THE ELECTRICAL AND MAGNETIC PROPERTIES OF MAGNETITE AT THE LOW TEMPERATURE PHASE TRANSITION

by

JOHN DAVID NICHOLAS CHEEKE B.A.Sc, The University of British Columbia, 1959

A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF APPLIED SCIENCE

in the Department

of Physics

We accept this thesis as conforming to the required standard

THE UNIVERSITY OF BRITISH COLUMBIA October, 1961

In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of

the requirements f o r an advanced degree at the U n i v e r s i t y of

B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y

a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission

f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be

granted by the Head of my Department or by h i s representatives.

It i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r

f i n a n c i a l gain s h a l l not be allowed without my written permission.

David J. N. Cheeke

Department of Physics

The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada.

May 26, 1961

i i

ABSTRACT

The ele c t r i c a l and magnetic properties of magnetite from 100-300°K have been studied. Specimens were obtained from natural crystals of local origin and fabricated in the, form of rectangular bars. When this was not possible, due to excessive cracks in the crystal, the ele c t r i c a l properties of the resulting irregularly shaped disc were measured by use of the Van der Pauw theorem.

Observation of the temperature variation of the r e s i s t i v i t y showed that the r e s i s t i v i t y increased as the tempera­ture* was decreased and that i t increased by a factor 2k over a 10°K temperature interval when the specimen was cooled below 1 1 5 ° ^ . This phenomenon i s related to an order-disorder transition at this temperature, i n which the crystalline symmetry changes from cubic to orthorhombic.

One specimen did not show the r e s i s t i v i t y transition, probably due to excessive impurity content, which would prevent ordering from occurring. The results are explained qualitatively by the Verwey model, which postulates a conduction mechanism involving the jumping of electrons between octahedral sites of the Fe^O^ unit c e l l , as opposed to the conventional band type of conduction. An attempt was made to measure the Hall mobility of the specimens, but a Hall voltage was not detected within the

p

sensitivity of the apparatus. An upper limit of 1 cm /volt-sec. for the mobility was established.

i i i A negative magnetoresistance effect was observed in

both specimens and was measured as a function of temperature for both transverse and longitudinal magnetic fi e l d s . A pronounced minimum at the transition temperature was observed for one specimen, while the other showed no change here. The AC permeability was observed over the transition region and again a sharp decrease was observed on cooling through the transition. Normal induction curves were obtained for the specimen at various fixed temperatures from the permeability data, from which i t was deduced that the specimen was much harder to saturate below the transition. The permeability on warming was found to be independ­ent of the magnetic state of the sample while cooling through the transition, and i t was again observed that one crystal did not undergo the phase transition at 115°K.

iv

TABLE OF CONTENTS

Page INTRODUCTION

1. Objectives of present work and extent realised .. 1 2. Summary of the properties of the ferrites 2

(a) Crystal structure 2 (b) Cation distribution over the i n t e r s t i t i a l

sites 3 (c) Low temperature transition 5 (d) Ordering mechanism at the transition 8 (e) Magnetic interaction . 8

1. ELECTRICAL CONDUCTIVITY 1.1. EXPERIMENTAL 11

1.1.1. Sample fabrication 11 1.1.2. Measurement techniques 12

(a) Two-terminal measurements 12 (b) Four-terminal measurements 16 (c) E l e c t r i c a l equipment 18

1.1.3* Description of apparatus 20 1.1.If. Temperature variation of the DC

r e s i s t i v i t y 23 1.2. THEORETICAL 25

1.2.1. Verwey model for ele c t r i c a l conduction in Fe^O^ .... 25

1.2.2. Experimental results and the Verwey model 27

1.2.3. Possible application of the band theory and the quantum mechanical description of the Verwey model 28

2. ELECTRON MOBILITY 2.1. EXPERIMENTAL 32

2.1.1. Equipment used 32

v Page 2.1.2. Measurement procedure 32 2 • 1 • 3 • T)i S C 11S S i On O f X*6SllXt/S • • o o * o * * o » o e e o e « e * > e * « 3̂

2.2. THEORETICAL 35 2.2.1. Upper limit for mobility 35 2.2.2. The possibility of a Hall effect for a

hopping conduction mechanism 36 3. MAGNETORESISTANCE

3 • 1 • EXPERIMENT A.Ij o e * 0 o o * e o o * o « * * e » o o o o o o o o e o e e e « o « « * « « o 3̂ 3• I • 1 o Pr^Ixniirisiry ronisriis © • • • • o o © » o « » o e « » « » o © o » o 3̂ 3-1.2. Room temperature measurements 39 3.1.3* Temperature runs at constant magnetic

f i e l d ^0 3.I.1*. Constant temperature measurements in the

transition region *+3 3 • 2 • THEORETICAL o « e « » e o o e e o * » o o « « 0 o a * * * o o « * 0 « o o o e o « « * » * * "̂3

3.2.1. Discussion of results 4̂-3 k. MAGNETIC PROPERTIES

4̂" 0 I © EXPERIMENTAL e © © « * 0 o a Q O o o 0 O o e e a © o e s o o o © * > o © o o © o o o o © © ^7

*+.l.l. Definition of terras and demagnetisation JOI* OC © Ci 111? 6 e c o o e e o o » o e e e o e * o e e e c » o e e e « o « e e o o 4̂" 7

W.1.2. Specimen orientation and measurement lf.l.3« Normal induction curves and B-H loops at

C OriS t cl-tl t ts6Qlj36X*£t t i l l " 6 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 a

h.l,h. Temperature variation of M, at constant magnetic f i e l d 53

D 2 • THEORETIC AXj • * o © o o » o * » « * © © o o o o o o o o o o o « » » o © e o o © 0 0 O 0 » 5̂ ^ ^ . l . Relation of core permeability to the

inductance of a c o i l 55 .̂2.2. Discussion of results and previous work ... 59

APPENDIX. THE VAN DER PAUW THEOREM 62

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LIST OF FIGURES

Figure T i t l e Facing Page 1 Temperature dependence of r e s i s t i v i t y ,

Sample B 24 2 Temperature dependence of r e s i s t i v i t y ,

Sample A 25

3 Magnetoresistance at room temperature, Sample B 39

4 Transverse magnetoresistance 41 5 AC permeability at room temperature 50

6 Normal induction curves 51 7 Normal induction curves in transition region,

Sample B 52

8 Hysteresis loop, Sample B 53

9 Temperature variation of AC permeability .... 51*

10 Temperature variation of AC permeability .... 55

v i i

ACKNOWLEDGMENTS

I wish to thank Professor R.E. Burgess for his guidance throughout the course of this work, and for his valuable comments and constructive criticism during the preparation of the thesis.

The assistance of the National Research Council of Canada through a Postgraduate -Studentship, Bursary and Summer Supplement i s gratefully acknowledged.

I also wish to acknowledge the assistance of the Defense Research Board in providing many of the research f a c i l i t i e s used in this work, of Mr. J. Lees of the department for providing dewars for the low temperature work, and of the Geology Department for the loan of magnetite crystals.

INTRODUCTION

Objectives of Present Work and Extent Realised These may be classified as:

To observe the 120°K r e s i s t i v i t y change in magnetite to determine i f the results are compatible with the previous findings and the Verwey hypothesis. This was successfully accomplished in the case of one crystal, while the other did not show a transition.

To measure the electron mobility as a function of temperature to determine the conduction mechanism operative in magnetite. This was not possible as.the specimens did not show a Hall voltage within the sensitivity of the measuring circuit used. The upper limit for the mobility was established as

p

1 cm /volt-sec.

To observe the magnetoresistance over the transition region to see i f i t i s affected by the ordering process. This was confirmed experimentally for the crystal which showed the transition.

To determine the change in the kC permeability AA* over the transition, i t s dependence upon magnetic prehistory and to determine whether such measurements are an effective means of observing such a phase transition. This was accomplished, and i t was seen that the crystal which showed no anomaly in other physical properties displayed a small peak in the i n i t i a l permeability.

2

2. Summary of the Properties of the Ferrites

(a) Crystal Structure Magnetite (Fe^O^) is the prototype of the class of

ferrites which has the spinel structure. This structure cr y s t a l l ­izes in the cubic system, the unit c e l l containing eight formula units of the molecule MeFe20l+; here Me stands for a divalent

o metal ion (generally 0 .5 - 1.0 A ionic radius, e.g., Mn, Co, Ni,

++ Cu.) which in the case of Fe^Oi^ i s the l ° n Fe • The other iron

+++ ions in the formula are Fe . Extensive X-ray investigations have revealed that the 0~ ions form an f.c.c. close packed l a t t i c e , while the 8 divalent and 16 trivalent ions are distribut­ed over the available tetrahedral and octahedral i n t e r s t i t i a l s (64 and 32 sites per unit c e l l respectively). From a plan view of the structure i t i s seen that the tetrahedral ions (on A sites) have k nearest 0~ neighbours, while the octahedral ions (on B sites) have 6 nearest 0 neighbours. The structure may also be described as that of the diamond lattice i f each carbon atom i s replaced by an Fe^O^ molecule (Bragg 1915).

In real spinel crystals the oxygen ions do not form an exact f.c.c. structure; for large cations the tetrahedral sites (being smaller than the octahedral) are often too small to accept the metal ions, with the result that the neighbouring anions are displaced away from the site to make way for the cation, thus slightly distorting the l a t t i c e . The distortion i s described in terms of an oxygen parameter u (Verwey 19^7) which may be determined by X-ray or neutron diffraction.

3 (b) Cation Distribution Over the I n t e r s t i t i a l Sites For a simple f e r r i t e we may write the general formula

M e x F e l - x ^ M e l - x F el+x^ t h e i o n s i r i s < l u a r e brackets occupy­ing octahedral sites, the other cations being in the tetrahedral positions. As before, Me + + represents a divalent metal ion. For the normal structure, we have x = ft, so that a l l f e r r i c (trivalent) ions occupy octahedral sites; for the inverse structure x = Q, and here the f e r r i c ions are equally divided be­tween the two i n t e r s t i t i a l positions. At low temperatures this i s an accurate description of a simple f e r r i t e ; i t Is found that only Zn and Cd f e r r i t e are normal, a l l others having the inverse structure. However, at higher temperatures the structure i s in general intermediate between the normal and inverse; the parameter x i s also dependent upon the cooling conditions during formation of the f e r r i t e . "In practice an unlimited number of solid solutions of two or more simple ferrites are also possible but these are not of direct interest in the present study.

Information regarding the cation distribution over the i n t e r s t i t i a l positions has been obtained by.

(a) X-ray scattering, as the scattering factor per atom is determined by the number of electrons per atom. However, the scattering factors of two different valence states of the same element are almost identical as are those of atoms with almost the same number of electrons per atom; hence in these cases (Fe^O^ and transition metal ferrites) the ionic distribution i s d i f f i c u l t to determine by this method.

h

(b) Neutron diffraction, which does not have the disadvantages listed above, and i s hence particularly useful for determin­ing the structure of the transition metal fe r r i t e s . Thus Shull (Shull et a l 1 9 5 D f i r s t showed that has the inverse structure by neutron diffraction.

