Intrinsic Point Defects: Vacancies - University of...

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Intrinsic Point Defects: Vacancies

Energy and entropy of vacancy formationConfigurational and thermal entropyEquilibrium vacancy concentrationDivacancies and vacancy aggregatesSources and sinks of vacanciesTime to reach equilibriumMethods of probing vacancy concentrationVacancies and thermal expansionVacancies and heat capacity

References:Allen & Thomas, Ch. 5, pp. 249-257Kelly, Groves, Kidd, Ch. 9, 261-289Swalin, Chs. 11 and 12, pp. 259-277


Equilibrium vacancy concentrationA distinctive characteristic of point defects is that they can be present at substantial concentrations under conditions of thermodynamic equilibrium. They are sometimes called “thermodynamically reversible defects” - they can be controlled by thermodynamic parameters - T and P.

Thermodynamic equilibrium under conditions of constant P and T: G = H – TS → min

To find the equilibrium vacancy concentration, we have to find the change in G as a result of the introduction of n vacancies into a crystal with N lattice sites:

nnf

n STHGGG Δ−Δ=−=Δ 0

fnf hnH Δ=Δ

Δhf is the enthalpy of vacancy formation - lattice relaxation makes it substantially (~3-4 times) smaller than the heat of vaporization vacancy formation by diffusion from the surface

E.g., for Au: Δhf ≈ 0.97 eV, latent heat of vaporization Δhs ≈ 2.6 eV, cohesive energy Ec ≈ 3.8 eV


What is entropy?

Introduced in the 2nd law of thermodynamics to quantify irreversibility of a process:2nd Law: There exist a state function, the entropy S, which for all reversible processes is defined by dS = δqrev/T and for all irreversible processes is such that dS > δq/T, or in general, dS ≥ δq/T.

The entropy of a system plus its surroundings (together forming “the universe” – an isolated system - δq = 0) never decreases and increases in any irreversible process.

Physical interpretation of entropy: In statistical thermodynamics entropy is defined as a measure of randomness or disorder.

“Any method involving the notion of entropy, the very existence of which depends on the second law of thermodynamics, will doubtless seem to many far-fetched, and may repel beginners as obscure and difficult of comprehension.”

Willard Gibbs

How to quantify “disorder”?


The entropy is related to the number of ways the microstate can rearrange itself without affecting the macrostate.

S = kB ln Ω

A macroscopic state of a system can be described in terms of a few macroscopic parameters, e.g. P, T, V. The system can be also described in terms of microstates, e.g. for a system of N particles we can specify coordinates and velocities of all atoms.

The 2nd law can be stated as follows: The equilibrium state of an isolated system is the one in which the number of possible microscopic states is the largest.

How to quantify “disorder”? - microstates and macrostates

Ω is the number of microstates, kB is the Boltzmann constant (it was first introduced in this equation), and S is the entropy.

The 2nd law can be restated again: An isolated system tends toward an equilibrium macrostate with maximum entropy, because then the number of microstates is the largest.

What is entropy?


Configurational entropy and thermal (vibrational) entropyVibrational entropy Sv: Entropy associated with lattice vibrations (the number of microstates Ω can be thought as the number of ways in which the thermal energy can be divided between the atoms).The vibrational entropy of a material increases as the temperature increases and decreases as the cohesive energy increases.The vibrational entropy plays important role in many polymorphic transitions. With increasing T, the polymorphic transition is from a phase with lower Sv to the one with lower vibrational frequencies and higher Sv, e.g. fcc → bcc (ΔHtr > 0 can be compensated by -TΔS < 0 ).

Configurational entropy Sc: Entropy can be also considered in terms of the number of ways in which atoms themselves can be distributed in space, e.g. entropy of mixing in quasi-chemical model.

