Heitjans P., Karger J. (Eds.). Diffusion in Condensed Matter: Methods, Materials, Models
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3 Diffusion Studies of Solids by Quasielastic Neutron Scattering 161
41. A. Kollmar, B. Alefeld. In: Proceedings of the Conference on Neutron Scatter-
ing, Gatlinburg, Tenn. USA 1976, ed by R.M. Moon (National Techn. Infor-
mation Service, U.S. Dept. Of Comm., Springfield 1976) pp 330-336.
42. M. Prager, W. Press, B. Alefeld, A. H¨uller: J. Chem. Phys. 67, 5126 (1977)
43. R.E. Lechner, H.-J. Bleif, H. Dachs, R. Marx, M. Stahn: Solid State Ionics
46, 25 (1991)
44. C.J. Carlile, M.A. Adams: Physica B 182, 431 (1992)
45. V.F. Sears: Adv. Physics 24, 1 (1975)
46. R.E. Lechner, G. Badurek, A.J. Dianoux, H. Hervet, F. Volino: J. Chem. Phys.
73, 934 (1980)
47. M. Russina, F. Mezei, R.E. Lechner, S. Longeville, B. Urban: Phys. Rev. Lett.
84, 3630 (2000)
48. C. Lamers, O. Sch¨arpf, W. Schweika, J. Batoulis, K. Sommer, D. Richter:
Physica B 180&181, 515 (1992)
49. J.C.Cook,D.Richter,O.Sch¨arpf, M.J. Benham, D.K. Ross, R. Hempelmann,
I.S. Anderson, S.K. Sinha: J. Phys.: Condensed Matter 2, 79 (1990)
50. K. Sk¨old, G. Nelin: J. Phys. Chem. Solids 28, 2369 (1967)
51. J.M. Rowe, J.J. Rush, L.A. de Graaf, G.A. Ferguson, Phys. Rev. Lett. 29,
1250 (1972)
52. Y.Li,G.Wahnstr¨om: Phys. Rev. Lett. 68, 3444 (1992); Y. Li, G. Wahnstr¨om:
Phys. Rev. B 46, 14528 (1992)
53. W. Gissler, G. Alefeld, T. Springer: J. Phys. Chem. Solids 31, 2361 (1970)
54. V. Lottner, J.W. Haus, A. Heim, K.W. Kehr: J. Phys. Chem. Solids 40, 557
(1979); V. Lottner, A. Heim, T. Springer: Z. Phys. B 32, 157 (1979)
55. H. Dosch, F. Schmid, P. Wiethoff, J. Peisl: Phys. Rev. B 46, 55 (1992)
56. S. Emin: Phys. Rev. B 3, 1321 (1971)
57. Y. Li, G. Wahnstr¨om: Phys. Rev. B 51, 12233 (1995); see also B. von Sydow,
G. Wahnstr¨om: Phys. Rev. B 53, 3171 (1996)
58. H. Metzger, J. Peisl, J. Wanagel: J. Phys. F 6, 2195 (1976); H. Dosch, J. Peisl:
Phys. Rev. Lett. 56, 1385 (1986)
59. J. Buchholz, J. V¨olkl,G.Alefeld:Phys.Rev.Lett.30, 318 (1973)
60. E. Wicke, H. Brodowsky, H. Z¨uchner: Top. Appl. Phys. 29, 73 (1978)
61. J. V¨olkl, G. Alefeld. In: Diffusion in Solids,edbyA.S.Nowick,J.J.Burton
(Academic Press, London 1975) p 232; Top. Appl. Phys. 28, 321 (1978)
62. J. T¨opler, E. Lebsanft, R. Sch¨atzler:J.Phys.F8
, L25 (1978)
63. E. Lebsanft, D. Richter, J.M. T¨opler: J. Phys. F 9, 1057 (1979)
64. R.C. Bowman, W.E. Tadlock: Solid State Commun. 32, 313 (1979)
65. T.K. Halstead, N.A. Abood, K.H.J. Buschow: Solid State Commun. 19, 425
(1976)
66. D. Richter, R. Hempelmann, L.A. Vinhas: J. Less-Common Met. 88, 353
(1982)
67. R.C. Bowman, J.R. Johnson: J. Less-Common Met. 73, 254 (1980)
68. R. Hempelmann, D. Richter, A. Heidemann: J. Less-Common Met. 88, 343
