Diffusion Solids Fundamentals Diffusion Controlled Solid State Episode 1 Part 5 pps

Ed., Atomic Defects in Metals, Landolt-B¨ornstein, New ries, Group III: Crystal and Solid State Physics, Vol.. 22: Semiconductors, Subvolume B: Impurities and Defects in Group IV Element

5.4 Intermetallics 87

in Chap 20 Some intermetallics are ordered up to their melting ture, others undergo order-disorder transitions in which an almost randomarrangement of atoms is favoured at high temperatures Such transitions

tempera-occur, for example, between the β  and β phases of the Cu–Zn system or

in Fe–Co There are intermetallic phases with wide phase fields and otherswhich exist as stoichiometric compounds Examples for both types can even

be found in the same binary alloy system For example, the Laves phase inthe Co-Nb system (approximate composition Co2Nb) exists over a composi-tion range of about 5 at %, whereas the phase Co7Nb2 is a line compound.Some intermetallics occur for certain stoichiometric compositions only Oth-ers are observed for off-stoichiometric compositions Some phases compensateoff-stoichiometry by vacancies, others by antisite atoms

Thermal defect populations in intermetallics can be rather complex and

we shall confine ourselves to a few remarks Intermetallic compounds arephysically very different from the ionic compounds considered in the previoussection Combination of various types of disorder are conceivable: vacanciesand/or antisite defects on both sublattices can form in some intermetallics

As self-interstitials play no rˆole in thermal equilibrium for pure metals, it isreasonable to assume that this holds true also for intermetallics

To be specific, let us suppose a formula AxBy for the stoichiometric pound and that there is a single A sublattice and a single B sublattice This

com-is, for example, the case in intermetallics with the B2 and L12 structure (seeFig 20.1) The basic structural elements of disorder are listed in Table 5.3

A first theoretical model for thermal disorder in a binary AB lic with two sublattices was treated in the pioneering work of Wagner andSchottky [2] Some of the more recent work on defect properties of inter-metallic compounds has been reviewed by Chang and Neumann [42] andBakker[43]

intermetal-In some binary AB intermetallics so-called triple defect disorder occurs These intermetallics form VA defects on the A sublattice on the B rich side

and AB antisites on the B sublattice on the A rich side of the stoichiometriccomposition This is, for example, the case for some intermetallics with B2structure where A = Ni, Co, Pd and B = Al, In, Some other inter-metallics also with B2 structure such as CuZn, AgCd, can maintain highconcentrations of vacancies on both sublattices

Table 5.3 Elements of disorder in intermetallic compounds

88 5 Point Defects in Crystals

Triple defects (2V A + AB ), bound triple defects (VA A B V A) and vacancy pairs (V A V B) have been suggested by Stolwijk et al [46] They can form

according to the reactions

The physical understanding of the defect structure of intermetallics isstill less complete compared with metallic elements However, considerableprogress has been achieved Differential dilatometry (DD) and positron an-nihilation studies (PAS) performed on intermetallics of the Fe-Al, Ni-Al andFe-Si systems have demonstrated that the total content of vacancy-type de-fects can be one to two orders of magnitude higher than in pure metals [44,45] The defect content depends strongly on composition and its temperaturedependence can show deviations from simple Arrhenius behaviour According

to Schaefer et al [44] and Hehenkamp [45] typical defect concentrations

in these compounds near the solidus temperature can be as high as severalpercent

Because Si and Ge crystallise in the diamond structure with coordinationnumber 4, the packing density is considerably lower than in metals Thisholds true also for compound semiconductors Most compound semiconduc-tors formed by group III and group V elements like GaAs crystallise in thezinc blende structure, which is closely related to the diamond structure Semi-conductor crystals offer more space for self-interstitials than close-packedmetal structures Formation enthalpies of vacancies and self-interstitials insemiconductors are comparable In Si, both self-interstitials and vacanciesare present in thermal equilibrium and are important for self- and solute dif-fusion In Ge, vacancies dominate in thermal equilibrium and appear to bethe only diffusion-relevant defects (see Chap 23 and [47, 50])

5.5 Semiconductors 89Semiconductors have in common that the thermal defect concentrationsare orders of magnitude lower than in metals or ionic crystals This is a con-sequence of the covalent bonding of semiconductors Defect formation ener-gies in semiconductors are higher than in metals with comparable meltingtemperatures Neither thermal expansion measurements nor positron annihi-lation studies have sufficient accuracy to detect the very low thermal defectconcentrations.