T1 < T2

T1 = T2

Sv ↑ Sc ↑


Equilibrium vacancy concentrationnn

fn STHGGG Δ−Δ=−=Δ 0

fnf hnH Δ=Δ

Δsv is change in the vibrational entropy due to the introduction of one vacancy -related to changes in the vibrational frequencies of atoms surrounding the vacancy

nv

nc

n SSS Δ+Δ=Δ vnv snS Δ=Δ

ncvf

n STsThnGGG Δ−Δ−Δ=−=Δ )(0

0c

nc

nc SSS −=Δ )ln( nB

nc kS Ω=

0)1ln( :0 0 === Bc kSn

)ln( :1 1 NkSn Bc ==12 )]1(21ln[ :2 cBc SNNkSn >>−==

22 )]2)(1(61ln[ :3 cBc SNNNkSn >>−−==

… in general, the number of distinct configurations for n vacancies is

( )( ) ( ) ( )!nN!n!N1nN...2N1NN

!n1

−=+−−−


Equilibrium vacancy concentration

( )!!!ln

nNnNkS B

nc −=

NNlnN!Nln −≈

( ){ } ( ) ( ) ( ){ }=−+−−−+−−∂∂

≈−−−∂∂

=∂∂ nNnNnNnnnNNN

nknNnN

nk

nS

BB

nc lnlnln!ln!ln!ln

Let’s first consider n

S nc

∂∂ using Stirling formula for big numbers:

( ) ( ){ } ( ) ( )( ) ( ) =⎟

⎠⎞

⎜⎝⎛ −

=⎭⎬⎫

⎩⎨⎧

−−−

−−+−−=−−−−∂∂

nnNk

nNnNnNnknNnNnnNN

nk BBB ln1ln1lnlnlnln

⎟⎠⎞

⎜⎝⎛−≈⎟

⎠⎞

⎜⎝⎛ −

Nnk

nNk BB ln1ln

{ } 0)ln()( =+Δ−Δ=−Δ−Δ∂∂

=∂Δ∂

==

NnTksThTSsThnnn

GeqBvf

nn

ncvf

nn eqeq

ncvf STsThnG Δ−Δ−Δ=Δ )(

Equilibrium concentration of vacancies can be found from: 0=∂Δ∂

= eqnnnG

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ=

Tkh

ks

Nn

B

f

B

veq expexp



⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ=

Tkh

ks

Nn

B

f

B

veq expexp

The equilibrium vacancy concentration is defined by the balance between

and

nneq

ΔG

)( vf sThn Δ−Δ

)ln( nBnc TkST Ω−=Δ−

)( vf sThn Δ−Δ ncSTΔ−

−∞→⎟⎠⎞

⎜⎝⎛≈

∂Δ−∂

→→ 00

ln)(

Nn

B

Nn

nc

NnTk

nST

steep dependence of the entropy term at small n/N→ defect-free crystals do not exist (nanoparticles?)

neq

Tm

T

ln(neq)

1/Tm

1/T


Energy of vacancy formation Δhf

Rcccf EEREEh Δ−=−−=Δ 2

Rough estimation of Δhf based on an assumption that cohesive energy Ec is equal to the sum of bond energies between adjacent atoms.

The accuracy of experimental and theoretical evaluation of Δhf is limited, with substantial variability in reported data.

By moving an atom to the surface: (1) bonds of an atom that was at the vacancy site are broken (adding Ec), (2) some of the bonds of the atoms surrounding the vacancy are broken (adding another Ec), (3) atom is moved to the surface and restores some of the bonds (removing Ec), (4) atoms around the vacancy are relaxed (reducing energy by ΔER):

⎟⎠

⎞⎜⎝

⎛−= ∑

i

atomisolid

atomsc EENE 1

16874.63

2.1c - 3.6d

Si

0.088.94.284.443.493.813.39Ec (eV)b

8436531808172613561336933Tm (K)a

0.06-0.09e3.71.51.61.220.970.68Δhf (eV)a

ArWα-FeNiCuAuAlmaterial

a Allen & Thomasb Kittel, Introduction to Solid State Physicsc Bracht et al., PRL 91, 245502, 2003d Dannefaer, Mascher, Kerr, PRL 56, 2195, 1986e Schwalbe, PRB 14, 1722, 1976

Δhf is mostly the energy of electron density rearrangement around the vacancy, with the elastic energy making relatively small contribution


Vibrational entropy of vacancy formation ΔsvΔsv is local and associated with the introduction of a single vacancy, which is different from ΔScthat depend on the total number of defects in the crystal.