(1982)
69. R. Hempelmann, D. Richter, R. Pugliesi, L.A. Vinhas: J.Phys. F 13, 59 (1983)
70. J. T¨opler, H. Buchner, H. S¨aufferer, K. Knorr, I. J. Prandl: J. Less-Common
Met. 88, 397 (1982)
71. D. Richter, T. Springer: Phys. Rev. B 18, 126 (1978)
72. H. Grabert, H. Wipf: Festk¨orperprobleme 30, 1 (1990)
162 Tasso Springer and Ruep E. Lechner
73. D. Steinbinder, H. Wipf, A.-J. Dianoux, A. Magerl, K. Neumaier, D. Richter,
R. Hempelmann: Europhys. Lett. 16, 211 (1991)
74. G. G¨oltz, A. Heidemann, H. Mehrer, A. Seeger, D. Wolf, Philos. Mag. A 41,
723 (1980)
75. D. Wolf: Phys. Rev. B 15, 37 (1977)
76. D. Wolf: Solid State Commun. 23, 853 (1977)
77. M. Ait-Salem, T. Springer, A. Heidemann, B. Alefeld, Philos. Mag. A 39, 797
(1979)
78. G. Vogl, W. Petry, Th. Flottmann, A. Heiming: Phys. Rev. B 39, 5025 (1989)
79. W. Petry: Phase Transitions 31, 119 (1991); W. Petry, A. Heiming, C. Herzig,
J. Trampenau: Defect and Diffusion Forum 75, 211 (1991)
80. H. R. Schober, W. Petry, J. Trampenau: J. Phys.: Condens. Matter 4, 9321
(1992)
81. H. Feldmann, R. E. Lechner, D. Wilmer, Mat. Res. Soc. Symp. Proc. 756,
EE2.4.1 (2003)
82. M. Spaeth, K. D. Kreuer, J. Maier, C. Cramer: J. Solid State Chem. 148, 169
(1999)
83. R. E. Lechner: Solid State Ionics 61, 3 (1993)
84. M. H. Dickens, W. Hayes, P. Schnabel, M. T. Hutchings, R. E. Lechner, B.
Renker: J.Phys. C 16, L1 (1983)
85. P. Schnabel, W. Hayes, M. T. Hutchings, R. E. Lechner, B. Renker: Radiation
Effects 75, 73 (1983)
86. M. B´eni`ere, M. Chemla, F. B´eni`ere: J. Phys. Chem. Solids 40, 729 (1979)
87. B. Wegener, R. Essmann, J. Bock, H. Jacobs, P. Fischer: Eur. J. Solid State
Inorg. Chem. 29, 1217 (1992)
88. F. Altorfer, W. B¨uhrer, B. Winkler, G. Coddens, R. Essmann, H. Jacobs:
Solid State Ionics 70/71, 272 (1994)
89. K. Funke: Prog. Solid St. Chem. 22, 111 (1993); K. Funke, Z. Phys. Chem.
188, 243 (1995)
90. K. Funke, T. Kloidt, D. Wilmer, C.J. Carlile: Solid State Ionics 53–56, 947
(1992)
91. R.E. Lechner: Ferroelectrics 167, 83 (1995)
92. E. H¨uckel: Z. Elektrochem. 34, 546 (1928)
93. J.D. Bernal, R. H. Fowler: J. Chem. Phys. 1, 515 (1933)
94. H.G. Hertz: Z. Phys. Chem. Neue Folge 135, 89 (1983)
95. H.G. Hertz, B.M. Braun, K.J. M¨uller, R. Maurer: J. Chem. Educ. 64, 777
(1987)
96. K. D. Kreuer. In: Proton Conductors, Chemistry of Solid State Materials 2,
ed by Ph. Colomban (Cambridge University Press, Cambridge 1992)
97. R.E. Lechner, Th. Dippel, R. Marx, I. Lamprecht: Solid State Ionics 61
,47
(1993)
98. H. Arribart, Y. Piffard: Solid State Commun. 45, 571 (1983)
99. A. V. Belushkin, C. J. Carlile, L. A. Shuvalov: J. Phys.: Condens. Matter 4,
389 (1992)
100. Ph. Colomban, G. Lucazeau, R. Mercier, A. Novak: J. Chem. Phys. 67, 5244
(1977)
