Kinetics of Materials - R. Balluff_ S. Allen_ W. Carter (Wiley_ 2005) Episode 5 pot

The dominant mechanism depends on a number of factors, including the crystal structure, the nature of the bonding in the host crystal, relative differences of size and electrical charge

160 CHAPTER 7 ATOMIC MODELS FOR DIFFUSION

Using the standard thermodynamic relation [ ~ G / B P ] T = V and realizing that the pressure dependence o f l n v will be relatively very small, we may write t o a good ap- proximation

(a) What is the most likely expected total displacement after a large number

(b) What is the standard deviation of the total displacement?

Solution

of diffusional jumps?

(a) The expected total displacement will be zero because there is no correlation be- tween successive jumps-after a jump the interstitial loses its memory of its jump and makes its next jump randomly into any one o f its nearest-neighbor sites (b) The distribution of displacements will be Gaussian (Eq 7.32) and the standard deviation will be the root-mean-square displacement given by Eq 7.35 as m

7.4 Suppose the random walking of a diffusant in a primitive orthorhombic crystal where the particle makes N1 jumps of length a1 along the XI axis, NZ jumps

of length a2 along the xz axis, and N3 jumps of length a3 along the 2 3

axis The three axes are orthogonal and aligned along the crystal axes of the orthorhombic unit cell and the diffusivity tensor in this axis system is

(a) Find an expression for the mean-square displacement in terms of the

numbers of jumps and jump distances

(b) Find another expression for the mean-square displacement in terms of the

three diffusivities in the diffusivity tensor and the diffusion time Your

answer should be analogous to Eq 7.35, which holds for the isotropic

where A = constant The mean-square displacement is then

7.5 Suppose a random walk occurs on a primitive cubic lattice and successive jumps are uncorrelated Show explicitly that f = 1 in Eq 7.49 Base your

argument on a detailed consideration of the values that the cosOi,i+j terms assume

Solution Because all jumps are o f the same length,

7.6 For the diffusion of vacancies on a face-centered cubic (f.c.c.) lattice with lattice constant a, let the probability of first- and second-nearest-neighbor

jumps be p and 1 - p , respectively At what value of p will the contributions

to diffusion of first- and second-nearest-neighbor jumps be the same?

Solution There is no correlation and, using Eq 7.29,

CHAPTER 8

DIFFUSION IN CRYSTALS

The driving forces necessary to induce macroscopic fluxes were introduced in Chap- ter 3 and their connection to microscopic random walks and activated processes was discussed in Chapter 7 However, for diffusion to occur, it is necessary that kinetic mechanisms be available to permit atomic transitions between adjacent locations These mechanisms are material-dependent In this chapter, diffusion mechanisms

in metallic and ionic crystals are addressed In crystals that are free of line and planar defects, diffusion mechanisms often involve a point defect, which may be charged in the case of ionic crystals and will interact with electric fields Addi- tional diffusion mechanisms that occur in crystals with dislocations, free surfaces, and grain boundaries are treated in Chapter 9

8.1 A T O M I C M E C H A N I S M S

Atom jumping in a crystal can occur by several basic mechanisms The dominant mechanism depends on a number of factors, including the crystal structure, the nature of the bonding in the host crystal, relative differences of size and electrical charge between the host and the diffusing species, and the type of crystal site pre- ferred by the diffusing species (e.g., anion or cation, substitutional or interstitial)

Kinetics of Materials By Robert W Balluffi, Samuel M Allen, and W Craig Carter 163

Copyright @ 2005 John Wiley & Sons, Inc

8.1.1 Ring Mechanism

A substitutional atom (indicated by shading in Fig 8.1) may jump and replace an adjacent nearest-neighbor substitutional atom In the rang mechanasm the substi- tutional atom exchanges places with a neighboring atom by a cooperative ringlike rotational movement

The atomic environment during a vacancy-exchange mechanism can be illus- trated in a three-dimensional cubic lattice Figure 8.3 shows an atom-vacancy exchange between two face-centered sites in an f.c.c crystal The migrating atom

( A in Fig 8.3) moves in a (110)-direction through a rectangular “window” framed

by two cube corner atoms and two opposing face-centered atoms The f.c.c crys- tal is close-packed and each site has 12 equivalent nearest-neighbor sites [l] In

Figure 8.2: Vacaiicy mecliaiiism for diffiisioii of substitutional atonis

‘Vacancies will always exist in equilibrium in a crystal because their enthalpy of formation can always be compensated by a configurational entropy increase a t finite temperatures (see the deriva- tion of Eq 3.65) Therefore, vacancies function as a component that occupies substitutional sites

and t,he vacant site (joined by the bold rectangle) form a "window" 1234 through which the

