

2283.

186

DENSIDADE DE ESTADOS ELETRÔNICOS EM VIDROS METÁLICOS

SÔNIA FROTA-PESSOA

Instituto de Físisa, Universidade de São Paulo, CP.20516»

São Paulo, SP.

INTRODUÇÃO - O conhecimento da densidade de estados eletrô-

nicos em geral, e da densidade de estados na energia de Fe_r

ml em particular» ê crucial para um melhor entendimento do

comportamento dos materiais. Freqüentemente a diferença e£

tre as propriedades de uma mesma liga, na forma cristalina

e na forma amorfa pode ser atribuída a diferentes densida-

des de estados eletrônicos para a liga nessas duas formas.

Assim, dentro do curso de vidros metálicos, essa parte será

dedicada ao estudo de densidade de estados eletrônicos em me

tais amorfos, com especial ênfase para ligas amorfas de me-

tal de transição.

1. Comentários gerais - Um cálculo exato da densidade de es_

tados pára -mate ri a is amorfos, ao nível de sofisticação atí£

gtdo em materiais cristalinos não é factível. A própria es_

trutura amorfa é de difícil determinação. Além disso os m£

todos mais poderosos para o cálculo de densidade de estados

são implementados no espaço recíproco que no amorfo é esseji

cialmente perdido. Felizmente, devido a diversidade de ar-

ranjos locais existentes no amorfo, a curva da densidade de

estados como função da energia é suave (ver fig. I) e pode-

mos usar com bons resultados métodos de pouca resolução em

energia, que não seriam úteis no caso cristalino.

Antes de prosseguir gostaria de dividir os metais em

duas categorias; os metais simples cujos elétrons de valên-

ela são s_ e £, e metais mais complicados envolvendo metais

de transição.

1.a. Metais Simples - Para esses sistemas a densidade de e$

tados é usualmente bem representada pelo modelo do elétron

livre* . Esse modelo funciona melhor para metais simples

amorfos que para cristalinos, devido a ausência de planosli.)de Bragg. Várias teorias de sucesso* ' (como a teoria de

Zlman para condutividade) tomam por base o modelo de elétron

livre para representar metais simples amorfos.

187

(b)

1.b. Sistemas envolvendo metais de transição - Como elétrons

_d não são bem representados por um modelo de elétrons livres

a situação neses materiais é mais complicada e um cálculo

de bandas mais

sofisticado se

faz necessário

A fig. I i lus- (o)

tra a densida-

de de estados

N ( E ) , tomada

por átomo do

material, para

a Landa d em

um material

cristalino

(fig. la) e em

um material a-r

morfo (fig.1b

A área sob as

curvas é a me_s

ma uma vez que em ambos os casos cabem 10 elétrons por át£

mo na banda d^ Para ilustrar o comportamento de uma liga

binaria, na fig. 2 mostramos uma comparação entre a densi-

dade de estados cal-

0.1 Ryd 0.1 Ryd

£o.» 2

Ii

m

]uiZ

1

ii

|

11/'

i— Zr3 Cu

—1 -

-5

Fig. 2 E (eV)

photoelectron espectroscopy). ,6 ,7 ,8)

culada para Zr-Cu nu-ma estrutura cristalj_na densa ( l inhacheia) e para um sis-tema amorfo<x-Zrg7Cu-, * ' (1 inhaponti1hada).2.a. Medidas de H(E)-2?_ XPS (X-ray photo-electron espectroscopy)e UPS (u l t rav io let

As técnicas de XPS e UPS dão uma idéia da forma deN(E)

como função da energia para energias menores que a energia

de Fermi E^. As duas técnicas diferem essencialmente pela

188

energia utilizada, a primeira usando raio X e a segunda uti-

lizando luz na região do ultra-violeta. Embora os resulta-

dos de XPS e UPS retratem a grosso modo a densidade de esta

dos do material, existem distorções pois os elementos de n»£

triz para transições óticas podem ser fortemente dependen-

tes da energia. 5 Normalmente espectros XPS são mais con-

fiáveis pois distorções que favorecem energias na vizinhan-

ça da energia de Fermi sio comuns quando são usados fotons

de mats baixa energia.

Por outro lado a res£

luçio (possibilidade

de distinguir estrutu

ras finas em N(E)) ê

melhor em medidas fej_

tas por UPS. Na fig.3

mostramos resultados

de UPS*6* para várias

concentrações da liga

amorfa de ZrCu.

2.b. Medidas de N(Ef) - Calor específico a baixas tempera-

turas, ' ' medidas de H , contra temperatura T em super-(11 in) c í

condutores* ' ' e medidas de susceptibi 1 idade de Pauli.i) Calor específico e baixas temperaturas - Em metais nor-

mais a baixas tempera

turas o calor espec£

fico C é da forma:

8 7 e sFig. 3

3 2 1 0 eV

BINDING ENERGY

C - Y T • 0 TJ (DPodemos avaliar

á densidade de esta

dos na superfície die

Fermi N(Ef) através

da constante y. 0

coeficiente 8 nos dá

a temperatura de

Debye 8Q, relaciona,

do com as freqüên-

cias de fonos no nu»

terial. Usualmente

V

10OO

IOO0

v •o

m

4

•

* : '

Fia

0W

5f

. 4

i

to i<

-Vi,

•

189

os dados experimentais sao arranjados em um gráfico de C/T

contra T como na fig. 4. Para um metal normal os pontos

em geral se alinham em uma reta de coeficiente angular B. 0

valor da constante y é dado pelo ponto de intersecçlo da re

ta com o eixo vertical. Na fig. k mostramos dados experimen(9) para uma amostra de a-2r7,Ni2.. Notetais de Ravex et.ai

que além da reta esperada, observamos um pico a baixas tern

peraturas. Esse pico está associado a uma transiçio super-

condutora, com temperatura de transição T ^3K, sofrida pe-

lo material. Para obter y e 6, ignoramos esta estrutura e e>c

trapolamos a reta observada na regiio normal T > T , para re

giões de T < T (parte pontilhada na figura). Se y é medido2 c

em mJ/mole-K podemos obter a densidade de estados na supejr

ffcie de Fermi N (E.) em estados/eV-ãtomo-spin através da

relação N*(Ef) = 0,212Y. A densidade de estados N*(E~) é

chamada densidade de estados vestida e incluí efeitos ass£

ciados com a dinâmica da rede. A densidade de estados

N(E-) obtida a partir de cálculo de bandas se relaciona com

N (E,) através da constante de acoplamento eletron-fonon ^

na forma dada abaixo:

M*(Ef) = N(E f). ( 1 + X) (2)

t interessante notar que ^ ê en geral a mesma constante

de acoplamento que aparece em supercondutividade.

ii) Medidas de JíC2_contra T - Medidas de calor especffico

são bastante trabalhosas, e es-

pecialmente difíceis de obter

em filmes finos. No entanto, no

caso de materiais superconduto-

res, N*(Ef) pode ser obtida de

forma simples através de medi-

das do campo crítico superior

Fig.5

H 2 como funçio da temperaturana vizinhança de T ' usand

a relação obtida a part i r de

teoria de Ginzburg-Landau gen£

ral i z a d a ( U ) dada abaixo:

x-0.13 \ V0.41V0.37 \0.24

T(K)

. N * ( E f ) ( 3 )

190

A resistividade normal pode ser estimada se a geometria da

amostra for conhecida. Como a constante de Boltzmann, a

carga e e a velocidade da luz c são constantes bem conheci-

das, H(E,) pode ser facilmente determinada. Note que esse

procedimento não fornece o valor de 9_. Na fig. 5 ntostra-(12)mos curvas de H . contra T amostras de ot-ZrRh

iíi) Medidas de susceptibi1idade - A contribuição de Paulí

para a susceptibi1 idade magnética fornece diretamente N(E_}

No entanto medidas de suscept i bi 1 idades são pouco usadas pj»

ra a determinação de N(E,) por ser extremamente difícil se_

parar a contribuição de Pauli das outras contribuições.

3. Densidade de estados local - Muitas vezes no caso de ma_

teriais amorfos, é* conveniente definir uma densidade dê es_

tados local N. (E) em torno de um átomo, de forma que N. (E)

dE dê o número de estados eletrônicos entre E e E+dE asso-

ciados com o átomo considerado. Sob esse ponto de vista a

densidade de estados N(E) do material pode ser considerada

como uma média sobre as densidade de estados locais dos vã

rios átomos que constituem o material. Se temos um número

ji de átomos na liga podemos escrever:

N(E) = l N[(E)/n

i « t o d o sos átomos

Para uma liga binaria, onde temos n. átomos do tipo J

e n . átomos do tipo 2^, a densidade de estados N(E) pode ser

escrita em termos das densidade de estados locais de cada

átomo como:

n.N(E) - l N[(E) + l N[(E) - n ^ U ) + n2N2(E) (5)

i 3 átomos i « átomostipo 1 tipo 2

onde, em analogia com a definição anterior, tomamos N . ( E ) ,

a densidade de estados para átomos do tipo 1 como uma média

sobre as densidades de estados locais dos n^ átomos desse

tipo.

Num composto cristalino onde todos os átomos do tipo I

são equivalentes N. (E) é dado pela própria densidade de es_

tados local. No caso amorfo no entanto, átomos de um mesmo

191

tipo podem ter densidades de estados locais distintas (ver

fig. 8), por se encontrarem cercados por arranjos locais dj^

ferentes.

Notando que c,«n,/n ê a concentração de átomos do tipo

1 na liga binaria podemos finalmente escrever:

N(E) - CjMÊ) + c2N2(E) (6)

onde N.(E) e N_(E) são definidos como uma média sobre densj[

dades de estados locais para átomos do tipo 1 e do tipo 2

respect ivãmente.

k. Modelos de banda rígida para sistemas binários - Os mo-

delos mais simples usados para representar a densidade de

estados numa liga binaria são baseados no modelo de banda

rígida* 1". Nesse modelo a densidade de estados da liga me

tâlica ê escrita em termos das densidades de estados dosrae

tais puros N° (E) e N^ (E) como:

N(E) » CiNf(E)+c2Nj(E) (7)

por exemplo, se estamos estuda'-?* ^ W g a de ZrCu, Nû(E)

seria a densidade de estados do Cu puro e N*° (E) a densi-

dade de estados do Zr puro.

0 modelo de banda rígida

embora simples ê inapro-

priado, principalmente no Fig.6

caso em que as bandas dos

materiais puros estão ceji

tradas em torno de ener-

gias E° e E°, (ver figura) *__

muito afastada entre si, Ej' EJ

caso bastante comum em li-

gas amorfas. Para salien-

tar as falhas do modelo dividiremos, por uma questão de cl£

reza suas origens em duas partes:

I- o modelo falha porque ignora interação entre átomos de

tipos diferentes presentes no composto.

li- mesmo que essa interação fosse nula o modelo continua

a fornecer resultados espúrios uma vez que não considera o

fato de que um átomo tem um menor número de vizinhos de sua

192

espécie no bínirio que no metal puro.

Podemos agora tentar ver como seria modificada a densi-

dade de estados se tentássemos corrigir as falhas do modelo

acima para obter uma banda mais realística para a liga

Vamos primeiramente ignorar a interação entre átomos distin^

tos (isso pode ser feito se E ]> > E «) e tentar corrigir a

segunda falha ( i í ) . Podemos visualizar as bandas d_ de um

metal dè transição dentro de um formalismo tipo "tight-bin-

ding". Nesse formalismo a largura da banda depende das in-

tegrais de "overlap" entre

os elétrons do átomo consj^

derado e seus vizinhos. C£ F i g . 7

mo os átomos de um dado tj_

pò no binârio tem menos v_i

zinhos que no metal puro

(lembre que só estamos con

siderando interação entre

átomos do mesmo tipo e po_r

tanto vizinhos do tipo opojs

to não conta!) as bandas

associadas a cada tipo de átomo no binário devem ser mais

estreitas que as obtidas pelo modelo de banda rígida (ver

figura 7 ) . Queremos agora descrever o efeito da interação

entre os átomos de espécie oposta sobre as densidades da es_

tados N.(E) e N 2 ( E ) . A inclusão dessa interação resulta n £

ma hibridízação que permite qie existam elétrons localiza-

dos em átomos do tipo 1 (portanto contribuindo para N.(E))

com energias características do átomo de, tipo 2. Na figura

£ Ilustramos a densidades de estados Nj (E) e N_(E) de um bj_

nãrio usando o modelo de banda rígida. He figura 7 ê feito

um esboço do que se espera para essas densidades de estados

para o mesmo binário num modelo.mais realístico. Alguns rno

dêlos mais sofisticados para tratar o amorfo são discutidos

a segui r. ......-.,...,.•

5. Métodos teóricos roajs usuais - Os métodos teóricos usar

dos para calcular densidades de estados em metais amorfos

podem ser arbitrariamente divididos em dois grupos princi-

pais* 0 primeiro deles engloba métodos que simulam a liga

amorfa através de estruturas mais tratáveis (cristais, pe-

193

quenos aglomerados etc..) Entre esses se destacam:

i) Os que simulam o amorfo através de estruturas periódicas

de empacotamento denso e utilizam métodos de espaço recíprc»

co, exatos e confiáveis para achar a densidade de esta-

ii) Os que usam uma rede cristalina e simulam desordem ape_

nas quanto ao tipo de átomos que ocupa cada ponto da ride.