(c) Saturation magnetisation at low temperatures, which may be interpreted in terms of the Ne'el theory of ferrimagnetism to give the occupancy of A and B sites (Ne'el, 19*+8).

The cation distribution in the spinel structure i s that configuration which gives a minimum to the free energy A = E - TS of the crystal. The internal energy E i s the sum of three components:

(i) E^, the electrostatic energy due to Coulomb forces between each ion and a l l other ions.

( i i ) E p, the elastic energy, due to Born repulsion forces between the ions.

( i i i ) E g, the electronic bonding and exchange energy.

In the h i s t o r i c a l development (Verwey and Hsaijman, 19*+7) a purely ionic model was considered, so that the term E g was neglected. X-ray analysis showed that many of the predicted stable configura­tions were observed, but not in a l l cases. In particular, the normal structure was predicted for Fe^O^, while i t has been seen that a stable inverse structure i s actually observed. Verwey

5 concluded that s t r i c t l y ionic considerations could not explain this preferential occupation of a particular set of sites by some ions. It was later shown (Goodenough and Loeb 1955) that this preference of fer r i c ions for the tetrahedral sites in magnetite could be explained in terms of covalent bonding of these fe r r i c ions with neighbouring oxygen ions.

(°) Low Temperature Transition There i s abundant experimental evidence that Fe.^0^

undergoes a phase transition in the neighbourhood of 120°K that i s not observed in other ferrites with a similar structure. Changes in the following physical properties suggest that the transition can be related to an ordering phenomenon. However, i t has not yet been clearly established whether the transition i s of the f i r s t or second order, and an attempt to classify i t w i l l be made on the basis of these observations;

(i) E l e c t r i c a l r e s i s t i v i t y . Measurement of the e l e c t r i ­cal r e s i s t i v i t y of pure synthetic single crystals of Fe.^0^ o n

cooling through the transition region (Calhoun 1951*) reveal that i t increases by a factor of 90 at 115°K in a temperature interval of 1°K. However many independent measurements of natural single crystals (including the present study) and of some synthetic ones (Domenicali 1950) reveal that the width of the transition region i s often as large as 10°K. The r e s i s t i v i t y transition can be explained satisfactorily i n terms of ionic ordering on the octahedral sites (section 1.2.1).

6

( i i ) Magnetic properties. Measurements for both natural (Okamura 1932, L i 19^-0) and synthetic (Domenicali 1950, Calhoun 195*+) crystals indicate that the magnetisation undergoes a sharp discontinuity at 120°K, which in most cases i s confined to a region 2-1+°K wide. The i n i t i a l permeability (Snoek, 19h7)

has an equally sharp transition here. Calhoun (195^) has shown that the magnetic symmetry follows the crystalline symmetry over the transition and hence can be related to the same ordering phenomenon.

( i i i ) Crystal structure. X-ray analysis (Abrahams and Calhoun 1953)? strain gauge measurements (Bickford 1953) and neutron diffraction (Hamilton 1958) indicate that the crystal structure changes from cubic to orthorhombic on cooling through 120°K. Goodenough (1955) has shown that such a transition can be explained in terms of ionic ordering on the octahedral sites, as postulated i n the Verwey model (section 1 . 2 . 1 ) . He has shown that the transition i s to an orthorhombic rather than a tetragonal structure i f the degree of cation anion covalent bonding on the octahedral sites i s small, which i s known to be true for Fe^O^.

(iv) Thermal expansion. Only two measurements of this property are recorded and these are of the linear expansion co­effici e n t in specific crystallographic directions (Okamura 1932,

Domenicali 1950). In both cases a sharp anomalous expansion on warming through the transition i s noted, the transition region being 1 or 2°K wide for synthetic crystals (Domenicali 1950) and slightly larger for a natural specimen (Okamura 1932).

7

(v) Specific heat at constant pressure (Cp). This has been measured for natural crystals only (Millar 1929, Okamura 1932).

In both cases a f i n i t e A point transition i s observed at 115°K with a transition region 10 - 15°K wide.

Since the basic issue i s whether the ordering at 120°K occurs abruptly or continuously, a transition region of 2°K i s arb i t r a r i l y assigned as the maximum allowable limit for a f i r s t order transition. The existence of a latent heat or a supercooling phenomenon could be accepted as definite evidence of a f i r s t order transition. A. definitely asymmetric peak in Cp may be considered characteristic of a second order transition.

The above observations and the present investigation indicate that natural magnetite crystals exhibit a second order phase transition below 120°K. The observed transition regions for properties ( i ) , ( i i ) and (iv) above were in most cases well above the 2°K criterion, and there i s no recorded evidence of a latent heat or supercooling phenomenon. The specific heat data i s especially convincing; a A transition at least 10°K wide was observed, the slow decay on the high temperature side indicating a high degree of short range order here. This conclusion i s consistent with electrostatic considerations as noted in the following section.

The measurements for synthetic crystals are not conclusive. The transition i s always sharper here and in nearly a l l cases the stated transition limits are below or roughly the same as

8 the 2°K criterion. Specific heat measurements would be helpful in resolving this issue. While no definite conclusion i s possible i t i s suggested that, since synthetic crystals have higher chemical purity than most natural crystals, a very pure specimen would exhibit a f i r s t order transition.

(d) Ordering Mechanism at the Transition Previous calculations have shown that the gain in

electrostatic energy upon ordering from the completely disordered state i s 1.71 eV per molecule (de Boer et al 1950)• However i t i s easy to see that the thermal energy corresponding to a transition temperature of 120°K i s about 10 eV. Assuming that the electrostatic contribution to the lattice energy i s the dominant term, this discrepancy can be explained only by postulating a high degree of short range order above the transition. That the transition can take place at such a low temperature i s attributed to electronic diffusion, the ionic mobilities being too low. Ease of electron interchange i s possible only when the octahedral sites are occupied by ions of the same element but of differing valency. Since this condition i s satisfied only i n the case of Fe^O^ this may well explain why the orthorhombic transition i s not observed in other f e r r i t e s .

(e) Magnetic Interaction In order to explain the observed saturation magnetisa­

tion of the ferrites, Ne"el (1948) proposed a negative exchange interaction between ions on the A and B sites so that the spins are oppositely aligned. The term ferrimaghetism i s used to

9

describe the case in which a net spin results. Due to the large separation of the metal ions a direct exchange interaction i s un­l i k e l y . Anderson (1950) has shown that a superexchange interaction, in which the oxygen ion between neighbouring A and B ions participates, describes the situation adequately. By the use of Hund's rules this interaction predicts parallel alignment of A

1 k and B sublattices i f the metal ions have a d to d configuration

•5 9

and anti parallel alignment for ad-' to d / occupation of the metal ion d shells. A modification of superexchange, semi-covalent exchange, was suggested by Anderson and investigated by Goodenough and Loeb (1955) . in which anti-parallel alignment i s always predicted. Since there are cases in which parallel alignment i s found (MnO) but others i n which the predicted superexchange interaction i s not found (Ca Mn 0^) i t i s clear that neither theory i s able to predict the correct interaction for a l l oxides. Both, however, predict anti-parallel alignment for magnetite in agreement with experimental results.

Zener has postulated a double exchange interaction (Zener 1 9 5 D which may explain the ferromagnetic alignment of a few oxides (e.g., LaMnO^) and which at the same time predicts a high conductivity. The interaction i s most favourable in cases where equivalent lattice sites are occupied by ions of the same element but of differing valency which are separated by anions. It i s not known whether the interaction i s responsible for the alignment of spins on the octahedral sites of magnetite, as this i s predicted by the dominant A-B interaction caused by either

10

semieovalence or super exchange. The double exchange interaction may provide a mechanism for the conductivity of magnetite as proposed by the Verwey model, although the indirect exchange interactions are weakest for a cation anion bond of 90°, which is the case here.

11

1. ELECTRICAL CONDUCTIVITY

1.1 EXPERIMENTAL

1.1 .1 . Sample Fabrication Two natural single crystals of magnetite were obtained

on loan from the Geology Department. For purposes of identifica­tion, the crystals were designated as:

Crystal A - [ill? parting. #17215 Al6. Blue River, B.C.

Crystal B - Octahedral parting. Origin unknown.

Light grinding of exposed surfaces indicated that both crystals had large networks of external cracks. Thus care was taken in choosing the most homogeneous face for purposes of obtaining samples. Slices 2 mm. thick were then cut from each crystal by means of the crystal wire cutting saw of the department's infra-red group. From the crystallographic parting of the crystals, both slices were known to be parallel to (111)

planes.

Further hand grinding and etching indicated that the cracks i n i t i a l l y observed extended into the body of the orystals,

so that this undesirable feature of the samples could not be

eliminated. Etching was done for one minute in a solution of

one normal HC1 at 70°C; other common acids and alkalis

(NaOH, cold HC1, R^SO^ and aqua regia) were ineffective. Specimens

12 were, cut in the form of rectangular bars from the slices. In many cases, due to the brittleness of the material, existence of prominent cleavages or large cracks, the specimens broke during fabrication. The irregular specimen thus obtained could be used for conductivity and mobility measurements, however, by virtue of the Van der Pauw theorem (Appendix ).

On completion of cutting and grinding the specimens were washed in water; the remaining abrasives, grease and organic materials were removed by rinsing in acetone. The samples were then etched, as outlined above, in order to obtain a homo­geneous surface. Traces of the etching solution were removed by rinsing in a weak alkaline solution, and after rinsing in d i s t i l l e d water and methanol (as a mild organic solvent) the specimen was used for measurements.

1.1.2. Measurement Techniques (a) Two-terminal Measurements Preliminary tests were carried out to determine, whether

two-terminal r e s i s t i v i t y measurements would be possible with the Fe^O^ specimens. The properties of a suitable contact for this purpose should include:

(a) Stability under variations of time, mechanical stress, and temperature from 100-300°K.

(b) Formation of a constant low resistance ohmic contact with Fe^O^ so that Rsample » Rcontact.

13

Following Van Vitert's (Van Vitert and Flashen 1956) re­sults with high r e s i s t i v i t y f errites, 3 types of contact material were tried:

(i) 50-50$ indium gallium (In-Ga) alloy, ( i i ) Graphite,

( i i i ) Silver micropaint.

A. bar of Ga metal was obtained from the Metallurgy Department, and chips of this were heated with an equal weight of In pellets in a porcelain crucible until molten. The molten alloy was then smeared directly onto the ends of the bar used for the test. Graphite contacts were applied simply by rubbing the ends of the specimen with a soft drawing pencil. For silver micropaint contacts, a number of low resistance paints produced by the Micro Circuits Co. were available; of these #3C15, an air drying paint was found to have a very low resistance, and i t had the advantage of ease of application since i t did not require baking. For highest conductivity, however, the paint was applied by brush, air dried for twenty minutes and baked at 200°F for one to two hours.