101. J.D. Axe, L.M. Corliss, J.M. Hastings, W.L. Roth, O. Muller: J. Phys. Chem.
Solids 39, 155 (1978)
102. J.C. Lass`egues, M. Fouassier, N. Baffier, Ph. Colomban, A.J. Dianoux: J.
Physique 41, 273 (1980)
3 Diffusion Studies of Solids by Quasielastic Neutron Scattering 163
103. N. Baffier, J.C. Badot, Ph. Colomban: Solid State Ionics 2, 107 (1980)
104. Th. Dippel, K. D. Kreuer: Solid State Ionics 46, 3 (1991)
105. A.T. Howe, M.G. Shilton: J. Solid State Chem. 34, 149 (1980)
106. K.D. Kreuer, A. Rabenau, R. Messer: Appl. Phys. A 32, 45 (1983)
107. K.D. Kreuer, M. Hampele, K. Dolde, A. Rabenau: Solid State Ionics 28-30,
589 (1988)
108. M. Pionke, T. Mono, W. Schweika, T. Springer, H. Schober: Solid State Ionics
97, 497 (1997)
109. M. Stahn, R.E. Lechner, H. Dachs, H.E. Jacobs: J. Phys. C 16, 5073 (1983)
110. K.-D. Kreuer: Chem. Mater. 8, 610 (1996); C. Herzig: Diffusion in Metals and
Alloys, ed by F.J. Kedves, D.L. Beke (Trans. Tech. Publ., Switzerland 1983)
111. Ch. Karmonik, R. Hempelmann, Th. Matzke, T. Springer: Z. Naturf. 50a,
539 (1995)
112. R.E. Lechner, G. Schuck, K. Langer: J. Phys. Soc. Jpn. 70, 274 (2001) Suppl.
113. G. Schuck, R.E. Lechner, K. Langer: Applied Physics A 74 [Suppl.], S1098
(2002)
114. A.I. Baranov, I.P. Makarova, L.A. Muradyan, A.V. Tregubchenko, L.A. Shu-
valov, V.I. Simonov: Sov. Phys. Crystallogr. 32, 400 (1987, AIP:1988)
115. A. Pawlowski, Cz. Pawlaczyk: Ferroelectrics 81, 201 (1988)
116. I.P. Makarova, I.A. Verin, N.M. Shchagina: Sov. Phys. Crystallogr. 31, 105
(1986)
117. A. Bohn, R. Melzer, R. Sonntag, R.E. Lechner, G. Schuck, K. Langer: Solid
State Ionics 77, 111 (1995)