Esses sistemas podem ser tratados através de cálculos tipoCPA.(17,18)

iii) Os que simulam amorfos através de aglomerados com pou-

cos átomos embebidos num meio efetivo usando métodos apro-

priados para a soluçio desse problema.

Por outro lado temos métodos mais simples, baseados n j

ma expansão da função de onda em termos de orbitais locali-

zados (LCAO), que podem ser implementados no espaço real e

procuram tratar a estrutura amorfa de forma mais realfstica.

Dentro desse espírito se destacam dois métodos o método dos

momentos ' ' e o método de recorrência. * Esses mé

todos fornecem a densidade de estados local e são bastante

semelhante em princípio diferindo apenas no processo usado

para, a partir da Hami1 toniana, obter a densidade de esta-

dos. Em ambos os casos, para apl'car o método é necessário

que se conheça a Hamiltoniana do sistema. Normalmente é t£

mada uma Hamiltoniana tipo (LCAO) em forma matricial obtida,

como veremos adiante através de parametrizaçio a cálculos

cristalinos. Para obter a densidade de estados para a ban-

da jd de um "cluster" amorfo de N átomos devemos construir

una matriz Hamiltoniana de dimensões 5Nx5N, uma vez que o

nível d é 5 vezes degenerado. Essas matrizes, normalmente

muito grandes, (é comum encontrarmos cálculos para "clusters"

de até 1000 átomos) são bastante esparças uma vez que den-

tro do formalismo "tight-binding" cada átomo interage ape-

nas com seus vizinhos próximos.

Para ilustrar a construção da matriz Hamiltoniana no

formalismo (LCAO) vamos considerar como exemplo o caso de 3

átomos distribuídos como mostrado abaixo, supondo a exístên

cia de um orbital localizado por átomo. Nesse caso tomamos

por base o conjunto { $. , $ *f <|>, } onde <f>. representa o es_;

tado localizado no átomo i. Considerando apenas interação

194

entre primeiros vizinhos a Hami1toniana fica:\

el

* 3 " "I *" ^ °0 0 e3

(8)

onde e. = < Í>.|H| $.> e t.. = < $.|H|$.> e nessa aproxima-

ção t.. » 0 se i e j nio são primeiros vizinhos. Em geral

podemos identificar os termos diagonais da Hamiltoniana com

energias efetivas associadas com os orbitais local izados ,eji

quanto que os termos não diagonais t.. constituem as chama-

das integrais de "hopping". No caso de estarmos interessa-

dos no calculo de uma banda ji para um ":!uster" de N átomos

para construir a Hamiltoniana devemos tomar por base um co_n

junto de 5N funções localizadas constituidos pelos 5 orbi-

tais d de cada átomo. Ainda assim os elementos da diagonal

serão energias efetivas dos orbitais localizados e os fora

da diagonal integrais de "hopping". São esses elementos que

devemos determinar.(24 25)

6. Parametrizaçao da Hamiltoniana * - Nos cálculos de

densidade de estados pelo método de momentos ou pelo método

de recorrência devemos fornecer a Hamiltoniana do sistema.

Se tentarmos calcular as energias e as integrais de "hopping"

diretamente supondo que as funções localizadas sejam fun-

ções atômicas e o potencial envolvido uma superposição poteji

ciais atômicos, os resultados podem ser insatisfatórios.

Isso porque embora seja apropriado tomar funções bases loca

lizadas para tratar o metal, elas podem não ser estritamen-

te atômicas, o mesmo acontecendo com os potenciais. Por ijs

so, se existe disponível um cálculo das bandas de energias

para um composto cristalino para o mesmo material que quere

mos estudar na forma amorfa, é mais confiável obter a Hamilí 2 5T

toniana através de parametrizaçao como descrito abaixo:

- Supomos que a estrutura de bandas do sistema cristalino

possa ser obtida corretamente, não só através do método exai

to usado no cálculo original, mas também através de um for-

ma li smo LCAO, desde que os orbitais localizados e os poten-

ciais usados na expansão sejam apropriados.

- Achamos então soluções tipo LCAO para as bandas do com-

195

posto cristalino, mas como não conhecemos a forma apropria_

da para os orbitais e potenciais deixamos as energias efe-

tivas e. e as integrais de "hopping" t.. como parâmetros a

serem determinados.

- Estes parâmetros são então escolhidos de forma que os

resultados LCAO para as bandas do composto cristalino repro

duzam os da melhor forma possível resultados do cálculo exa

to.

- As integrais de "hopping" assim obtidas são caracterís-

ticas da rede cristalina considerada e não podem ser usadas

para o material na forma amorfa. No entanto para orbitais

á_ essas integrais podem, em primeira ordem, ser expressas

em termos de 3 integrais moleculares (ddo, ddir e dd6) e da(25)

geometria da rede. Como a geometria da rede cristalina

é conhecida podemos usar as integrais de "hopping" já detejr

minadas para obter as integrais moleculares. Como as inte-

grais moleculares só dependem do tipo de átomos envolvidos

e da distância entre eles, elas podem (com correções apro-

priadas para variação da distância entre os átomos) ser u-

sadas para caracterizar a interação entre os átomos também

no caso amorfo.

- Finalmente conhecendo as integrais moleculares e a es-

trutura (posição dos átomos) do "cluster" amorfo considera

do, podemos construir as integrais de "hopping'1 e assim oj>

ter uma Hamiltoniana para caracterizar a liga amorfa.

Obter a Hamiltoniana pode ser como vimos acima um pro-

cesso trabalhoso. Em alguns casos no entanto (ligas biná-

rias de metal de transição por exemplo) podemos obter bons

resultados para a liga binaria usando apenas integrais mo-

leculares obtidas para os metais puros. Quando isso acon-

tecer o procedimento se simplifica pois podemos em muitos(26)casos usar integrais moleculares estão tabeladas .

7. Alguns resultados - Para finalizar apresentamos aquí al-

guns resultados obtidos pelo método de recorrência para a

densidade de estados eletrônicos em ligas amorfas. ' ' Co

mo ressaltamos anteriormente o método de Recorrência forne-

ce a densidade de estados 1oc.il em torno de cada átomo. No

entanto como vimos na secção 5» conhecendo as densidades de

estados locais podemos obter as densidades de estados N|(E)

196

e N-(E) para

cada um dos

componentes da

11 ga, bem co-

mo a densida-

de de estados

da liga N(E).

Nas figs.

8a e 8b mos-

tramos a de£

sidade de es_

tados local

para dois ã-

tomos de Cu

e dois de Zr

para um clu£

ter com per-

to de 600 !t£

trios simuian-

I>.DC

«5

40

20

1 A

- , i \'VV

I» V/ / v

(a) Cu

-0.4

Fig. 8

-0.3 -0.2 -0.1

E(Ry)

1

•en -0.2

Fig.9ElRy)

Fig.10

197

a-Zr^CuçQ. A diferença entre as densidades de estados p£

ra dois átomos do mesmo tipo no "cluster11 e bastante sign£

ficativa e se deve ao fato de que os átomos em questão es-

tão cercados por diferentes arranjos locais.

Na fig. 9 ê mostrada a densidade de estados N(E) calcu

lada para a-Zr^Cu-,, bem como as densidades de estado

M C u(E) (linha cheia) e N Z r(E) (linha pontilhada) para cada

componente da liga. Resultados para a-Zr^Ni,, são mostra

das de forma análoga na fig. 10. A densidade de estados

N(E) mostrada na parte superior das figuras inclui, além da

banda <1, contribuições da banda js.

Na figura 11 comparamos a densidade de estados N(E) caj

culada para duas composições de oc-ZrCu com dados de UPS .

(b)

Fig.ll

(c)

cQ.

eoo

O)

O)

BVi

UJ

2

1

0

I

I•i

J. f• 1rIji

i

i

i

— i ' i

\ fVI

\'wY >1 \\ \\ \\ \

\ \

i

'41 C U 5 9

%7 CU33

__

—-—r^

I

E(eV)

198

A densidade de estados calculada para o Cu fcc também ê mo£

trada na figura lia.

Finalmente na fig. 12 a densidade de estados na energia

de Fermi calculada para duas composições de a-ZrCu ê compa-

rada com valores de N(E,) obtidos a partir de dados experi-

mentais de Minnige-

rade et ai* 2 7*. Na

figura valores cal-

culados sio indica-

dos por círculos e£

curecidos enquanto

que os dados experj_

mentais sio repre-

sentados por quadra oj . «• - "" cãdos. Aqui convém •%

ressaltar que medi- 2

das de calor especj_

fico e H - contra T

fornecem N*(E C), a

densidade de esta-

dos vestida relacio

nada com N(E-) através da constante de acoplamento eletron-

-fonon A. Uma estimativa do valor de X pode ser feita, u-Í28)

sando a equação de McMillanv se conhecermos a temperatu-

ra de Debye 6. e a temperatura crítica supercondutora T do

material. Os pontos experimentais da figura 12, foram obtj_

dos a partir de N*(E_) usando valores estimados para X.

Devemos ressaltar que nem sempre a concordância entre

resultados teóricos e experimentais, principalmente no que

diz respeito a N(E_) é tio boa quanto a mostrada aqui. No

caso de a-Zr^.N!-. por exemplo a densidade de estados na

calculada energia de Fermi é quase duas vezes maior que a(12)obtida a partir dos dados experimentais de Altounían etar

para fitas fabricadas por "mel t-spinning". 0 problema no eji

tanto não está necessariamente associado ao cálculo de

N(E-). Antes de calcular a densidade de estados devemos e£

colher um "cluster" que seja representativo da estrutura da

liga. A discordância entre os resultados teóricos e experj_

mentais podem ser fruto da escolha de um " cluster" inapro

199

priado. Nesse contexto convém ressaltar que a estrutura

amorfa nio ê única, sendo amorfa qualquer disposição nio

cristalina dos átomos. £ interessante notar que ligas de

mesma composição podem exibir diferentes densidades de es-

tado N*(E_), dependendo da técnica de fabricação. Em a-ZrNi

por exemplo ligas obtidas por "sputtering"*°* ' tem N*'EF'

bastante mais elevado que as obtidas por mel t-spinningv '*" .

REFERENCIAS

1) G. BERGMANN, Phisics Reports (Physics Letters C) 27, 159

(1976).

2) G. BUSCH and H.-J. GUNTHERODT, in "Solid State Physics"

Vol. 29, (197*).

3) V.L. MORUZZI, C D . GELATT, Jr. and A.R. WILLIAMS, "Caleu

lated Properties of Ordered Alloys" - a ser publicado.

k) SÔNIA FR0TA-PESS0A, "Cálculo de Densidade de Estados em

Ligas Binârías Amor fas de Metal de Transição; uma apltc£

ção a a-ZrCu e a-ZrNi" - Tese de Livre-Docência - USP

(1983); Phys. Rev. B2£, 3753 (1983).

5) P. OELHAFEN, in "Glassy Metals II", ed. H. Beck and H.

-J. GUNTHERODT, Topics in Current Physics (Springer

Verlag, New York) - (a ser publicado).

6) P. OELHAFEN, E. HAUSER, H.-J. GUNTHERODT and K.H.

BENNEMANN, Phys. Rev. Lett. j«3., 1 13^ (1979) -

7) A. AMAMOU and G. KRILL, Solid State Co mm. 2£, 957 (1978).

8) A. AMAMOU, Solid State Comm. 3j}, 1029 (1980).

9) A. RAVEX, J.C. LASJAUNIAS and 0. BÉTHOUX, Physica 107B,

367 (1981).

10) D.M. KR0EGER, C.C. KOCH, J.O. SCARBROUGH and C.G.

MCKAMEY, Phys. Rev. JJ29, 1199 (1984).

11) F.P. MISSELL, S. FROTA-PESSÕA, J. WOOD, J. TYLER and

J.E. KEEM, Phys. Rev. £2_7, 596 (1983).

12) Z. ALTOUNIAN and J.O. STROM-OLSEN, Phys. Rev. B27,

J>U»9 (1983).

13) W.L. JHONSON, J. Appl. Phys. ££, 1557 (1979).

200

14) 0. RAINER, G. BERGNANN and U. ECKHARDT, Phys. Rev.

B8, 5321» (1973).

15) A.R. MIEDEMA, J. Phys. F4_, 120 (1974).

16) V.L. MORUZZI , P. OELHAFEN, A.R. WILLIAMS, R. LAPKA,

H.-J. GUNTHERODT and J. KUBLER - a ser publicado.

17) G.M. STOKS, W.M. TEMMERMAN and B.L. GYORFFY, Phys. Rev.