For each of the three contact types, phosphor bronze ribbon was pressed against the coated surface to form a spring contact. In no case, however, was a contact resistance as small as the sample resistance obtained. A typical result, after numerous t r i a l s , was:

lit For a rectangular bar of crystal B, with dimensions

1 cm. x 3 .85 mm. x 2 .0 mm. and j? = .090 ohm-cm. (as determined by subsequent ^-terminal and Van der Pauw measurements).

Rsample = £1 ~ 1*13 ohms. A

Two-terminal resistances for the bar were:

(a) In-Ga contact: 10 ohms. (b) Graphite contact: 25 ohms. (c) Silver paint contact: 6 ohms.

No change in total resistance with reversal of current was observed, so that i t could be assumed that the contacts had identical properties, but were not necessarily ohmic. The above values were observed to change significantly with time and slight mechanical shock; hence these contacts were considered quite unsuitable for reliable two-terminal measurements.

A last experiment with contacts for Fe^O^ was an attempt to produce a suitable welded contact. Thin aluminum (Al) wire was used for this purpose. To one end of a bar of Fe^O^ was attached a silver paste and phosphor bronze spring contact, the other contact being made by springing Al wire against the surface. The specimen was connected in series with a 200 ohm variable resistor and connected to the 100 volt DC line. A steady stream of nitrogen was directed at the specimen while current was passed, the time and magnitude of the latter being varied until the wire end and the contact region were glowing red

15 hot. This procedure was carried out a number of times without great success, although in two cases a two-terminal resistance of 15 ohms was obtained. Judging by the appearance of the contact, excessive oxidation of the aluminum appeared to be taking place, even in the presence of an inert atmosphere. The contact was also extremely fragil e , an expected result since the high melting point of Fe^O^ (1700°K) prevented true fusion of the two materials from occurring. Physical strength of the bond could be greatly increased by coating i t with glyptal and by making the connection on a rough clean surface of the Fe^O^.

To reduce the degree of oxidation of the metal and possible damage to the specimen from prolonged local heating (and possible change of composition) an attempt was made to pulse a high current through the specimen in a short time. Aluuf condenser was shunted across the sample, switch and series resistor Rs, the optimum value of Rs being determined by experiment. It was found necessary to reduce Rs to such a small value (<10 ohms) that when melting of the wire did occur a large spark and almost instantaneous vapourisation of the wire end resulted.

The successful welded contacts were thus found to be inferior to those made from In-Ga and silver paste in mechanical s t a b i l i t y and resistance. A. further unfortunate property was that the contact resistance was observed to increase slowly with time to very large values (> 500 ohms.), while the maximum increase of contact resistance for the previous cases was 50$.

As a result, the welded contacts were considered unsatisfactory.

16 As a result of these experiments the two-terminal

measurements were abandoned, and the more reliable four-terminal techniques were employed. As found above, the silver micropaint contacts were found to be superior to a l l others tested in st a b i l i t y , resistance and ease of application; these properties made them quite suitable for four-terminal measurements.

(b) Four-terminal Measurements As stated in section 1.1.1, in some cases i t was found

necessary to alter the specimen shape from that of a rectangular bar due to excessive cracking of the sample; this was true for the sample cut from crystal A, and for such an irregularly shaped body the Van der Pauw theorem for measuring r e s i s t i v i t y was employed. A satisfactory bar shaped sample was obtained from crystal B so that the conventional k probe method could be used as well for i t .

Validity of the Van der Pauw theorem was verified independently (see Appendix ) by measurements on aluminum sheet f o i l , but comparison between the four probe and Van der Pauw techniques was also made on the sample from crystal B to determine the effect of the cracks known to be present in the specimen. Such inhomogeneities would invalidate the conventional method as, for the potential difference between two points on the sample we may write \/^-j£.d2 = -p>x.^ . For an inhomogeneous sample J i s generally an unknown function of position, so that f>

cannot be determined in this way.

17 A cracked specimen may likewise invalidate the Van der

Pauw method. The theorem holds for a homogeneous body of uniform thickness which i s singly connected (i.e., there are no isolated holes). Cracks in the central region of the sample thus violate the f i r s t and last condition, as such a gap represents a violent inhomogeneity in p. Cracks which come in from one edge and extend to the f u l l depth of the sample do not invalidate the theorem, as this merely makes the body of irregular shape: this assumes there are no internal contacts of the crack edges which would create isolated holes. However, such cracks which do not go to the f u l l depth of the specimen, and a l l others not listed in the acceptable class above would violate the homogeneity condition. A practical d i f f i c u l t y in applying the theorem to these Fe^O^ specimens became apparent during the measurements; often one of the transfer resistances was extremely small (un-measurable with any degree of accuracy), thus makingpindeterminate. Such effects are expected in a cracked specimen (see Appendix ) and they could be avoided only by re-orienting the probes or by remov­ing the major inhomogeneities by reduction of the specimen size. This was done until the ratio of the transfer resistances was < 10.

A specimen from sample B was obtained in which these effects were not evident. Comparison of four-probe and Van der Pauw measurements on i t are given below.

(i) Four-probe method. Large area contacts were placed on the ends of the bar, and two potential rings at an average spacing of 2.5 mm. were painted on the surface in the central

18

region. For I = 10 ma and V = 2.62 mV, and for a cross sectional area of 3-85 mm. x 2.06 mm. we have JD = .08 ohm-cm.

( i i ) Van der Pauw method. For a current I = 5 ma and k a r b i t r a r i l y spaced probes making contact with thin silver paint strips down the side of the specimen, the two average transfer voltages were:

VABico = h.h5 mV

KD,CB = 5.20 mV.

For a sample thickness of d = 2.06 mm., this gives:

J /?£? £ T \ R A D , C S )

where-^has been defined in the Appendix. Noting that -f(Z) = j^(l/Z), substitution yieldsJ>- .090 ohm-cm. The per cent deviation of the two methods for this specimen i s thus 12 .5$. This i s quite reasonable in view of the experimental error involved in each method (see section 1 .1 .3 (c)) and the assumptions made above which w i l l not be s t r i c t l y valid.

For sample A, using I = 5 ma, and for d = 2 .0 mm. the average transfer voltages for1 a Van der Pauw measurement were 0.75

and 3-75 mVolts, giving JD - .33 ohm-cm.

(°) E l e c t r i c a l Equipment The Van der Pauw method for measuring r e s i s t i v i t y was

used in most cases, and a switchbox with a rotary switch for the contacts and reversing switches for I and V terminals was

19

constructed, so that a l l possible combinations of the four probes could be used. In this way, spurious readings due to the rec t i f i c a t i o n of 60 Cps or thermal voltages could be detected, the condition for an acceptable voltage reading being that i t reversed with reversal of the I and V terminals, and that the same reading was obtained when the probes were used in their complement­ary positions.

A current generator was set up to supply a constant current to the sample, which did not vary with changes in the body or contact resistance. The John Fluke Model k06 DC power supply (500 volts at 100 ma) with a General Radio decade resistance box (type l1+32-P) in series was used for this purpose. When possible (due to wattage limitations of the series resistor and desired value of current) the power supply was operated at maximum output voltage so that the largest possible series resistance could be used (100 kohms to 1 megohm). In a l l cases this series resistance was at least 100 times larger than the sample and contact resistance. The current was measured by a Hickok model Ik milliammeter.

Voltages were measured directly by the Hewlett Packard' Model 425 A DC Vacuum Tube Voltmeter. For greater accuracy in measuring changes of voltage with temperature and magnetic f i e l d , a bias unit was installed in the switchbox, consisting of two 1.5

volt dry cells and a potential divider with a low-and- high range. This was put in series with the sample voltage so that any desired fraction of the latter could be bucked out.

20

The absolute accuracy for a single measurement at constant temperature could be calculated by.

Per cent error = ammeter error (1$) + voltmeter error (3%) - k%.

The same figure applies to the relative error of any two readings for a temperature run, as a maximum variation of the current of 1% (due to change of sample resistance) i s possible.

The same current generator and voltmeter were used for the conventional four-probe measurements, the only difference being that the selector switch was fixed in the appropriate position and only the I and V reversing switches were used.

Possible rectification of 60 Cps voltages was avoided by shielding a l l e l e c t r i c a l equipment and the sample and by using co axial cable for the leads. A l l shields were connected to the negative side of the voltmeter which was then grounded to a water pipe. The input to the voltmeter was preceded by a low pass RC f i l t e r to bypass a very low frequency signal which had been observed in the output.

1.1.3• Description of Apparatus The sample was secured to a small bakelite sheet by

Goodyear Pliobond glue, the sheet being bolted to the sample holder; this allowed easy interchange-ability of samples in the holder, and ensured that they had the same orientation with respect to the apparatus (in particular to a DC magnetic fie l d ) during different sets of measurements. The holder was fastened on

21

the end of a long hollow bakelite tube, the whole assembly then being fit t e d into an inner dewar. Fine wire leads (B^S gauge #40, Formel covered) were soldered to the contacts, and together with the thermocouple leads were run up through the inside of the tube to external measuring equipment. Fine wire was chosen In order to minimise the flow of heat conducted down the leads to the sample.

Throughout the experiments the sample temperature was monitored by use of one or more copper-constantan thermocouples; the reference junction was inserted in a bath of melting ice, the other being glued securely to one face of the sample. The resulting thermal emf was measured by a Rubicon students' potentiometer. Preliminary experiments were done in which the maximum thermal gradient along the sample under various cooling and warming rates was determined. For a cooling or warming rate of 1°K per minute or less, the thermal gradient along the length of the specimen was <.25°K per cm., an expected result due to the cylindrical symmetry of the sample's environment. The transverse thermal gradient, however, was considerably larger. Near the transition region cooling rates of 0.5°K per minute and 1°K per minute produced gradients of 1»75°K per cm. and 4°K per cm. respectively. For a specimen 4 mm. wide this would mean an average transverse temperature difference of 0.75°K and 2°K respectively under these cooling conditions.

Since the effect of thermal gradients would be to mask the transition, as different parts of the specimen would pass

22

through the transition region at different times, care was taken to reduce the cooling rate as much as possible. (<0.5°K per minute through the transition region.) Also, as the potential probes were mounted on the same side of the sample, and the longitudinal temperature gradient was small, these measures effectively eliminated the problem of thermal gradients for the r e s i s t i v i t y measurements.

The sample was immersed in a thermal bath in order to exchange heat with the sample while i t remained a sensibly constant temperature medium. An organic f l u i d , 2-methy1-butane, was used for this purpose. The choice was by no means arbitrary, as this i s the only commercial o i l available which has a freezing point (113°K) appreciably below 120°K the magnetite transition temperature, and which has a boiling point above room temperature. Non-toxicity and freedom from explosion hazard are also obvious requirements.