118. R. Melzer, R. Sonntag, K.S. Knight: Acta Cryst. C 52, 1061 (1996)
119. R. Melzer, T. Wessels, M. Reehuis: Solid State Ionics 92, 119 (1996)
120. R.E. Lechner, G. Schuck, K. Langer, Th. Dippel, W. Renz, M. Casciola: Solid
State Ionics (2003) to be published
121. R.E. Lechner, R. Melzer, J. Fitter: Physica B 226, 86 (1996)
122. R.E. Lechner: Solid State Ionics 145, 167 (2001)
123. A. Avogadro, M. Villa: J. Chem. Phys. 66, 2359 (1977)
124. C.A. Sholl: J. Phys. C 14, 447 (1980)
125. P.H. Fries: Mol. Phys. 48, 503 (1983)
126. A. Schirmer, P. Heitjans: Z. Naturforsch. 50a, 643 (1995)
127. G. Schuck, R. Melzer, R. Sonntag, R.E. Lechner, A. Bohn, K. Langer, M.
Casciola: Solid State Ionics 77, 55 (1995)
128. R.E. Lechner: Solid State Ionics 77, 280 (1995)
129. A.J. Dianoux, F. Volino, H. Hervet: Mol. Phys. 30, 1181 (1975)
130. R.E. Lechner, N.A. Dencher, J. Fitter, G. B¨uldt, A.V. Belushkin: Biophys.
Chem. 49, 91 (1994)
131. R.E. Lechner, J. Fitter, Th. Dippel, N.A. Dencher: Solid State Ionics 70/71,
296 (1994)
132. G. Careri, M. Geraci, A. Giansanti, J.A. Rupley: Proc. Natl. Acad. Sci. USA
82, 5342 (1985)
133. N.A. Dencher, J. Fitter, R.E. Lechner. In: Hydration Processes in Biology,ed
by M.-C. Bellissent-Funel (IOS Press 1999) p 195
134. J. Fitter, R.E. Lechner, N.A. Dencher: J. Phys. Chem. B 103, 8036 (1999)
135. R. Henderson, P.N.T. Unwin: Nature 257, 28 (1975)
136. J.A. Rupley, L. Siemankowski, G. Careri, F. Bruni: Proc. Natl. Acad. Sci.
USA 85, 9022 (1988)
164 Tasso Springer and Ruep E. Lechner
137. R. Kutner: Phys. Lett. 81A, 239 (1981); see also K.W. Kehr, R. Kutner, K.
Binder: Phys. Rev. B 23, 4931 (1981)
138. S.K. Sinha, D.K. Ross: Physica B 149, 51 (1988)
139. P.C. Clapp, S.C. Moss: Phys. Rev. 171, 754 (1968); ibid. Phys. Rev. 171 764
(1968)
140. R. Hempelmann, D. Richter, D.A. Faux, D.K. Ross: Z. Phys. Chem. Neue
Folge 159, 175 (1988)
141. P.G. De Gennes: Physica 25, 825 (1959)
142. B.A. Dasannacharia, K.R. Rao: Phys. Rev. 137, 417 (1965)
143. J. V¨olkl, Ber. Bunsen-Ges. Phys. Chem. 76, 797 (1972)
144. M.J. Gillan, D. Wolf: Phys. Rev. Lett. 55, 1299 (1985)
145. D. Wilmer, K. Funke, M. Witschas, R.D. Banhatti, M. Jansen, G. Korus, J.
Fitter, R.E. Lechner: Physica B 266, 60 (1999)
146. D. Wilmer, H. Feldmann, J. Combet, R.E. Lechner: Physica B 301, 99 (2001)
147. D. Wilmer, H. Feldmann, R.E. Lechner: Phys. Chem. Chem. Phys. 4, 3260
(2002)
148. A. Kvist, A. Bengtzelius. In: Fast Ion Transport in Solids, ed by W. van Gool
(North-Holland, Amsterdam 1973) p 193
149. A. Lund´en: Solid State Ionics 28-30, 163 (1988)
150. A. Lund´en: Solid State Ionics 68, 77 (1994)
151. V.F. Sears, Can. J. Phys. 45, 237 (1967)
152. D.A. Neumann, J.R.D. Copley, R.L. Cappelletti, W.A. Kamitakahara, R.M.
Lindstrom, K.M. Kreegan, D.M. Cox, W.J. Romanow, N. Coustel, J.P. Mc-
Cauley Jr, N.C. Maliszewskyj, J.E. Fischer, A.B. Smith III: Phys. Rev. Lett.
67, 3808 (1991)
4 Diffusion in Semiconductors
TehYuTanandUlrichG¨osele
4.1 Introduction
This chapter discusses diffusion phenomena in the semiconductors Si, Ge,
and GaAs. Silicon and GaAs are the two main materials used in fabricating
electronic and optoelectronic devices. Diffusion in Ge will also be mentioned
for the reason that it is the simplest case among semiconductors. Diffusion
processes are used in doping a semiconductor with n-type and p-type dopant
atoms to produce pn-junctions for device operations. The n- and p-type
dopants are specific kinds of substitutional impurity species producing the
electric carriers electrons (e) and holes (h), respectively, in a semiconductor.