A atom must, pass The A atom is centered in unit-cell face 2356 The vacancy is centered

in unit-cell face 2378

a hard-sphere model in which nearest-neighbor atoms are in contact, the atoni must "squeeze" through a window that is about 27% smaller than its diameter The potential-energy increases required for such distortions create the energetic migration barriers discussed in Section 7.1.3

8.1.3 lnterstitialcy Mechanism

A substitutional atom can migrate to a neighboring substitutional site by the two- step process illustrated in Fig 8.4 The first step is an exchange with an interstitial defect in which the migrating substitutional atom becomes the interstitial atom.2 The second step is to exchange the migrating atom with a neighboring substitu- tional atom This mechanism is only possible when substitutional atoms can occupy interstitial sites This cooperative and serpentine motion constitutes the intersti- tialcy mechanism, and when large normally substitutional atoms are involved, can occur with a much lower migration energy than the interstitial mechanism (see below)

Interstitialcy migration depends on the geometry of the interstitial defect How- ever, an a priori prediction of interstitial defect geometry is not straightforward

in real materials For an f.c.c crystal, a variety of conceivable interstitial defect candidates are illustrated in Fig 8.5 The lowest-energy defect will be stable and predominant For example in the f.c.c metal Cu the stable configuration is the (100) split-dumbbell configuration in Fig 8.5d [ 3 ]

The (100) split-dumbbell defect in Fig 8.5d, while having the lowest energy of all interstitial defects, still has a large formation energy (Ef = 2.2 eV) because of the

large amount of distortion and ion-core repulsion required for its insertion into the close-packed Cu crystal However, once the interstitial defect is present, it persists until it migrates to an interface or dislocation or annihilates with a vacancy The 21nterstitial point defects involving normally substitutional atoms will always exist (although typically at very low concentration) a t equilibrium in a crystal a t finite temperatures because as

in the case of vacancies described above, their enthalpy of formation can always be compensated

by a configurational entropy increase

Figure 8.5: Geonietric corifiguratioris for a self-interst,itial defect atom in an f.c.c crystal:

(a) oct,ahedral site, (b) tetrahedral site, ( c ) (110) crowdion, (d) (100) split, dumbbell

(e) (111) split, (f) (110) split crowdioii [2]

activation energy for migration (Em = 0.1 eV) is small compared to Ef because little additional distortion is required for its serpentine motion, which is illustrated

in Fig 8.6 It therefore migrates relatively rapidly

durnbbells [one of which is sliowii iii (b)] atrid into four others creatirig [OlO] diirnbliells

Diffusiorial niigratioii of a [loo] split-duiiibbell self-iiiterstitial iii ail f.c c The durnbbell in (a) rail juirip int,o four iiearest-iieighbor sites: c,rwtiiig [OOl]

8.1.4 Interstitial Mechanism

An interstitial atom can simply migrate between interstitial sites as in Fig 8.7 The interstitial atom must attain enough energy to distort the host crystal as it migrates between substitutional sites This mechanism is expected for small solute atoms that normally occupy interstitial sites in a host crystal of larger atoms Diffusion by the interstitial mechanism and by the interstitialcy mechanism are quite different processes and should not be confused Diffusion by the vacancy and interstitialcy niechanisms requires the presence of point defects in the system whereas diffusion by the ring and interstitial mechanisms does not

8.1.5 Diffusion Mechanisms in Various Materials

Diffusion of relatively small atoms that normally occupy interstitial sites in the sol- vent crystal generally occurs by the interstitial mechanism For example, hydrogen atoms are small and migrate interstitially through most crystalline materials Car- bon is small compared to Fe and occupies the interstitial sites in b.c.c Fe illustrated

in Fig 8.8 and migrates between neighboring interstitial sites

Migration of atoms that occupy substitutional sites may occur through a range

of mechanisms involving either vacancy- or interstitial-type defects In f.c.c., b.c.c., and hexagonal close-packed (h.c.p.) metals, self-dzffuszon occurs predominantly

by the vacancy mechanism [4, 51 However, in some cases self-diffusion by the

X

Figure 8.8: 1nterst)itial sites for C atjoins iii b.c c Fe (a) Tlie intrrstitial sites liwe point-group syriiirietry 4/mmm, and tlie orient,at ions of tlie fourfold axes are indicxtrti by the shorter, grey spokes on tlie symbols (b) Noiiicnclat,ure uscd in t,he model for diffusion