Lett. k\_, 339 (1978).

18) A. BANSIL, Phys. Rev. B££, 4025 (1979); B2J), 4035

(1979).

19) B. DELLEY, D.E. ELLIS and A.J. FREEMAN, Journal de

Physique £8, *»37 (1980).

20) R.H. FAIRLIE, W..M. TEMMERMAN and B.L. GYORFFY, J. Phys.

F: Met. Phy=. JJ2, 1641 (1982) .

21) F. CYROT-LACKMANN, Journal de Physique CJ_, 67 (1970).

22) S.N. KHANMA and F. CYROT-LACKMANN, Phys. Rev. B Z\_,

1412 (1980).

23) R. HAYDOCK, V. HEINE and J. KELLY, J. Phys. C (Solid

State Phys.) j>, 7.345 (1972).

24) R. HAYDOCK, in Solid State Physics, Vol. 35, ed. H.

Ehrenreich, f. Seitz and D. Turnbull (Academic Press,

New York, 1S80).

25) J.C SLATER and G.F. KOSTER, Phys. Rev. 94, 1498

(1954).

26) W.A. HARRISON, "Electronic Structure and Properties

of Solids" (W.H. Freeman and Co., San Francisco, 1980).

27) G. von MINMIGERODE and K. SAMWER, Physica (Utrecht)

108B-C, 1217 (1981).

28) W.L. MCMILLAN, Phys. Rev. J_62, 331 (1968).

29) F.P. MISSELL and J.E. KEEM, a ser publicado.

Este Trabalho foi subvencionado pela FAPESP, CNPq e FINEP.

201

Real Space Renormalization Techniques

for Disordered Systems.

Enrique V. Anda

Instituto de Física,

Universidade Federal Fluminense

CP. 296 Niterói 24210 R.J. Brasil*

Summary

Real Space renormalization techniques are applied to study

different disordered systems, with an emphasis on the under-

standing of the electronic properties of amorphous matter,

mainly semiconductors.

Work partially supported by Brazilian agencies CNPq and FINEP

202

Introduction

Real Space renormalization methods have permitted important

progress in the understanding of critical properties near a

phase transition. They were mainly applied to the study of

thermal properties of a great variety of different physical

systems. In the last years these scaling ideas have been applied

successfully to analyse dynamical properties, through the

calculation of the density of states of various elementary

Í1 2 3}

excitations present in condensed matterv * ' . These tech-

niques have been mainly used to study systems êscribed by

linear chain tight binding hamiltonians, although they have

recently been extended to treat Bethe and Husimi cactus

lattices<4'5).

Its simplicity makes it suitable to study ill-condensed

matter properties (diagonal, non-diagonal, configurational and

topological disordered systems). It has been shown that the

method permits., the application of different approximated

schemes to treat the disorder which on one hand they are more

accurate that the traditional C.P.A. at least for lattices with

small coordination number Z, and in the other more efficient in

terms of computational time than the recursion method and

cluster numerical simulation.

The intention of this seminar is to analyse the main ideas

involved, the potentialities of this renormalization method

when applied to study dynamical properties of different systems,

with an emphasis on the understanding of the electronic proper-

ties of amorphous matter, mainly semiconductors.

I - One-dimentional systems

The tight binding linear chain is the simplest possible

203

system to which this method has been applied and its description

is particularly usefull to understand the ideas involved.

The local density of states of a linear chain hamiltonian

of identical atoas given by ' :

H - V E CfioCjo (I.I)(ij)

can be obtained decimating the local green function equation

GIo> (1.2)Goo -where

G00 '

A #» K • "

go

< «

+ go V (cio

o

This results in a renormalization of the locator go and the

propagator V at each step.

The imaginary part of the fixed point obtained for the

locator corresponds to the electronic density of states of the

linear chain.

The renormalization process defines a Poincari map for

the variable E ( N ) - » ( N )/V ( N ) given by

E(N) . (EC«-1V_ 2 (I.4)ÍN) fit)

where u and V are the renormalized frequency and hopping

integral respectively.and N labels the iteration.

This mapping processes a chaotic region for |u|<2

which corresponds to the support of the electronic density of

states and a non-chaotic tegion |u|>2 where there are no eigeti

states.

Defining a density of visits D(u) as the number of visits

to the interval (u,u+du>) after N-*» iterations of the map (I.A)

204

it is possible to show that D(u) coincides with the density

(3)of states of the linear chain ,

It is interesting to note that the map defined in (1.4)

is a particular case of a more general map, extensivelly

studied by Feingenbaun within the context of the theory of

chaos(6).

Although the density of states for the ordered linear chain

is a very veil known object, it constitutes an excellent exam-

ple to develop renormalization techniques, with the idea that

this scheme could in principle be extended to more complicated

situations, for instance, two or three dimensional systems.

The more interesting aspects of this formalism are obtained

when they are applied to disorder systems. For the case of the

linear chain there have been several proposals to include the

disorder, which has been in all cases treated approximate-*

ly •-» » .in order to apply the formalism, the renormalization

relations for the locator and the propagator have to be

configurationally independent, which in fact they are not. Ás a

consequence, an approximation has to be done. Some authors have

taken the configurational average over the contribution of t!-e

riecimatec? points to the self energy in the first iteration or

in the second . Better results are obtained averaging the

diagonal green function corresponding to the decimated point

at each stage* '.

Although these results are approximated they are more

accurate than thotse resultant from mean field approximations

like C.P.A. with the additional advantage in terms of numerical

calculation that they are not self consistent. This is a

Consequence of the fact that in low dimensional systems, Z be-

205

ing small, the configurational fluctuations neglected in mean

field theories are very important. The renormalization tech-

niques are capable of reproducing the main features of the

density of states (peaks and gaps) because, even if in each

iteration the information of the decimated points are averaged,

the disorder is maintained at each step» which implies that

fluctuations of all orders are included. Very recently a

method has been proposed which consists essentially in deci-

mating clusters which size is frequency dependent, so as to

permit the density of states for the disorder linear chain to

converge, numerically speaking, to the exact result .

The applicability of real space renormalization techniques

are not restricted to systems described by nearest-neighbour

tight binding hamiltonians. All physical quantities satisfying

an equation with the structure given in (1.2) are good candi-

dates to he studied by this formalism. This is the case of the

(8)Kronig and Penney modelv ' described by a potential given by

MVOO - g I fi(x-nai (1.5)

n«Q

If the functions

n An ex"+Bn e l l tJ t (1.6)

are taken to be the basis to represent the hamiltonian, the

following recursion relation is satisfied

where

' 2(cosk+eo Zjç^) (I.8a)

(2lE/h)1/2 (I.8b)

206

which is similar to equation (1.2) and can be decimated to ob-

tain the eigenfunctions and eigenvalues of the problem.

For the case of an ordered linear chain the renormalization

relation is similar to equation (1.4)

* - a2 - 2 (1.9)

n n-1

If we were to impose periodic boundary conditions to a chain

of M atoms, the last decimation before eliminating all points

of the chain corresponds to the number N * £uM/lu2 with aLi

final result for ax. - 2 which is the fixed point of theNL

mapping (1.9). This relation stablishes that the correct setof starting values for a (k(E)) has to be such as to satisfy

that after N iterations the a •+ a «2. The startinge ° NL

values fulfilling this condition determines the k's and the

eigenvalues of the finite chain. It is easy to realize that all

values of a that reach the fixed point a "2 after a number° NL

N<N_ óf iterations belong to the correct starting set. Theit

distribution of these values T-.'hich are contained within the

interval (-2,2) is equivalent to the density of visits obtained

2 -1/2for the case of the nearest neighbour linear chain D(a)"(4-a ) ,

As a consequence the density of states for a Kronig Penney model

can be written immediately as being

PKp(E) - /Í2<(a-2(cos k(E) • *Q

(1*10)

Generally speaking the existence of boundary conditions in

a finite linear chain determines which is the value for the

a., coefficient. To obtain the starting set a and from it theNL °

eigenvalues and eigenfunctions, it is useful to iterate the

inverse Poincarl mapping N. times

207

taking all possible combinations of + and - signa». This could

be an efficient procedure to study the behaviour of small

particles and its spectroscopic properties as a function of

II - The Bethe Lattice

the extention of the method described in the previous

sections for two and three dimensions presents' an essential

difficulty. For lattices of dimensionality greater than one the

decimation of the nearest-neighbours change the structure of

the equation of motion for the green function not allowing an

assignment between parameters so as to define the renormalization

relations. In the case of a square lattice, for instance, one

point in the lattice linked to its four nearest-neighbours is

after the first decimation linked to eight lattice points.

Although it is possible to define several approximated cell to

cell renormalization approaches the results obtained ar«

essentially incorrect because they are not able to reproduce

adequately the density of states of the ordered system. For the

case of ordered lattices a similar sxfeeme. to the one applied to

the linear ordered chain can be used* operating in the recipr£

cal space. For a square lattice it can be shown that the

density of states can be obtained, just by simply calculating

the density of visits of a superposed mapping of two summed

independent variables . However these ideas can not be

extended to the more interesting situation of a disordered

lattice for which it is not passible to define a reciprocal

s p a c e . • • • - • ' . • • • . : , • • • . . • - , - . •

Within the context of the decimation procedure applied to

208

linear chains, it can be defined a renormalization approach

(4 5)applicable to Bethe lattices * . Even if they are objects far

from any real lattice they can reproduce the local environment

correctly for systems of two or three dimensions. Topologically

speaking the linear chain and the Bether lattice are equivalent

(a linear chain is a Bethe lattice of Z » 2). The localized

density of states of an ordered Bethe lattice is identical to

the density of states at the surface of a semi-infinite linear

chain(10) defined by

where

ts - h V (II.2a)

t - /z-1 V (II.2b)

and V is the non diagonal element of the real Bethe lattice.

The semi-infinite linear chain can be decimated similarly to

the infinite linear chain. In this case two frequencies

u • g,t and u • gt have to be defined satisfying differ-

ent renormalization relations. In terms of these variables a

two dimensional mapping is determined by

-J - 4 ( N" 1 }" 2 (II.3a)

-J-- -g(""l) a»/*"1*" Z/U-l) (II.3b)

Although an extention of the result obtained for the linear

chain, would permit us to suppose thpt the density of visits

of the variable w , D(u ) , corresponds to the density of8 o

states of the Bethe lattice, this is not so. The region where

209

chaotic behaviour it found for the napping (II.3a) coincides

with the support of the density of states of the Bethe lattice

( |«J|</Z-1). However, this is not true for (II.3b).

A lore interesting approach for the decimation of the

Bethe lattice is based on a formalism which permits to treat

the disordered lattice (in principle any kind of disorder)

using the same ideas that were successful for the one dimension-

al case.

The formalism is based on a diagrammatic scheme which re-

constructs an arbitrary Bethe lattice from a conveniently.

defined linear chain, which is taken to be the basic diagram-

(4 5)matic element * . It can be seen that the Bethe lattice is

obtained as a limit of an iterative procedure. At each step of

this procedure all free vertices of an infinite linear chain

are linked to other semi-infinite chain of the same structure.

It can be shown that this renormalization process defines

for the case of a pure Bethe lattice a Poincaré mapping given

by

(II.4)

where »_ is the frequency starting value. The fixed point of

this renormalization process is

Z« • (Z-2)/Ü 2 -

* •

In fig.1 u* is represented as a function of u_. The well known

density of states of the Bethe lattice can be obtained from

l»g(- - 1J*). In fig. 1 the path by the iteration process is

1/2shown as dotted lines. In the region (<D_ |<2(Z-1) <•>* has an

imaginary part. As this renormalization scheme dresses the

210

vertices by linear chains at each step, u gains an imaginary

(N)part only when it satisfies the relation w <2, value from

which it converges to the complex fixed point.

This is very instructive to understand how is the process

of convergence for the more interesting case of a disorder

Bethe lattice.

Ill - Amorphous Semiconductors

An obvious extention of the method analysed in Section II

can be applied for the case of a disordered Husimi cactus ,

It is generally accepted that an important topological proper-

ty of amorphous tetrahedral bonded semiconductor lattice is the

existence, due to disorder, of odd as well as even rings, the

smallest of which involves five bonds, statistically

distributed*12*.

The impossibility of satisfying the antibonding electronic

state for a closed loop of odd number of atoms produces an

erotion of the states near the antibonding band edge. This is

(12 13)an important topological disorder effect, between others ' ,

because it changes the gap value of the semiconductor. It can

be qualitatively studied in a Husimi cactus, mixing with

arbitrary concentration rings of five and six number of bonds

as it is shown in fig. 2, with concentration x and (1-x)

respectively.

The electronic density of states for this system is shown

in fig. 3. It corresponds to a one orbital first-neighbour

tight binding hamiltonian. Two main features are worthwhile

mentioning. For intermediate concentrations of five and six-

fold rings, the density of states is smoother in comparison

with the result gained for a pure cactus. This is consistent

211

with photoemission measurements of amorphous silicon which

shows a curve without the characteristic peaks of the crystal-

(14)line structure . The result suggests that the Bethe lattice,

being a ringless lattice with a smooth density of states, could

be considered as an average, a sort of mean field approximation

for the ring statistics.