Diffusion processes are also involved in the removal of detrimental metallic
impurities, in silicide formation and in thermal SiO
2
growth in fabricating
devices using Si.
4.2 Diffusion Mechanisms and Point Defects
in Semiconductors
Impurity atoms may dissolve in a semiconductor by occupying interstitial (i)
or substitutional (s) sites, or both. As in metals (see Chap. 1), atoms of i
species in semiconductors migrate directly between the appropriate intersti-
tial sites, which is described well by Fick’s second law
∂C
∂t
= D
∂
2
C
∂x
2
, (4.1)
where D is the diffusivity and C is the concentration. The diffusivity of an i
species is characterized by a single activation enthalpy.
The semiconductor crystal host atom self-diffusion and the diffusion of s
impurity species utilize native point defects. While the dominant native point
defect species in metals are vacancies (V ), both V and self-interstitials (I)
contribute in semiconductors. To diffuse, an impurity s atom may directly ex-
change or migrate as a pair or complex with an I or a V , but only the exchange
mechanism is operative for the host crystal atom self-diffusion, which is slower
than that of a s species utilizing the same point defect species as diffusion
166 Teh Yu Tan and Ulrich G¨osele
vehicles via the pairing mechanism. Furthermore, in a semiconductor both
V and I can be in several different charge states with their concentrations
dependent upon the semiconductor doping level and temperature. These fac-
tors lead to complications in the Fick’s second law description of self-diffusion
and diffusion of substitutional impurities in that the diffusivity shall consist
of the contributions of all the involved point defect species which may not be
a constant at a given temperature, and may also not reveal a single activa-
tion enthalpy over a temperature range. Furthermore, some impurity species
are interstitial-substitutional (i-s) species which dissolve predominantly on
substitutional sites but their diffusion is actually due to the rapid migration
of those occupying interstitial sites and the interchange between atoms oc-
cupying the two different types of sites. The diffusivity in the Fick’s second
law description of the diffusion process of the i-s species is an effective diffu-
sivity D
eff
which may exhibit a strong dependence on its own concentration.
These aspects will be examined with the diffusion phenomena of the specific
semiconductor.
4.3 Diffusion in Silicon
Because of its use in fabricating integrated circuits, Si is the most important
electronic material presently used and is likely to keep that position in the
future. Diffusion processes are essential in the electronic device fabrication
processes. Figure 4.1 shows the diffusivities of the most important impurities
and that of Si self-diffusion. Borrowing the knowledge from metals, diffusion
in Si had been first thought to be dominated by V -related mechanisms. In
1968 Seeger and Chik [1] suggested that in Si both V and I contribute to
self- and dopant diffusion processes. The problem of the dominant point de-
fect species in Si constituted a controversy for more than 20 years. Finally,
during the 1980s [2], it was generally accepted that both I and V have to
be taken into account, so as to consistently interpret the results of a large
number of different experiments. The main indications for the involvement
of I came from experiments performed under non-equilibrium native point
defect conditions, such as the influence of surface oxidation or nitridation on
dopant diffusion. In this regard, diffusion of the i-s species such as Au or Pt
was also crucial. What is still uncertain is the diffusivity and the thermal
equilibrium concentration values of I and V , to be mentioned later.
4.3.1 Silicon Self-Diffusion
Because I and V both contribute, the transport of Si self-atoms under ther-
mal equilibrium conditions is governed by the uncorrelated self-diffusion co-
efficient
D
SD
= D
I
c
eq
I
+ D
V
c
eq
V
, (4.2)
4 Diffusion in Semiconductors 167
Fig. 4.1. Asurvey
of diffusivity values of
the most important
impurities in Si and of
Si self-diffusion.
