Diffusion Solids Fundamentals Diffusion Controlled Solid State Episode 1 Part 4 potx

Uncorrelated random walk: Let us consider for the moment a random walker that executes a sequence of jumps in which each individual jump isindependent of all prior jumps.. For example, i

4.1 Random Walk and Diffusion 61

The first term contains squares of the individual jump lengths only The

double sum contains averages between jump i and all subsequent jumps j.

Uncorrelated random walk: Let us consider for the moment a random

walker that executes a sequence of jumps in which each individual jump isindependent of all prior jumps Thereby, we deny the ‘walker’ any memory.Such a jump sequence is sometimes denoted as a Markov sequence (memory

free walk) or as an uncorrelated random walk The double sum in Eq (4.20)

terms contain memory effects also denoted as correlation effects For a Markov

sequence these average values are zero, as for every pair x i x j one can find

for another particle of the ensemble a pair x i x j equal and opposite in sign.Thus, we get from Eq (4.20) for a random walk without correlation

2 random =

The index ‘random’ is used to indicate that a true random walk is considered

with no correlation between jumps

In a crystal lattice the jump vectors can only take a few definite values For

example, in a coordination lattice (coordination number Z), in which neighbour jumps occur (jump length d with x-projection d x), Eq (4.21) re-duces to

nearest-

R2 random

62 4 Random Walk Theory and Atomic Jump Process

Table 4.1 Geometrical properties of cubic Bravais lattices with lattice parameter a

Bardeen and Herring in 1951 [7, 8] recognised that this can be

ac-counted for by introducing the correlation factor

f = lim

n →∞

2

2 random = 1 + 2 limn →∞

n−1 i=1

1 In a non-dilute interstitial solution correlation effects can occur, because some

of the neighbouring sites are not available for a jump

4.1 Random Walk and Diffusion 63

f x= lim

n →∞

2

2 random = 1 + 2 limn →∞

n−1 i=1

is zero

Diffusion in solids is often defect-mediated Then, successive jumps occurwith higher probability in the reverse direction and the contribution of thedouble sum is negative Equation (4.27) also shows that one may define the

correlation factor as the ratio of the diffusivity of tagged atoms, D ∗, and

a hypothetical diffusivity arising from uncorrelated jump sequencies, Drandom,

Table 4.2 Correlation effects of diffusion for crystalline materials

f = 1 Markovian jump sequence

No diffusion vehicle involved: direct interstitial diffusion

f < 1 Non-Markovian jump sequence

Diffusion vehicle involved: vacancy, divacancy, self-interstitial, mechanisms

64 4 Random Walk Theory and Atomic Jump Process

4.2 Atomic Jump Process

In preceding sections, we have considered many atomic jumps on a tice Equally important are the rates at which jumps occur Let us take

lat-a closer look to the lat-atomic jump process illustrlat-ated in Fig 4.3 An lat-atommoves into a neighbouring site, which could be either a neighbouring va-cancy or an interstitial site Clearly, the jumping atom has to squeeze be-tween intervening lattice atoms – a process which requires energy The en-ergy necessary to promote the jump is usually large with respect to the

thermal energy kB T At finite temperatures, atoms in a crystal oscillate

around their equilibrium positions Usually, these oscillations are not lent enough to overcome the barrier and the atom will turn back to its ini-tial position Occasionally, large displacements result in a successful jump

vio-of the diffusing atom These activation events are infrequent relative to

the frequencies of the lattice vibrations, which are characterised by the bye frequency Typical values of the Debye frequency lie between 1012 and

De-1013s−1 Once an atom has moved as the result of an activation event, the

energy flows away from this atom relatively quickly The atom becomes activated and waits on the average for many lattice vibrations before itjumps again Thermally activated motion of atoms in a crystal occurs in

de-a series of discrete jumps from one lde-attice site (or interstitide-al site) to thenext

The theory of the rate at which atoms move from one site to a ing one was proposed by Wert [9] and has been refined by Vineyard [10].Vineyard’s approach is based upon the canonical ensemble of statistical me-chanics for the distribution of atomic positions and velocities The jumpprocess can be viewed as occurring in an energy landscape characterised by

neighbour-Fig 4.3 Atomic jump process in a crystalline solid: the black atom moves from an

initial configuration (left) to a final configuration (right) pushing through a point configuration (middle)

saddle-4.2 Atomic Jump Process 65

the difference in Gibbs free energy G M between the saddle-point barrier and

the equilibrium position (Fig 4.3) G M is denoted as the Gibbs free energy

of migration (superscript M ) of the atom It can be separated according to

where H M denotes the enthalpy of migration and S M the entropy of tion Using statistical thermodynamics, Vineyard [10] has shown that the jump rate ω (number of jumps per unit time to a particular neighbouring

migra-site) can be written as

con-The theory of the rates at which atoms move from one lattice site toanother covers one of the fundamental aspects of diffusion For most practicalpurposes, a few general conclusions are important:

1 Atomic migration in a solid is the result of a sequence of localised jumpsfrom one site to another

2 Atomic jumps in crystals usually occur from one site to a neighbour site Molecular dynamic simulations mostly confirm this view.Multiple hops are rare, although their occurrence is indicated in somemodel substances Jumps with magnitudes larger than the nearest-neighbour distance are more common on surfaces or in grain boundaries(see Chap 32)

nearest-3 The jump rate ω has an Arrhenius-type dependence on temperature as

indicated in Eq (4.31)

4 The concept of an atomic jump developed above applies to all the

diffu-sion mechanisms discussed in Chap 6 Of course, the values of G M , H M,

66 4 Random Walk Theory and Atomic Jump Process

and S M depend on the diffusion mechanism and on the material underconsideration:

For interstitial diffusion, the quantities G M , H M , and S M pertain to thesaddle point separating two interstitial positions For a dilute interstitialsolution, virtually every interstitial solute is surrounded by empty inter-stitial sites Thus, for an atom executing a jump the probability to find

an empty site next to its starting position is practically unity

For vacancy-mediated diffusion, G M , H M , and S M pertain to the point separating the vacant lattice site and the jumping atom on itsequilibrium site

saddle-5 There are cases – mostly motion of hydrogen in solids at low tures – where a classical treatment is not adequate [16] However, the endresult of different theories including quantum effects is still a movement

tempera-in a series of disttempera-inct jumps from one site to another [17] For atomsheavier than hydrogen and its isotopes, quantum effects can usually bedisregarded

For more detailed discussions of the problem of thermally activated jumpsthe reader may consult the textbook of Flynn [3] and the reviews byFranklin [11], Bennett [12], Jacucci [13], H¨anngi et al [15] Pon-tikis [14] and Flynn and Stoneham [17]

Molecular dynamic calculations become increasingly important and areused to check and to supplement analytical theories Atomistic computersimulations of diffusion processes have been reviewed, e.g., by Mishin [18].Nowadays, simulation methods present a powerful approach to gain funda-mental insight into atomic jump processes in materials The capabilities ofsimulations have drastically improved due to the development of new simula-tion methods reinforced by increased computer power Reliable potentials ofatomic interaction have been developed, which allow a quantitative descrip-tion of point defect properties Simulation of atomic jump processes havebeen applied to ordered intermetallic compounds, surface diffusion, grain-boundary diffusion, and other systems The challenge is to understand, de-scribe, and calculate diffusion coefficients in a particular metal, alloy, or com-pound In some – but not all – of the rare cases, when this has been done,the agreement with experiments is encouraging

References

1 A Einstein, Annalen der Physik 17, 549 (1905)

2 M van Smoluchowski, Annalen der Physik 21, 756 (1906)

3 M van Smoluchowski, Z Phys 13, 1069 (1912); and Physikalische Zeitschrift

17, 557 (1916)

4 C.P Flynn, Point Defects and Diffusion, Clarendon Press, Oxford, 1972

5 P H¨anggi, P Talkner, M Borkovec: Rev Mod Phys 62, 251 (1990)

References 67

6 J.R Manning, Diffusion Kinetics for Atoms in Crystals, D van Norstrand

Comp., Inc., 1968

7 J Bardeen, C Herring, in: Atom Movements, A.S.M Cleveland, 1951, p 87

8 J Bardeen, C Herring, in: Imperfections in Nearly Perfect Solids, W Shockley

(Ed.), Wiley, New York, 1952, p 262

9 C Wert, Phys Rev 79, 601 (1950)

10 G Vineyard, J Phys Chem Sol 3, 121 (1957)

11 W.M Franklin, in: Diffusion in Solids – Recent Developments, A.S Nowick,

J.J Burton (Eds.), Academic Press, Inc., 1975, p.1

12 C.H Bennett, in: Diffusion in Solids – Recent Developments, A.S Nowick, J.J.

Burton (Eds.), Academic Press, Inc., 1975, p.74

13 G Jacucci, in: Diffusion in Crystalline Solids, G.E Murch, A.S Nowick (Eds.),

Academic Press, Inc., 1984, p.431

14 V Pontikis, Thermally Activated Processes, in: Diffusion in Materials,

A.L Laskar, J.L Bocqut, G Brebec, C Monty (Eds.), Kluwer Academic lishers, Dordrecht, The Netherlands, 1990, p.37