There is an erotion of the density states near the low

energy edge. The edge has not a perceptible dependence upon

concentration up to values of x< 0.9, above which, it changes

rapidly. This is related with the presence of a percolating

path of six-fold rings that are the only ones that contribute

to the lower energy tail, being xs0«9 tîe percolation threshold.

In our case this band edge is not a Lifshiftz point due to the

infinite dímesionality of the Bethe lattice.

This calculation shows that the topological disorder relatec

with the ring statistics can be represented, in a first approxi-

mation by a Bethe lattice. Using this lattice our purpose now

is to study the effect of impurities typically Hydrogen in

Silicon, which introduces a completely different type of dis-

order into the system.

In order to study the real density of states of amorphous

Silicon we include four sp orbitais corresponding to the

valence states of the Silicon atom. A simplified version of a

tight binding hamiltonian which describes some of the main

features of a four orbital first neighbour tight binding

random network has been proposed by Wearie and Thorpe1 . They

have shown that the density of states of their simplified

hamiltonian can be mapped into a density of states of an S band

system with the same topological structure. However this

212

hamiltonian has some important short-comings. Two of the p

atomic orbitais do not hybridized to constitute a molecular

orbital, which in the density of states appears as two delta

functions at th? top of the valence and conduction band. This

problem can be overcome describing the system by an sp

Hamiltonian which includes all possible nearest neighbour

hopping integrals . Unfortunately taking these more general

Hamiltonian, there is no mapping of a four orbital density of

states into a one-orbital density of states. In order to be

able to apply the decimation technique used for the Bethe lattice,

we have to generalize the procedure for the case of four orbitais

per site. This can be done by simply rewriting the renormalized

equations in terms of 4 x 4 matrices

Many different theoretical approaches have been used to

study the electronic properties of hydrogenated amorphous

Silicon. They are essencially two different types of theories:

a) those which strictly speaking are only applicable to systems

with zero concentration of impurities as for instance the Cluster

Bethe lattice^18'19). The Hydrogen, sitted in a different

possible configuration is surrounded by an appropriate environ-

ment of pure material. This structure is treated exactly and

the rest of the lattice is represented by a Bethe lattice.

Similar procedures have been proposed using quantum chemistry

methods , These theories are unable to study the effects

of inter-impurity interference which certainly is present in

Si:H which can have up to 207. of Hydrogen; b) the well known

mean field approaches, typically C.P.A., which have the short-

coming that for systems with low coordination number (Silicon

has Z-4) do not include properly compositional fluctuations.

213

(2This method has been applied to hydrogenated crystalline Silicon

It has the difficulty that for lo« concentration of Hydrogen

the density of states reproduces the Van Hove singularities as

peaks, absent in an amorphous structure.

Defects of various Si-H bonding conformations have been

(22 23)suggested to explain some of the experimental results *

Unfortunately these results remains controversial. Ultraviolet

photo emission spectra for a Si:H reflects the appearence of

C 24)three characteristic peaks C, D, and Ev ' that are not present

in a:Si as it is shown in fig. 4. However this structure has

not been founded in recent measurements which shows other peaks

localized at different values.

We will not discuss which should be the most appropriate

hydrogen conformation inside an amorphous Silicon substrate.

Instead we will adopt the simplest one, within the context of

our formalism, in order to use it as an example to show all the

possibilities of the decimation procedure. We take a monohydride

substitutional four fold site impurity that has been studied

Í21 25)by different authors ' . This hydrogenated vacancy is such

that the four dangling bonds directed into a vacancy are

saturated by a Hydrogen atom. Due to its structure it can be

treated as a substitutional impurity by defining effective sp

orbitais adequately combining the four hydrogenic s orbitais.

(21)Taken the matrix element used by Papaconstantopulos et al ,

applied to a first-neighbour Bethe lattice, it is possible to

obtain the density of states shown in fig. 5 and 6 for different

concentrations of impurities. The results show the well known

gap enhancement produced by the presence of Hydrogen and the

existence of an antibonding peak very near the bottom of the

band. The split off peak at the bottom of the valence band»

corresponding to the bonding level, has been seen recently in

(25)photoemission experiments . A shoulder near the top of the

valence band, is transformed into.a peak with a clear anti-

resonance besides it which increases with hydrogen content,

showing the importance of the interference between the impurities

as concentration increases. No other intermediate peaks appear

in the valence band region.

From these results we conclude that a monohydride substitu-

tional four fold site impurity can not explain the peaks C, D

and E seen in photoemission experiments. As has already be

mentioned the existence of these peaks is controvertial and

probably depends upon the way in which the sample is prepared.

Conclusions

Real space renormalization techniques show to be a powerful

method to calculate the density of states of different elementary

excitations in solids. It is particularly interesting when

applied to disorder systems because it permits the formulation

of a simple approximated scheme to treat the disorder which

incorporates compositional fluctuations at all orders, or even

to obtain exact numerical results in a more efficient way in

terms of computational time than numerical simulation, Unfortunat

ly, up to the moment it has not been possible to extend the

method using a simple formalism to study two or three dimensional

lattices. The method is restricted to linear chains, Bethe or

Husimi cactus lattices which are not able to reproduce a real

system due to its topology. Caskets admit the application of

(24)decimation te .uniques and they have much more conectivity

215

that the Bethe lattice. Research oriented to the application

of these structures to describe amorphous semiconductors is

now in progres.

216

References

01. Gonçalves da Silva CET and Koiller K 1981 Sol. St. Comm.

40, 215-19.

02. Koiller B, Robbins MO, Davidovich MA and Gonçalves da Silva

CET 1983 Sol. St. Coram. 45, 955-9.

03. Oliveira PM, Continentino MA and Anda EV 1983, Phys. Rev. B

£9_, 2808, 1984.

04. Albuquerque J 1984 to be published.

05. Anda EV, Makler SS, Continentino MA and Oliveira PM to be

published in Journal of Phys. C 1984.

06. Feingenbaum. Los Alamos Science 1, 4 (1980).

07. Makler SS and Anda EV to be published in J. of Phys. C 1984.

08. Jose J 1982 Lecture Notes, XIX Latin American School of

Physics Cali, Colombia.

09. Monsivais M and Jose J to be published 1984.

10. Thorpe MF 1981 "Excitation in Disordered Systems" ed Thorpe

MF (Plenum Press NY) 85-107.

11. Balseiro C private coimnunication.

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13. "The Physics of Amorphous Solids" R Zallen 1983, John Wiley

and Sons, N.Y.

14. B yon Roedern, L Lexg, M Cordona and FW Smith 1979 Philos.

Mag. B 4_0, 433.

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16. Singh J 1981 Phys. Rev. B 23, 4156-4168.

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217

18. Joannopoulos JD and Tndurain F 1974, Phys. Rev. B JJ), 5164-73,

19. Tndurain F and Joannopoulos JD 1976 Phys. Rev. B l± 3569-3577.

20. R Barrio and J Tagflena-Martlnez in this Proceedings (Simpósio

Latino Americano de Física dos Sistemas Amorfos).

21. Papaconstantopoulos DA and Economu EN 1981 Phys. Rev. B 24,

7233-7246.

22. Ching VY, Laa DJ and Lin CC 1979 Phys. Rev. Lett. 42,805-808.

23. Allan DC and Joannopoulos JD 1980 Phys. Rev. Lett 44, 43-47.

24. Robertson J 1983 Phys. Rev. B 28, 4658-70.

25. Wesner D and Eberbart W 1983 Phys. Rev. B 28, 7087-93.

26. Domany E, Alexander S, Bensimon D and Kadanoff LP Phys. Rev.

B 28, 3110, 1984

218

ftt«L

I

/

i i i• i

?r

Renormaliration process for

of an ordered Bethe Lattice.

Fig

the

. 1

self

Wl

energy

Fig. 2

Husimi cactus Lattice formed by a random

distribuition of hexagons and pentagons.

219

>'\

Fig. 3

A/ *

! /•

0 +

V

^

Density of states for one orbital binary alloy

Husimi cactus composed by hexagons and pentagons

with concetration x and 1-x respectivelly.

Fig. 4

Ultraviolet photoemission spectra for a-Si(dashed line) and hidrogenated a-Si (solidline) ref.14 .

220

Fig. 5

tf

.10 -C O € «O

Density of states of pure Si.

•.I

Fig. 6

Aft-I* «10

Total density of states for the alloy

at Ch= 0,1 .

221

RESEARCH ON HYDROGENATED AMORPHOUS SILICON

AND RELATED ALLOYS FOR PHOTOVOLTAIC APPLICATIONS

Ivan E. CharobouleyronPhotovoltaic Conversion LaboratoryInstitute of Physics, P.O.Box 6165UNICAMP-Campinas, SP.,13100 Brasil

INTRODUCTION

In this paper we present an overview of the main

activities on amorphous materials of photovoltaic interest

performed at our laboratory. The research group started in

1980 and was supported from the beginning by the Universi-

dade Estadual de Campinas (UNUCAMP), the Companhia Energé-

tica de São Paulo (CESP) and the Financiadora de Estudos e

Projetos (FINEP). The funding level varied from year to

year. A rough estimate gives 100.000 US$ per year approxi-

mately. Since 1980 the group has grown to its actual size-

Researchers, post graduate studentes and technicians were

incorporated during the past years. The minimum infrastructure

necessary to work in amorphous semiconductors was locally

built or imported and, taking advantage of the good existing

facilities at UNICAMP, many research activities were under-

taken in collaboration with other groups. In order to over-

come the difficulties associated to the relative isolation

we experience in the field we established a scientific

exchange program with some foreign laboratories and

specialists. We benefited from visits to these laboratories

and we also received some outstanding experts that stay

between us for periods of varying length. '"

222

THE PHOTOVOLTAIC CONVERSION LABORATORY TODAY

The main research activities of our group concern

amorphous semiconductors, other activities currently

developed relate to poiycrystalline silicon solar cells and

M-I-S structures on single crystal and poly silicon.

PERSONNEL: Prof.: Ivan Chambouieyron (Head of the

Group), Fernando Alvarez, Rene Brenzikofer, Jorge I. Cisne-

ros and Sergio Koehlecke; six post graduate students, two

of them working for their Ph.D. and the remaining four for

their M.Sc. degree and three technicians.

FACILITIES

Two capacitively coupled glow discharge reactors were

locally built. The first one is a stainless steel, 30 cm

diamter, reactor having a parallel field configuration (1).

The second one is a pyrex, 20 cm diameter, reactor possessing

a cross field configuration (2). Each reactor has recently

been equiped with two mechanical pumps and a diffusion pump.

They operate in a way similar to other reactors (mainly

Japanese). One rotary + diffusion pump being used to reach

a good background vacuum and devised to pump only air. The

second rotary pump work when si lane or dopant gases are

present. This system prevents oxygen and silane mixtures

to occur that are harmuful to the oil of the rotary pumps.

Both reactors allow to deposit a-S1:H films in 2Mx2"

areas with good uniformity. More recently an RF sputtering

and electron beam (Leybold Heraus Z-400) system was installed.

It 1s now used for studies on transparent electrodes,

223

Metallization, or growing a-Si:H and other compound amorphous

semiconductor layers. Studies on the properties of amorphous

and polycrystalline SnO2 have been made during the whole

period, the material being deposited by the chemical spray

method. Finally some facilities exist to measure electrical

and optoelectronic properties of the films, to deposit

metallic contacts, to anneal the samples, etc., etc.

RESEARCH ACTIVITIES

a-Si:H

a-Si:H films were grown to characterize the reactors

behavior. The measurements made to determine the properties

and quality of the material include,among others:

•Growth rate as a function of si lane pressure, gas flux

and RF power. Silane is normally helium diluted in a

proportion varying from 25 to 75%. Few studies on film

properties with varying substrate temperature were under»

taken. We chose deposition temperatures in the neighborhood

of 300OC. Our working conditions give growth rates of ca.

lX/sec.•Transport properties. Electrical properties of the

samples are determined by measuring dark conductivities as

a function of temperature in the temperature range 300-500K

and under vacuum conditions. Activation energies for undoped

and doped samples are deduced from °à vs 1/T plots. Photo-

conductivity of amorphous samples is also measured under

vacuum as a function of both photon flux and temperature

(90 - 350K). They provide useful information concerning the

224

nature and density of recombination and trap levels. We are

presently making a system able to mesure thermopower on our

amorphous films.

Doped samples are currently made by adding controlled

amounts of phosphine or diborane to the main si lane flux.

Till now we used mechanical flowmeters. Electronically

controlled mass flowmeters were bought and will be installed

during the present year.

•Optical properties. We determine the optical prpperties

of our films by transmission measurements in a visible-near

IR spectrophotometer (3). Infrared studies are made on

samples deposited onto high purity polycrystalline silicon

wafers* giving useful information concerning the amount of

hydrogen included in the material.