Δsv can be roughly estimated by considering Einstein model of a solid proposed to explain low-T behavior of heat capacity (deviation from Dulong – Petit law).

Einstein model: a solid is as an ensemble of independent quantumharmonic oscillators all vibrating at the same frequency υ. Quantum theory gives the energy of ith level of a harmonic quantum oscillator as εi = (i + ½) hν where i = 0,1,2…, and h is Planck’s constant.

For a quantum harmonic oscillator the Einstein-Bose statistics must be applied (rather than Maxwell-Boltzmann statistics and equipartition of energy for classical oscillators). For temperatures higher than the Debye temperature, one can show that

hν

⎟⎠⎞⎜

⎝⎛

ν==⎟⎠⎞⎜

⎝⎛

ν= ∑ hTk

hTkkS B

i i

BBv ln3Nkcrystal]perfect for [ln B

In a crystal with n vacancies:n×z atoms have modified frequency ν’N-n×z atoms have the unperturbed frequency ν

z - coordination number vacancymass reduced

constant force21π

=υ


Vibrational entropy of vacancy formation Δsv

∑ ⎟⎠⎞⎜

⎝⎛

ν=i i

BBv h

TkkS ln

In a crystal with n vacancies:n×z atoms have modified frequency ν’ and N-n×z atoms have the unperturbed frequency ν

⎟⎠⎞⎜

⎝⎛

ν+⎟⎠⎞⎜

⎝⎛

ν−= 'lnln)3( hTknzkh

TkknzNS BB

BB

nv

⎟⎠⎞⎜

⎝⎛

ν= hTkS B

v ln3Nk B0In a perfect crystal:

( ) ( )zBBvnv

nv v

vnkvvnzkSSS 'ln'ln0 ==−=Δ

per vacancy: ( )zBvv vvkSs 'ln1 =Δ=Δ

for vacancies, ν’ < ν 0>Δ vs

for interstitials, ν’ > ν 0<Δ vs

generation of vacancy - interstitial pairs (Frenkel defects) can result in partial cancellation of Δsv

The value of Δsv ≈ kB for fcc metals and, in general, reflects the spatial “size” of the point defect - more atoms have modified frequencies around a large defect



⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ=

Tkh

ks

Nn

B

f

B

veq expexp

from Allen & Thomas

168736531808172613561336933Tm (K)

2211Δsv/kB

2.1a - 3.6b3.71.51.61.220.970.68Δhf (eV)

SiWα-FeNiCuAuAlmaterial

a Bracht et al., PRL 91, 245502, 2003b Dannefaer, Mascher, Kerr, PRL 56, 2195, 1986

1exp assuming - =⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ

B

v

ks

Nneq

eV 1~fhΔ

3- 1~/ Bv ksΔ

for metals


Divacancies and vacancy aggregates

⎟⎟⎠

⎞⎜⎜⎝

⎛ −Δ−=⎟

⎟⎠

⎞⎜⎜⎝

⎛ Δ−∝

Tkhh

Tkh

nB

vbf

B

vfv

eq

222 2

expexp

vbf

vf hhh 22 2 −Δ=Δ

If two vacancies join together and form a divacancy, the total number of “broken bonds”decreases and the energy of the divacancy is smaller than the sum of the enthalpies of the individual vacancies by binding enthalpy hb

2v (aka binding energy):

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−∝⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ−⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ=

Tkh

Tkh

ksNn

B

f

B

f

B

vveq expexpexp1

⎟⎟⎠

⎞⎜⎜⎝

⎛ −Δ∝

Tkhh

nn

B

vbf

veq

veq

2

2

1

expmetals) fccfor eV 2.01.0 (e.g., since 22 −≈>Δ v

bv

bf hhhthe fraction of single vacancies decreases as T increases. but even at T ≈ Tm, most of the vacancies are the individual ones: m

veq

veq TTNnNn ≈−≈−≈ −−−− at 1010/ and 1010/ 562341


Generation of vacancy aggregates upon quenching

Thus, in contrast to the equilibrium conditions, when the fraction of single vacancies increases as T decreases, the fast cooling may decrease the fraction of single vacancies as the excess vacancies form divacancies and larger clusters.