Pub-15 P H¨anngi, P Talkner, M Borkovec, Rev Mod Phys 62, 251 (1990)

16 J V¨olkl, G Alefeld, in: Diffusion in Solids – Recent Developments, A.S Nowick,

J.J Burton (Eds.), Academic Press, Inc., 1975

17 C.P Flynn, A.M Stoneham, Phys Rev B1, 3966 (1970)

18 Y Mishin, Atomistic Computer Simulation of Diffusion, in: Diffusion Processes

in Advanced Technological Materials, D Gupta (Ed.), William Andrews, Inc.,

2005

5 Point Defects in Crystals

The Russian scientist Frenkel in 1926 [1] was the first author to introducethe concept of point defects (see Chap 1) He suggested that thermal agita-tion causes transitions of atoms from their normal lattice sites into interstitialpositions leaving behind lattice vacancies This type of disorder is nowadaysdenoted as Frenkel disorder and contained already the concepts of vacanciesand self-interstitials Already in the early 1930s Wagner and Schottky [2]treated a fairly general case of disorder in binary AB compounds consideringthe occurrence of vacancies, self-interstitials, and of antisite defects on bothsublattices

Point defects are important for diffusion processes in crystalline solids.This statement mainly derives from two features: one is the ability of pointdefects to move through the crystal and to act as ‘vehicles for diffusion’

of atoms; another is their presence at thermal equilibrium Of particularinterest in this chapter are diffusion-relevant point defects, i.e defects whichare present in appreciable thermal concentrations

In a defect-free crystal, mass and charge density have the periodicity ofthe lattice The creation of a point defect disturbs this periodicity In metals,the conduction electrons lead to an efficient electronic screening of defects

As a consequence, point defects in metals appear uncharged In ionic crystals,the formation of a point defect, e.g., a vacancy in one sublattice disturbs thecharge neutrality Charge-preserving defect populations in ionic crystals in-clude Frenkel disorder and Schottky disorder, both of which guarantee global

charge neutrality Frenkel disorder implies the formation of equal numbers

of vacancies and self-interstitials in one sublattice Schottky disorder

con-sists of corresponding numbers of vacancies in the sublattices of cations andanions For example, in AB compounds like NaCl composed of cations andanions with equal charges opposite in sign the number of vacancies in bothsublattices must be equal to preserve charge neutrality Point defects in semi-conductors introduce electronic energy levels within the band gap and thuscan occur in neutral or ionised states, depending on the position of the Fermilevel In what follows, we consider at first point defects in metals and thenproceed to ionic crystals and semiconductors

Nowadays, there is an enormous body of knowledge about point defectsfrom both theoretical and experimental investigations In this chapter, we

70 5 Point Defects in Crystals

provide a brief survey of some features relevant for diffusion For more prehensive accounts of the field of point defects in crystals, we refer to thetextbooks of Flynn [3], Stoneham [4], Agullo-Lopez, Catlow andTownsend[5], to a review on defect in metals by Wollenberger [6], and toseveral conference proceedings [9–12] For a compilation of data on point de-fects properties in metals, we refer to a volume edited by Ullmeier [13] Forsemiconductors, data have been assembled by Schulz [14], Stolwijk [15],Stolwijk and Bracht [16], and Bracht and Stolwijk [17] Proper-ties of point defects in ionic crystals can be found in reviews by Barrand Lidiard [7] and Fuller [8] and in the chapters of Beni`ere[18] andErdely[19] of a data collection edited by Beke

com-5.1 Pure Metals

5.1.1 Vacancies

Statistical thermodynamics is a convenient tool to deduce the tion of lattice vacancies at thermal equilibrium Let us consider an elemental

concentra-crystal, which consists of N atoms (Fig 5.1) We restrict the discussion to

metallic elements or to noble gas solids in which the vacancies are in a singleelectronic state and we suppose (in this subsection) that the concentration

is so low that interactions among them can be neglected At a finite

temper-ature, n 1V vacant lattice sites (monovacancies, index 1V ) are formed The

total number of lattice sites then is

N  = N + n

The thermodynamic reason for the occurrence of vacancies is that the Gibbs

free energy of the crystal is lowered The Gibbs free energy G(p, T ) of the

Fig 5.1 Vacancies in an elemental crystal

5.1 Pure Metals 71

crystal at temperature T and pressure p is composed of the Gibbs function

of the perfect crystal, G0(p, T ), plus the change in the Gibbs function on forming the actual crystal, ∆G:

G(p, T ) = G0(p, T ) + ∆G , (5.2)where

∆G = n1V G F 1V − T Sconf. (5.3)

In Eq (5.3) the quantity G F

1V represents the Gibbs free energy of formation

of an isolated vacancy It corresponds to the work required to create a cancy by removing an atom from a particular, but arbitrary, lattice site andincorporating it at a surface site (‘Halbkristalllage’) Not only surfaces alsograin boundaries and dislocations can act as sources or sinks for vacancies

va-If a vacancy is created, the crystal lattice relaxes around the vacant site andthe vibrations of the crystal are also altered The Gibbs free energy can bedecomposed according to

G F 1V = H 1V F − T S F

into the formation enthalpy H F

1V and the formation entropy S F

1V The last

term on the right-hand side of Eq (5.3) contains the configurational entropy

Sconf, which is the thermodynamic reason for the presence of vacancies.

In the absence of interactions, all distinct configurations of n1V vacancies

on N  lattice sites have the same energy The configurational entropy can be

expressed through the equation of Boltzmann

where W1V is the number of distinguishable ways of distributing n1V

mono-vacancies among the N  lattice sites Combinatoric rules tell us that

W 1V = N

!

The numbers appearing in Eq (5.6) are very large Then, the formula of

Stirling, ln x! ≈ x ln x, approximates the factorial terms and we get

ln W 1V ≈ (N + n 1V ) ln(N + n 1V)− n 1V ln n 1V − N ln N (5.7)Thermodynamic equilibrium is imposed on a system at given temperatureand pressure by minimising its Gibbs free energy In the present case, thismeans

The equilibrium number of monovacancies, n eq 1V, is obtained, when the Gibbs

free energy in Eq (5.3) is minimised with respect to n 1V, subject to the

constraint that the number of atoms, N , is fixed Inserting Eqs (5.5) and (5.7)

72 5 Point Defects in Crystals

into Eq (5.3), we get from the necessary condition for thermal equilibrium,

This quantity also represents the probability to find a vacancy on an arbitrary,

but particular lattice site In thermal equilibrium we have C 1V eq ≡ n eq

dence of C 1V eq is primarily due to the formation enthalpy term in Eq (5.11)

We note that the vacancy formation enthalpy is also given by

con-5.1.2 Divacancies

Divacancies (2V) are point defects that form in a crystal as the simplestcomplex of monovacanies (1V) This is a consequence of the mass-actionequilibrium for the reaction

The probability that a given lattice site in a monoatomic crystal is vacantequals the site fraction of monovacancies Let us suppose that a divacancyconsists of two monovacancies on nearest-neighbour lattice sites For non-interacting monovacancies, the probability of forming a divacancy is propor-

tional to (C1V)2 For a coordination lattice (coordination number Z) the

1 Concentrations as number densities are given by C

1V N , when N is taken as the

number density of atoms

5.1 Pure Metals 73

equilibrium fraction of divacancies C 2V eq that form simply for statistical sons is given by Z2(C 1V eq)2 However, there is also a gain in enthalpy (and en-tropy) when two vacancies are located on adjacent lattice sites Fewer bonds

rea-to neighbouring area-toms must be broken, when a second vacancy is formed next

to an already existing one Interactions between two vacancies are accounted

for by a Gibbs free energy of binding G B

2V > 0 the interaction is attractive and binding occurs, whereas for

G B 2V < 0 it is repulsive Combining Eq (5.14) with the mass-action law for

tration (see Fig 5.2) With increasing G B

2V the equilibrium concentration ofdivacancies increases as well

The total equilibrium concentration of vacant lattice sites, C V eq, in thepresence of mono- and divacancies (neglecting higher agglomerates) is then

C V eq = C 1V eq + 2C 2V eq (5.16)For a typical monovacancy site fraction in metals of 10−4 near the melting

temperature (see below), the fraction of non-interacting divacancies would be

Fig 5.2 Arrhenius diagram of equilibrium concentrations of mono- and

divacan-cies in metals (schematic)