of interstitial atonis in b.c.c Fe discussed in Section 8.2.1 Three different types of sit,es are present: sites 1 2 and 3 have nearest-neighbor Fe atonis lying along 2 y and z rcymtiwly

interstitialcy mechanism contributes a small amount to the overall diffusion (see Section 8.2.1) In Ge, which has the less closely packed diamond-cubic structure, self-diffusion occurs by a vacancy mechanism In Si (which like Ge has covalent bonding), self-diffusion occurs by the vacancy mechanism at low temperatures and

by an interstitialcy mechanism at elevated temperatures [6-81 In ionic materials, diffusion mechanisms become more complex and varied Self-diffusion of Ni in N i 0

occurs by a vacancy mechanism; in Cua0 the diffusion of 0 involves interstit,ial defects [9] In the alkali halides, vacancy defects predominate and the diffusion of both anions and cations occurs by a vacancy mechanism However: the predominant defect is not easy to predict in ionic materials For example, vacancy-interstit,ial pairs dominate in AgBr and the smaller Ag cations diffuse by an interstitialcy mechanism (see Section 8.2.2)

Solutes that normally occupy substitutional positions can migrate by a vari- ety of mechanisms In many systems they migrate by the same mechanism as for self-diffusion of the host atoms However, the details of migration become more complex if there is an interaction or binding energy between the solute atoms and point defects-this is described in Section 8.2.1 for vacancy-solute-atom binding Certain solute a t o m can migrate by more than one mechanism For example while

Au solute atoms in Si are mainly substitutional, under equilibrium conditions: a rel- atively small number of Au atoms occupy interstitial sites The rate of migration of the interstitial Au atoms is orders of magnitude faster than the ratme of the substitu- tional Au atoms, and the small population of interstit,ial Au atoms therefore makes

an important contribution to the overall solute-at,om diffusion rate [6 81 The so- lute atoms transfer from substitutional sites to interstitial sites by either kick-out or dissociative mechanisms (Fig 8.9) In the kick-out mechanism, an interstitial host atom, H I , pushes the substitutional solute atom, Ss into an interstitial position and simultaneously takes up a substitutional position according to the reaction

8.2 ATOMIC MODELS FOR DlFFUSlVlTlES 169

00

00

00 0 00-0

Figure 8.9:

interstitial site by Transfer (a) the kick-out rriechanisni arid (b) t,lw dissoriative mechanism of a soliitr at,oni (filled at,orii) from a subst,it,utional site to ail

In the dissociative mechanism, a substitutional solute atom enters an interstitial site leaving a vacancy, V , behind according to the reaction

These reactions are reversible This dual-sit,e occupancy leads to complicated solute diffusion behavior and has been described for several solut,e species in Si [4, 6, 81 There is no compelling evidence that the ring mechanism in Fig 8.1 contributes significantly to diffusion in any material

8.2 ATOMIC MODELS FOR DlFFUSlVlTlES

Atomic models for the diffusivity can be constructed when the diffusion occurs

by a specified mechanism in various crystalline materials A number of cases are

considered below

8.2.1 Metals

Diffusion o f Solute Atoms by the Interstitial Mechanism in the B.C.C Structure The general expression that connects the jump rate I? the intersite jump distance, r , and the correlation factor, Eq 7.52 then takes the form

(8.3) Because each interstitial site has four nearest-neighbors, the jump rate r is given

by 4r', where I" has the form of Eq 7.25.3 If a is the lattice constant for the b.c.c

unit cell in b.c.c Fe, then r = a12 and Eq 8.3 yields

(8.4)

,'The quantity r', introduced in Section 7.1.1, is the jump rate of an atom from one specified site

to a specified neighboring site r is the total jump rate of the atom in the material If the atom

is diffusing among equivalent sites in a crystal where each site has z equivalent nearest-neighbors, then r = zr'

where the weakly temperature-dependent terms have been collected into D,” Be- cause D; is relatively temperature independent, the Arrhenius form of Eq 8.4 indicates a thermally activated process The enthalpy of migration, H m , is the activation energy, E , for the interstitial diffusion For C in Fe, 0; = 0.004 cm2 s-l and Hm= 80.1 kJ mol-’ [lo] This experimental value of D; is consistent with the value predicted by Eq 8.4 for a = 2.9 x m, I/ = 1013 s-l, and S” = lk/atom

The relationship between jump rate and diffusivity in Eq 8.3 can be obtained by

an alternate method that considers the local concentration gradient and the number

of site-pairs that can contribute to flux across a crystal plane A concentration gradient of C along the y-axis in Fig 8.8b results in a flux of C atoms from three distinguishable types of interstitial sites in the cy plane (labeled 1, 2, and 3 in