where D
I
and D
V
are respectively the diffusivities of I and V ,andc
eq
I
and c
eq
V
are respectively the I and V thermal equilibrium concentrations normalized
by the appropriate site densities into atomic fractions. As has been men-
tioned, native point defects may exist in several charge states. The observed
doping dependence of group-III and V dopant diffusion (Sect. 4.3.3) indicates
the contributions of neutral, positively charged, negatively and doubly neg-
atively charged native point defects, but it is not known whether all these
charge states occur for both I and V . Taking all observed charge states into
account we may write D
I
c
eq
I
as
D
I
c
eq
I
= D
I
o
c
eq
I
o
+ D
I
+
c
eq
I
+
+ D
I
−
c
eq
I
−
+ D
I
2−
c
eq
I
2−
, (4.3)
and an analogous expression for V .ThequantityD
I
c
eq
I
comprises the sum of
the concentrations of self-interstitials in the various charge states according
to
c
eq
I
= c
eq
I
o
+ c
eq
I
+
+ c
eq
I
−
+ c
eq
I
2−
. (4.4)
Therefore, the diffusivity D
I
is actually an effective diffusion coefficient con-
sisting of an weighted average of the diffusivities in the different charge states.
The same holds analogously for c
eq
V
and D
V
.
Self-diffusion in Si is studied by measuring the diffusion of Si isotope
tracer atoms D
T
which differs slightly from D
SD
of (4.1), since it contains
the correlation factors f
I
and f
V
in the expression D
T
= f
I
D
I
c
eq
I
+ f
V
D
V
c
eq
V
(see Chap. 1 for details). In the diamond cubic lattice, f
V
=0.5, and f
I
≤ 1
168 Teh Yu Tan and Ulrich G¨osele
(depends on the unknown self-interstitial configuration). Measured results for
D
T
are usually fitted to the expression D
T
= D
o
exp(−Q/k
B
T ), where D
o
is a pre-exponential factor, Q is the activation enthalpy, k
B
is Boltzmann’s
constant and T is absolute temperature. A summary of these results is shown
in Fig. 4.2 and elsewhere [3]. Tracer measurements do not allow to separate
I and V contributions to self-diffusion. Such a separation became possible by
investigating the diffusion of Au, Pt and Zn in Si, described in some detail
in Sect. 4.3.2. These experiments allowed a fairly accurate determination of
D
I
c
eq
I
but only a crude estimate of D
V
c
eq
V
derived from a combination of
different types of experiments [2, 3]. The results are
D
I
c
eq
I
=9.4 × 10
−2
exp(−4.84 eV/k
B
T )m
2
s
−1
, (4.5)
D
V
c
eq
V
=6× 10
−5
exp(−4.03 eV/k
B
T )m
2
s
−1
. (4.6)
The values of (4.5) and (4.6) are within the experimental range of the mea-
sured Si tracer diffusion coefficient D
T
. The doping dependence of Si self-
diffusion [4] allows to conclude that neutral as well as positively and nega-
tively charged point defects are involved in self-diffusion, but the data are
not accurate enough to determine the individual terms of (4.3) or the anal-
ogous expression for vacancies. Since D
T
as well as D
I
c
eq
I
and D
V
c
eq
V
each
consist of various terms, their representation in terms of an expression of the
type of (4.5) and (4.6) can only be an approximation holding over a limited
temperature range. In Sect. 4.3.5 we will discuss what is known about the
individual factors D
I
, c
eq
I
, D
V
and c
eq
V
.
4.3.2 Interstitial-Substitutional Diffusion: Au, Pt and Zn in Si
Both Au and Pt can reduce minority carrier lifetimes in Si because their
energy levels are close to the middle of the band gap. They are used in power
devices to improve the device frequency behavior. In contrast, Au and to
a lesser extent Pt are undesirable contaminants in integrated circuits and
hence have to be avoided or removed. For these reasons, the behavior of
Au and Pt has been investigated extensively. Zinc is not a technologically
important impurity in Si, but scientifically it served as an example with
diffusion behavior in between substitutional dopants and Au and Pt in Si.
These impurities are i-s species dissolved mainly on substitutional sites
(A
s
) but accomplish diffusion by switching over to an interstitial configuration
(A
i
) in which their diffusivity D
i
is extremely high. The change-over from
i-site to s-site and vice versa involves native point defects. For uncharged
species the two basic forms of the i-s change-over processes are the kick-out
(KO) mechanism involving I [4,5]
A
i
⇔ A
s
+ I, (4.7)
and the much earlier proposed dissociative mechanism or Frank-Turnbull
(FT) mechanism involving V [6]
4 Diffusion in Semiconductors 169
Fig. 4.2. Tracer self-diffusion coefficients of silicon as a function of reciprocal tem-
perature. Partly from [4].