Point defects in semiconductors can be neutral and can occur in variouselectronic states This is because point defects introduce energy levels into theband gap of a semiconductor Whether a defect is neutral or ionised depends

on the position of the Fermi level as illustrated schematically in Fig 5.10

A wealth of detailed information about the electronic states of point defects

in these materials has been obtained by a variety of spectroscopic means andhas been compiled, e.g., by Schulz [14]

Let us consider vacancies and self-interstitials X ∈ (V, I) and suppose that both occur in various ionised states, which we denote by j ∈ (0, 1±, 2±, ) The total concentration of the defect X at thermal equilibrium can be written

as

C X eq = C X eq0 + C X eq1++ C X eq 1− + C X eq2++ C X eq 2− + (5.42)Whereas the equilibrium concentration of uncharged defects depends only

on temperature (and pressure), the concentration of charged defects is ditionally influenced by the position of the Fermi energy and hence by the

ad-doping level If the Fermi level changes due to, e.g., background ad-doping the

concentration of charged defects will change as well

The densities of electrons, n, and of holes, p, are tied to the intrinsic carrier density, ni, via the law of mass action relation

Fig 5.10 Electronic structure of semiconductors, with a defect with double

ac-ceptor character (left) and donor character (right)

90 5 Point Defects in Crystals

Then, Eq (5.42) can be rewritten as

neg-Furthermore, the ratio n/nivaries with temperature because the intrinsiccarrier density according to

den-of n i at different temperatures are determined mainly by the band gap

en-ergy E g of the semiconductor For a given background doping concentration

the ratio n/ni will be large at low temperatures and approaches unity athigh temperatures Then, the semiconductor reaches intrinsic conditions Theband gap energy is characteristic for a given semiconductor It increases inthe sequence Ge (0.67 eV), Si (1.14 eV), GaAs (1.43 eV) The intrinsic carrierdensity at a fixed temperature is highest for Ge and lowest for GaAs Thus,doping effects on the concentration of charged defects are most prominentfor GaAs and less pronounced for the elemental semiconductors

Let us consider as an example a defect X which introduces a single X1− and a double X2− acceptor state with energy levels E

X 1− and E X 2−above thevalence band edge Then, the ratios between charged and uncharged defectpopulations in thermal equilibrium are given by

where Ef denotes the position of the Fermi level The degeneracy factors

g X 1− and g X 2− take into account the spin degeneracy of the defect and thedegeneracy of the valence band The total concentration of point defects inthermal equilibrium for the present example is given by

References 91Diffusion in semiconductors is affected by doping since defects in variouscharge states can act as diffusion-vehicles Diffusion experiments are usually

carried out at temperatures between the melting temperature Tmand about

0.6 T m As the intrinsic carrier density increases with increasing temperature,doping effects in diffusion are more pronounced at the low temperature end

of this interval One can distinguish two types of doping effects:

– Background doping is due to a homogeneous distribution of donor or

ac-ceptor atoms, that are introduced during the process of crystal growing.Background doping is relevant for diffusion experiments, when at the dif-fusion temperature the carrier density exceeds the intrinsic density.– Self-doping is relevant for diffusion experiments of donor or acceptor ele-

ments If the in-diffused dopant concentration exceeds either the intrinsiccarrier density or the available background doping, complex diffusion pro-files can arise