•Devices. Some devices of Schottky type and p-i-n solar

cells were fabricated in 1982. Figure 1 shows the structure

of those devices that possessed low conversion efficiency

Corning glass (7059)

T1 (2500 Jt)

P* (800 %) n* (300 %)

V»O.(75O X)

J-Sl.M(1)(6000Ü)

225

(~1 - 2%). We are presently undertaking a program aiming to

produce p-i-n solar cells having more than 4% conversion

efficiency in areas larger than 1 cm .

a-SiNx_:H

In order to reduce the optical losses produced by the

top layer of p-i-n solar cells we undertook a systematic

study of the properties of off-stoichioraetric silicon

nitride compounds. Samples are prepared by the simultaneous

plasma decomposition of SiH4/N gaseous mixtures. The optical

gap of glow discharge a-SiNY:H samples prepared in that way

increases with r.f. power densities and depends on silane

IN

ü

c

O)

o

it ti "

PHOTON ENERGY

>

FIO. 2 (a * * ) 1 " vs photon energy for different a-Si, N , . , :H layers obtainedfrom the O D of a SiH« and N , mixture of fixed compoaition (Sir!*/(N,) - 0.33. AD sample» are nonintentkmally doped and i f power density isindicated on each curve.

FIO. 3 (a) Room-temperature conductivity vs gaseous doping mixture forsamples grown under conditions identical of those of curve C. Fig. Z Theeffect of optical gap shrinking due to boron doping is alto indicated, (b) Darkconductivity activation energies vs gaseous doping mixture. It is found ex-perimentally that the conductivity varies with temperature according toa « e9 cxp(-£,/kT) in the 300-JOO-K temperature range.

-t

-6

-1

•to

-12

t

1.0

0J0.80.70.10.5

1

p — 1 - *•> |• • '

i

- I'I

"*Air*

* 1 « 1i

- (I

T• I•j

1,1

*

•

•

-

-

i i |

Ib)1 1

- 23

• U

?o

O)1.1 UJ

- 1.7

0 -t .4 .8 - « - » • * 0i ii t

"i rPM,

log gas. imp. ratio

226

to nitrogen gaseous ratio. At low r.f. power densities the

plasma will essentially break silane molecules, the

dissociation energy of N2 molecules being 2 to 3 times

larger than that of the silane molecule. The material does

not differ much from a-Si:H. Higher power densities produce

more nitrogen radicals in the plasma increasing the number

of nitrogen atoms incorporated into the network. Figure 2

shows the effect of increasing r.f. power on the optical

gap of samples produced by a SiH^/N2 gaseous mixture of

fixed composition (4).

Doping experiments were done in samples grown at fixed

silane dilution and r.f. power. The results for phosphorus

and boron doping on samples having an optical gap of nearly

2eV, are shown in figure 3 were room temperature conductivities,

optical gaps and activation energies are plotted as a(4)function of dopant gas concentration in the reactor chamber .

The nitrogen concentration of those layers was AES determined

at SERI, Colorado (USA), and estimated to be 33Í 2 at. %.

The photoconductivity of such samples was studied as a

function of photon flux and under ELH - 100 mW/cmz

illumination ' '. For undoped and lightly boron doped

samples a supralinear dependence of the photocurrent on

light intensity was found within the illumination range

- 1 0 - 10 photons cm s . Light boron doping

unsensitizes the material while heavy doping sensitizes

it again. The opposite behavior is found with phosphorus

doping. The supralinear behavior is interpreted assuming

two types of defect centers having energies that are

227

located above and below the dark quasi Fermi level and that

possess a large difference in their electron capture cross

section.

Figure 4 shows, for a set of samples, the photocurrent

as a function of photon flux for monochromatic light

excitation of -2.2eV. Figure 5 shows dark and photoconduc-

tivities as a function

if.

Fig:4. *8i»x:H •«•ple«. Photocurrenta« • function of photon flux for mono-ehroaatic light excitation of c«. 2.2 eV.

of r.f. power density

for undoped a-SiN :H

samples. Infrared

studies were performed

on samples grown onto

silicon substrate (6).

From them it is possible

to determine» not only

the hydrogen content

in the material, but

the nitrogen content

as well. Independent

AES measurements are

used for calibration

purposes.

SnOg

In solar cells, and opto-electronic devices in general,

1t is necessary to minimize the losses coming from the

optical reflection at the semiconductor air interface. To

that aim transparent layers, having an appropriate thickness

228

b°

and refractive index, are deposited. If the transparent

coating is conductive its use has a twofold purpose: to

minimize optical losses and to allow a thinner top electrode

in p-i-n structures. Tin oxide films deposited by the

chemical spray

method (hydrolysis

of SnCl4) possess

both properties and

we studied the

electrical behavior

of such films as a

function of deposition

temperature and doping.

Low deposition (Td£

280<>C) temperatures

produce amorphous

\3

-19

POWER (WcrrT)

5- Filled squares: roo» temperature dark conductivity

•s • function of RF power density for undoped

staples. Optn squirts: roo» teapereture photo-

conductivity on ELH 111u»1n«t1on (100 «H/c»2).0pen

trUnglcs: ratio of roo» te»p»raturt photoconductivity to

dark conductivity «s • function of RF power density.

films while polycrys^

talline layers are

obtained when the

reaction takes place

at higher temperature^),

In the case of non-intentionally doped films, large

conductivity variations are obtained with deposition

temperatures. They come from oxygen vacancies and chlorine

inclusion into the network liberated during the incomplete

hydrolisis reaction* '. Our work on tin oxide films refers

to conductivity studies^9). Hall efect(10), composition*8)

and the amorphous-polycrystalline transition (See figures

229

'eu

a:

CH

LO

RIN

E C

ON

CE

N1

-

500 400 300 25C)

I I I I

Aboof «tol /

• ,

Kone etoi. /^-

* /

r i gOH >

• • • •

9 ™ #

1 I 1

—

. -

1

- 1 0 s

"Io

- . O 2

1.2 1.4 1.6 1.8 2.0

Fig. 6 Chlorine concenirttion vs. inverse deposition temperature for samples grown at four differenttemperatures. On the same figure measurements by Aboaf ei al. , Carlson and Kane el al. areindicaicd. Relative O H and H contem (arbitrary uniu) are also shown for the same samples.

6 and 7. It is worth to mention that the temperature of the

process for tin oxide film deposition is compatible with

amorphous silicon solar cell technology and that the spray

method has the advantages of easy of operation, reprodudbility,

low cost and no-scaling-up area problems. The properties of

fluorine doped tin oxide layers are presently studied and

the main results will soon be published.

FUTURE WORK

We are presently starting a series of projects concern-

ing amorphous semiconductors and devices:

230

SnO

200 300 400 500

SUBSTRATE TEMPERATURE I°C) 200 300 4 0 0 500

FIG. 7. Variations of room temperatureconductivity, electron Hall mobility, andcarrier concentration as a function of sub-«rate temperature (solution flunes: T 1 m l /min.Q'iml/minJ.

a) Amorphous silicon alloys a-SixGe. :H and a-SixSn-t :H

deposited by RF sputtering techniques;

b) a-SnOg and a-ITO deposited by electron beam techniques;

c) High efficiency, medium area, a-Si:H solar cells

(p-i-n structures obtained by glow discharge);

d) Schottky type and SIS (a-Si :H/SnO2) cells;

e) New characterization techniques including Raman

spectroscopy,photoluminescence and thermopower.

CONCLUSION

The Laboratory of Photovoltaic Conversion at UNICAMP is

actively engaged in a research program concerning amorphous

semiconductors and devices. We presently have the necessary

231

infrastructure and personnel and we hope to contribute in

the near future to this interesting and expanding field

of research.

REFERENCES

1. F. Alvarez, Rev. Bras. FTs., 12, 4, 832 (1982).

2. R. Brenzikofer, unpublished.

3. J.I. Cisneros, E.L. Carpi, F. Alvarez and I. Chambo£

leyron, to be published.

4. F. Alvarez, I. Chambouleyron, C. Constantino and J.

I. Cisneros, Appl. Phys. Letters, 44, 1, 116 (1984).

5. F. Alvarez and I. Chambouleyron, Solar Energy Materials

(in press).

6. I. Chambouleyron, F. Alvarez, C. Constantino and J.I.

Cisneros, Proc. 5th E.C. Photovoltaic Solar Energy

Conference, Athens Greece, 1983 (in press).

7. I. Chambouleyron, C. Constantino, D. Jousse, R. Assumg

ção and R. Brenzikofer, J. Phys. (Paris), £2, C-4,

1009 (1981).

8. I. Chambouleyron, C. Constantino, H. Fantini and M.

Farias, Solar Energy Materials, 9, 2, 127 (1983).

9. 0. Jousse, C. Constantino and I. Chambouleyron, J.

Appl. Phys., £4, 1, 431 (1983).

10. D. Jousse, J. Non-Cryst. Solids, 59 & 60, 637 (1983).

Ill

A GENERALIZED SLATER-PAULING CONSTRUCTION FOR MAGNETIC AMOR-

PHOUS ALLOYS

A. P. Malozemoff, A. R. Williams and V. L. Momzzi

IBM T. J. Watson Research Center

Yorktown Heights, NY 10S98

ABSTRACT:

The concept of magnetic valence and a generalized Slater-Pauling construction permit a

consistent interpretation of tbe ferromagnetism of both transition metal and metalloid alloys

with Fe, Co and Ni. Amorphous alloys like CoB, CoSi, FeB and FeSi are revealed to be

strong ferromagnets with a constant number of majority-spin sp electrons over a certain

composition range.

233

We review our recent progress in developing a coherent picture of the zero-temperature

magnetism of transition-metal alloys.1*7 The essential ideas are extremely simple and apply to

both crystalline and amorphous alloys.

We focus on an AtxBx alloy whose atom-averaged moment in Bohr magnetons is the

difference of the atom-averaged number of spin-up and spin-down electrons:

= ^ - J V * (1)

We define the average valence as the atom-averaged number of electrons outside the last filled

shell:

Za*e - ZA(\ - x) + ZBx = JV* + JV* (2)

Now fr is in most cases less precisely known than N' for reasons to be discussed below; so

eliminating it in Eqs. 1 and 2 we find Friedel's8 simple but profound result

Have = M f ' - Z ^ (3)

Now we can break up Ar into its sp and d components. In two limiting cases, Ny is

known quite precisely. First, in what we call the "common-band" limit of strong ferromagne-

tism, the d-bands of the host (A) and solute (B) merge, and, for the majority-spin electrons,

these "commmon" bands lie entirely below the Fermi level, so that AfJ is precisely five. An

example is Ni^_xCux. Second, in what we call the "split-band" limit of strong ferromagnetism,

the d-bands of the alloy subdivide or "split" into two groups, one with wave functions

concentrated on the host atoms and one with wave functions concentrated on the solute

atoms. The meaning of "strong" for these split-band magnets is that the position of the Fermi

level lies in the gap separating the two sets of band states. An example of such a split-band

system is shown in Fig. 1, where we see that the d states of YCo^ segregate into two groups

separated by a region of very low state density ("gap") of approximately 2 eV. The hybridi-

234

zation of host and solute states results in the almost perfect maintenance of local neutrality.

Nevertheless, as long as the gap between the subbands remains well-defined, quantum

mechanics dictates that the number of states in each subband is conserved. In the case of our

example, YCo^ shown in Fig. 1, the atom-averaged number of up-spin d electrons N* is

therefore simply the number associated with Co, namely five times the concentration factor

1-x with x * 0.25. To summarize, in spite of hybridization and local neutrality, the number of

up-spin d electrons per atom is, to a very good approximation, simply five or zero.

The common- and split-band limiting cases are actually quite often realized in Fe, Co and

Ni alloys, and can usually be predicted from well known band-structure systematics. For

example, the upper edge of the d-bands is known to shift systematically downward in energy

as one proceeds either to the right along a row of the periodic table or up along a column.6

Thus> for early transition-metal alloys1 such as Y, Zr or Ti with Fe, Co or Ni, one can

anticipate the split-band limit (e.g. N'dY » 0, N'dFe « 5), while for late transition-metal

alloys5 involving Cu, Zn or Pd solutes, for example, one can anticipate the common-band limit

(e.g. N\CU = 5). Metalloid alloys,4 involving B, Al and Si, for example, correspond to the

split-band limit, because their d-states lie very high in energy (e.g. N'dB * 0). The number

of up-spin d electrons is an integer multiple of five (and therefore easily predicted) only when

the Fermi level lies above the up-spin common bands or in the gap of the split bands. This is

usually, but not always, the case. It is hot the case, for example, in Fe-rich alloys, where the

intra-atomic exchange interaction is not quite strong enough to fully occupy the Fe-dominated

up-spin d bands. Even for Fe alloys, however, when the solute concentration is sufficient,

intra-atomic exchange again wins over inter-àtomic hopping, and strong magnetism (fully

occupied Fe-derived up-spin d bands) and simply predicted magnetization return. (The origin

of this behavior is discussed in Sec. II of Ref. 4.)- Thus, even in an amorphous alloy with

different degrees of hybridization in different local environments, the simple d-state sum rule

holds, as long as the local density of states has a clear-cut gap or top and the lower-lying d

bands are fully occupied.