1. R.F. Peart: Phys. Stat. Sol. 15, K119 (1966)
2. R.N. Ghostagore: Phys. Rev. Lett. 16, 890 (1966)
3. J.M. Fairfield, B.J. Masters: J. Appl. Phys. 38, 3148 (1967)
4. J.J. Mayer, H. Mehrer, K. Maier: Inst. Phys. Conf. Ser. 31, 186K (1977)
5. L. Kalinowski, R. Seguin: Appl. Phys. Lett. 35, 171 (1980)
6. J. Hirvonen, A. Antilla: Appl. Phys. Lett. 35, 703 (1979)
7. F.J. Demond, S. Kalbitzer, H. Mannsperger, H. Damjantschitsch:
Phys. Lett. 93A, 503 (1983)
A
i
+ V ⇔ A
s
. (4.8)
Both mechanisms are schematically shown in Fig. 4.3. The KO mechanism
is closely related to the interstitialcy mechanism. The main difference is that
the foreign atom, once in an interstitial position, remains there for only one
step in the interstitialcy mechanism and for many steps in the KO mecha-
nism. In contrast, the FT mechanism and the vacancy exchange mechanism
are qualitatively different. For the vacancy exchange mechanism an increase
in V concentration enhances the diffusivity, whereas for the FT mechanism
an increase in V concentration rather decreases the diffusivity of the substi-
tutional species.
For a detailed description of i-s atoms diffusing via the KO or FT mecha-
nism, it requires to solve a coupled system of three partial differential equa-
tions describing diffusion and reaction of A
i
,A
s
and the native point defect
170 Teh Yu Tan and Ulrich G¨osele
(a) Frank-Turnbull Mechanism
(b) Kick-Out Mechanism
V
I
A
i
A
s
A
i
A
s
Fig. 4.3. Schematic represen-
tation of the Frank-Turnbull
mechanism (a) and the kick-
out mechanism (b).
involved, V or I [4]. But for cases of in-diffusion from the surface the profile
shapes are described well by an effective diffusivity D
eff
obtained for use in
the Fick’s second law representation, (4.1), of the A
s
diffusion process. This
requires to assume that the local equilibrium conditions in accordance with
reactions (4.7) and (4.8) are
C
s
C
I
/C
i
= C
eq
s
C
eq
I
/C
eq
i
, (4.9)
C
i
C
V
/C
s
= C
eq
i
C
eq
V
/C
eq
s
, (4.10)
where C
s
and C
i
are respectively the concentrations of the s and i atoms
of the i-s species, holds for the KO and FT mechanisms, respectively. For
the KO mechanism, if the in-diffusion of A
i
is slow enough to allow the Is
generated to migrate out to the surface to keep their thermal equilibrium
concentration (i.e., D
i
C
eq
i
D
I
C
eq
I
holds), then the effective diffusivity of
A
s
is given by
D
eff
(i)
= D
i
C
i
/C
eq
s
, (4.11)
provided C
eq
s
C
eq
i
holds, which is generally the case. Here the lower index
in bracket indicates the rate limiting species. If on the other hand the in-
diffusion of A
i
is so fast that the generated I cannot escape quickly enough
to the surface (i.e., D
i
C
eq
i
D
I
C
eq
I
holds) a supersaturation of I will develop
and further incorporation of A
s
is limited by the out-diffusion of the generated
Is to the surface. This leads to an effective A
s
diffusivity D
eff
(I)
given by
D
eff
(I)
=(D
I
D
eq
I
/C
eq
s
)(C
eq
s
/C
s
)
2
. (4.12)
For the FT mechanism, sufficiently slow in-diffusion D
i
C
eq
i
D
V
C
eq
V
leads
to the same D
eff
(i)
as given by (4.11). An effective A
s
diffusivity D
eff
(V )
controlled