Fig 8.8) The sites are assumed to be occupied at random with small relative populations of C atoms that can migrate between nearest-neighbor interstitial sites

If c’ is the number of C atoms in the cy plane per unit area, the carbon concentration

on each type of site is c’/3 Carbon atoms on the types 1 and 3 sites jump from

plane Q to plane at the rate ( c 1 / 3 ) I ” The jump rate from type-2 sites in plane

Q to plane p is zero The contribution to the flux from all three site types is

If c is the number of C atoms per unit volume, c = 2c’/a, and therefore

Therefore, the net flux is

Comparison of Eq 8.8 with the Fick’s law expression,

Droduces

(8.9)

(8.10) The total jump frequency for a given C atom is r = 4r’, and therefore

DI=-=- (8.11)

which is identical to Eq 8.3 The same result would have been obtained with the

cy and /3 planes chosen at any arbitrary inclination in the Fe crystal because DI is isotropic in all cubic crystals (see Exercise 4.6)

8.2: ATOMIC MODELS FOR DlFFUSlVlTlES 171

Self-Diffusion by the Vacancy Mechanism in the F.C.C Structure Each site on an f.c.c lattice has 12 nearest-neighbors, and if vacancies occupy sites randomly and have a jump frequency rv,

If the vacancies are in thermal equilibrium, XV = XEq, where, according to Eq 3.65,

(8.18) where 5'6 and H b are the vacancy vibrational entropy and enthalpy of formation, re~pectively.~ Using Eqs 8.13 and 8.15

*D = fa2v ,(SF+SC)/~,-(HF+HG)/(~T)

(8.19)

- *Doe-E/("T) where *DO fa2v e ( S F + S $ ) / k

and the activation energy is given by

Equation 8.19 contains the correlation factor, f , which in this case is not unity since the self-diffusion of tracer atoms by the vacancy mechanism involves corre- lation Correlation is present because the jumping sequence of each tracer atom produced by atom-vacancy exchanges is not a random walk This may be seen by 4The diffusion of vacancies is uncorrelated for the same reasons given above for diffusion of the interstitial atoms After each jump, a vacancy will have the possibility of jumping into any one

of its 12 nearest-neighbor sites with equal probability

5Because Gfv is the free energy to form a vacancy exclusive of the configurational mixing entropy (see Section 3.4.1), the only entropy included in Sf in the relation Gfv = Hf -TSf is the thermal vibrational entropy

considering a tracer atom immediately after a jump The vacancy with which it

just exchanged will be one of its 12 nearest-neighbors For its next jump, the tracer

atom can either jump back into the vacancy with which it has just exchanged, jump into another vacancy which happens to be present in another nearest-neighbor site,

or wait for another vacancy to arrive at a nearest-neighbor site into which it jumps Because the first possibility is most probable, the atom jumping is nonrandom The second jump is correlated with respect to the first because the first jump creates a situation (i.e., the existence of a nearest-neighbor vacancy) that biases the second jump

A rough estimate for f can be obtained based on the number of nearest-neighbors and the probability that a tracer atom which has just jumped and vacated a site will return to the vacant site on the vacancy's next jump A vacancy jumps randomly

into its nearest-neighbor sites, and the probability that the return will occur is l / z This event will then occur on average once during every z jumps of an atom For

each return jump, two atom jumps are effectively eliminated by cancellation, and the overall number of tracer-atom jumps that contribute to diffusion is reduced by

the fraction 2/25 According to Eq 8.3, *D is proportional to the product rf, and

since the number of effective jumps is reduced by 2/2, f can be assigned the value

f M 1 - 2/25 = 0.83 for f.c.c crystals More accurate calculations (see below) show that f = 0.78

To find a more accurate value of f , Eq 7.49 is applied [4, 11-13] If all displace-

ments are of equal length,

(8.21)

The quantity (cosOi,i+j) can be evaluated with the aid of the law of cosines from spherical trigonometry:

(8.22)

where a is the angle between the two planes defined by the successive jump vectors

r', and < + I and the successive jump vectors r',+1 and r'i+z For cubic crystals, contributions from angle a will be canceled by those from angle (180' - a ) on the

average Therefore, the last term in Eq 8.22 containing cosa will average to zero:

(cos 6 ' i , i + z ) = (cos &,i+l cos Q i + l , i + Z ) (8.23)

The average cosine of the angle between successive jumps must be the same for all pairs of successive jumps.6 Therefore,