A larger dewar surrounded the inner one described, the lower end being necked down for subsequent magnetic measurements. The outer dewar was supported in a bakelite mount, the mount being held to the magnet table by four brass studs so that i t s elevation could be altered at w i l l . The inner dewar was supported by a clamp and ring stand, so that i t could be raised, lowered or removed from the assembly with relative ease. During the experiments the outer dewar was f i l l e d with liquid nitrogen to any desired level. It was found unnecessary to silver the dewar, as wrapping the dewar in aluminum f o i l did not decrease

23 the boiling off rate of the nitrogen. Hence heating due to radiation could be disregarded, the main source of heat influx being conduction through the very narrow wall gap at the lower end of the dewar, and heat influx to the bath down the leads and supporting mount of the sample. The resultant loss of liquid nitrogen did not pose any problems, however, as a slowly varying bath temperature, (rather than a constant temperature) was desired, and this was automatically attained as the level of liquid nitrogen i n the outer dewar dropped. It was also found that the bath and sample temperature could be controlled easily by varying the vacuum between the walls of the inner dewar (by means of a CENCO HYVAC rotary pump) or by adjusting the depth of immersion of the inner dewar in the liquid nitrogen. During a cooling run increasing the vacuum tended to thermally isolate the bath from the nitrogen, thus decreasing the cooling rate. The latter could be increased by adding liquid nitrogen or changing the immersion depth. Careful balancing of these vari­ables allowed any desired rate of cooling or warming or the attainment of a constant temperature in the sample for several minutes.

Temperature Variation of the DC Resistivity Samples cut from crystals A and B were mounted in

the low temperature apparatus described previously and the temper­ature variation of the r e s i s t i v i t y was measured from 80-300°K. Cooling and warming rates of 1,0 to 1.5°K per minute were used over this range, except for the transition region, in which these

FIGURE /

FACING PAGE £4

. TEMPERATURE DEPENDENCE OE RES/ST/V/71T SAMPLE B

2k

were reduced to 0.25°K per minute.

The results for specimen B have been plotted in f i g . 1. A well defined transition was observed at T = 113»5°K (chosen arb i t r a r i l y as the inflection point of the curve In j^v.s 10^/T°K). When cooling through the transition region (which extended from T = 120.5°K to T = 108.5°K) there was an increase in the r e s i s t i v i t y by a factor 2k. Further measurements indicated that there was negligible thermal hysteresis between cooling and warming runs (<3% deviation). There was some scatter in the data just above the below the transition, but these small deviations were not in general reproducible, and may be attributed to different thermal gradients in the sample on different runs. No change in the transition point was noted when the specimen was cooled through the transition regi on in an annealing DC magnetic f i e l d of 8000 oersteds in a direction perpendicular to the plane of the sample. However the r e s i s t i v i t y of the sample below the

transition was increased after such annealing; at 78°K, ^ V ^ ^

l . k .

If, to a f i r s t approximation, the data i s assumed to f i t j3=f>0eQ//*r (so thatcr= o-0 e Q/MT) then the activation energy for

the conductivity cr may be found directly from the slope of the curve ln^pvs JO-^T°K i n f i g . 1. It i s readily seen that

O - kAQfx.p) _ A G ?-5 \og,oP) p l r = -06Sel/ fbr T<?5°/< W " ACl/T) ~ I l,600A(l/Tj e "

= O a 5 e \ / for T > 2 0 0 ° K

Since the curve i s very non-linear immediately above and below the transition region, the above values of the activation energy

/""'/GO/IE £

FAC/MG fAGE £5-

TEMPERATURE DEPENDENCE OE PES/ST/l/ITY

/O3/T°/<

* * £ i £ $ 1d j? i2 7s >

(corresponding to nearly complete order and disorder respectively) are of more significance than the local slopes near the transition.

These results are in reasonable agreement with previous investigations (Domenicali 1950) for natural crystals of magnetite, although the transition temperature i s somewhat lower, the range somewhat greater and the relative change in r e s i s t i v i t y less than have been observed in synthetic crystals (Verwey 19^1, Calhoun 195^). Such deviations between different crystals could easily be caused by impurities.

A similar temperature run for the specimen from crystal A i s shown in f i g . 2. No transition was observed, and a much smaller dependence of J> upon temperature was noted. The slope of the curve In J> vs 10^/T°K at 115°K gives an activation energy Q = .0¥+ eV.

1 .2. THEORETICAL

1 . 2 . 1 . Verwey Model for E l e c t r i c a l Conduction in Fe.^0^

In order to explain why magnetite has a high electrical conductivity at room temperature, i t was assumed (Verwey and de Boer 1936) that the 8 Fe ions and 8 of the 16 Fe ions of the unit c e l l were randomly distributed over the 16 available octahed­r a l i n t e r s t i t i a l sites. Hence equivalent lattice sites would be occupied by ions of differing valency, which could allow easy electron interchange between them.

26 The model predicts the highest conductivity for pure

stoichiometric magnetite, and varying degrees of non-stoichiometry should upset the i o n i c configuration on the octahedral s i t e s , causing an increase i n the r e s i s t i v i t y and a c t i v a t i o n energy. This has been confirmed by experiment (Verwey and Haaijman 19^1)•

Addition of cation impurities (Mn,Co, N i , Zn) to the octahedral s i t e s likewise increases the r e s i s t i v i t y ; i n terms of the model these impurities impede the electron transfer process between ir o n ions, which i s the p r i n c i p l e ' of controlled valency semi­conductors.

The r e s i s t i v i t y d i s c o n t i n u i t y of magnetite at 120°K was also incorporated i n the model. I t could be explained by proposing an electron ordering process, i n which the random d i s t r i b u t i o n of F e + + and F e + + + ions on the octahedral s i t e s was replaced by an ordered array of ions. Hence below the t r a n s i t i o n , interchange of electrons becomes more d i f f i c u l t r e s u l t i n g i n a much lower conductivity; the dominance of electron jumps between nearest neighbours on the octahedral s i t e s over a l l other paths i s i m p l i c i t l y assumed here. By observing the t r a n s i t i o n for a number of sintered bars of Fe^O^ of d i f f e r i n g non-stoichiometry, Verwey showed that i t i s lowered i n temperature and the r e l a t i v e decrease i n conductivity reduced for increasing non-stoichiometry. A 1%

addition of oxygen ions was found to suppress the t r a n s i t i o n e n t i r e l y ; addition of impurities (Miles et a l , 1957) which p r e f e r e n t i a l l y occupy the octahedral s i t e s likewise reduces the t r a n s i t i o n by impeding the ordering; addition of 5 to 10% cation

27 impurity (Mn, Co, etc.) was found sufficient to suppress the transition entirely.

The model predicts an isotropic conductivity above the transition and anisotropic conductivity for the ordered state. Verwey (1947) proposed that the crystal orders in such a way that successive (001) layers of octahedral sites were occupied alternately by F e + + and F e + + + ions, which would produce maximum conductivity in the <001> direction below the transition. By cooling a crystal in a magnetic f i e l d parallel to this direction, and thus establishing the C axis for the structure below the transition, he was able to show that the conductivity i s highest in this direction and least in a direction perpendicular to i t . A similar degree of anisotropy was also confirmed by Calhoun (195*+)•

1.2.2. Experimental Results and the Verwey Model Observations for sample B are in agreement with previous

investigations for natural crystals (section 1.1.4) and they may be interpreted satisfactorily in terms of the Verwey hypothesis. That the DC r e s i s t i v i t y (.08 ohm-cm) at room temperature i s much larger than that of carefully prepared synthetic crystals ( .005

ohm-cm) may be attributed to impurities, non-stoichiometry or macroscopic imperfections in this natural crystal. As in a l l previous observations a definite increase in the activation energy for conductivity i s observed below the transition.

That the r e s i s t i v i t y discontinuity was not observed at a l l i n sample A can only be explained by a high impurity content or a deviation from stoichiometry; i n the previous section i t was

28

noted that addition of a small degree of such defects to a pure specimen i s sufficient to mask the transition. The DC r e s i s t i v i t y (.33 ohm-cm) and activation energy (.0*+ eV) at room temperature are also considerably larger than for a crystal which does show a transition.

Evidently neither previous investigations or the present study of the conductivity of Fe^O^ near the transition provide conclusive proof of the validity of the Verwey hypothesis; i t i s seen later, however, that the band theory i s by no means so successful in explaining the observed properties of magnetite.

1.2.3* Possible Application of _the Ba-nd Theory and Quantum Mechanical Description of the Verwey Model

While most known experimental results can be explained at least qualitatively by the Verwey model, the exact mechanism of el e c t r i c a l conduction in the ferrites i s s t i l l unclear. Quantum mechanically the Verwey hypothesis i s best incorporated in the Heitler London (H-L) method, and whether this or the Bloch scheme i s most applicable to a given solid i s not always apparent. Recent theoretical calculations (Yamashita I960) indicate that magnetite may f a l l mid-way between the structures which may be treated by the Bloch and H-L formulations.

In the band theory of solids a free electron i s represented by a Bloch wave with wave rector k travelling through the crystal. It moves in the periodic potential of the la t t i c e , which gives rise to allowed and forbidden energy bands when the

29

electron wave function i s made to satisfy the Schrodinger equation. The electron la t t i c e interaction Is treated as a perturbation which produces scattering and hence causes transitions from states k to k.

There are serious objections to the application of this model to semiconductors such as the metallic oxides, in particular to the ferri t e s . Here the lattice spacing i s large and the potential barrier between neighbouring ions i s very high. Consequently the conduction band ( i f one exists) i s very narrow and the mobility of the carriers i s low.

o

For a sufficiently low mobility ( < 5 cm /sec-volt) calculations indicate that the mean free path i s of the same order or less than the lattice parameter (Joffe i 9 6 0 ) . Then the quantities k and v (the d r i f t velocity) which are defined for motion over many lattice parameters no longer apply. As Yamashita does ( i 9 6 0 ) , we may set the condition for applicability of the band theory as:

(energy band width). T »"h , where T i s the mean free time. For a band width~ h K/£m where m* i s the effective mass, and k 0 ~ T r / a , then yet- = e "r//7? * » A-eaa/jr1\ .

Here a i s the lattice parameter, which we take as 4-A. Substitution gives for the lower limit of the mobility to which

2 the band theory should apply 2 cm /volt-sec. Since the observed

2

mobility for magnetite and most oxides i s below 1 cm /volt-sec, the band theory should be inapplicable.

3 0

Further evidence that the band theory i s not relevant to e l e c t r i c a l conduction in the oxides i s given by NiO (Morin 1 9 5 ^ 3 • Band theory predicts that NiO should be a good conductor

g due to the 3 d configuration of the outer electron shell, implying a partially f i l l e d energy band. However experimentally pure stoichiometric NiO has a r e s i s t i v i t y 1 0 J 1 ohm-cm; i.e., i t i s a good insulator. Also, the mobility i s observed to increase with temperature, a fact inconsistent with the band theory. Only one known set of results (section 2 . 2 . 1 ) indicates that the band theory may apply to the oxides.