The vacancy aggregates may grow into voids, prismatic dislocation loops or (for fcc metals) stacking fault tetrahedra.

Fast quenching can create a strong supersaturation vacancies - in the absence of nearby (within the diffusion length) sinks, the vancancies will meet each other and form divacancies, trivacancies, and larger clusters - this will reduce the enthalpy of the system.

CBA

Cottrell, Phil. Mag. 6, 1351, 1961TEM of stacking fault tetrahedra in quenched Au


Sources and sinks of vacancies

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ=

Tkh

ks

Nn

B

f

B

veq expexpLaw of conservation does not apply to vacancies - they appear and disappear with temperature

different defects have different activationenergies for vacancy formation but this does not affect the equilibrium vacancy concentration - from thermodynamics point of view, the process of vacancy formation is irrelevant. Equilibrium corresponds to reversible transfer of vacancies between the material and “vacancy vapor” or vacuum.

elastic fields from various defects may affect the local equilibrium vacancy concentration - the equilibrium corresponds to the equity of chemical potentials at and in the vicinity of different crystal defects.

(1) free surfaces - original model by Frenkel (1945)

(2) grain and phase boundaries(3) dislocations with an edge component(4) 2D vacancy discs(5) micro-pores

from Swalin


Time to equilibrium

vacancy generation kinetics in polycrystalline gold samples quickly heated by electrical pulses from 436 to 653ºC

[Seidman and Balluffi, Phys. Rev. 139, A1824, 1965]

dislocations play an important role as vacancy sources

eqnn

for random distribution of sources/sinks, the relaxation time after sadden change of temperature:

vADL2

≈τ

<L2> - mean squared vacancy diffusion path from/to the source/sinkDv - vacancy diffusion coefficientA - geometrical factor

quenching to T when Dv is sufficiently large for vacancies to reach each other, but not the sinks, local equilibrium may correspond to formation of vacancy clusters and pores


Methods of probing vacancy concentration

Experimental:(1) Electrical resistivity(2) Thermal conductivity(3) Thermopower measurements(3) Thermal expansion(4) Heat capacity(5) Positron-annihilation spectroscopy

Computational:(1) Atomistic simulations with semi-empirical interatomic potentials(2) Ab-initio (DFT-based) electronic structure calculations

Y. Kraftmakher, Phys. Rep. Rev. Sect. Phys. Lett. 299, 79, 1998


Probing equilibrium vacancy concentration: Simmons & Balluffi, 1960

Simmons & Balluffi Al: Phys. Rev. 117, 52, 1960Ag: Phys. Rev. 117, 600, 1960Au: Phys. Rev. 125, 862, 1962Cu: Phys. Rev. 129, 1533, 1963

measure change in bar length with camera:

L L + ΔLT ↑

v

generation of a vacancy = an extra lattice site in the crystal

measure change in lattice constant with XRD:

relaxationexth

v

VV

VV

Nn

LL

VV Δ

+Δ

+=Δ

=Δ

..

3

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ+

Δ=

Δ

relaxationexth VV

VV

aa

..31

average dilatation due to the vacancies


Probing equilibrium vacancy concentration: Simmons & Balluffi, 1960

⎟⎠⎞

⎜⎝⎛ Δ

−Δ

=aa

LL

Nnv 3

relaxation and thermal expansion effects cancel

1.7×10-4Pb7.5×10-4Na1.9×10-4Cu7.2×10-4Au1.7×10-4Ag9.4×10-4Al

nv/N at Tmmaterial

relaxationexth

v

VV

VV

Nn

LL Δ

+Δ

+=Δ

..