(cos 6'i,i+z) = (cos6'i,i+l cos 6'i+1,i+2) + (sin6'i,i+l sin 6'i+l,i+2 cos a )

( c o s ~ ~ , ~ + ~ ) = ( C O S B ~ , ~ + ~ ) ( C ~ S ~ ~ + ~ ~ + ~ ) = ( c ~ s ~ ) ~ (8.24)

where cos 6' denotes the angle between successive jumps By induction,

3

(8.25)

(COS ei,i+3) = (COS ei,i+2) (COS ei+2,i+3) = (COS e)

(COS ei,i+j) = (COS e ) j

6The immediate surroundings after each jump must be the same (excepting a change of orientation

of the vacancy-atom pair); consisting of the atom with a nearest-neighbor vacancy next t o it The average of what happens t o produce the next jump must be the same for all pairs of jumps (in their respective orientations)

8.2: ATOMIC MODELS FOR DlFFUSlVlTlES 173

Putting Eq 8.25 into Eq 8.21 yields

on the accuracy with which (cos8) can be determined

Equation 8.29 can be used to obtain another approximation for f by employing

an estimated value of (cos8) To estimate this quantity, consider an atom that has just exchanged with a vacancy; there is a probability l / z that the vacancy’s next exchange will be with the same atom and, therefore, the probability that cos8 = -1 is l / z or (cos8) = -l/z If the vacancy separates from the particular atom, that particular atom cannot migrate until it obtains another (or the same) vacancy as a neighbor; the contribution to (cos 8) from these displacements will be small compared to -l/z, and therefore Eq 8.29 can be written

1 - l / z 2 - 1

l + l / z - z + 1

and f = 0.85 for f.c.c crystals, which is close to the previous estimate

An accurate determination of f can be obtained by considering all contributing vacancy trajectories to determine (cos8) by use of Eq 8.29 [13] For f.c.c., the accurate value o f f is found to be 0.78; thus, correlations affect the diffusivity value

by about 22% in Eq 7.52.7 Correlations can have a considerably larger effect on the diffusivity for substitutional solute atoms by the vacancy mechanism

For the vacancy self-diffusion mechanism in many metals, experimental values of

*Do are approximately 0.1-1.0 cm2 s-l, which correspond to physically reasonable values of the quantities in *Do according to Eq 8.19: f x 1, a x 3.5 x lo-’’ m,

v x 1013 s-l, and (S& + SF) x 2k In metals, as in many classes of materials, the

7A calculation of f in a two-dimensional lattice that takes into account multiple return vacancy trajectories appears in Exercise 8.8

activation energy for vacancy self-diffusion, E in Eq 8.20, scales with the melting temperature because the crystal binding energy correlates with both melting tem- perature and vacancy formation energy Activation energies are typically 0.5-6 eV (1 eV = 96.46 kJ mol-l) and *Do is often nearly the same for materials with the same crystal structure and bonding [4]

Vacancy formation and migration energies, such as H; and H F , have been obtained by independent experiments For example, to obtain H,, equilibrium vacancy concentrations can be measured from simultaneous thermal expansion and lattice expansion during quasi-equilibrium heating [14] and by positron annihila- tion [15] The vacancy migration rate can be determined by measuring the decay of

a supersaturated population of quenched-in vacancies to their equilibrium popula- tion in order to measure H F [16] The results of these independent determinations are generally consistent with the measured values of the substitutional diffusivity activation energies inferred from Eq 8.19 [4]

In Section 3.1.1, self-diffusion was analyzed by studying the diffusion of radioac- tive tracer atoms, which were isotopes of the inert host atoms, thereby eliminating any chemical differences Possible effects of a small difference between the masses

of the two species were not considered However, this difference has been found

to have a small effect, which is known as the isotope effect Differences in atomic masses result in differences of atomic vibrational frequencies, and as a result, the heavier isotope generally diffuses more slowly than the lighter This effect can-if migration is approximated as a single-particle process-be predicted from the mass differences and Eq 7.14 If ml and m2 are the atomic masses of two isotopes of the same component, Eqs 7.13 and 7.52 predict the jump-rate ratio,

f

Jump rate and diffusivity scale inversely with the square root of atomic mass How- ever, if migration involves many-body effects and collective motion, the assumptions leading to Eq 8.31 are no longer valid and this model must be discarded