For such substances in which the electrons or holes have a large effective mass and there i s a strong electron lattice interaction the H-L model seems more suitable. Here the stationary state for an electron i s a localised wave function at a given ion, and the effect of the other ions i s treated as a perturbation. Thus there exists a transition probability W per unit time that the electron w i l l jump to a vacant state in a neighbouring ion. In the case of magnetite, this model incorporates the essential features of the Verwey hypothesis. To determine whether the model applies, following Yamashita ( i 9 6 0 ) we require that the lifetime of a localised state i s much longer than the period of the latt i c e vibration, i.e., that v » w . We may use the Einstein relation to write the mobility as

= ea2W . kT

With T = 300°%.< a = k A. W = 1 0 J - 3 s e c - 1 .

31 p

This gives /OC^ 0.5 cm /volt-sec. as the upper l i m i t of the

mobility for which the model holds. Since this i s an order of

magnitude c a l c u l a t i o n , neither the H-L modelnor the band theory

necessarily apply for m o b i l i t i e s ^ which l i e i n the region p

0.1</^<10 cm /volt-sec. Experimental r e s u l t s indicate that for

magnetite JA, l i e s within this range. Further t h e o r e t i c a l and

experimental work would be necessary to determine whether the

H-L model could be extended to apply to larger m o b i l i t i e s before

i t s a p p l i c a t i o n to the case of magnetite could be f u l l y j u s t i f i e d .

32

2. ELECTRON MOBILITY

2.1 EXPERIMENTAL

2 . 1 . 1 . Equipment Used The e l e c t r i c a l measuring equipment was the same as

that used in section 1. The precautions against 60 cps noted there were essential for the mobility observations, as measurement of very small voltages was necessary here.

The sample holder was f i t t e d in the inner dewar which was placed between the polepieces of the laboratory Newport electromagnet so that the plane of the sample was perpendicular to the f i e l d . With high permeability conical polepieces and the minimum possible gap ( 2 . 2 cm.) the maximum available f i e l d was 10 koersteds. For these and a l l subsequent magnetic measurements, f i e l d strengths were measured by use of the Radio Frequency Laboratories model 1295 Gaussmeter (3$ accuracy).

2 . 1 . 2 . Measurement Procedure A sample 1 mm. thick in the form of a rectangular bar

was cut from crystal B; in view of the brittleness and d i f f i c u l t y of fabrication of the material, cutting of thinner slices was unfeasible.

Area current electrodes were attached to the ends of the specimen and Hall electrodes were placed on opposite sides of the sample at the centre. Ideally the Hall electrodes should be placed on the same equipotential but in practice i t was not

33 possible to set up physically stable contacts with an asymmetry voltage less than a few hundred microvolts. A maximum current of 10 ma was passed; for larger currents the voltage between the Hall probes (as observed on a sensitive scale of the voltmeter by use of the bias unit) was observed to d r i f t at a rate which would seriously limit the minimum observable voltage change on applying a f i e l d . This d r i f t was attributed to localised heating at the contacts which would set up a temperature gradient in the sample. Significant random fluctuations were also observed to be superimposed on the voltmeter d r i f t at currents larger than 10 ma.

Upon application of a magnetic f i e l d + H a voltage change which was an even function of H was observed; this was caused by the magnetoresistance effect on the asymmetry voltage,, Both of these were bucked out by the bias unit, and the voltmeter was set on the most sensitive scale ( l O ^ V f u l l scale) to observe an odd component in the voltage change,, However, to the limiting accuracy of the instrument on this range (+ lyUV) no odd component was observed.

For the dimensions of the specimen used (length §• width ^ 2 . 5 ) the shorting effect of the current contacts upon the Hall voltage can be neglected. Then the upper limit on the Hall mobility for this specimen i s given by:

^ = °l Ri = '°B<ft ̂ ± = 1 cm 2/volt-sec.

3*+ Where o-u - conductivity ^ 10 (ohm-cm)-^".

Ru = Hall c o - e f f i c i e n t ^ i n cm-Vcoulomb. V H = Hall voltage, 1 uv. t = sample thickness =0.1 cm. H = magnetic f i e l d = 10,000 oersteds. I L = current = 10 amperes.

2.1.3* Discussion of Results It i s evident that considerable improvements would be

necessary in order to observe the Hall effect in these Fe^O^ speci­mens. Two major limitations in the method used were: (i) Deriva­tion of the bias voltage from a source independent of the sample current; i t is thus to be expected that there w i l l be considerable fluctuations in the difference between bias and asymmetry voltages when i t i s observed in fine detail, ( i i ) A maximum allowable current of 10 ma. Both of these restrictions could be removed by use of an el e c t r i c a l balance system, in which one of the Hall probes was replaced by two probes (one on either side of the equipotential line used) connected by a high resistance potential divider. The position of the tap could then be adjusted so that the difference in potential between i t and the other Hall probe was zero. This would eliminate the asymmetry voltage and the accompany­ing magnetoresistance effect. Also the current could be increased to much larger values than before, as the asymmetry and bucking voltages would now be derived from the same current, thus eliminating fluctuation and d r i f t problems.

. . . . . . 35

As discussed in section 1 .2 .3 i t i s the determination of the temperature dependence of the mobility which i s of greatest interest. However, i t i s doubtful that this could be accomplished by a DC method, even with the above balancing scheme, due to the spurious voltages of thermoelectric and galvanomagnetic origin which would appear between the Hall probes. An AC method (to eliminate these effects and provide possible amplification of the Hall voltage) coupled with an e l e c t r i c a l balance technique would be the most promising system. However, a previous attempt to do this (Lavine 1958), using a more temperature dependent bucking system, was unsuccessful, due to small fluctuations in temperature. In this connection, the use of constant boiling point cooling solutions would be preferable to a slow cooling run.

2 . 2 . THEORETICAL

2 . 2 . 1 . Upper Limit for Mobility The measured value for the upper limit of for these

natural crystals was 1 cm /volt-sec. By the Verwey model, i f ++

we assume that n = carrier concentration = number of Fe ions 22 - ̂

per molecule, then n = 1.35 x 10 cm ° for a material density of 5 ' 2 grams per c.c. Using this value of n for an order of magnitude calculation, and with cr = 10 (ohm-cm)-"'" we find

/ 2

M- -v/rte=- .01 cm / v o l t - s e c , which i s well below the observed upper limit.

However, recent results are in disagreement with this upper limit. Samokhvalov (I960) has investigated the properties

36

of a magnetite single crystal near room temperature. The observed room temperature mobility was 6 cm /volt-sec; this disagrees with previous measurements (Lavine 1958) and from the discussion of section 1 .2 .3 would make the H-L model invalid for these specimens. Samokhvalov obtained a positive Hall co-efficient (indicating that the carriers are holes) which again disagrees with Lavine^s results. Also, the mobility was observed to decrease with temperature increase, i n accordance with the band theory. While the temperature dependence ofJM had not been otherwise established, a l l other results (section 1) support the Verwey model. Determination of the correct model thus awaits further measurements, and since the properties of natural crystals are known to vary widely, these should be made with synthetic specimens.

2 . 2 . 2 . The Possibility of a Hall Effect for a Hopping Conduction Mechanism We are concerned with the ordinary Hall effect (for

magnetic fields above the saturating f i e l d for the specimen) in which the Hall voltage i s normally derived from a Lorentz force on the carriers due to an applied magnetic f i e l d .

Sewell (I960) considers a two dimensional mechanistic model in which electrons jump from one site to another over the potential h i l l between the sites. Tunnelling processes are excluded on the basis of energy conservation in an applied electric f i e l d ; this i s true for direct tunnelling between di-srete local­ised states, but phonon activated tunnelling cannot be excluded

37 - {• • i

on these grounds. Sewell then shows that a magnetic field' exerts no net velocity to a carrier perpendicular to i t s line of travel, assuming that the electron starts and finishes at rest. This, however, i s not sufficient to show that there i s no Hall voltage. Experimentally, Read and Katz (I960) have observed a Hall effect for NaCl crystals; here ions are jumping from f i l l e d sites to la t t i c e vacancies over a potential h i l l , a situa­tion which may be treated approximately by the mechanistic model above.

Such a model i s too crude to apply to the case of magnetite. The electron wave function may be spread out over a distance comparable to the lattice spacing; thus the assumption that an electron starts with zero velocity at a given site and follows a known trajectory to another i s quite unrealistic. The potential barrier between sites i s large and the force on'the electron due to this potential gradient i s expected to be much larger than the Lorentz force due to a magnetic f i e l d . As for the case of magnetoresistance the problem must be approached quantum mechanically rather than classically; determination of the effect of a magnetic f i e l d upon the transition probability in the H-L model would seem to be the most promising approach.

A;

i

38

3 . MAGNETORESISTANCE

3 . 1 . EXPERIMENTAL

3•1•1• Preliminary Remarks The apparatus and samples used for the magnetoresistance

measurements were identical to those used for the r e s i s t i v i t y , as described in section 1 . 1 .2 . The ends of sample B were coated with silver micropaint to form broad area current contacts (necessary for the longitudinal measurements) and potential probes were attached close to these so as to observe a maximum in magnitude of voltage change on application of a DC magnetic f i e l d . With no f i e l d applied, the voltage (V) observed across the potential probes was exactly opposed by the required bias voltage. The voltage change ( A V ) caused by the f i e l d was then observed on the appropriate sensitive scale of the meter. For a fixed probe orientation A V / V =Af>/p.

Preliminary experiments were done to determine the effect of the magnetic state of the sample on the voltage change produced by a given f i e l d . For a longitudinal orientation of the sample with respect to an external magnetic f i e l d Ho, i t was found that the results for Af>/f> as a function of Ho were scattered and non-reproducible i f the sample was in an unknown magnetic condition prior to the measurement. However, reproducible results were obtained when the following procedure was employed; with the specimen i n i t i a l l y demagnetised,^plf> was measured as a function of Ho for successively increasing values of Ho. In this way the

A MAGNETORESISTANCE AT ROOM TEMPERATURE SAM PL £ 3

// KO£RST£DS

magnetic condition of the sample was always described by the same point on the normal induction curve for a given Ho. As stated in section ^t, the demagnetisation correction has been neglected for this orientation, so that Ho i s assumed to be equal to H, the true internal f i e l d in the specimen.

The precaution of demagnetising and measuringAV for monotonically increasing H was also used for the transverse observations, although the dependence ofA^pupon the sample's magnetic prehistory was almost negligible. With the plane of the sample normal to Ho, i t i s assumed that the demagnetisation factor D = kTJ , so that the induction B inside the specimen = Ho. Since magnetoresistance effects derive from a Lorentz force, which depends upon B rather than H, the dependence of the transverse only upon Ho thus agrees with the expected result. Since B is the relevant parameter here, the transverse magnetoresistance measurements to be described have been plotted as a function of Ho rather than the internal f i e l d H.

3 . 1 . 2 . Room Temperature Measurements Aplp as a function of Ho at room temperature has been

plotted in f i g . 3j for a current of 2 ma. Values of Af> for + 1 = 1 ,10,20 and ^0 ma were also recorded; deviations ofA^'from the curves of f i g . 3 were in a l l cases < 10$ in magnitude, and since they were of a random nature they could be attributed to experimental error rather than to any systematic dependence of the magnetoresistance upon the current. Repeated measurements i n ­dicated that neither the transverse or longitudinal effect showed

ho . .. any tendency to level off at fields above magnetic saturation but that they became linearly increasing functions of Ho.