References

1 J.I Frenkel, Z Physik 35, 652 (1926)

2 C Wagner, W Schottky, Z Physik Chem B 11, 163 (1931)

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

4 A.M Stoneham, Theory of Defects in Solids, Clarendon Press, Oxford, 1975

5 F Agullo-Lopez, C.R.A Catlow, P Townsend, Point Defects in Materials,

Academic Press, London, 1988

6 H.J Wollenberger, Point Defects, in: Physical Metallurgy, R.W Cahn, P.

Haasen (Eds.), North-Holland Publishing Company, 1983, p 1139

7 L.W Barr, A.B Lidiard, Defects in Ionic Crystals, in: Physical Chemistry –

An Advanced Treatise, Academic Press, New York, Vol X, 1970

8 R.G Fuller, Ionic Conductivity including Self-diffusion, in: Point Defects in

Solids, J M Crawford Jr., L.M Slifkin (Eds.), Plenum Press, 1972, p 103

9 A Seeger, D Schumacher, W Schilling, J Diehl (Eds.), Vacancies and

Inter-stitials in Metals, North-Holland Publishing Company, Amsterdam, 1970

10 N.L Peterson, R.W Siegel (Eds.), Properties of Atomic Defects in Metals,

North-Holland Publishing Company, Amsterdam, 1978

11 C Abromeit, H Wollenberger (Eds.), Vacancies and Interstitials in Metals and

Alloys; also: Materials Science Forum 15–18, 1987

12 O Kanert, J.-M Spaeth (Eds.), Defects in Insulating Materials, World

Scien-tific Publ Comp., Ltd., Singapore, 1993

13 H Ullmaier (Vol Ed.), Atomic Defects in Metals, Landolt-B¨ornstein, New ries, Group III: Crystal and Solid State Physics, Vol 25, Springer-Verlag, Berlinand Heidelberg, 1991

Se-14 M Schulz, Landolt-B¨ornstein, New Series, Group III: Crystal and Solid State

Physics, Vol 22: Semiconductors, Subvolume B: Impurities and Defects in

Group IV Elements and III-V Compounds, M Schulz (Ed.), Springer-Verlag,

1989

92 5 Point Defects in Crystals

15 N.A Stolwijk, Landolt-B¨ornstein, New Series, Group III: Crystal and Solid

State Physics, Vol 22: Semiconductors, Subvolume B: Impurities and Defects

in Group IV Elements and III-V Compounds, M Schulz (Ed.), Springer-Verlag,

1989, p 439

16 N.A Stolwijk, H Bracht, Landolt-B¨ornstein, New Series, Group III: Condensed

Matter, Vol 41: Semiconductors, Subvolume A2: Impurities and Defects in

Group IV Elements, IV-IV and III-V Compounds, M Schulz (Ed.),

Springer-Verlag, 2002, p 382

17 H Bracht, N.A Stolwijk, Landolt-B¨ornstein, New Series, Group III: Condensed

Matter, Vol 41: Semiconductors, Subvolume A2: Impurities and Defects in

Group IV Elements, IV-IV and III-V Compounds, M Schulz (Ed.),

Springer-Verlag, 2002, p 77

18 F Beni`ere, Diffusion in Alkali and Alkaline Earth Halides, in: Diffusion

in Semiconductors and Non-metallic Solids, Landolt-B¨ornstein, New Series,Group III: Condensed Matter, Vol 33, Subvolume B1, D.L Beke (Vol.Ed.).Springer-Verlag, 1999

19 G Erdelyi, Diffusion in Miscelaneous Ionic Materials, in: Diffusion in

Semi-conductors and Non-metallic Solids, Landolt-B¨ornstein, New Series, Group III:Condensed Matter, Vol 33, Subvolume B1, D.L Beke (Vol.Ed.) Springer-Verlag, 1999

20 A Seeger, H Mehrer, Analysis of Self-diffusion and Equilibrium Measurements, in: Vacancies and Interstitials in Metals, A Seeger, D Schumacher, J Diehl

and W Schilling (Eds.), North-Holland Publishing Company, Amsterdam,

1970, p 1

21 R Feder, A.S Nowick, Phys Rev 109, 1959 (1958)

22 R.O Simmons, R.W Balluffi, Phys Rev 117, 52 (1960); Phys Rev 119, 600 (1960); Phys Rev 129, 1533 (1963); Phys Rev 125, 862 (1962)