Diffusion of Solute Atoms by Vacancy Mechanism in Close-Packed Structure Diffu- sion of substitutional solutes in dilute solution by the vacancy mechanism is more complex than self-diffusion because the vacancies may interact with the solute atoms and no longer be randomly distributed If the vacancies are attracted to the solute atoms, any resulting association will strongly affect the solute-atom diffusivity The effect is demonstrated in a simple manner in two dimensions in Fig 8.10, which shows an isolated solute atom with a vacancy occupying a nearest-neighbor site [4] Three jump frequencies are considered: the intrinsic host-vacancy jump rate, the solute-vacancy jump rate, and the jump rate for a vacancy and a host atom that also has a neighboring solute, When there is an attractive interaction between a solute atom and the vacancy (a negative binding energy), is decreased because of the increase in activation energy of the jump due to the binding energy between the vacancy and the solute atom The activation energies for the remaining two types of exchange are not influenced by the binding energy, and two extremes can be considered:

a 2 ATOMIC MODELS FOR DIFFUSIVITIES 175

Figure 8.10: A solute atoiri (darker shading) with a nearest-neighbor vacancy iii a close- packed at,oniic plnric 'I'hc vacancy itlid its three different nearest-neighbor types exchange places with differiiig j i u r i p freqiiencies

Case A is an example of strong correlation Since the jump rate rlfsSv is relatively small, the vacancy remains bound to the solute atom for relatively long periods and the solute atom and bound vacancy exchange positions repeatedly at the rate

rkSv, which is relatively high in comparison to rhos However, eventually the vacancy will exchange with a host atom that is a nearest-neighbor of the solute atom (at the rate and a new mode of oscillation of the solute atom is established with the bound vacancy in a new nearest-neighbor site This allows the solute atom

to occupy a new site outside the first oscillating mode If this occurs repeatedly, the solute atom can occupy new sites and execute long-range migration by a sort

of tumbling motion of the oscillating mode's axis The effective jump frequency of the solute atom during the period when the vacancy is bound to the solute atom

is then and the self-diffusivity of the solute atom during the time that the vacancy is bound to it can be written as

at equilibrium is

$(bound) = e-GG/(kT)Xeq V ( free) (8.33) where Gb, is the binding energy (negative when attractive) of the vacancy to the solute atom and Xbq(free) is the fraction of free vacancies in the bulk crystal The number of bound vacancies (per unit volume) in a system where the solute concentration is cs is therefore 12csp7( bound), and since 12p7 (bound) << 1, the probability of finding more than one vacancy bound to a solute atom at any time

is very small The fraction of solute atoms with a bound vacancy is then approx- imately 12p"vqbound) Over a long period of time, the fraction of time that any solute atom has a vacancy bound to it is then also given by 12p7(bound) The effective jump rate of the solute during this long time period is therefore lower than

the jump rate when a bound vacancy is present by the factor 12py(bound), and putting this result into Eq 8.32 and using Eqs 3.63 and 8.33, the solute diffusivity over a long period of time, including many H f V jumps, is

(8.34) The self-diffusivity of solute atoms is then proportional to the rate at which bound vacancies circulate around them rather than the rate at which they exchange with vacancies

In Case B, vacancies are again bound to solute atoms for relatively long periods

of time However, a bound vacancy will spend most of its time circling around a stationary solute atom by making a large number of V O S jumps, although it will occasionally make an S S V jump, allowing the solute atom to occupy a new site Repetition of this process leads to long-range migration of the solute atom Using

an analysis similar to the above, the solute self-diffusivity is then

DifFusion of Self-Interstitial Imperfections by the lnterstitialcy Mechanism in the F C C

Structure For f.c.c copper, self-interstitials have the (100) split-dumbbell configu- ration shown in Fig 8.5d and migrate by the interstitialcy mechanism illustrated in Fig 8.6 The jumping is uncorrelated,8 (f = l), and a / f i is the nearest-neighbor distance, so

These defects will always be present at thermal equilibrium, but their concentra- tions will be very small because of their high energy of formation They can also

be created by nonequilibrium processes such as irradiation [3]

Self-Diffusion by the lnterstitialcy Mechanism If their formation energy is not too large, the equilibrium population of self-interstitials may be large enough to con- tribute to the self-diffusivity In this case, the self-diffusivity is similar to that for self-diffusion via the vacancy mechanism (Eq 8.19) with the vacancy formation and migration energies replaced by corresponding self-interstitial quantities The 8After each jump the (100) dumbbell has an equal probability of making any of eight different jumps Its next jump is therefore made at random

8 2 ATOMIC MODELS FOR DIFFUSIVITIES 177

correlation factor f-similar to self-diffusion by the vacancy mechanism-is less than unity because the atom jumps produced by the interstitialcy mechanism are correlated