Measurements below room temperature were mainly confined to those for the transverse orientation only, for two reasons. F i r s t l y , due to possible inhomogeneities in the sample, i t was f e l t that the assumption that the flow lines were parallel to the f i e l d in the longitudinal position was rather dubious. Hence a combination of both longitudinal and transverse effects might be observed in such measurements. One would expect such d i f f i c u l t i e s to be of lesser importance in the transverse position. Secondly, the value of B inside the sample could only be determined semi-quantitatively for the longitudinal position, while i t could be much more surely related to the applied f i e l d Ho for the transverse case. Also, in the former case B i s a function of position for non-ellipsoidal specimens, while in the transverse position i t i s , to a good approximation, constant throughout the specimen.

3«1'3« Temperature Runs at Constant Magnetic Field The temperature dependence of the magnetoresistance at

fixed fields (7-2, 8.0 koersteds) and a current of 2 ma. was determined in the range 77-300°K. Primary interest was placed on the behaviour near the transition region (i.e., 100-130°K). However, even at the lowest cooling and warming rates attainable (<̂ C . 25°K/min) at these temperatures, the variation in due to temperature changes during a single measurement time interval was a significant fraction of the change caused by the magnetic

A'/GUPE <?• AAC/MG PAGE 4/

TRA /VS VERSE MA GNE TO RES/S TA NCR /J = <9 X0£-/?STE£XS

SL

SAM PI £ A

SAMPLE B -0 COOL/MG X IA/ARMJNG

f i e l d (Af)H. The measurement accuracy (15$, as determined by deviations between two runs made under identical conditions) was thus considerably poorer than for the constant temperature case. This effect could not be eliminated by using a larger current (and thus increasingly^) asCAV^would also increase proportion­ately.

Temperature runs were made only for magnetic fields well above the saturating value. Data below saturation was not taken as (/\j>/p)H was an order of magnitude smaller here, and almost completely masked byfA^Lat the best possible cooling rates.

E.g., considering the part of the transition region for which the temperature co-efficient of the r e s i s t i v i t y i s least, at T = 125°K, (iff>)df/a/r =-0.1%. For a cooling rate of 5°K per hour T changes by 3 x 10~^°K in a measurement period of 2

/ -4 seconds. Hence ^p/f due to this temperature change ~ 5 x 10 This i s larger than (Af>/f>)H (or an appreciable fraction of) at field s below saturation for most temperatures in the transition region, making such data extremely unreliable. However, the behaviour at low fields can be obtained from the series of constant temperature measurements mentioned in the next section, where i t was seen that (Aj^Jh i s here a very slowly varying function of H and T.

The variation of the transverse magnetoresistance of sample B at Ho = 8 koersted i s shown in f i g . 4. Above and during the transition regipn there was no significant hysteresis effect

k2

between cooling and warming runs. Below the transition Ap/fdld.

show a dependence on the presence of an annealing magnetic f i e l d while cooling; on subsequent warming a specimen which had been so annealed showed a smallerAy^ below the transition and a smaller peak at i t . However, the data below 110°K was in a l l cases scattered and reproducible only to 20%. Hence small variations in the curve shape here are not significant. A. similar degree of non-reproducibility below the transition was noted in the magnetic measurements of section k.

In this and subsequent runs a sharp minimum in &p/p w a s

observed at ll1+i.l°K and a broad positive maximum occurred at 131°K. Also Ap/p vanished at T = 121+1°K. These two limits on the well defined negative peak of Ap/p correspond to the limits of the phase transition region as determined by the r e s i s t i v i t y measure­ments of section 1.

A temperature run of the longitudinal magnetoresistance at Ho = 8 koersted was also done. The temperature dependence was the same as for the transverse effect, and, as at room temperature, \Ap/p | was always slightly larger. A transverse temperature run for sample A ( f i g . 4.), indicated that the sharp anomaly found at the phase transition of the other specimen was not present here. The sign at Ap/p became positive at approximate­ly 155°K as for sample B; however, i t did not again become negative but rather appeared to decrease slowly with temperature. However, as A/^/jP was an order of magnitude less than for sample B, the per cent error of the run, especially in the transition

»*3

region, was very large (>25$). Also, since fewer measurements were taken than for the previous data, the run was really only qualitative in nature, the main purpose being to determine whether the low temperature magnetoresistance minimum was s t i l l present for an Fe^O^ specimen which did not show a r e s i s t i v i t y discontinuity at the transition temperature.

3•1•k. Constant Temperature Measurements in the Transition Region By the method of section l . l A a constant temperature

in the sample was attainable over a time interval of several minutes, making possible measurements of the magnetoresistance at any desired temperature. Observations were made at seven fixed temperatures in the upper half of the transition region and at seven in the lower half,for the transverse orientation only. Save for temperatures from 122 to 132°K the curves of &j>/J> vs Ho became linearly increasing above Ho = 2500 oersteds. From 122-132°K, however ,Ay?//£> became constant above H = 5000 oersteds. At a constant temperature of 122°K AJD showed a positive sign for fi e l d s below 6000 oersteds and a negative sign for higher f i e l d s .

3 . 2 . THEORETICAL

3 * 2 . 1 . Discussion of Results The measurements of section 3*1 indicate that both

transverse and longitudinal magnetoresistance effects become linear functions of H above saturation at a l l temperatures. That the longitudinal effect saturates at a lower f i e l d than the

$ transverse i s explained by the higher i n t e r n a l f i e l d i n the specimen

kk

for the former case. The sharp minimum observed in ^\J>/f for sample B at 113°K i s undoubtedly related to the r e s i s t i v i t y transition observed here; this i s seen experimentally by the fact that sample A, which showed no r e s i s t i v i t y change at the transition temperature did not show a minimum in the magnetoresist­ance..

Since the data below the transition was not reproducible in detail there are two main features of the temperature runs which must be explained. F i r s t l y , for both longitudinal and transverse positions the effect was seen to become positive in the range 121-155°K with a broad maximum at 131°K. Secondly, both effects showed a sharp minimum at 113°K within a temperature range of 10-15°K (approximately 109-121°K). Aside from the obvious correlation of the second to the order-disorder transition

«

there i s no explanation for the phenomena.

The experimental results have been confirmed by the work of Zotov ( i 9 6 0 ) , in which similar measurements were made at fiel d s of 20,000 oersteds. Two minima-, (at 95°K and 111.5°K) were noted, which Zotov attributes to preferential scattering of conduction electrons; i t i s clear that such an explanation cannot be incorporated i n the Verwey hypothesis as a band model is implied.

An explanation for the negative transverse and longi­tudinal magnetoresistance of magnetite i s also lacking. The effect i s characteristic of nearly a l l ferrites and transition

^5

metals and alloys. An explanation in terms of the s-d interaction appears successful for the ferromagnetic metals but this should not apply to the ferrites as i t derives fundamentally from the band theory. Ziman (I960) indicates that a decrease in resistance for alloys with transition element impurities can be brought about by polarisation of the spins of these impurities by an external f i e l d . Again the model depends upon scattering of electrons in the normal sense, and should not apply to the ferrites at a l l .

A band model which could lead to a magnetoresistance effect in magnetite has been given by Iudin (1958) although the theory i s presented in connection with Curie point phenomena. The principle i s that the conduction band structure for a fer r i t e may be changed by application of a magnetic f i e l d ; i.e., i t w i l l be s p l i t into two levels corresponding to + and - spirit, the energy difference being twice the Zeeman energy. Hence the electron concentration and r e s i s t i v i t y w i l l be changed. For the low temperature transformation in magnetite this could also be invoked for the hopping model; here a magnetic f i e l d may change the magnetic ordering thus leading to a change in n by altering the ionic surroundings of a given Fe ion. Hence a change in a rather t h a n ^ remains a possibility for the magnetoresistance

effect.

The most promising approach would appear to be to determine the effect of a large external f i e l d upon the transition

^6 probability for the H-L model, but such calculations clearly await a more sophisticated version of the model for magnetite.

h7

k. MAGNETIC PROPERTIES

k.l. EXPERIMENTAL

l+.l.l. Definition of Terms and Demagnetisation Procedure The normal induction curve represents the curve which

would be followed by a virgin sample (which had never before been subjected to a magnetic fie l d ) i f H were increased monotoni-cally from zero. It also represents the locus of the tips of a complete family of hysteresis loops. For real samples the curve may be obtained by use of the former definition with the specimen i n i t i a l l y demagnetised, and from the latter by measurements during the demagnetisation process. For the observations recorded here, an attempt was made to keep the sample in a magnetic condi­tion corresponding to a point on the normal induction curve at a l l times. The hysteresis loop, for a given Hmax, represents the locus of points generated by putting the material in a cyclic condition starting from any point on the normal induction curve; i.e., decreasing Hmax monotonically to zero, thence to -Hmax, again decreasing monotonically to zero, etc. The remanent induction Brem. for a given symmetric hysteresis loop i s defined as the induction in the specimen when H = 0. The co-ercive force Hcoer. for the same loop i s the f i e l d H which must be applied to reduce B from Brem. to zero.

Prior to any set of magnetic measurements on the sample a demagnetisation procedure was followed. The method used was:

(i) Erasing of magnetic prehistory by applying H'>Hsat. and larger than any previous f i e l d applied to the specimen.

( i i ) Subjecting specimen to a series of reversals of this f i e l d until a symmetric hysteresis loop has been established with H' as the end-point.

( i i i ) At either + or - H' reducing H' to H" and again cycling until in a symmetrical loop. If | H ' - H"| i s too large the loop w i l l become unsymmetrical, which may be detected by a galvanometer or inductance measurement (section 4.1.3).

(iv) Continuing this process until H" <smallest value to be used in subsequent measurements. When reducing H as in ( i i i ) , this was done at +H on one loop and at -H on the next, etc. This helps to offset the unsymmetrical condi­tion noted above.

(v) Before reducing H to a smaller value, the value of B can be measured to obtain a point on the normal induction curve as outlined in the section on the static method.

4.1.2. Specimen Orientation and Measurement Procedure The AC permeability^*and the magnetic induction B of

the specimens were measured only in a direction parallel to the long dimension of the sample. A measurement of ̂ *or B in the transverse direction would be d i f f i c u l t and inaccurate; almost complete demagnetisation occurs in this orientation, so that a c o i l about the specimen with axis perpendicular to i t would hardly

9̂ be affected by the presence of the core at a l l . For the specimen dimensions used, i t i s d i f f i c u l t to wind such a c o l l with N > 2 5

turns. Thus L(inductance with core) and Lj[air core inductance) are about the same value and quite small ( <2CvaH), necessitating a high frequency measurement to distinguish between them. For reasons to be discussed, high frequency measurements are undesir­able for this material.

