Diffusion Solids Fundamentals Diffusion Controlled Solid State Episode 2 Part 7 docx

23.2 Germanium 401 Fig. 23.3. Doping dependence of Ge self-diffusion according to Werner et al. [12] – Further information about the nature and properties of point defects in- volved in the diffusion process has been deduced from the effect of pressure on diffusion (see Chap. 8). Measurements of Ge self-diffusion under pres- sure are reported in [12]. The activation volumes are comparatively small and vary from 0.24 to 0.41 atomic volumes (Ω) as the temperature in- creases from 876 to 1086 K (Fig. 23.4). Values of the activation volume for self-diffusion of gold are shown for comparison [14]. These larger val- ues are typical for vacancy-mediated diffusion in close-packed metals. The lower values for Ge support the concept that self-diffusion in Ge occurs via vacancies, which are more relaxed or ‘spread-out’ than in close-packed materials. The positive sign very likely excludes self-interstitials as the de- fects responsible for Ge self-diffusion. For an interstitialcy mechanism the activation volume should be negative (see Chap. 8). The experimental activation volume increases with temperature. This in- crease can be attributed to the fact that neutral and negatively charged vacancies with differnet activation volumes contribute to self-diffusion. The activation volume of the neutral vacancy contribution (∆V V 0 =0.56 Ω) is larger than that of the negatively charged vacancy (∆V V − =0.28 Ω) [12]. The experimental activation volume is an aver- age value weighted with the relative contributions of the two types of vacancies to the total self-diffusivity; ∆V =∆V V 0 D ∗ V 0 D ∗ +∆V V − D ∗ V 0 − D ∗ . (23.13) Since the contribution of charged defects becomes more significant with decreasing temperature, the effective activation volume decreases. 402 23 Self-diffusion in Elemental Semiconductors Fig. 23.4. Activation volumes of Ge self-diffusion according to Werner et al. [12]. For comparison activation volumes of Au self-diffusion are also shown [14] 23.3 Silicon Natural silicon has three stable isotopes with the following abundances: 92.2 % 28 Si, 4.7 % 29 Si, and 3.1 % 30 Si. Self-diffusion studies on silicon are aggravated by the fact that the only accessible radioisotope, 31 Si, has a half- life of 2.6 hours. This limits the time for a diffusion experiment. Nevertheless, in view of the great importance of silicon as base material for microelectronic devices several attempts have been made to apply tracer methods using either the radioisotope 31 Si, the stable enriched isotope 30 Si, or isotopically con- trolled heterostructures. Ghoshtagore [15] evaporated enriched 30 Si layers onto Si wafers, performed neutron activation of the sample after diffusion an- nealing and subsequent chemical sectioning for profile analysis. Peart [16] evaporated the radiotracer 31 Si and utilised mechanical sectioning. Fa irfie ld and Masters [17] studied 31 Si diffusion in intrinsic and doped Si and sec- tioned the samples by chemical etching. Mayer et al. [18] and Hettich et al. [19] sputter-deposited thin layers of neutron activated Si (contain- ing 31 Si) and used sputter sectioning for profile analysis. Hirvonen and Anttila [20] and Demond et al. [21] implanted a layer of 30 Si and studied its diffusional broadening by a proton beam utilising the resonant nuclear re- action 30 Si(p, γ) 31 Si. Kalinowski and Seguin [22] evaporated layers of 30 Si and studied in-diffusion profiles by SIMS. The study by Bracht et al. [23] employed stable silicon isotope heterostructures with highly enriched 28 Si 23.3 Silicon 403 Fig. 23.5. Self-diffusion coefficients of Si measured by tracer methods [15–23] layers. The heterostructures were grown by chemical vapour deposition on natural Si wafers. Diffusion profiles of 29 Si and 30 Si isotopes were determined by SIMS. The data of the various authors are shown in Fig. 23.5. The results are less consistent than those on Ge self-diffusion (see Fig. 23.2) and far less consistent than those on self-diffusion