235

Now it becomes advantageous to write Eq. 3 in the form

I) (4)

The atom-averaged moment is seen to be a sum of two terms, one being two times the number

of up-spin sp electrons and the other being an atom average over the quantity

2mi - 2tfJ,.-Z, (5)

for atom i . We call this quantity the "magnetic valence".2 Since, as shown above, N'djis

usually an integer, Zmi is usually an integer depending on the valence of each constituent. For

example, it is 2, 1 and 0 for Fe, Co and Ni respectively, reflecting their increasing valence and

their common ability to bind five up-spin d electrons. It is -1 and -2 for Cu and Zn respec-

tively, -3 for Y, B and Al, -4 for Zr, Ti, Si and Ge, etc. These negative numbers reflect a

valence that exceeds the ability to bind d electrons. Negative values for the magnetic valence

measure the tendency of these solutes to reduce the magnetic moment. In certain cases, like

Rh for example, with an electronic valence of 9, the d-band position relative to Ni or Co is

neither clearly in the split-band nor in the common-band limits. Thus, one might expect the

moment to lie somewhere between that predicted from ZmRh = 1 and ZmHh = - 9, as turns

out to be the case.5

What is the physical significance of the magnetic valence? It is just the number of holes

in a low-lying d-band which are left after filling it with the electronic valence. Fe, Co and Ni

give non-negative values because they have no more electrons than d-states. But atoms with

no low-lying d-bands give a negative value because the excess of electrons will lead to a

reduction in the number of holes of the host d-band. It should be noted that this effect does

not imply change transfer from one atom to another, because hybridization compensates for

any apparent charge transfer.

236

The utility of the magnetic-valence concept arises from the consistent treatment of all

solutes, whether transition metal or metalloid. By contrast, the traditional Slater-Pauling

construction9'10 which plots average moment versus average electron-to-atom ratio, cannot

conveniently incorporate the metalloid alloys. Using the magnetic valence concept, we can

generalize the Slater-Pauling construction to all alloys2-4 by simply plotting average moment

vs. average magnetic valence as in Fig. 2 for crystalline TVi,_xCux and NiixSilxl (we have

corrected the moment for a shift in the g-factor away from 2). According to Eq. 4, if 2N]p is

constant, this "generalized Slater-Pauling construction" should yield a line of slope 1 intersect-

ing the moment axis at 2JV .

The crucial question for this approach is whether or not 2N'sp should be constant. For

alloys consisting exclusively of transition metals, this is a plausible assumption. It was

originally explained in terms of the "rigid-band model".910 but it is clearly more general, since

N\p is an integrated quantity not depending necessarily on the shape of the band. Data like

that shown in Fig. 2a on Nii_xCux demonstrate the validity of this assumption. For alloys

containing metalloids, however, it is much less obvious why N\p should be constant as a

function of composition. For while 2N\p is approximately 0.6 (or more precisely O.SS) for

most purely transition-metal alloys like Nix_xCux, one might expect 2N\p to be 3 for trivalent

metalloids, 4 for a tetravalent metalloids, etc., simply because of charge neutrality. And yet

the data of Fig. 2b on Nix_xSix, for example, show that, while not precisely constant, the

deviation in 2N\p at x « 0.1 {Zmavt = -0 .4) is only 0.05 rather than the 0.34 ( « (0.55 x

0.9-l-4 x 0.1)-0.55) one might have expected.

This mystery was solved by Terakura and Kanamori'2 who identified the physical

mechanism that holds 2N\p constant in nickel-metalloid alloys to be a d-sp hybridization

effect. This "Fano anti-resonance" interaction causes a strong depression in the sp-state

density at energies near the top of the d band. This depression acts like a gap, preventing

metalloid states from passing through the Fermi energy and thus from increasing 2N]p. We

237

want to emphasize that the surprising constancy of .v]_ is not inconsistent with local neutrality.

For example, the metalloid Si in magnetic alloys does have four (strongly hybridized) s and p

electrons, and is approximately neutral. The implication of the Fano-gap-induced constancy of

N]p is that local neutrality is achieved in such systems by the polarization of existing states,

(e.g. of sp states on the neighbors) not by the occupation of new ones.

The constancy of JV*, means that magnetic valence continues to be a useful concept in

alloys containing metalloids, and that the magnetization of such alloys can be understood

without a detailed knowledge of the atomic geometry. Because of the importance of the

split-band and the Fano gaps, we have called a theory based on the constancy of 2N]p a

"bandgap theory" for strong ferromagnetism.4

While Terakura and Kanamori accounted successfully for dilute Ni-alloys on the basis of

single impurity calculations, it remained unclear for many years whether the same mechanism

would apply to more concentrated alloys. Indeed, most amorphous alloys were stable only at

large solute concentrations, typically around 20% metalloid.

The discovery that the Terakura-Kanamori effect can persist to higher concentrations

came from our band calculations on 3:1 ordered compounds like that shown in Fig. 3, which

revealed a characteristic minimum in the sp-state-density near the Fermi energy (shown by the

arrow). The appearance of this minimum in both bec and fee calculations suggested an

independence of structure, which might extend to amorphous materials. The presence of such

a minimum is a requisite for holding 2N\p constant, as we have seen above. A second

discovery was that the value of 2N\p deduced from the calculated moments of spin-polarized

calculations for 3:1 structures varied from 0.8 for bec structures to more than 1 for fee

structures.4 This suggested that 2N\p might be different from the value of ~0.6 found for

most transition metal-transition metal alloys.

238

Why have we done ordered-compound calculations rather than using more realistic

amorphous structures?1316 The simplification achieved by treating the periodic replication of a

single local atomic arrangement instead of a random array of atomic arrangements allows us to

perform calculations in which the physical quantities of greatest importance are treated

accurately. In the present context these are the atomic volume and charge. Because our

compound calculations determine both the volume and the self-consistent charge distribution

that minimize the calculated total energy, these effects tend to be accurately described. It

would be enormously more difficult to treat these effects with comparable accuracy for a

random system. Furthermore, as we have seen, crucial to the understanding of alloy magneti-

zation is the constancy of the number of s-p electrons. The compound geometry therefore

plays an important role in not restricting us to models, such as tight-binding theory, in which

the s-p electrons are ignored. Finally, the analysis of photoemission data for several amor-

phous alloys17-1* suggests that the close-packed fcc-like compound geometry used in our

calculations models the amorphous alloy sufficiently well to describe the chemical bonding

responsible for the principal variations of the state density within the d band.

Nevertheless, until both bonding and structural aspects can be treated with equal com-

pleteness, we must turn to experiment as the final arbiter. By making generalized Slater-

Pauling plots, we can identify where 2N]p holds constant and where by implication, the

Terakura-Kanamori "bandgap" mechanism applies. We have reviewed existing data1'5 and

find a large number of systems following Eq. 4 with constant 2/vj. over at least some range of

concentration. A good example is amorphous Cof_xBx shown in Fig. 4. Amorphous CoSi,

CoSn, CoP, FeB and FeSi also show such regions. However, some alloys do not;4 these

include amorphous FeC, FeSn, FeGe and FeP. It will be interesting to trace down the reasons

why 2N]p is constant in certain cases and not in others. Moment insensitivity to metallurgical

conditions can be expected in regions where 2N\p is constant, but not where it varies as a

function of composition. The existence of a minimum in the sp-density also has implications

239

for conduction and structural stability which remain to be explored. These are topics for

further work.

The authors acknowledge the key input and earlier collaboration of K. Terakura in this

work.

240

References

1. A. P. Malozemoff, A. R. Williams, K. Terakura, V. L. Moruzzi and K. Fukamichi, J.

Magn. and Magnetic Materials 35, 192 (1983).

2. A. R. Williams, V. L. Moruzzi, A. P. Malozemoff and K. Terakura, IEEE Trans.

Magn. 19, 1983 (1983).

3. V. L. Moruzzi, A. R. Williams, A. P. Malozemoff and R. J. Gambino, Phys. Rev.

B.28, 5511 (1983).

4. A. P. Malozemoff, A. R. Williams and V. L. Moruzzi, Phys. Rev. B 29, 1620 (1984).

5. A. R. Williams, A. P. Malozemoff, V. L. Moruzzi and M. Matsui, J. Appl. Phys. 55,

2353 (1984).

6. V. L. Moruzzi, J. F. Janak and A. R. Williams, Calculated Electronic Properties of

Metals (Pergamon Press, New York, 1978).

7. A. R. Williams, J. Kübler and C. D. Gelatt Jr., Phys. Rev. B 19, 6094 (1983).

8. J. Friedel, Nuovo Cimento, Suppl. to Vol. Ill, 287 (1958); also in Metallic Solid

Solutions, ed. J. Friedel and A. Guinier (W. A. Benjamin, Inc., New York, 1963).

9. J. C. Slater, Phys. Rev. 49, 537 (1936).

10. L. Pauling, Phys. Rev. 54, 899 (1938).

11. J. Crangle and M. J. C. Martin, Phil. Mag. 4, 1006 (1959).

12. K. Terakura and J. Kanaraori, Prog. Theor. Phys. 46, 1007 (1971).

13. R. P. Messmer, Phys. Rev. B23, 1616 (1981).

241

14. T. Fujiwara, J. Phys. K 12. 661 (19X2).

15. S. S. Jaswa! and W. Y. Ching, Phys. Rev. B26, 1064 (1982).

16. S. Frota-Pessoa, Phys. Rev. B28, 3753 (1983).

17. P. Oelhafen, V. L. Moru/zi, A. R. Williams, D. S. Yee, J. J. Cuomo, U. Gubler, G.

Indlekofer and H.-J. GUntherodt, Sol. St. Comm. 44, 1551 (19H2).

18. V. L. Moruzzi, P Oelhafen, A R. Williams, R. Lapka, H.-J. Güntherodt and J.

Kübler, Phys. Rev. B 27, 2049 (1983).

19. H. Watanabe, H. Morita, and H. Yamauchi, IEEE Trans. Magn. 14, 944 (1978).

20. R. Hasegawa and R. Ray, J. Appl. Phys. 50, 1586 (1979).

21. T. R. McGuire, J. A. Aboaf and E. Klokholm, IEEE Trans. Magn. 16, 905 (1980).

These data lie slightly lower than those of other workers (Refs. 17 and 18). We

tentatively attribute this difference to systematic error in determining sample thickness.

22. M. C. Cadeville and E. J. Daniel, J. Phys. (Paris) 27, 449 (1966).

-5 0 5

ENERGY (IN eV RELATIVE TO Ef )

10

Fig.l - Parameter-free, self-consistent, spin-polarized

energy-band calculation of majority and minori-

ty-spin state densities for Co.Y in the CuÂu

structure. This is an example of a split-band

strong magnet with a well-defined gap separa-

ting the lower energy Co band from the higher

energy Y band. The calculated moment of 0.8 y_

average over all atoms of the compound agrees

well with the value for amorphous Co_Y, which

also has an fee-like nearest neighbor environment.

243

-1.0 -0.8 -0.6 -0.4 -0.2 0

(a) ATOM-AVERAGED MAGNETIC VALENCE

- l . o -o.e -o.e -0.4 -0.2 0ATOM-AVERAGED MAGNETIC VALENCE

Fig.2 - Generalized Slater-Pauling construction (average

magnetic moment vs average magnetic valence) for

data of Crangle and Martin for crystalline a)

NiCu and b) NiSi alloys. The moment has been co-

rrected for the shift in g-factor from 2 to 2.2,

which is assumed to be constant as a function of

composition. Along the 45° slope, 2N is cons-

tant as a function of concentration. The Si alloy

approximates this condition in spite of the large

difference in the number of sp electrons for pure

Ni (0.55) and pure Si (4) .

to

i<ena.«oI/)

b O 5

tNtHÜY I IN EV RELATIVE TO EF )

10

Fig..3 - Calculated paramagnetic (non-spin-polarized) sp

and d state-densities on the cobalt site of Co_Si4 3

in the Cu-Au fee structure. Note the difference

in scales for the sp and d densities. The arrow

indicates the gap discussed in the text.

IT

>3 2.5

ÊS 2.0O

sc 1 S

Ctt^ 1.0

0.5SO

v fee xtal, Hasegawa and Hay+ cpd, Cadeville and Danielo amorph. Hasegawa and Kayo amorph, UcGuire et al.A amorph, Hatanabr et al.

- 2 - 1 0 1

ATOM-AVERAGED MAGNETIC VALENCE

Fig.4 - Generalized Slater-Pauling plot for CoB alloys

with data from references 19-22. The 45° line

shows that 2N is constant in the amorphous alloy.