3

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ+

Δ=

Δ

relaxationexth VV

VV

aa

..31

determined from length-lattice parameter measurements


Probing equilibrium vacancy concentration: Heat capacity

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−⎟⎟

⎠

⎞⎜⎜⎝

⎛ ΔΔ=Δ=Δ

Tkh

ksNhhnH

B

f

B

vAffeq expexp

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ=

Tkh

ks

Nn

B

f

B

veq expexp

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ=⎟

⎠⎞

⎜⎝⎛∂Δ∂

=ΔTk

hks

Tkh

kNTHC

B

f

B

v

B

fBA

Pp expexp

2

extra molar enthalpy due to the vacancies:

Avagadro number

R

difficult to separate the increase in the specific heat due to anharmonicity of interatomic interactions from the contribution of point defects


Atomistic simulation of vacanciesAtomistic simulations with semi-empirical interatomic potentials

where

Pair potentials: The total potential energy of the system of N atoms is ( )∑ ∑

>

=i ij

ij2N21 rU)r,...,r,r(Urrr

ijij rrrrr

−=repulsion

attraction

equilibrium

ri rj

ijr

2U

Examples of commonly used pair potentials:

Hard/soft spheres – the simplest potential without any cohesive interaction

( )⎩⎨⎧

>≤∞

=0

02 0 rrfor

rrforrU

ij

ijij

Ionic – Coulomb interaction of charges

( )n

ijij r

rrU

−

⎟⎟⎠

⎞⎜⎜⎝

⎛=

02 - soft

- hard

( )ij

jiij r

qqrU =2

Lennard-Jones – van der Waals interaction in inert gases and molecular systems

( )⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛ σ−⎟

⎟⎠

⎞⎜⎜⎝

⎛ σε=

612

2 4ijij

ij rrrU

Morse – similar to but is a more “bonding-type” potential compared to Lennard-Jones [Morse, Phys. Rev. 34, 57, 1930]

( ) [ ])()(22

00 2 rrrrij

ijij eerU −α−−α− −ε=


Atomistic simulation of vacancies

[R. A. Johnson,J. Phys. F: Met. Phys. 3, 295, 1973]

REh cf −=Δ

Problems with pair potentials:

Pair potentials do not have environmental dependence (e.g. atom in the bulk is too similar to the atom on the surface or near a defect site). In reality, the strength of the “individual bonds” should decrease as the local environment becomes too crowded due to the Pauli’s principle, but pair potentials do not depend on the environment and cannot account for this decrease.

Pair potentials do not account for directional nature of the bond. Covalent contributions (d orbitals) of the transition metals can not be described. Pair potentials work better for metals in which cohesion is provided by s and p electrons.

Quantitative problems:

The vacancy formation energy is significantly overestimated by pair potentials (Δhf ≈ Ec with pair potential, Δhf≈ 0.21Ec for Au, Δhf ≈ 0.35Ec).

The ratio between the cohesive energy and the melting temperature, Tm, is underestimated by as much as 2-3 times. Metals have some “extra cohesion” that is less effective than pair potential in keeping the system in the crystalline state.

small toois R

Δhf and Ec cannot be simultaneously reproduced with pair potentials


Atomistic simulation of vacanciesSolution - to introduce environmental dependence through the concept of local density

Embedded Atom Method (EAM)

( ) ∑≠

+=ij

ij2iii )(rU21ρFE∑=

iitot EE ∑

≠=

ijijji )(rfρ

embedding energy

i

iρ i

jijrrtwo-body term

Electron gas

[Foiles, Baskes, Daw, Phys. Rev. B 33, 7983, 1986][Johnson, Phys. Rev. B 37, 3924, 1988][Raeker, DePristo, Intl. Rev. Phys. Chem. 10, 1, 1991]

Direct electronic-structure (quantum-mechanics-based) calculations of interatomic interaction can be performed in so-called ab-initio atomistic simulation - Schrödinger equation for electrons is solved within the Density Functional Theory for a given set of positions of nuclei. In this case there is no need to assume a particular U(ri), it is calculated during the simulation.

EAM potential accounts for the dependence of the interatomic interactions on local atomic density and allows one to reproduce both Δhf and Ec

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