23 R.O Simmons, R.W Balluffi, Phys Rev 125, 862 (1962)

24 A Seeger, J Phys F Metal Phys 3, 248 1973)

25 R.O Simmons, R.W Balluffi, Phys Rev 117, 52 (1960)

26 G Bianchi, D Mallejac, Ch Janot, G Champier, Compt Rend Acad

Science, Paris 263 1404 (1966)

27 B von Guerard, H Peisl, R Sizmann, Appl Phys 3, 37 (1973)

28 H.-E Schaefer, R Gugelmeier, M Schmolz, A Seeger, J Materials Science

Forum 15–18, 111 (1987)

29 M Doyama, R.R Hasiguti, Cryst Lattice Defects 4, 139 (1973)

30 P Hautoj¨arvi, Materials Science Forum 15–18, 81 (1987)

31 H.-E Schaefer, W Stuck, F Banhart, W Bauer, Materials Science Forum

15–18, 117 (1987)

32 W.M Lomer, Vacancies and other Point Defects in Metals and Alloys,

Institute of Metals, 1958

33 J E Dorn, J.B Mitchell, Acta Metall 14, 71 (1966)

34 G Berces, I Kovacs, Philos Mag 15, 883 (1983)

35 C Tubandt, E Lorenz, Z Phys Chem 87, 513, 543 (1914)

36 J Maier, Physical Chemistry of Ionic Solids – Ions and Electrons in Solids,

John Wiley & Sons, Ltd, 2004

37 A.R Allnatt, A.B Lidiard, Atomic Transport in Solids, Cambridge University

Press, 1993

38 R Friauf, Phys Rev 105, 843 (1957)

References 93

39 R Friauf, J Appl Phys Supp 33, 494 (1962)

40 J.H Westbrook, Structural Intermetallics, R Dariola, J.J Lewendowski,

C.T Liu, P.L Martin, D.B Miracle, M.V Nathal (Eds.), Warrendale, PA,TMS, 1993

41 G Sauthoff, Intermetallics, VCH Verlagsgesellschaft, Weinheim, 1995

42 Y.A Chang, J.P Neumann, Progr Solid State Chem 14, 221 (1982)

43 H Bakker, Tracer Diffusion in Concentrated Alloys, in: Diffusion in Crystalline

Solids, G.E Murch, A.S Nowick (Eds.), Academic Press, Inc., 1984, p 189

44 H.-E Schaefer, K Badura-Gergen, Defect and Diffusion Forum 143–147, 193

(1997)

45 Th Hehenkamp, J Phys Chem Solids 55, 907 (1994)

46 N.A Stolwijk, M van Gend, H Bakker, Philos Mag A 42, 783 (1980)

47 W Frank, U G¨osele, H Mehrer, A Seeger, Diffusion in Silicon and

Germa-nium, in: Diffusion in Crystalline Solids, G.E Murch, A.S Nowick (Eds.),

Academic Press, Inc., 1984, p 64

48 K Girgis, Structure of Intermetallic Compounds, in: Physical Metallurgy,

R.W Cahn, P Haasen (Eds.), North-Holland Physics Publishing, 1983, p 219

49 M Schulz (Vol.-Ed.), Impurities and Defects in Group IV Elements, IV-IV

and III-V Compounds, Landolt-B¨ornstein, Group III, Vol 41, Subvolume A2,Springer-Verlag, 2002

50 T.Y Tan, U G¨osele, Diffusion in Semiconductors, in: Diffusion in

Con-densed Matter – Methods, Materials, Models, P Heitjans, J K¨arger (Eds.),Springer-Verlag, 2005