245

AMORPHOUS MAGNETISM

H. R. Rechenberg

Instituto de FTsica - USP

1. INTRODUCTION

In these notes we will briefly examine the consequences of dis-

order on the magnetic properties of solids. In this context, the word

"disorder" is not synonimous to structural amorphicity; chemical dis-

order can be achieved e. g. by randomly diffusing magnetic atoms on a

nonmagnetic crystalline lattice. The name Amorphous Magnetism must be

taken in a broad sense.

The immediate consequence of structural and/or chemical disorder

is a great variety of atomic environments, which in turn has three

effects: •

-a distribution of atomic moments 14;;

-a distribution ef exchange interactions J..;

-a distribution of local anisotropy energies and easy axes.

Many types of magnetic behavior result from the Interplay of

these factors. For the convenience of discussion, we shall classify

disordered materials in three major categories, according to the con-

centration x of magnetic atoms:

a) 10 £ x < 1: ferromagnetic or noncollinear spin structures;

b) 10~ ;£ x £ 10 : cluster magnetism (mictomagnetism);

c) x;£ 10 : spin glasses.

From the viewpoint of chemical composition and preparation meth-

od, several broad families of materials have been investigated to date:

246

a) Amorphous transition meta)-metalloid alloys (metallic glasses).

The basic formula is T«.M..t where T stands for a combination of 3d

metals (Fe, Co, Ni, Cr ...) and M is a combination of metalloids (B,

P, Si, C . . . ) , of course including the binary case. They are most often

prepared by the melt-spinning technique, and many are technologically

important.

b) Amorphous rare earth-transition metal alloys, mostly binaries;

prepared by vapor deposition, sputtering, melt spinning, etc.

c) Glasses in the strict sense, containing transition metals or

rare earths as relatively concentrated impurities; most often prepared

by cooling from the melt.

d) Crystalline metallic alloys or insulating compounds, in which

disorder is introduced by substituting a magnetic element by a non-mag-

netic one or vice versa.

e) Amorphous oxides, halides, sul fides, etc. have been less stu-

died because of preparation difficulties. Some interesting materials,

such as ferric hydroxide gel, can be found in nature.

2.FERR0MAGNETISM

As mentioned above, disorder introduces a distribution P(j) of

exchange interactions. If this distribution is relatively narrow ano

concentrated on positive J values, spins will align parallel to each

other at low temperatures. Ferromagnetic order without a periodic lat-

tice presents no conceptual difficulties, although the possibility of

its existence had not been proposed until I960 (by Gubanov). As we

shall see, its properties differ only marginally from those of crys-

talline ferromagnets.

Representative examples are: most 3d-meta!loid metallic glasses;

247

many 3d-W5d alloys; some 3d-4f alloys. As in the crystalline case,

insulators are rarely ferromagnetic. One striking exception is FeF«,

which is an antiferromagnet in crystal form and a ferromagnet in the

amorphous state. This is a good example of the sensitivity of superex-

change parameters (including their sign) to metal-ligand-metal bond

angles.

In amorphous GdCo- and GdFe_, the Gd moments are aligned anti-

parallel to the transition metal moments, which are ferromagneticaily

coupled among themselves. In such cases i t is appropriate to speak of

ferr{magnetism.

Some characteristic features of amorphous ferromagnetIsm are

listed below.

a) There is a well-defined Curie temperature, with "normal"

critical behavior ( f> - 0.32 for Fei,oNll»oPii,B6' °'1'1 f o r

Fe32Ni36Cpi4P12B6 • " ) -

b) Reduced (M/M vs. T/T ) magnetization curves are flattened

with respect to Briliouin or to crystalline Fe or Ni curves. This can

be ascribed to the P(J) distribution, whose width can be roughly esti -

mated from experimental curves.

c) Mttssbauer spectra have broad lines, reflecting the distribution

of atomic moments at low temperatures, otherwise also the P(J). In the

case of Fe, a quadrupole splitting is often observed at T > T but

not in the six-line spectra below T ; this is explained by the vanish-

ing, on the average, of the factor (3 cos S - 1) which appears in the

expression for hyperfine levels when magnetic and electric quadrupole

interactions are simultaneously present. Since 6 is the angle between

the local electrostatic z axis and the magnetization direction, the

fact that {cos 0 > « 1/3 confirms that the z axis varies ran-

248

domly from site to site, as is expected in an amorphous material.

d) Detailed analysis of the magnetization vs. temperature in me-

tallic glasses, in the range 0 ^ T £o.3 T , reveal a T^ behavior,

which is evidence for the existence of spin waves. However, the norma1-

ized coefficient of the 1 term turns out to be l» to 5 times larger

than In crystalline Fe or Ni. This would imply a significant reduction

of the spin wave stiffness coefficient, whose origin is still obscure.

An alternative interpretation has been proposed whose central Idea

is that amorphous magnets can be well described by a Heisenberg model

of short-range exchange interactions between localized spins. As is

well known, such s model is highly inadequate for dealing with crys-

talline metallic ferromagnets.

The last mentioned feature is perhaps the most fundamental differ-

ence between amorphous and crystalline ferromagnets. Of course, there

are also the important differences in permeability, losses etc., which

render metallic glasses so attractive for many applications, but these

will not be discussed here.

3.NONCOLLINEAR SPIN STRUCTURES

Let us now examine the case where exchange interactions are pre-

dominantly negative, i.e., tending to favor antiparallei spin arrange*

ments. In the simplest situation, we can imagine a crystal lattice to

be subdivided Into two interpenetrating sublattices A and B, such that

all nearest neighbors of an A spin are on the B sub lattice and vice

versa. The ground-state spin configuration is then the classical anti-

ferromagnetic one, in which the spins are collinear but point in oppo-

site directions, in a 1:1 proportion.

Such a subdivision is not always possible. In an FCC lattice,

249

nearest neighbors of a given site may be nearest neighbors of each

other; thus there will be three spins on the vertices on an equilateral

triangle, and it is obviously impossible to have three pairs of anti-

para U el spins. Energy minimization necessarily requires a noncollinear

arrangement»

This ts the simplest example of what is nowadays known as "frus-

tration". This concept acquires its deepest significance in the spin

glass problem, to be discussed later on.

Frustration is certainly favored when the exchange interactions

can be randomly positive or negative. This occurs especially in the

case of magnetic impurities diluted in a nonmagnetic metal, since then

the exchange interactions are of the oscillatory RKKY type:

cos (2k_r..)

J (r ) - A E-4J 3

4r ) 3

where k_ is the Fermi wavevector of the electron gas. Computer simula-

tions have shown that P(J) is then essentially symmetric around J = 0.

Another source of directional disorder is the single-site, "crys-

tal-field" anisotropy, which is particularly important for rare earths.

With this term included (in its simplest, uniaxial form), the magnetic

hamiltonian is written as

D.S7 - 2. J(r..)S.«S.

where not only the strength D. but also the z direction of the easy

axis varies from site to site.

When anisotropy is the dominant factor (|D| > /Jj), we expect

that each spin gets locked along its own easy axis, with a +z or -z

orientation determined by exchange interaction with its neighbors.

250

The resulting structure is then, ideally, that named "speroroagnetic"

by Coey: spins freezed in random directions, without long-range corre-

lations. This type of magnetic ordering can also occur with negligible

an I sotropy, as a result of competition between exchange interactions

of both signs (case of metallic spin glasses) or even in the case of

a single, negative J value, as a result of frustration associated with

topological disorder (examples are amorphous FeF. and Fe(OH).). In

this case It can be said that speromagnetism is the amorphous version

of antiferromagnetism, which obviously cannot exist in the absence of

a crystalline lattice.

Other, more complicated noncollinear arrangements are possible.

Fuller discussions can be found in Refs. (2, 3).

4.CLUSTER MAGNETISM

We will now discuss disordered materials in which the concentra-

tion of magnetic atoms is below the percolation limit. This means that

ferromagnetic, speromagnetic etc. order can still exist below a certain

temperature, but the spatial extent of this order will now be limited

to small regions within the sample. Such ordered domains or clusters

behave essentially as independent entities. As a consequence of this

fine subdivision, completely new features will show up in the magnetic

behavior, both in bulk and in microscopic (e. g., MBssbauer) experi-

ments. These are closely related to superparamagnetism, a class of

phenomena known long before the appearence of amorphous magnetism. Work

in this area was initiated by Nêel in 1949. For a recent review on su-

perparamagnetism, see Ref. (4).

Consider an assembly of ferromagnetic particles so small that the

usual subdivision into domains will be energetically unfavorable. For

251

iron, this means a critical diameter of about 15 nm. Each particle will

2 3then have a resultant magnetic moment of 10 -.10 Bohr magnetons. In

an external field, this moment will rotate as a whole, since individual

spins are kept parallel to each other by exchange coupling.

Even if the intraparticle spin coupling is of antiferromagnetic

nature, a net moment will result, since sublattice cancellation or spe-

romagnetic randomness, whatever the case, can never be perfect in a

small particle.

Each particle will usually have an anisotropy energy whose origin

may be magnetocrystal line, dipolar, shape, etc. An energy barrier E

thus has to be overcome if the net spin is to flip between different

easy directions. Due to small ness of the particles, however, E can be

very small, not greatly exceeding kT; spontaneous thermal fluctuations

will then occur, with a relaxation time X that is a very steep function

of temperature.

The outcome of any oper intent will then be governed by the compa-

rison between T and r , the intrinsic time constant of that experi-

ment. (For static magnetization measurements, T 2f1 - 10 sec; for

mAC susceptibility, Xm « f t / 1 £T io"5 - 10"2 sec; for 57Fe Mtfssbauer,

r í IO"' sec; for neutron scattering, t CIO" 1 1 sec.)m ^ m

The temperature at which C« C >s called the blocking tempera-

ture Tg. I t must be emphasized that a T_ is defined for a given experi-

mental technique and for a given particle size ( i . e . , energy E).

I f T y T_, or X 4C T » magnetization or susceptibility experi-

ments wil l exhibit the equilibrium behavior (Langevin, Curie-Weiss . . )

of an assembly of giant spins. In Hbssbauer spectra, on the other hand,

magnetic hyperfine structure wil l appear washed out by too rapid fluc-

tuations. This is the so-called "superparamagnetic regime".

252

If T < L or T » t (the "blocked regime"), a) the magnetiza-B m

tion will respond very slowly to a sudden change of the external field,

giving rise to phenomena like thermoremanence, etc.; b) spins will be

unable to follow an oscillating magnetic field, thus exhibiting an es-

sentially zero AC susceptibility; c) Müssbauer spectra will show normal

hyperfine structure.

The most conspicuous sign of a blocking temperature is a peak in

the AC susceptibility, separating the two regimes. This may sometimes

be confused with an antiferromagnetic transition. In that case, it

suffices to repeat ths measurements at a different frequency, thereby

changing the measurement time; a shift of the peak temperature is an

unequivocal proof of its relaxational nature.

Turning back to amorphous magnetism, superparamagnetic behavior

is frequently observed in a great variety of disordered materials.

Whenever chemical clustering of magnetic atoms occurs, small-particle

effects such as those described above will be present. This happens,

more often than not, in crystalline transition metal-noble metal alloys,

even at high dilution and despite great metallurgical care.

Chemical clustering, however, is not a necessary condition for the

formation of magnetic clusters. As the temperature is lowered, random-

ly positioned spins may progressively build themselves into locally

correlated regions which can then rotate as a whole. This process will

be assisted by the concentration fluctuations that are always present

on purely statistical grounds. Upon further lowering of temperature,

clusters will grow either by incorporation of new spins or by coalescen-

ce of adjacent groups. As a result, the average cluster moment will be

temperature dependent.

Many transition-metal containing glasses are good examples of this

253

behavior. Alumino-silicate glasses ( MO.Al.O-.SiO., M « Mn or Co ) have

been extensively investigated during the last decade . Measurements

of heat capacity, sound velocity, Mbssbauer spectroscopy, muon depolari-

zation rates, frequency and field dependence of AC susceptibility, mag-

netization decay have yielded results consistent with the superpara-

magnetic interpretation. More recently, measurements of AC susceptibili-

ty in small DC fields have provided evidence for a strong temperature

dependence of the average cluster moment . As to the kind of order

within clusters, it is presumably speromagnetic, since exchange inter-

actions in these glasses are strongly negative.

Strictly speaking, Neel's theory of superparamagnetism should

not be applied, for quantitative purposes, to systems other than physi-

cally well-defined and noninteracting magnetic particles. Neglect of

interactions is sometimes a major shortcoming. On the other hand, the

simple Arrhenius1 law used to relate the relaxation time to particle vo-

lume and temperature has proven inadequate in some cases. Nevertheless,

the physical picture provided by Neel's theory is often the only sim-

ple way to rationalize experimental results on amorphous sysitems.

5.SPIN GLASSES

(For a good recent review, see Ref. (7).)

The spin glass problem is a fashionable and fascinating research

topic. This statement raises two questions: What is a spin glass? Why

is there a problem?