Hence the longitudinal direction was chosen for the c o i l axis. A cylindrical four-layer c o i l with 100 turns of #̂ •0 B and S gauge wire was wound and pressed to an oval shape so that i t would f i t closely about the central region of the specimen. The inductance of the c o i l was determined by a bridge measurement to be described, the low value of the inductance (approximately 150 ̂ H) making an operating frequency at least as high as 10 kc/sec. desirable. The AC magnetic f i e l d set up in the core by the bridge oscillator could be considered as a , perturbation to the larger DC magnetic f i e l d Ho which was applied by the electromagnet, Ho thus determining the working point on the B-H curve. Using the assumptions of section k.2.1 we consider the DC inductance of the c o i l to be proportional to^u- of the core at the value of Ho chosen. By choosing an operating frequency well below the self resonance frequency of the c o i l the c o i l capacit­ance and skin effect could be neglected so that the measured inductance may be assumed equal to the DC value. The magnetic loss factor of the core was seen to be significant above 100 kc/sec. so that i t could be neglected at the chosen operating frequency of 12 .5 kc/sec.

/TAC/A/G AAGE SO

50

The inductance measurements were made with the General Radio model I6O3-A Z-Y Bridge. The instrument was convenient in that i t measured the unkriown reactance as a series inductance and resistance. The stated measuring accuracy i s better than 2%.

The General Radio type 1001-A Standard Signal Generator was used as the source, the operating frequency being modulated by an internally generated 400 cps signal for detection purposes. The generator was checked against a reliable frequency standard (the Hewlett Packard model 523 C/D Electronic Counter) and over a time interval of several hours the frequency was found to be stable 1 part in 10^. The Airmec Type C864 Receiver was used as the detector for the bridge.

^ • 1 . 3 . Normal Induction Curves and B-H Loops at Constant Temperature

(a) Inductance Method The procedure follows exactly that outlined in the

previous remarks, except for the presence of an externally applied DC magnetic f i e l d . The c o i l and sample were mounted i n the low temperature sample holder. The specimen was then demagnetised in the standard way, and the inductance of the c o i l was measured for different values of H increasing monotonically from zero. This was continued until saturation was reached, at which time ^t*=land the inductance of the c o i l became a constant. The resulting dependence of>£*upon H i s shown i n f i g . 5 for both samples.

Using the approximations set forth in section 4.2.1

EAC/A/G PAGE S/

51

the normal induction curve of B vs H can now be obtained by integrating the above curves directly. This has been done by measuring the area under the curves of f i g . 5. (by planimeter) for various values of H and thus obtaining the appropriate value of B. The resulting normal induction curves are shown in f i g . 6.

The hysteresis loops forttfo values of Hmax for sample B were obtained in identical fashion. In this case, however, H was increased monotonically from H = 0 (with sample i n i t i a l l y de­magnetised) to the desired value of Hmax, and then cycled continuously to -Hmax and back to +Hmax, a symmetrical B-H curve thus being obtained; the room temperature hysteresis loop of sample B for Hmax>Hsat is shown in f i g . 8. It i s seen that

Remanent = k^5° S a u s s a n d Hcoercive = l 2 ^ ° oersteds, for Bmax = 4850 gauss.

(b) Static Method This was used as a check on the above results; i t

should be noted that the same assumptions regarding uniformity of B and the presence of air gaps around the core must be made, so that while both methods should yield the same value of B for a given H, this observed induction w i l l be different from the true one.

The c o i l used for the inductance measurements was also used here, the two leads being connected directly to a suitable galvanometer as defined below. The measurement of induction was made during the demagnetisation procedure; at a time when the

A-/GUPE 7 PACJA/S PAGE.

3 AGAUSS

4-NORMAL INDUCTION CURVES /N TRANSITION REG/ON

SAMPLE B

A/ KOEPSTEDS

52 specimen was in a symmetric hysteresis loop,, a fast reversal from -Hmax to +Hmax produced a deflection in the galvanometer, one-half of which gives a point on the normal induction curve corresponding to this value of Hmax. The sample was then cycled in a hysteresis loop corresponding to a smaller Hmax and the whole procedure was repeated. In the actual measurement procedure, due to mechanical d i f f i c u l t i e s i n producing a sufficiently fast reversal with the magnet power supply, the measurement was done in two parts, the deflection from -Hmax to H = 0 being added to that for H = 0 to +Hmax.

Choice of a suitable galvanometer was made as follows. On the sudden collapse of the flux linking the c o i l , the time constant of the growth current i s given by2T = -^-seconds. We require that the period T of the galvanometer satisfy T so that the deflection varies as the total charge flowing through the galvano­meter. For the ci r c u i t of c o i l and galvanometer we have Z c / y ^

= cJft/dt where R i s the total resistance i n the ci r c u i t . Integrat­ing from the start to finish of the growth current, this gives q = = AT=A where A. i s the area of the core. Thus, for a suitable value of R the galvanometer w i l l act b a l l i s t i c a l l y .

For the scalamp galvanometer used T = 2 seconds and R = 1400 ohms. For the c o i l Lmax ^ 700/^H and (R o) min. = 25 s s ohms. Hence the time constant for the growth current w i l l be at

-6 most T = 750 x 10 ~ . 5^secs. Thus the galvanometer i s quite

1425 ' adequate for this purpose.

FAC//VS FAGE SS

HYS TEPES/S LOOP •SA A4£>Z £7 £

GAUSS

T=7S °/<

£000

/7 0£-/?STF£>S

53

The hysteresis loops were not traced out in this way, as the required small fast changes in H would have been d i f f i c u l t to attain. Moreover the method was used mainly as a check, and since agreement between the normal induction curves obtained by the two methods was excellent, further observations were made using one method only.

The normal induction curves for sample B at other fixed temperatures were observed by the inductance method as shown in f i g . 7» The hysteresis loop for Hmax = Hsat. was also measured at liquid nitrogen temperature. The fixed temperatures were chosen just above and below the transition region, the main object being to determine the change in Hsat. here. The results indicate that a magnetic f i e l d approximately 50$ larger was required to saturate the specimen below the transition, while about double the f i e l d was required at liquid nitrogen temperature.

4 . I . H . Temperature Variation of ytc at Constant Magnetic Field (a) Sample B The sample was de-magnetised at room temperature in the

usual way. The variation of the i n i t i a l permeability JJ^ (i.e., with H = 0) with temperature was f i r s t observed ( f i g . 9 ) . There was excellent agreement between cooling and warming runs (maximum deviation = 3 $ ) and negligible thermal hysteresis occurred over the transition. The magnetic transition was thus seen to be a sharp well reproduced phenomenon. Below the transition/4 was essentially constant as T was decreased to 77°K.

y ^ / G L / P S 9

TEMPERATURE VAR/A TION OE AC PERMEABILITY SAMPLE: a

9o TAC 7J3 TSO 773 : liS T °A

5h

The specimen was then warmed in a f i e l d H Q = 7 .9 koersted. As expected, the observed j^* was equal to 1 for a l l temperatures from 77°K to room temperature; measurements of the normal induc­tion curve in the previous section indicated that the f i e l d i s above the saturating f i e l d at a l l temperatures and hence the transition i s completely suppressed.

The sample was demagnetised at room temperature and again cooled to liquid nitrogen temperature in zero f i e l d . The variation of on warming for a f i e l d below saturation at a l l temperatures was then observed ( f i g . 9 ) . Again, as expected, the absolute value of yOL above and below the transition was much

(it-smaller than for H 0 = 0 and the relative change in M- over the transition was -greatly rettueed. Subsequent cooling in the same f i e l d revealed a pborfcr reproducibility than found previously, but the general behaviour at the transition was the same. As in the

*

following measurements, a change in the absolute value of M- below the transition was observed.

To determine the effect of the magnetic state of the sample while i t was cooled through the transition upon the subsequent magnetic properties, the sample was now cooled to 77°K in H Q = 8 koersteds. Upon subsequent warming in H Q = 0 ( f i g . 10)

the behaviour of jua during and above the transition was found to be identical in every way to the previous results in which the sample had been cooled in zero f i e l d . A very gradual increase in sU-o with decreasing temperature was noted below the transition;

AAC/A/G PAGE SS

TEMPERATURE VARIATION OE AC PERMEABILITY

~9d To Tsb Tsd TTO l90~

55

small fluctuations from a constant value had been observed in cases where cooling in H Q = 0 had been used.

(b) Sample A The detailed analysis carried out above was not consider­

ed pertinent to the properties of interest of this specimen, the main issue in question being whether or not i t showed a magnetic transition below 120°K. For this purpose a cooling run in H 0 = 0

from room temperature to 85°K was carried out ( f i g . 10). It i s seen that the usual maximum in y U 0 above the transition occurred here, but that there i s only a very slight decrease la y U 0 in the transition region.

4 .2. THEORETICAL

^"2.1. Relation of Core Permeability to the Inductance of a Coil The response of a magnetic medium to small AC magnetic

fields i s given by the AC permeability^- . If damping processes are present the core induction w i l l no longer follow the applied f i e l d , so that there w i l l be a phase lag between them. Regardless of the nature of the damping we express this by writing ~JM°

/U. i s in general a tensor for a single crystal, and i s only specified for a ferromagnetic material when H and B r e m are given.

//

The loss component^ for the core of a c o i l represents an effective series resistance added to the DC resistance. Experimentally (section 4.1.2) the AC resistance of a typical c o i l decreased as the frequency was lowered, and below 15 kc/sec. the c o i l resistance was equal to the DC resistance. Hence for

56

frequencies below this we may set JZ"<^R0/TJJL0. Then ^u. = ^ 6 6 ,

and the Induction i s able to follow small changes in the applied f i e l d without phase displacement so that /sL = dB/dH, the slope of the normal induction curve.

Consider a solenoid of length J . cm. with N turns (n turns per cm.) wound closely about a magnetic cylinder. The fi n i t e thickness of the winding i s neglected so that the cross sectional area of the core (Acm ) and solenoid are assumed to be identical. The f i e l d vectors 13, H" and "M w i l l be assumed as constant in magnitude and directed parallel to the solenoid axis (only rigorously true for an in f i n i t e solenoid and a scalar yU, ). The above also implies a constant demagnetising factor D, which w i l l only hold for an in f i n i t e core or ellipsoid. If an AC current i flows through the windings then

L d i / d t = d0/dt (1) and using

H = H 0 - D M . • • (2) this gives

I - 4TTN% ^6* (7) .

o

If there i s a significant air gap of area A cm. between the core and the c o i l , then the total flux linking the N turns i s

0 = 0air + 0core

Proceeding exactly as above we then obtain

L ~ ~~£~ 1 + CD/4TT)(^--1) + ' which assumes that H .,, = 4TTQL.

air

57

The assumptions used in deriving (4) above make the result valid only for an in f i n i t e solenoid or a toroid. Evidently the applied f i e l d H 0 was not uniform in the core; the c o i l used did not even approximate a long solenoid and i t was pressed to an oval shape so as to f i t more closely about the specimen. In practice then, H 0 w i l l be a very complicated function of position, the value for a cylindrical short c o i l being given by Bozorth ( 1 9 5 D .