in metallic elements (see, e.g., [24]). This is surprising since in most of the above mentioned studies virtually dislocation-free Si single crystals of high purity were used. The reason for these discrepancies are not completely clear. Oxygen is one of the main im- purities in both Czochralski-grown and float-zone single crystals. It is well- known that oxidation at the surface or oxide formation or dissolution causes deviations of intrinsic point defects from their thermal equilibrium concen- trations. Implantation damage may be an additional reason for discrepancies in those experiments where 30 Si was implanted prior to diffusion. All earlier experiments either suffer from the short half-life of 31 Si or from the natural abundance of stable 30 Si, when enriched 30 Si was used as tracer. 404 23 Self-diffusion in Elemental Semiconductors Drawbacks of the earlier experiments have been avoided in the already mentioned study with isotope heterostructures by Bracht et al. [23]. Nei- ther a short-lived radioisotope nor the background of the natural 30 Si limited the experiment. An influence of the surface can very likely be also excluded, since diffusion is observed near an interface of the isotope heterostructure. Even at the lowest temperature no doping effects could be detected. Thus, it is believed that these data reflect intrinsic behaviour of Si self-diffusion. Val- ues of the self-diffusion coefficient cover about seven orders of magnitude and the widest temperature range of all studies. Accordingly, the self-diffusivity of intrinsic silicon is described by [23] D ∗ =0.53 exp  − 4.75 eV k B T  m 2 s −1 . (23.14) Measurements of the tracer self-diffusion coefficient alone are not definitive in establishing the self-interstitial and vacancy contributions to self-diffusion. Additional information is needed and can be obtained, for example, from the analysis of self- and foreign-atom diffusion experiments involving non- equilibrium concentrations of intrinsic point defects. Below, we summarise the main conclusions that have been drawn from such experiments concerning Si self-diffusion [25]. The native point defect which predominantly mediates Si self-diffusion is the self-interstitial. This conclusion is consistent with experimental data for the transport product C eq I D I deduced from diffusion of hybrid foreign ele- ments (see Chap. 25). The self-interstitial contribution, C eq I D I , is well-known from diffusion studies of hybrid foreign atom diffusers (Au: [27], Zn: [28]). Therefore, one can extract the vacancy term, C eq V D V in Eq. (23.7), from thetotalself-diffusivity.FromananalysisofZndiffusioninSiBracht et al. [28] obtain: C eq I D I =0.298 exp  − 4.95 eV k B T  m 2 s −1 . (23.15) Using this expression, a fit of Eq. (23.7) to the most reliable self-diffusion data [23] yields the vacancy transport product as: C eq V D V =0.92 × 10 −4 exp  − 4.14 eV k B T  m 2 s −1 . (23.16) The temperature dependences of the transport products, C eq I D I and C eq V D V , are shown in Fig. 23.6 and compared with the self-diffusion data [23]. The agreement between D ∗ and f I C eq I D I + f V C eq V D V (taking into account the correlation factors) implies that self-diffusion is mediated by both self- interstitials and vacancies. C eq I D I equals C eq V D V at about 890 ◦ C and not at temperatures between 1000 and 1100 ◦ C as had been assumed earlier [4, 5]. At temperatures above 890 ◦ C the self-interstitial contribution domi- nates, whereas at lower temperatures vacancy-mediated diffusion dominates. 