In 1972, Cannella and MydoSh made the first observation of a

sharp, cusp-like maximum In the At susceptibility of some AuFe alloys

at low concentrations. Such an anomaly strongly suggested a phase tran-

sition. Conventional types of magnetic ordering could, however, be

254

ruled out because of the low concentration of magnetic impurities (the

critical concentration for ferromagnetism in Au-Fe is \6% Fe). The

"spin-glass" transition was then assumed to be a cooperative freezing

of spins in random directions, analogous to speromagnetism, occurring

at a well-defined freezing temperature T,.

For seme time it was believed that the essential ingredient for

a spin glass was the RKKY interaction, which is capable of coupling iso-

lated spins separated by rather long distances. Various RKKY, or "ca-

nonical", spin glasses have been investigated, including ûMn, AgMn,

AuCr, AuMn, etc. The same mechanism is responsible for the coupling be-

tween rare-earth moments in La. Gd Al, and similar alloys. However,

spin glass behavior has also been observed in concentrated amorphous

alloys such as Gd.-Al,-, Ni-qP.-Bg, FeqQZr10, etc., where exchange in-

teractions are short-range (cf. Section 2) and spin-spin coordination

is high. Finally, there is the striking case of the insulating com -

pounds Eu. Sr S, which show cluster magnetism for x - 0.13» ferro -

magnetism for x ,> 0.50 and spin-glass behavior in the intermediate

range. For this material, it is by now rather clear that the mechanism

responsible for spin-glass behavior is the competition between posi-

tive (nearest-neighbor) and negative (second-neighbor) exchange inter-

actions.

That the spin-glass transition is one of a rather special nature

became clear when it was realized that the heat capacity has no anoma-

ly at all; the magnetic contribution to it shows, at best, a rounded

maximum well above T-. Magnetic entropy at T, usually has less than

half the total value Nk In (23 + 1), implying considerable short-range

magnetic order above T,.

Another universal feature of spin glasses is the onset of time

2S5

effects (magnetic viscosity, thermoremanence ...) just below T,. In

other words, the ordered state seems to be in metastable equilibrium.

Much theoretical effort has been made during the last decade in

order to clarify the spin glass phenomenon. The question of whether a

a phase transition really occurs has not been settled yet. One essen-

tial point is that the time effects observed below T, are hard to re-

concile with the idea of the ordered phase as a state of thermodynami-

cal equilibrium.

It is not surprising that an alternative school of thought has

emerged, which discards the very idea of a phase transition at T,, in

favor of an interpretation based on Niel's theory of superparamagne-

tism. Indeed, the susceptibility peak, time effects, the absence of a

heat capacity anomaly and so on fit naturally into the simple model of

a progressive blocking of small clusters of correlated spins upon tem-

perature lowering.

The existence of magnetic clusters above T, can indeed be inferred

from neutron scattering, heat capacity and electrical resistivity mea-

surements. The crucial question now is: can the increase of the relaxa-

tion time of these clusters alone explain the observed susceptibility

cusp? It is often argued that rather artificial energy barrier distri-

butions would be needed to produce a sharp cusp. On the other hand,

the actual sharpness of a susceptibility maximum is often hard to as-

sess experimentally, due to the rounding-off effect of a finite mag-

netic field.

The frequency dependence test has not been conclusive either.

Some alloys show a distinct peak shift, but at least for AgMn none

was observed over five decades of measuring frequency. The range of

experimental time constants can be considerably expanded by the inclu-

256

sion of neutron scattering experiments, but their interpretation may

be rather subtle.

Metastabi1ity effects below T f are probably a manifestation of

the high degeneracy of the ground state. It is quite conceivable that

the configuration into which the spins actually freeze at low tempera-

tures is not unique; it surely does correspond to an energy minimum,

but other configurations are equally possible, which are also energy

minima, separated from the chosen one by energy barriers. These are

then responsible for the slow response to a change in external para-

meters, as the magnetic field. The ground state degeneracy has a close

connection with the frustration concept». In Hurd's words , "a frus-

trated system is one which, not being able to achieve a state that sa-

tisfies entirely its microscopic constraints, possesses a multiplicity

of equally unsatisfied states".

The picture that will eventually emerge for the spin glass phe-

nomenon must be more complex than any of the models we have been dis-

cussing, but it will almost certainly include elements of every one.

(8)As an example, it has been suggested that simultaneous freezing

and growth of magnetic clusters may result, at T « T,, in the appear-

ance of an infinite cluster, such a "cluster percolation" being a

true cooperative effect or phase transition. The infinite cluster, in

turn, would still contain weak links due to frustration, so that ther-

mally activated processes would occur, much in the spirit of Neel's

model.

257

REFERENCES

(1) S. N. KAUL: Phys. Rev. B 27, 5761 (1983)

(2) J. M. D.COEY: J. Appl. Phys. 49, 1646 (1978)

(3) C. M. HURD: Contemp. Phys. 23, 469 (1982)

(4) J. L. DORMANN: Rev. Phys, Appllquêe J6, 275 (1981)

(5) H. R. RECHENBERG, A. H. OE GRAAF: J. Phys. C J£, L397 (1980)

(6) E. NUNES FILHO, H. R. RECHENBERG: to be published

(7) P. J. FORD: Con temp. Phys. 23., 141 (1982)

(8) J. A. HYDOSH: J. Hagn. Magn. Mater. Ji-Ji, 99 (1980)

258

Reentrant Ferromagnetism

M. A. Continentino

Instituto de Física

Universidade Federal Fluminense

Caixa Postal 296 - Niterói - RJ - Brasil

We briefly review the problem of reentrant ferromagnetism

with particualr emphasis on the controversy between coexistence

or reentrant behaviour. We discuss a cluster and a glassy

model which are relevant for the understanding of the proper-

ties of these materials. We find that for a ferromagnetic in-

stability to occur it is essential the existence of non-col*

linearity or an anisotropic coupling between spins.

259

Recently much interest has been devoted to the phenomenon

of reentrant ferromagnetism, that is of systems which exhibit

a ferromagnetic to spin glass transition on cooling . In spite

however of experimental evidence, provided- for example through

a scaling analysis of the magnetization as a function of tempe£

ature in small fields , there is still a controversy if all,

or any of, the systems included under this designation do real-

ly have a ferromagnetic instability below a given temperature .

This controversy can be expressed as a duality involving co-

existence versus reentrant behaviour.

From the theoretical side we can get support for both

point pf views.

In fact Gabay 'and Toulouse have studied a vector model of

a spin glass and found a new mixed phase where long range

ferromagnetic order of the longitudinal component of the

magnetization coexists with spin gla^s ordering of the trans-

verse components. On the other hand a random field type of

approach to these systems suggests taat the ferromagnetic

phase should become unstable as the transverse components

freeze with a spin glass ordering .

Cluster Model

Many of the systems which exhibit reentrant type of behaviour-

are dilute ferromagnets with a concentration of magnetic atoms

a little above the percolation threshold . These systems can be

imagined as consisting of an infinite ferromagnetic island,

where spin waves propagate, coexisting with finite clusters

of magnetic atoms randomly distributed in the non-magnetic .

medium ' . For simplicity we shall consider just the two

260

lowest energy configurations of the magnetic clusters and

couple them to,the spin waves of the infinite ferromagnetic

island. The Hamiltonian for this system is

H « I e. a^.a. + E JafJ<yV (1)

k k k k a,6

where e. • Dk is the energy of the spin waves of the in-

finite clusters. The Fauli matrices oa are associated with thea

finite clusters and the S are the components of a spin in

the ferromagnetic agglomerate. We considered an anisotropic

coupling between the infinite and finite clusters for general-

ity. If we look for the effect on the spin waves of this

coupling we find that these excitations have their energy

renormalized and are damped. The renormalized energy E. and

dampi ng T. are given by

E k - efc -

8SN/-T 2^T 2 f x, . it, .«I 8SN _ 2_ f z, > zjj—(J

1 +J2 )Re[x (o)-x (w)J+ s — J2 Re[x (o)-xo o

_ _ 8SN ( . 2^ T 2 . _ x , W 8 S N .T^ • j — CJj- +J 2 )Imx Cu)+]j— ^ 1

o o

where u • e^/h, M and NQ are the number of finite clusters

and spin in the infinite «lusters respectively. xZ(<») and

X*(w) are the longitudinal and transverse dynamic

susceptibilities of the finite clusters and are given in

rsf. 7. We took for simplicity JoB - Jj 6aB and Ja0 - J2 for

a i 0. The theory presented above is in fact a random field

type of theory where we took into account the dynamic nature

of these fields. They are due to the finite clusters and in

concentrated ferrotnagnets with frustration they arise from

261

the "frustration islands" which are agglomerates of spins loosel

connected to the ferromagnetic medium due to frustration effect

In the random field model Ja8 - 0, (J°°)2 - J.2 and

(J ) • J2 for o 4 8. In this case the energy splittings of

the' finite clusters are zero that is the configurations are

degenerate. For this situation the static shift is zero and one

gets for the energy and damping of the spin waves

2SN (J2*2J 2) U)2T2 •l zk

k N 2 2O kT (i) T + 1

- . 2SN <Jl 2^J 22> ,t

O tií T TI

where T - to exp(V/kT) with T>- the inverse of an attempt

frequency and V an activation energy. From the expressions

above we can obtain the lineshift and linewidth measured in a

ferromagnetic resonance experiment on reentrant ferromegnets,

although for detailed comparison they should be averaged over

th.e distribution of activation energies. We point out that

there is no ferromagnetic instability in this random field

model which is in fact consistent with coexistence between

ferromagnetism and freezing of finite clusters. The spin wave

energy shift is always positive and the magnons become stiffer

due to their interaction with the finite clusters. When these

clusters freeze, that is for <DT>>1, appears a gap in the spin

wave spectrum. This gap exists even if the coupling between

the finite and infinite clusters is isotropic (Ja0-O, a + 6;

J°° - J,V a) and may be considered as an isotropic "anisotropj

field acting on the infinite cluster. This result does not

contradict Goldstone's theorem since the w - 0 mode is not

shifted.

262

Glassy Model

The glassy model ic appropriate to describe ferromagnets

with frustration which have a large number of nearly degener-

ate equilibrium configurations. These configurations are local

minima of the classical oagnetic energy and are specified by a

set of angles describing the orientations of the spins. The

metastable ferromagnetic states are accessible to one another

through quantum mechanical tunne tunnellling or thermal

activation and most probably the transition between them

involves the rearrangement i-. the directions of a small number

of spins. The Hamiltcnian describing the glassy ferromagnet

is8

H » t e.aife, + ôcrz + La*: + Kozs"

The first term describe;, the spin waves which are the ex-

citations on the local ninima and furthermore we assume they

have the sane dispersici relation in ell metastable configur-

ations, e represents the energy difference between metastable

states lying on eppogita sides of a double well potential and

A allows for quantum mechanical tunnelling between these

states. The lest term in equation above is the lowest order

term on spin wave operators coupling these excitations to

tunnelling modes. It a.-ises if ve consider an anisotropic

interaction between r.pins nr non-colline&rity between them .

The effect of raetastibiliiy on spin vave propagation is

again to renomalí-c anJ dar? these excitations. In this case

we can obtain a êrrorjssuetic instability due to the dynamic

transverse randca field arising fro.?, the fourth term on the left

hand side of the ec.uatioa above. This transverse field

263

increases as the temperature is lowered due tt> an increase in

the susceptibility of the tunnelling modes. Eventually it

erodes the spin wave gap, due to the dipolar interactions,

and gives origin to a soft spin wave node which causes the

8instability of the ferromagnetic phase .

Conclusions

We have presented two models which we believe are

relevant for the understanding of reentrant ferromagnetism.

Ve found that both situations of coexistence and reentrance

can occur in magnetic materials which are generally called

reentrant ferromagnets. It is essential for the appearance

of a ferromagnetic instability, the existence of an aniso-

tropic coupling between spins.

264

References

1. S.M. Shapiro, C.R. Fincher, Jr., A.C. Palumbo and R.D. Parks,

J. Appl. Phys. _52_, 1729 (1981); J.W. Lynn, R.W. Erwin,

J.J. Rhyne and H.S. Chen ibid. 52_, 1738 (1981).

2. T. Yeshorom, M.B. Salamon, K.V. Rao and H.S. Chen, Phys.

Rev. B24, 1536 (1981).

3. A.P. Morani, Phys. Rev. B28, 432 (1983).

4. M. Gabay and G. Toulouse, Phys. Rev. Lett. 4_7, 201 (1981).

5. K. Motoya, S.M. Shapiro and T. Muraoka, Phys. Rev. B28,

6183 (1983).

6. B.R. Coles, B.V.B. Sarkissian and R.H. Taylor, Phil. Mag.

B37, 489 (1978).

7. M.A. Continentino, J. Phys. C: Solid State Physics 16,

L71 (1983).

8. M.A. Continentino, Phys. Rev. B27, 4351 (1983).

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