For example, the (100) split-dumbbell configuration of the self-interstitial defect

in Cu has a formation energy that is considerably larger than the vacancy formation energy [3] However, the relatively small population of equilibrium self-interstitials may contribute significantly to the self-diffusivity because the activation energy for interstitial migration is considerably lower than that for vacancy migration (as described in Section 8.1.3)

8.2.2 Ionic Solids

Diffusion in ionically bonded solids is more complicated than in metals because site defects are generally electrically charged Electric neutrality requires that point de- fects form as neutral complexes of charged site defects Therefore, diffusion always involves more than one charged speciesg The point-defect population depends sen- sitively on stoichiometry; for example, the high-temperature oxide semiconductors have diffusivities and conductivities that are strongly regulated by the stoichiom- etry The introduction of extrinsic aliovalent solute atoms can be used to fix the low-temperature population of point defects

Intrinsic Crystal Self- DifFusion A simple example of intrinsic self-diffusion in an ionic material is pure stoichiometric KC1, illustrated in Fig 8 1 1 ~ As in many al- kali halides, the predominant point defects are cation and anion vacancy complexes (Schottky defects), and therefore self-diffusion takes place by a vacancy mechanism For stoichiometric KC1, the anion and cation vacancies are created in equal num- bers because of the electroneutrality condition These vacancies can be created

Figure 8.11: (a) Rocksalt st,riictiire of KCI and AgHr with (100) planes delineated (b) Schottky defect on t~ (100) plaiie in KC1 coriiposed of anion vacancy arid cittion va.caricy

( c ) Freiikel defect on a (100) plaiie in AgHr composed of cation self-iriterstitirtl and cation vacancy

'For general discussions, see Kingery et al [17] or Chiang et, al [ls]

by removing K+ and C1- ions from the bulk and placing them at an interface or dislocation, or on a surface ledge as illustrated in Fig 8.12 A vacancy on a K+

cation site will have an effective negative electronic charge and a vacancy on a C1- site will have a corresponding effective positive charge This defect creation is a reaction written in Kroger-Vink notation as

KG + Cl& = V& + Vb1 + KZ + Cl&

null = Vl, + V&

In the Kroger-Vink notation used here, the subscript indicates the type of site the species occupies and the superscript indicates the excess effective charge associated with the species in that site [17] A positive unit of charge (equal in magnitude to the electron charge) is indicated by a dot ( 0 ) superscript, a corresponding negative charge by a prime (') superscript, and zero charge (a neutral situation) by a times ( x ) superscript

The equilibrium constant, Keq, for Eq 8.37 is related to the free energy of formation, G$, of the Schottky pair

or

(8.39) where the a's are the activities of the anion and cation vacancies For dilute con- centrations of the vacancies, activities are equal to their site fractions by Raoult's law,

where the square brackets indicate a site fraction Equation 8.40 is a general mass- action law for the combined anion and cation vacancy site fractions Furthermore, electrical neutrality requires that

Combining Eqs 8.40 and 8.41,

(8.41)

(8.42) The vacancy populations enter the expressions for the self-diffusivity of the K+ cations and C1- anions Starting with Eq 7.52 and using the method that led to

Eq 8.19 for vacancy self-diffusion in a metal,

(8.43)

Figure 8.12:

crystal sites to ledges a t the surface

Creation of Schottky defect by transfer of anion and cation from regular

8 2 ATOMIC MODELS FOR DlFFUSlVlTlES 179

where g is a geometrical factor and the correlation factor, f , has a value slightly less than unity The activation energy for the self-diffusion is therefore

A similar expression applies to C1 self-diffusion on the anion sublattice

Self-diffusion of Ag cations in the silver halides involves Frenkel defects (equal numbers of vacancies and interstitials as seen in Fig 8.11b) In a manner sim- ilar to the Schottky defects, their equilibrium population density appears in the diffusivity Both types of sites in the Frenkel complex-vacancy and interstitial- may contribute to the diffusion However, for AgBr, experimental data indicate that cation diffusion by the interstitialcy mechanism is dominant [4] The cation Frenkel pair formation reaction is

The activity of Ag;, is unity and, therefore,

The electrical neutrality condition constrains the two site fractions:

(8.45)

(8.46)

(8.47) The activation energy for self-diffusivity of the Ag cations by the interstitialcy mechanisms is the sum of one-half the Frenkel defect formation enthalpy and the activation enthalpy for migration,

E = H F + - H i

Extrinsic Crystal Self-Diffusion Charged point defects can be induced to form in an ionic solid by the addition of substitutional cations or anions with charges that differ from those in the host crystal Electrical neutrality demands that each addition results in the formation of defects of opposite charge that can contribute to the diffusivity or electronic conductivity The addition of aliovalent solute (impurity) atoms to an initially pure ionic solid therefore creates extrinsic defects.'O