23.3 Silicon 405 Fig. 23.6. Si self-diffusion coefficients (symbols) compared with the self-interstitial and vacancy transport products, C eq I D I and C eq V D V , according to Bracht et al. [23] Slightly different activation parameters for the transport products are re- ported in [6] based on earlier studies. The enthalpy and entropy of self-interstitial mediated diffusion obtained from Eq. (23.15) are: H SD I = H F I + H M I =4.95 eV and S SD I = S F I + S M I ≈ 13.2 k B . (23.17) Those for vacancy-mediated diffusion are H SD V = H F V + H M V =4.14 eV and S SD V = S F V + S M V ≈ 5.5 k B . (23.18) Theoretical calculations of intrinsic point defect properties are in good agree- ment with these values. Tang et al. [29] obtained from their tight-binding molecular dynamic studies H SD I =(5.18 ± 0.2) eV, H SD V =(4.07 ± 0.2) eV, and S SD I =14.3 k B . The first principles calculations of Bl ¨ ochl et al. [26] yield S F V =(5± 2) k B leaving for S M V values between 1 and 2 k B . In conclusion, we can state that experiments and theory suggest that Si self-diffusion is mediated by simultaneous contributions of self-interstitials 406 23 Self-diffusion in Elemental Semiconductors and vacancies. Self-interstitials dominate at high temperatures whereas va- cancies take over at lower temperatures. The doping dependence of Si self-diffusion [5] allows the conclusion that neutral as well as positively and negatively charged defects are involved in self-diffusion. However, the data are not accurate enough to determine the inidvidual terms in Eqs. (23.5) and (23.6). Since the total tracer diffusivity as well as the transport products consist of several terms, Eqs. (23.15) and (23.16) can only be approximations holding for a limited temperature range. References 1. S.M. Sze, Physics of Semiconductor Devices, Wiley, New York, 1967 2. S.T. Pantelides, Defect and Diffusion Forum 75, 149 (1991) 3. U. G¨osele, T.Y. Tan, Defect and Diffusion Forum 83, 189 (1992) 4. A. Seeger, K.P. Chik, Diffusion Mechanisms and Point Defects in Silicon and Germanium,Phys.Stat.Sol.29, 455–542 (1968) 5. 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, 1984 6. T.Y. Tan, U. G¨osele, Diffusion in Semiconductors,in:Diffusion in Condensed Matter – Methods, Materials, Models,P.Heitjans,J.K¨arger (Eds.), Springer- Verlag, 2005 7. H.LetawJr.,W.M.Portnoy,L.Slifkin,Phys.Rev.102, 636 (1956) 8. M.W. Valenta, C. Ramsastry, Phys. Rev. 106, 73 (1957) 9. H. Widmer, G.R. Gunter-Mohr, Helv. Phys. Acta. 34, 635 (1961) 10. R.Campbell,Phys.Rev.B12, 2318 (1975) 11. G. Vogel, G. Hettich, H. Mehrer, J. Phys. C 16, 6197 (1983) 12. M. Werner, H. Mehrer, H.D. Hochheimer, Phys. Rev. B 32, 3930 (1985) 13. H.D. Fuchs, W. Walukiewicz, E.E. Haller, W. Dondl, R. Schorer, G. Abstreiter, A.I. Rudnev, A.V. Tikhomirov, V.I. Oshogin, Phys. Rev. B 51, 16817 (1995) 14. M. Werner, H. Mehrer, in: DIMETA-82, Diffusion in Metals and Alloys,F.J. Kedves, D.L. Beke (Eds.), Trans Tech Publications, Diffusion and Defect Mono- graph Series No. 7, 392 (1983) 15. R.N. Ghostagore, Phys. Rev. Lett. 16, 890 (1966) 16. R.F. Peart, Phys. Status Solidi 15, K119 (1966) 17. J.M. Fairfield, B.J. Masters, J. Appl. Phys. 38, 3148 (1967) 18. H.J. Mayer, H. Mehrer, K. Maier, Inst. Phys. Conf. Series 31, 186 (1977) 19. G. Hettich, H. Mehrer, K. Maier, Inst. Phys. Conf. Series 46, 500 (1979) 20. J. Hirvonen, A. Anttila, Appl. Phys. Lett. 35, 703 (1979) 21. F.J. Demond, S. Kalbitzer, H. Mannsperger, H. Damjantschitsch, Phys. Lett. A 93, 503 (1983) 22. L. Kalinowski, R. Seguin, Appl. Phys. Lett. 35, 211 (1979); Erratum: Appl. Phys. Lett. 36, 171 (1980) 23. H. Bracht, E.E. Haller, R. Clark-Phelps, Phys. Rev. Lett. 81, 393 (1998) 24. H. Mehrer, N. Stolica, N.A. Stolwijk, Self-diffusion in Solid Metallic Elements, Chap. 2 in: Diffusion in Solid Metals and Alloys, H. Mehrer (Vol.Ed.), Landolt- B¨ornstein, New Series, Group III: Crystal and Solid State Physics, Vol. 26, Springer-Verlag, 1990, p.32 References 407 25. H. Bracht, MRS Bulletin, June 2000, 22 26. P.E. Bl¨ochl,E.Smargiassi,R.Car,D.B.Laks,W.Andreoni,S.T.Pantelides, Phys. Rev. Lett 70, 2435 (1993) 27. N.A. Stolwijk, B. Schuster, J. H¨olzl, H. Mehrer, W. Frank, Physica (Amster- dam) 116 B&C, 335 (1983) 28. H. Bracht, N.A. Stolwijk, H. Mehrer, Phys. Rev. B 52, 16542 (1995) 29. M. Tang, L. Colombo, T. Diaz de la Rubbia, Phys. Rev B 55, 14279 (1997) 24 Foreign-Atom Diffusion in Silicon and Germanium Diffusion of foreign atoms in Si and Ge is very important from a technological point of view. Due to its complexity it provides a challenge from a scientific