An estimation of the demagnetising factor of the core may be obtained by considering the limiting cases of circumscribed and inscribed cylinders of the same length. Demagnetising factors for cylinders have been given by Bozorth (1951) for magnetic fields applied along the axis. For a specimen 10 x 3«85 x 2.06 mm. we have (length/width) = 1+.85 and (length/width). n -2 . 6 0 . For B/H = 5 (the smallest value for which the curves are plotted) DATT i s seen to l i e between .03 and . 0 8 , so that D/hTT^.06.

This gives an approximate value of the demagnetising factor for the specimen in an external f i e l d (that applied by the electromagnet). Since the c o i l did not actually cover the f u l l length of the core G?c^8 mm.) the demagnetising factor for the f i e l d produced by the c o i l w i l l be slightly larger and has been estimated at D/̂ -TT ̂ . 0 8 .

There i s considerable uncertainty in these figures (+ 50$) as the specimen did not closely approximate a cylinder; a further un­certainty i s caused by the fact that the demagnetising factor w i l l vary from point to point in the specimen.

. 58 .

Since the AC permeability was always small (at room temperature^ m a x = 3«8) making the demagnetisation correction small, and in view of the many approximations which must be made in applying (k) to the c o i l the measurements were treated to a f i r s t approximation only and the demagnetisation correction was not made. This i s further justified by the large uncertainty in the correction factor. The air gap about the core has also been neglected in the treatment of the data, although the gap had an estimated area about half that of the core. The shape of the B-H loops suggests that the observed core permeability has been diluted considerably by the air gap. Again, a quantitative correction has not been made as the value of H in this region i s not known exactly, this determining the correction term in the expression for inductance.

The experimental values of were thus obtained by setting D'ATT= 0 and A Q = 0 in (h) which gives L/L 0 ~M where L Q

i s the inductance without a core (or alternatively the c o i l inductance above saturation). As seen above the results w i l l thus be of a semi-quantitative nature only. However since the correction factors mainly involve geometrical factors only (which w i l l be the same at a l l temperatures) the temperature dependence of i s expected to be considerably more accurate.

A precise quantitive determination of the AC permeabil­i t y would involve the following measurements:

(i) Use of ellipsoidal specimens or ones with length/width so large that the demagnetisation factor could be

59

neglected. A. toroidal core would be ideal for this purpose but the fabrication d i f f i c u l t i e s in the case of natural crystals would be considerable.

( i i ) For non-toroidal specimens the c o i l used should closely approximate a long solenoid to ensure a uniform f i e l d inside.

( i i i ) Use of an AC f i e l d produced by the c o l l to trace out the whole B-H loop, with a second air f i l l e d c o i l in opposition to cancel out air gap effects. This would be more accurate and convenient than applying a calculated correction factor of the form given in (4). The procedure would be d i f f i c u l t in that very high fields (at least 4 koersteds) would be needed to trace out the B-H loop to saturation at low temperatures. This would involve the use of large coils and correspondingly larger samples.

4.2.2. Discussion of Results and Previous Work The magnetic properties of magnetite below the

transition have been Investigated In detail (Calhoun 195^) and i t has been established that the magnetic symmetry follows the orthorhombic symmetry predicted by the Verwey hypothesis. The orthorhombic c axis becomes the direction of easy magnetisation below the transition, and this axis i s determined as the cube edge most closely aligned to a DC magnetic f i e l d applied while cooling through the transition. Calhoun has determined the anisotropy constants below the transition. In general, larger magnetic fields are required to saturate the crystal here than at

60

room temperature, and in some directions a f i e l d of 15,000

oersteds may be needed for this purpose. This may be explained by an increa&e in the anisotropy constants, although the results are complicated by twinning; i f the annealing f i e l d i s not applied very close to a single cube edge there may be two or more cube edges equally close to the f i e l d direction. As a consequence the orthorhombic c axis may not have the same orientation through­out the crystal.

The crystallographic direction along which the magnetic f i e l d was applied was not known for the present work; however almost double the f i e l d was required to saturate the specimen at liquid air temperature as compared to the room temperature value. The quantitative value of the saturating f i e l d i s of no s i g n i f i ­cance below the transition, as i t w i l l vary with direction, depending upon anisotropy and twinning effects.

A sharp decrease in the i n i t i a l permeability of sample B was observed at 115°K, the transition region extending from 109-121°K. The completion of the transition was especially abrupt and below this temperature /^„was constant. This i s an indication that the long-range ordering i s complete at 109°K. On the high temperature side of the transition a broad maximum in occurs at 133°K. Since the i n i t i a l permeability i s related to the anisotropy energy and the magneto-strdction, this effect may be explained by a vanishing of the anisotropy energy at 133°K, .which has been observed by Bickford (1950).

61 The temperature dependence of for sample A indicated

a small peak at 125°K which may again be attributed to the vanishing of the anisotropy energy. A. small but definite decrease

*

in was observed at 120°K, below which the results were quite scattered.

If this decrease represents a reduced magnetic transi­tion then i t appears that the measurement of I n i t i a l permeability i s a more reliable technique than r e s i s t i v i t y or magnetoresistance observations for determination of the transition temperature, since the latter quantities showed no change here. However, i t is quite possible that the decrease in yU.0 i s instead related to the usual peak above the transition; the absence of a transition in the other properties has been attributed to a large impurity content and i f this i s so the magnetic transition should appear below 115°K i f at a l l .

62 APPENDIX

THE VAN DER PAUW THEOREM

The theorem (Van der Pauw 1 9 5 8 ) enables measurement of the r e s i s t i v i t y of a disc of arbitrary shape. The only re­quirements for the sample and contacts are:

(a) The contacts are at the circumference of the sample. (b) The contacts are sufficiently narrow and extend the f u l l

depth of the specimen. (c) The sample i s homogeneous in thickness. (d) The surface of the sample i s singly connected.

If contacts satisfying the above conditions are attached to the edge of the sample (so that they appear as a point in a plan view of the disc) they shall be labelled as A, B, C and D in cyclic order. Then the resistance R A B C D i s defined as the potential difference between C and D (V Q-V C) per unit current which enters contact A and leaves contact B. Van der Pauw then shows that

thickness of the disc. (1)

where t sample r e s i s t i v i t y .

To solve (1) for p, the equation i s re-written in the form

(2) which shows f i s a function

6 3

which satisfies the implicit equation:

^ A ° , " — Pee, OA _ J_ arc cosh j e x p [ ( ] n £ ) / f ] I (3), where the positive value of the arc cosh function i s taken for R A 6 , C D > &ec,DA and the negative value for R B C i D A> R A B I C D 5 the theorem i s of course valid for any f i n i t e values of RAB,C D /R sc. OA but i t was f e l t that greater measuring accuracy would be achieved i f the ratio were kept within the limits of one to ten. This can be done simply by re-orienting the relative positions of the contacts.

The theorem was checked experimentally by r e s i s t i v i t y measurements of sheets of commercial purity (99.9$) aluminum f o i l 2 x 10~3 inches thick. A constant current of 1 ampere was passed through the sample, a six volt lead acid battery and a high wattage 20 ohm variable resistor being used as a current source. :

The voltage across the potential contacts was measured directly with the Hewlett Packard model ̂ 25A vacuum tube voltmeter. Point size contacts to the sample were made with phosphor bronze spring c l i p s . A switching arrangement as described i n section 1.1.2 was used so that a l l possible combinations of contacts could be used; this necessitated 2h measurements for a single determination of the r e s i s t i v i t y .

The r e s i s t i v i t y obtained for a six inch diameter circular disc for various probe configurations was checked by ten

-6

independent t r i a l s . The average value for j9 was 2.6h x 10 ohm-cm with a standard deviation of l.h x 10"̂ ohm-cm. As in section 1, the estimated measuring error i s + k%\ a l l of the above

64 readings were well within this departure from the mean value. The disc was then cut into a rectangular shape; the measured

-6 r e s i s t i v i t y was 2.68 x 10 ohm-cm, in good agreement with the previous measurement. Alteration of the specimen to a polygonal shape indicated further close agreement.

If one or more of the contacts i s of f i n i t e size or i s not situated at the circumference the measured value of the r e s i s t i v i t y w i l l be in error. For the case where these faults are confined to a single contact and the sample is a circular disc, Van der Pauw has shown that;

(i) If one contact has dimension dcm. along the circumference of the sample

^ f / f -dJi^D^lrxZ (4)

( i i ) If the contact has dimension d perpendicular to the circumference

(iii) If the contact i s of point size but i s displaced a distance d from the circumference

Af/f^ -cT/2D 2|n2 (6).

The above expressions hold for small values of d. D

However, even for d = 1 i t i s evident that equation (*+) could D 2

not be checked, as the expected error would be less than the measuring error.

65

For case ( i i ) above, with d = 1.9 cm and D = 4 .7 cm, i t was found that j> - 2.44 x 10 ohm-cm, giving Af>/f> = - 0 . 0 6 5 .

The theoretical error, obtained from equation (5) i s Aj^? =-0.058,

i n reasonable agreement with the experimental value.

Finally, for a displaced contact with d = 2 .0 cm and D = 4 .7 cm, f> = 2.3*+ x 10"^ ohm-cm, so thatA/^/p= - .115.

From equation (6) we expect an error Ay?/^ = - .130? again i n good agreement with the observed value. It i s thus seen that a f i n i t e contact along the edge of the sample (that most l i k e l y to occur in practice) introduces a negligible error to the observed r e s i s t i v i t y , but that a displaced contact (as case ( i i i ) ) w i l l decrease the apparent r e s i s t i v i t y quite significantly. If such faults are present in more than one contact the error w i l l be correspondingly larger (again, this i s only approximately true for d / / l ) .

D A f i n a l experiment was concerned with the effect of

cracks upon the observed r e s i s t i v i t y and the accuracy of the measurement. A sheet of metallic f o i l forms an ideal uniformly thick specimen for such measurements since a crack of any desired width or length can be created easily.

It was noted in section 1 .1 .2 that of a l l possible ways in which a specimen can be cracked there i s only one for which case the theorem should remain valid; this case only w i l l be investigated here.

66 It i s the simplest type to create and deal with

experimentally, and i t can be used to simulate anomalous effects which were observed in some of the magnetite specimens. The deformity to be observed i s a crack which intrudes part way into the specimen from the edge extending for the f u l l depth of the specimen and having no contact between the two exposed faces.

A thin crack of this type was cut midway between the pair of current and voltage contacts in a fixed probe configuration. As the length of the crack was increased i t was observed that the ratio of the transfer resistances also increased steadily, while the indicated r e s i s t i v i t y maintained a constant value. When the crack extended almost to the further edge of the sample the ratio was found to reach values above 5 0 , 0 0 0 . In this orientation i t i s easy to see that very l i t t l e current flows in the region in which the voltage probes are positioned; the latter are thus at nearly the same potential so that the transfer resistance R A B , C D

has a small value. It was also seen that by creating two or more of such cracks the transfer resistances could once more be made about equal. Hence a specimen which was highly irregular in this permissible fashion could s t i l l have a measurable r e s i s t i v i t y by the Van der Pauw technique.

67

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