For example, the self-diffusivity of K in KC1 depends on the population of both extrinsic and intrinsic cation-site vacancies Extrinsic cation-site vacancies can be created by incorporation of Ca++ by doping KC1 with CaCl2 and can be considered

a two-step process First, two cation vacancies and two anion vacancies form as illustrated in Fig 8.12.'' Second, the single Ca++ cation and two C1 anions from CaC12 are inserted into the cation and anion vacancies, respectively; electric neu- trality requires that each substitutional divalent cation impurity in KC1 be balanced 10Eztrinsic has the same meaning as in doped semiconductors

llThis process involves creation of additional sites in the crystal Cation and anion sites must be created in the same proportion as the ratio of cation to anion sites in the host crystal-in this case, 1:l These defects can also be formed at point-defect sources such as dislocations and grain boundaries (see Sections 11.4 and 13.4)

by the formation of a cation vacancy The cationic and anionic vacancy populations are related to the site fraction of the extrinsic Ca++ impurity,

The mass-action relationship in Eq 8.40 for the product of the cation and anion vacancy site fractions combined with Eq 8.49 yields

(8.50) The last term on the right-hand side of Eq 8.50 is the square of the cation vacancy site fraction in pure (intrinsic) KC1 Solving the quadratic equation for the cation vacancy site fraction yields

(8.51)

There are two limiting cases for the behavior of [V’,] according to Eq 8.51: Intrinsic: [V’,]pure >> [Cak], then [V’,] = [V’,]pure

Extrinsic: [V’,]pure << [Cak], then [V’,] = [Cak]

The intrinsic case applies at small doping levels or at high temperatures where the thermal equilibrium site fraction of the intrinsic cation vacancy population exceeds that due to the aliovalent solute atoms In this case, the effect of the added solute atoms is negligible The activation energy for cation self-diffusion is therefore the same as in the pure material and is given by Eq 8.44

The extrinsic case applies at low temperatures or large doping levels The site fraction of cation vacancies is equal to the solute-atom site-fraction and is therefore temperature independent In the extrinsic regime, no thermal defect formation is necessary for cation self-diffusion and the activation energy consists only of the activation energy for cation vacancy migration

The expected Arrhenius plot for cation self-diffusion in KC1 doped with Ca++

is shown in Fig 8.13 The two-part curve reflects the intrinsic behavior at high temperatures and extrinsic behavior at low temperatures

range

1 IT

Figure 8.13:

doped with Ca++ The intrinsic and extrinsic ranges have different activation energies

Arrhenius plot for self-diffusivity on the cation sublattice, *DK, in KC1

8 2 ATOMIC MODELS FOR DIFFUSIVITIES 181

Crystal Self- D i f h i o n in Nonstoichiometric Materials Nonstoichiometry of semicon- ductor oxides can be induced by the material's environment For example, materials such as FeO (illustrated in Fig 8.14), N O , and COO can be made metal-deficient (or 0-rich) in oxidizing environments and Ti02 and ZrOz can be made 0-deficient under reducing conditions These induced stoichiometric variations cause large changes in point-defect concentrations and therefore affect diffusivities and electri- cal conductivities

In pure FeO, the point defects are primarily Schottky defects that satisfy mass- action and equilibrium relationships similar to those given in Eqs 8.39 and 8.42 When FeO is oxidized through the reaction

X

2 each 0 atom takes two electrons from two Fe++ ions, as illustrated in Fig 8 1 4 ~ ~ according t o the reactions

Electrical neutrality requires that a cation vacancy be created for every 0 atom added, as in Fig 8.14b; this, combined with site conservation, becomes

(8.55)

1

2 2Fege + - 0 2 = 2Febe + 0; + V:e

which can be written in terms of holes, h, in the valence band created by the loss

of an electron from an Fe++ ion producing an Fe+ft ion,

1

hFe = FeFe - Fege Equation 8.56 predicts a relationship between the cation vacancy site fraction and the oxygen gas pressure The equilibrium constant for this reaction is important for oxygen-sensing materials:

The cation self-diffusivity due to the vacancy mechanism varies as the one-sixth

power of the oxygen pressure at constant temperature and the activation energy is

(8.60) The dominance of oxidation-induced vacancies creates an additional behavior regime The effect of this additional regime on diffusivity behavior is illustrated

in Fig 8.15 Other types of environmental effects create defects through other mechanisms and may lead to other behavior regimes