6 Diffusion Mechanisms

Any theory of atom diffusion in solids should start with a discussion of fusion mechanisms We must answer the question: ‘How does this particularatom move from here to there?’ In crystalline solids, it is possible to describediffusion mechanisms in simple terms The crystal lattice restricts the posi-tions and the migration paths of atoms and allows a simple description ofeach specific atom displacements This contrasts with a gas, where randomdistribution and displacements of atoms are assumed, and with liquids andamorphous solids, which are neither really random nor really ordered

dif-In this chapter, we catalogue some basic atomic mechanisms which giverise to diffusion in solids As discussed in Chap 4, the hopping motion ofatoms is an universal feature of diffusion processes in solids Furthermore, wehave seen that the diffusivity is determined by jump rates and jump distances.The detailed features of the atomic jump process depend on various factorssuch as crystal structure, size and chemical nature of the diffusing atom, andwhether diffusion is mediated by defects or not In some cases atomic jumpprocesses are completely random, in others correlation between subsequentjumps is involved Correlation effects are important whenever the atomicjump probabilities depend on the direction of the previous atom jump Ifjumps are mediated by atomic defects, correlation effects always arise Thepresent chapter thus provides also the basis for a discussion of correlationeffects in solid-state diffusion in Chap 7

6.1 Interstitial Mechanism

Solute atoms which are considerably smaller than the solvent atoms are corporated on interstitial sites of the host lattice thus forming an interstitialsolid solution Interstitial sites are defined by the geometry of the host lat-tice In fcc and bcc lattices, for example, interstitial solutes occupy octahedraland/or tetrahedral interstitial sites (Fig 6.1) An interstitial solute can diffuse

in-by jumping from one interstitial site to one of its neighbouring sites as shown

in Fig 6.2 Then the solute is said to diffuse by an interstitial mechanism

To look at this process more closely, we consider the atomic movementsduring a jump The interstitial starts from an equilibrium position, reachesthe saddle-point configuration where maximum lattice straining occurs, and

96 6 Diffusion Mechanisms

Fig 6.1 Octahedral and tetrahedral interstitial sites in the bcc (left) and fcc

(right) lattice

Fig 6.2 Direct interstitial mechanism of diffusion

settles again on an adjacent interstitial site In the saddle-point configurationneighbouring matrix atoms must move aside to let the solute atom through.When the jump is completed, no permanent displacment of the matrix atomsremains Conceptually, this is the simplest diffusion mechanism It is also de-

noted as the direct interstitial mechanism It has to be distinguished from the interstitialcy mechanism discussed below, which is also denoted as the indi- rect interstitial mechanism We note that no defect is necessary to mediate

direct interstitial jumps, no defect concentration term enters the ity and no defect formation energy contributes to the activation energy ofdiffusion Since the interstitial atom does not need to ‘wait’ for a defect toperform a jump, diffusion coefficients for atoms migrating by the direct in-terstitial mechanism tend to be fairly high This mechanism is relevant fordiffusion of small foreign atoms such as H, C, N, and O in metals and othermaterials Small atoms fit in interstitial sites and in jumping do not greatlydisplace the solvent atoms from their normal lattice sites

occurs by a direct exchange of neighbouring atoms (Fig 6.3), in which

two atoms move simultaneously In a close-packed lattice this mechanismrequires large distortions to squeeze the atoms through This entails a highactivation barrier and makes this process energetically unfavourable Theoret-ical calculations of the activation enthalpy for self-diffusion of Cu performed

by Huntington et al in the 1940s [1, 2], which were confirmed later bymore sophisticated theoretical approaches, led to the conclusion that directexchange at least in close-packed structures was not a likely mechanism

The so-called ring mechanism of diffusion was proposed for crystalline

solids by the American metallurgist Jeffries [3] already in the 1920s andadvocated by Zener in the 1950s [4] The ring mechanism corresponds to

a rotation of 3 (or more) atoms as a group by one atom distance The requiredlattice distortions are not as great as in a direct exchange Ring versions ofatomic exchanges have lower activation energies but increase the amount ofcollective atomic motion, which makes this more complex mechanism unlikelyfor most crystalline substances