point of view as well. Group-III elements B, Al, Ga and group-V elements P, As, Sb are a special class of foreign elements known as dopants. Dopants are easily ionised and act as donors or acceptors. Their solubility is fairly high compared to most other foreign elements except group-IV elements. Diffu- sion of dopants plays a vital rˆole in diffusion-doping to create p-n junctions of microelectronic devices. Diffusion also controls the incorporation of ‘un- wanted’ foreign atoms, e.g., of the metal atoms Fe, Ni, and Cu during thermal annealing treatments of device fabrication. A detailed understanding of the diffusion behaviour of unwanted foreign atoms is of technological significance to keep the contamination of electronic devices during processing to a harm- less state. Oxygen diffusion and growth of SiO 2 precipitates play a crucial rˆole in gettering processes of unwanted foreign elements. Other foreign atoms like Au in Si are used for tuning the minority-carrier lifetime. Dopants are incorporated in substitutional sites of the host lattice, some foreign elements dissolve in interstitial sites only, others are hybrid foreign elements, which are dissolved on substitutional and on interstitial sites. The very high mobility of the interstitial fraction can dominate diffusion of hy- brid elements. In addition, the interstitial-substitutional exchange reactions – either the dissociative or the kick-out reaction – can lead to non-Fickian dif- fusion profiles of hybrid diffusers. The diffusion behaviour of foreign elements is largely determined by the type of solution, i.e. whether they are located at substitutional or intersti- tial sites, or a mixture of both. In what follows, we consider first solubilities of foreign elements and their site preference. Then, we give a brief review of the diffusion of foreign elements in Ge and Si and classify them accord- ing to their site occupancy and diffusivity. The theoretical framework of the relatively complex diffusion patterns of hybrid solutes involving interstitial- substitutional exchange reactions is postponed to Chapt 25. 24.1 Solubility and Site Occupancy The solubility of a foreign element is the maximum concentration which can be incorporated in the host solid without forming a new phase. Solid sol- 410 24 Foreign-Atom Diffusion in Silicon and Germanium ubilities are temperature-dependent as represented by the solvus or solidus lines of the phase diagram. The solubility limit is defined with respect to a second phase. For most foreign elements in solid Ge or Si at high temper- atures equilibrium is achieved with the liquid phase. At lower temperatures the reference phase usually is the solid foreign element or a compound of the foreign element. When the foreign atom is volatile, the saturated host crystal is in equilibrium with a vapour. In this case, the solubility depends not only on the temperature but also on the vapour pressure. For solid-solid equilibria and for solid-vapour equilibria the temperature dependence of the solubility limit is usually described by an Arrhenius rela- tion containing a solution enthalpy. If a solid-liquid equilibrium is involved, the behaviour is more complex. The right-hand side of Fig. 24.1 shows the normal variation of a solubility giving rise to a maximum at the eutectic tem- perature. A frequently encountered case is the so-called retrograde solubility, illustrated on the left-hand side of Fig. 24.1. This phenomenon implies that the maximum solubility is achieved at a temperature which lies below the melting temperature of the host crystal but above the eutectic temperature. Below this maximum temperature, the solubility can often but not always be approximated by an Arrhenius relation. Solubility data of foreign elements have been collected for Si by Schulz [6], for Ge by Stolwijk [7] and have been updated for Ge by Stolwijk and Bracht [8] and for Si by Bracht and Stolwijk [9]. Depending on the foreign element, the solubility can vary over orders of magnitude: B, P, As in Si and Al, Ga, Sn in Ge can be incorporated