Numerical studies of anomalous fast diffusion in metallic alloys and semiconductors

www.elsevier.nlrlocaterapsusc

Numerical studies of anomalous fast diffusion in metallic alloys and semiconductors

A. Hasnaoui

^a,)

, A. Benmakhlouf

^a

, A. Hoummada

^a

, J.K. Naciri

^b

, A. Menai

^a

aU.F.R. de Physique des Particules et des Phenomenes de Transport, Faculte des Sciences Aın Chock, B.P. 5366 Maarif,´ ` ´ ¨ Casablanca, Morocco

bU.F.R. de Mecamique, Faculte des Sciences Aın Chock, B.P. 5366 Maarif, Casablanca, Morocco´ ´ ¨

Abstract

The so-called anomalous fast diffusion in metallic alloys and semi-conductors is often analysed within an interstitial–sub- stitutional model. The equations used for modelling these mechanisms are reaction–diffusion type whose analytical solutions are available only under drastic simplifications. The dissociative variety of this model is simulated using both finite

Ž . Ž . Ž

difference FD and Monte Carlo MC methods. In MC simulation, diffusion of different species interstitial impurities B ,_i substitutional B , and vacancies V and reaction jumps occur according to the suitable probabilities which are jump_s . frequencies-dependent. Whereas in FD method, an implicit scheme is used to solve the system of non-linear partial differential equations. In both cases, the finite source conditions have been considered. A good agreement between results obtained by the two methods is found. On the other hand, the double-stages of simulated profiles are found to have similar

Ž . Ž .

shapes to those obtained experimentally in Nb Co and in GaAs Zn . The first stage is well analysed by a Gaussian function, whereas the second one is well represented by an erfc type function. Furthermore, a detailed study of the two stages leads to a qualitative agreement with Stolwijk’s analysis in two limited cases where the diffusion is vacancy- or foreign interstitial-controlled. However, the effective diffusion coefficients present a quantitative departure from those obtained by Stolwijk’s expressions.q2000 Elsevier Science B.V. All rights reserved.

Keywords: Monte Carlo; Finite difference; Dissociative diffusion; Point defect; Profiles

1. Introduction

The long-range transport of predominantly substi- tutionally dissolved foreign atoms in systems like

) Corresponding author. Tel.: q212-2-23-06-80r82; fax:

q212-2-23-06-74.

Ž .

E-mail address: [email protected] A. Hasnaoui .

Ž . w x Ž . w x Ž . w x Ž

Nb Co 1 , GaAs Zn 2 , Ge Cu 3,4 and Pb noble

. w x

metals 5–8 is known to be governed by the disso- ciative mechanism introduced by Frank and Turnbull w x 9 . In this mechanism an impurity atom B can

Ž .

occupy a regular lattice site substitutional atom B

_s

Ž .

or an irregular lattice site interstitial atom B , while

_i

the diffusion occurs essentially by interstitial atom B

_i

migration in spite of the predominance of solute atoms B . The equilibrium between B and B atoms

_{s i s}

0169-4332r00r$ - see front matterq2000 Elsevier Science B.V. All rights reserved.

Ž .

PII: S 0 1 6 9 - 4 3 3 2 0 0 0 0 1 7 7 - X

and vacancies V is maintained following the reac- tions:

k_iv

B

ⁱ

q V m

_k

^B

^s

^{Ž . 1}

s

k and k

_{s iv}

represent respectively the creation and the annihilation frequencies or reaction rates of an inter- stitial impurity atom B and a vacancy V. The mass

_i

action law is given by:

k C

_{s s}⁾

s k C

_{iv i}⁾

C

_v⁾

Ž 1

^X

.

)

Ž .

where C

_p

p s i, s or v is the equilibrium concen- tration of specie p in at r at unit. If the C

_p⁾

concen- trations are expressed in cm

^y3

unit, the mass action law must be written as:

k C C

_{s 0 s}⁾

s k C

_{iv i}⁾

C

_v⁾

where C is the volume density of host atoms.

₀

Furthermore, the measures of diffusion under-

Ž . w x Ž . w x

taken on Nb Co 1 and GaAs Zn 2 lead to two stages penetration profiles characterised each one by an effective diffusion coefficient. Equations of diffu- sion which can model this mechanism have no ana-

w x

lytical solutions except in limited cases 10 for thin or thick deposits at the surface finite or infinite Ž source . The purpose of this work is to use numerical .

Ž Ž .

methods finite difference FD and Monte Carlo Ž MC .. to study this mechanism without needing to impose restrictive conditions.

2. Mathematical formulation and analytical solu- tions

To describe the coupled diffusion of the three species B , B and V involved in the dissociative

_{i s}

reactions Ž . 1 , the following system of non-linear

w x partial differential equations is invoked 11,12 :

E C

_i

E

²

C

_i

s D

_{i 2}

y k C C

_{iv i v}

q k C

_{s s}

Ž . 2

E t E x

E C

_v

E

²

C

_v

s D

v 2

y k C C

iv i v

q k C

s s

q k C Ž

v)

y C

v

.