Direct exchange and ring mechanisms have in common that lattice

de-fects are not involved The observation of the so-called Kirkendall effect in

alloys by Kirkendall and coworkers [5, 6] during the 1940s had an portant impact on the field (see also Chaps 1 and 10) The Kirkendall effectshowed that the self-diffusivities of atoms in a substitutional binary alloy dif-fuse at different rates Neither the direct exchange nor the ring mechanismcan explain this observation As a consequence, the ideas of direct or ringexchanges were abandoned in the diffusion literature It became evident that

im-98 6 Diffusion Mechanisms

Fig 6.4 Atom chain motion in an amorphous Ni-Zr alloy according to molecular

dynamics simulations of Teichler [13]

vacancies are responsible for self-diffusion and diffusion of substitutional lutes in metals in practically all cases Further historical details can be found

so-in [7]

There is, however, some renewed interest in non-defect mechanisms of

diffusion in connection with the enhanced diffusivity near phase transitions [8,9] For substitutionally dissolved boron in Cu there appears to be evidence

from β-NMR experiments for a non-defect mechanism of diffusion [10].

Collective mechanisms, which involve the simultaneous motion of

sev-eral atoms appear to be quite common in amorphous systems Moleculardynamic simulations by Teichler [13] as well as diffusion and isotope ex-periments on amorphous metallic alloys reviewed by Faupel et al [11, 12]suggest that collective mechanism operate in undercooled metallic melts and

in metallic glasses Such collective mechanisms involve the simultaneous tion of several atoms in a chain-like or caterpillar-like fashion An exampleobserved in molecular dynamic simulations of an amorphous Ni-Zr alloy isillustrated in Fig 6.4

mo-It appears that collective jump processes play also a rˆole for the motion

of alkali ions in ion-conducting oxide glasses [14] Finally, we note that terstitialcy mechanisms involving self-interstitials are collective in the sense

in-that more than one atom is displaced permanently during a jump event (seeSect 6.5)

6.3 Vacancy Mechanism

As knowledge about solids expanded, vacancies have been accepted as themost important form of thermally induced atomic defects in metals and ioniccrystals (see Chaps 5, 17, 26) It has also been recognised that the dominant

6.3 Vacancy Mechanism 99

Fig 6.5 Monovacancy mechanism of diffusion

mechanism for the diffusion of matrix atoms and of substitutional solutes in

metals is the vacancy mechanism An atom is said to diffuse by this

mecha-nism, when it jumps into a neighbouring vacancy (Fig 6.5) The constriction,which inhibits motion of an adjacent atom into a vacancy in a close-packedlattice is small, as compared to the constriction against the direct or ring ex-change Each atom moves through the crystal by making a series of exchangeswith vacancies, which from time to time are in its vicinity

In thermal equilibrium, the site fraction of vacancies in a monoatomic

crystal, C 1V eq, is given by Eq (5.11), which we repeat for convenience:

kB

exp

formation entropy and the formation enthalpy of a monovacancy, respectively.

Typical values for the site fraction of vacancies near the melting temperature

of metallic elements lie between 10−4 and 10−3 From Eqs (4.31) and (6.1)

we get for the exchange jump rate Γ of a vacancy-mediated jump of a matrix

atom to a particular neighbouring site

Γ = ω 1V C 1V eq = ν0exp



S F 1V + S M 1V

kB

exp

ω 1V denotes the exchange rate between an atom and a vacancy and ν0 the

pertinent attempt frequency H M

1V and S M

1V denote the migration enthalpy andeutropy of vacancy migration, respectively The total jump rate of a matrix

atom in a coordination lattice with Z neighbours is given by Γtot = ZΓ The

vacancy mechanism is the dominating mechanism of self-diffusion in metalsand substitutional alloys It is also relevant for diffusion in a number of ioniccrystals, ceramic materials, and in germanium (see Parts III, IV and V ofthis book)

In substitutional alloys, attractive or repulsive interactions between soluteatoms and vacancies play an important rˆole These interactions modify the