to several percent into the host crystals. Other dopant elements, such as Ga, Sb, Li in Si and As and Sb in Ge Fig. 24.1. Schematic phase diagram of a semiconductor and a foreign element (or a compound of a foreign element) illustrating the phenomenon of retrograde solubility 24.1 Solubility and Site Occupancy 411 have maximum solubilities in the range of 10 −3 atomic fractions. Noble metal impurities, iron group impurities, nickel group impurities, cobalt group impu- rities, and Zn have very low solubilities in the range of 10 −6 atomic fractions. Generally speaking, every foreign atom A may occupy both substitutional A s and interstitial A i sites in the lattice. The equilibrium solubility of either configuration, C eq s and C eq i , is determined by the pertinent enthalpy and entropy of solution, which refers to some external phase of the foreign atom. Dealing with equilibrium solubilities one should be aware of the the following features [10]: – For most foreign atoms one atomic configuration dominates in the well accessible temperature range of diffusion studies between about 2/3 T m and 0.9 T m : – Dopants are dissolved in substitutional sites and diffuse with the aid of intrinsic point defects. – Foreign atoms with interstitial site preference (C eq i  C eq s )diffuse via a direct interstitial mechanism. Examples are group-I and some group-VIII elements. – Foreign atoms with substitutional site preference (C eq i  C eq s ), but some minor interstitial fraction are interstitial-substitutional exchange diffusers (hybrid solutes). Examples are some noble metals and further elements mainly from neighbouring groups of the noble metals. – The equilibrium solubilities on substitutional and interstitial sites, C eq s and C eq i , depend sensitively on the solute, the solvent, and on tempera- ture. C eq s and C eq i can have a dissimilar variation with temperature. This leads to appreciable changes in the interstitial to substitutional concen- tration ratio C eq i /C eq s with temperature. There are some foreign elements for which the dissimilar variation and the decreasing solubility near the melting temperature leads to intersections of the solubilities of C eq s and C eq i . This is illustrated for the best studied example Cu in Ge in Fig. 24.2. – In general, foreign atoms in semiconductors carry electronic charges. Therefore, the position of the Fermi level affects their solubility. A con- comitant phenomenon is that the ratio C eq i /C eq s may change with back- ground doping, because of the different electronic structure of A i and A s . Dramatic effects have been observed for 3d transition elements in Si [11, 12]. For example, in intrinsic Si the predominant species of cobalt, Co i ,is a donor. By contrast, in heavily phosphorous doped Si acceptor-like Co s becomes more abundant. In what follows, we confine ourselves to intrinsic conditions. This im- plies that the intrinsic carrier concentration at the diffusion temperature, n i (T ), exceeds the maximum concentration of electrically active foreign atoms. This is usually fulfilled for host crystals having doping levels not higher than 10 18 cm −3 . 1 1 In shallow dopant diffusion experiments on, e.g., GaAs the relatively small n i (T ) in conjunction with the high solubility of dopants causes a shift of the Fermi level [...]... p .25 4 22 W Kaiser, Phys Rev B 105, 175 1 (19 57) 23 C Haas, J Phys Chem Sol 15, 108 (1960) 24 J.W Corbett, R.S McDonald, G.D Watkins, J Phys Chem Sol 25 , 873 (1964) 25 R.C Newman, Defect and Diffusion Forum 143–1 47, 9903 (19 97) 26 P.M Fahey, P.B Griffin, J.D Plummer, Rev Modern Physics 621 , 28 9 (1989) 27 S.M Hu, in: Diffusion in Semiconductors, D Shaw (Ed.), Plenum Press, London and New York, 1 973 , p 21 7 28 ... of partial differential equations for modelling interstitial-substitutional exchange diffusion in crystals is given by: ∂ 2 Ci ∂Ci = Di + k−1 Cs − k+1 Ci CV + k 2 Cs CI − k +2 Ci ∂t ∂x2 ∂ 2 Cs ∂Cs = Ds +k+1 Ci CV − k−1 Cs + k +2 Ci − k 2 Cs CI ∂t ∂x2 Ds ≈0 2 ∂CV ∂ CV = DV + k−1 Cs − k+1 Ci CV + KV ∂t ∂x2 1− ∂CI ∂ 2 CI = DI − k 2 Cs CI + k +2 Ci + KI ∂t ∂x2 CI eq − 1 CI CV eq CV (25 .3) (25 .4) (25 .5) (25 .6)... Crystalline Solids, G.E Murch, A.S Nowick (Eds.), Academic Press, 1984 3 H Bracht, Materials Science in Semiconductor Processing 7, 113 (20 04) 4 H Bracht, MRS Bulletin 20 00, p .22 5 N.A Stolwijk, Phys Rev B 42, 579 3 (1990) References 423 6 M Schulz, Silicon, in: Impurities and Defects in Group IV Elements and IIIV Compounds, Landolt-B¨rnstein, New Series, Group III: Crystal and Solid o State Physics, Vol 22 :... (1 973 ) 29 M Yoshida, E Arai, H Nakamura, Y Terunuma, J Appl Phys 45, 1489 (1 974 ) 30 U G¨sele, in: Microelectronic Materials and Processes, R.A Levy (Ed.), Kluwer o Academic, Dordrecht 1989, p 588 31 M Uematsu, J Appl Phys 62, 22 28 (19 97) 32 S.M Hu, Materials Science and Engineering R13, 105 (1994) 424 24 Foreign-Atom Diffusion in Silicon and Germanium 33 H Bracht, N.A Stolwijk, H Mehrer, Phys Rev B 52, ... [21 ] to be 1.03 eV kB T 1.03 eV = 1.53 × 1 0 27 exp − m−3 kB T eq Ci = 3 × 10 2 exp − (24 .1) Oxygen dissolves interstitially in a bond-centered configuration shown in Fig 24 .5 (see [22 24 ]) The oxygen diffusivity has been determined over wide 416 24 Foreign-Atom Diffusion in Silicon and Germanium Fig 24 .5 Bond-centered configuration of the oxygen interstitial in the Si lattice according to Frank et al [2] ... Semiconductors, Subvolume A2, Part α: Group IV Elements, M Schulz (Vol.Ed.), Springer-Verlag, 20 02, p 3 82 9 H Bracht, N.A Stolwijk, Silicon, in: Impurities and Defects in Group IV Elements, IV-IV and III-V Compounds, Landolt-B¨rnstein, New Series, Group o III: Condensed Matter, Vol 41: Semiconductors, Subvolume A2, Part α: Group IV Elements, M Schulz (Vol.Ed.), Springer-Verlag, 20 02, p 77 10 N.A Stolwijk,... Springer-Verlag, 1998, p 2 1 14 A van Wieringen, N Warmholtz, Physica 22 , 849 (1956) 15 E.R Weber, Appl Phys A 30, 1 (1983) 16 C.S Fuller, J.A Ditzenberger, Phys Rev 91, 193 (1953) 17 C.S Fuller, J.C Severiens, Phys Rev 96, 22 5 (1954) 18 E.M Pell, Phys Rev 119, 1014 (1960) 19 S Pearton, J.W Corbett, T.S Shi, Appl Phys A 43, 153 (19 67) 20 W Kaiser, H.L Fritsch, H Reiss, Phys Rev 1 12, 1546 (1958) 21 R.A Craven,... Si by Gosele et al [2] In the meantime, positive identifications of interstitial-substitutional exchange diffusion have been made in several other cases (see Sects 25 .2, 25 .3, and Chap 24 ) 25 .1 Combined Dissociative and Kick-out Diffusion This section is based on the assumption that both mechanisms operate simultaneously In two subsequent sections, we consider the two mechanisms 426 25 Interstitial-Substitutional... diffusion [ 27 , 26 , 13] Since dopant diffusion proceeds via the same mechanisms as self-diffusion, a key question is what activation enthalpies can be expected for the vacancy and for the interstitialcy mechanism: – The energetics of the vacancy mechanism of dopant diffusion for the diamond structure has been pointed out by Hu [ 27 , 28 ] An enthalpy diagram M of the vacancy diffusion process is displayed in Fig 24 .6... and Solid o State Physics, Vol 22 : Semiconductors, Subvolume B, M Schulz (Vol.Ed.), Springer-Verlag, 1989, p 20 7 7 N.A Stolwijk, Germanium, in: Impurities and Defects in Group IV Elements and III-V Compounds, Landolt-B¨rnstein, New Series, Group III: Crystal o and Solid State Physics, Vol 22 : Semiconductors, Subvolume B, M Schulz (Vol.Ed.), Springer-Verlag, 1989, p 439 8 N.A Stolwijk, H Bracht, Germanium, . 1990, p. 32 References 4 07 25 . H. Bracht, MRS Bulletin, June 20 00, 22 26 . P.E. Bl¨ochl,E.Smargiassi,R.Car,D.B.Laks,W.Andreoni,S.T.Pantelides, Phys. Rev. Lett 70 , 24 35 (1993) 27 . N.A. Stolwijk, B 116 B&C, 335 (1983) 28 . H. Bracht, N.A. Stolwijk, H. Mehrer, Phys. Rev. B 52, 165 42 (1995) 29 . M. Tang, L. Colombo, T. Diaz de la Rubbia, Phys. Rev B 55, 1 4 27 9 (19 97) 24 Foreign-Atom Diffusion in. (23 .7) , from thetotalself-diffusivity.FromananalysisofZndiffusioninSiBracht et al. [28 ] obtain: C eq I D I =0 .29 8 exp  − 4.95 eV k B T  m 2 s −1 . (23 .15) Using this expression, a fit of Eq. (23 .7)