E t E x

Ž . 3

E C

_s

E

²

C

_s

s D

_{s 2}

q k C C

_{iv i v}

y k C

_{s s}

Ž . 4

E t E x

Ž .Ž .

where C

_p

x, t p s i, v or s denotes the local concentration of specie p at the penetration depth x for the diffusion time t. The intrinsic diffusivities D

_p

depend only on temperature and thus are constant in a common isothermal diffusion experiment. The last term of Eq. 3 represents the vacancy concentration Ž . variation owing to internal sources and sinks dislo- Ž cations, surfaces . . . . .

Ž . Ž . The analytical solution of the system 2 – 4 cor- responding to a finite source in the two limited cases mentioned hereafter and under some simplifying hy-

w x pothesis is given by 10 :

C x ,t Ž . f C

s

Ž x ,t .

2 Q y x

²

s exp Ž . 5

' ^{p ( 4 D}

^exp

^t ž ^{4 D}

^exp

^t /

where Q denotes the total quantity of diffusing substance per unit area, and D

_exp

is the effective diffusion coefficient of the impurity whose expres- sion depends on the nature of the defect controlling the diffusion.

Ž . i In vacancy-controlled diffusion, the observed effective diffusion D

_exp

is expressed as:

2C

_v⁾

D

_{v 2}

4 D

_exp

t s 4 D

_effŽ1.

t s Ž t qt

₀

. Ž . 6 C 0,0

_s

Ž . t

₀

and the surface concentration of solute atom is:

t

₀

C 0,t

s

Ž . s C 0,0

s

Ž . Ž . 7 t q t

₀

where t

₀

is a parameter which depends on C

_v⁾

, D

_v

Ž .

and C 0, 0 .

_s

Ž . ii When the diffusion is governed by interstitial

Ž .

atoms, D

_exp

and C 0, t are given by:

_s

C

_i⁾

4 D

_exp

t s 4 D

_effŽ2.

t s 4

₎

D t

_i

Ž . 8 C

_s

and:

Q 1

C 0,t

s

Ž . s . Ž . 9

D ' p t

(

^exp

In the case of an infinite source condition, the pene- tration profile is an erfc type function, whereas the effective coefficient diffusion is constant for the two limited cases. In an interstitial governed diffusion, the effective diffusion coefficient is given by expres- sion 8 . Ž .

3. Description of the simulation methods

3.1. Finite difference method

To solve the system of non-linear partial differen- Ž . Ž .

tial Eqs. 2 – 4 , we use the FD method. The dis- critisation of these equations is done according to an implicit scheme which removes the stability con- straints of the explicit treatments presented in the

w x

literature 13 and allows time steps considerably larger. This technique require the knowledge of ini- tial and boundary conditions.

3.1.1. Boundary conditions

One supposes that the concentration of solute atoms deposited on the surface decreases with time only by the diffusion process to the interior of the sample, therefore the variation of the concentrations

Ž .

of the different species on the surface x s 0 is given by:

D D t

n p

nq1 nq1 nq1

C s C q C y C

Ž

^p

.

¹

Ž

^p

.

¹

_{D x}

²

ž Ž

^p

.

²

Ž

^p

.

¹

/

Ž 10 . Ž .

ⁿ

where Cp

_1,2

is the concentration of the specie p at

Ž .

^ieme

the plan 1,2 in the n time step.

At the other sample side Ž x s L, L is the thick- ness of the sample , there is no loss of matter; hence: .

™

grad C Ž . s 0. Ž 11 .

Ž

p

.

xsL

3.1.2. Initial conditions y x

²

0 )

C

_s

Ž x ,0 . s C exp

_s

ž ^{2 D}

₂

/ ; C

_v

Ž x ,0 . s C

_v

; x 12

Ž .

C x ,0

i

Ž . s 0 x Ž / 0 ; . C 0,0

i

Ž . s C

i⁰

Ž 13 . D is the width to the half height of the Gaussian- shaped initial distribution.

3.2. Monte Carlo method

The diffusion of interstitial atoms B , vacancies V

_i

Žinvolving the exchange with substitutional impurity

Scheme 1. a Jump frequencies for an interstitial atom in the neighbourhood of vacancy in an fcc lattice. The possible interstitial sites areŽ .

Ž . Ž . Ž

numbered in order of increasing distance from the vacancy open square . b Vacancy jumps near a substitutional solute atom B_s full

. Ž .

circle and host atoms A open circles in 2D representation.

Fig. 1. Normalised concentration profiles obtained for various diffusion times. The solid lines represent fits corresponding to Eq.

Ž14 . a FD result with D. Ž . is10^{y1 0} cm²rs, Dss1.6=10^y15

2 y12 2 y2 y1

cmrs, D_vs10 cmrs and k_ivs6.25=10 s . b MCŽ . result with v₀s0.2275, gsk_ivrk₀s1, k₀s1 and a is the lattice parameter.

atoms B

_s

or with lattice atoms A . and reactions involved in the equilibrium 1 are simultaneously Ž . taken into account in the MC simulation method which is done on a f.c.c. lattice containing 256,000 regular sites and equal number of irregular sites Ž octahedral sites . Whenever a defect B , B or V . Ž

_{i s}

. leaves the lattice, periodic boundary conditions are applied to bring the defect into the lattice. In each

Ž .

run walk , an initial configuration is generated by randomly distributing different species on lattice sites according to the relative concentration of each of

Ž .

them. Then a randomly chosen defect B , B or V

_{i s}

undergoes a test of a free jump or a reaction jump

according to the calculated probabilities which de- pend on the appropriate frequencies. The B atom

_i

free jumps occur with the frequency k . In the

₀

neighbourhood of a vacancy, B can react with this

_i

one to form a B atom according to the annihilation

_s

Ž .

probability p

_a

s k

_iv

r k

_iv

q 12 k

₀

or perform a reori- entation and a dissociation jumps with the probabil-

Ž .

ity 1 y p . The B atom can jump to a randomly

_{a s}

chosen site from the six nearest neighbour interstitial sites with the frequency k creating thus a B atom

_{s i}

and a vacancy V for meaning of k , k and k Ž

_{0 s iv}

see Scheme 1a . When the vacancy jump is fulfilled . according to the frequency v

₀

, the jump correspond- ing to an exchange with a B atom or with a solvent

_s

Fig. 2. Normalised concentration profiles relative to B atoms fulls Ž

. Ž . Ž . Ž

circles and to B atoms open circles . a FD simulation same as_i

. Ž . Ž .

in Fig. 1a ; b MC simulation same as in Fig. 1b .

Table 1

Ž . Ž . Ž . Ž

Comparison between the parameters 4 D t and 4 D t for various times. a FD results same as in Fig. 1a ; b MC results same as in Fig._{1 2} 1b.

Ž .a Time minŽ . 0.33 10 40 300 1200

2 y14 y12 y11 y10 y10

Ž .

4 D t cm1 9.01=10 3.73=10 1.52=10 1.17=10 4.96=10

2 y9 y7 y7 y6 y6

Ž .

4 D t cm₂ 8.69=10 2.33=10 8.11=10 2.49=10 3.22=10

y5 y5 y5 y5 y4

D₁rD₂ 1.03=10 1.60=10 1.87=10 4.70=10 1.54=10

Ž .b Time mcsŽ . 2000 6000 10,000 14,000

Ž2.

4 D t a₁ 0.128 0.286 0.686 1.152

Ž2.

4 D t a₂ 650 989 1265 1340

y4 y4 y4 y4

D₁rD₂ 1.97=10 2.89=10 5.42=10 8.59=10

atom A has to be done with frequency v

_B

or v

_A

Ž Scheme 1b . .

The diffusion profiles are obtained using the tech- w x

nique developed by Murch 14 in the vacancy diffu- sion case. All simulated profiles are averaged over a

Ž .

certain number of runs walks ‘‘n

_walk

’’, each one

Ž .

consisting of N Monte Carlo steps mcs which is a measure of diffusion time.

4. Results and discussion

The simulation parameters are chosen in order to reproduce the two stages profiles. A preliminary MC simulations using a realistic value of the initially concentration ratio C

_i⁰

r C

_s⁰

equal to 0.01 have been performed. The obtained profiles present very large fluctuations and require very long computing times.

However, when using the value C

_i⁰

r C

_s⁰

s 0.1 we found that the simulated profiles have the same shape and keep the same behaviour in spite of the difference between the two values. So, we use this later value to reduce computing time and fluctua- tions. Hence, the MC simulation results presented here are obtained for C

⁰

s 4.54 = 10

^y5

at r at, C

⁰

s

i s

y4

Ž

^{0 0}

.

⁾

4.54 = 10 at r at C r C s 0.1 and C s 4.22 =

i s v

10

^y4

at r at, which are the defect concentrations ini- tially distributed in the crystal bulk. According to the Murch technique, this allow to get the initial mean surface concentrations C

_i⁰

s 14.5 at r plan and C

_s⁰

s 135 at r plan for B and B atoms, respectively. To

_{i s}

make the simulation efficient, the various jump and reaction frequencies are in k unit. The ratio,

₀

v

_A

rv

_B

is fixed at 0.1 while n

_walk

s 500. In FD simulation, a

Ž .

dislocation-free crystal k s 0 is considered and the concentrations are taken in cm

^y3

unit for calculus

Fig. 3. Variation of the fit parameter 4 D t in respect to D for1 v

various times. Solid lines correspond to linear fits. a FD results;Ž . Ž .b MC results.

Fig. 4. Time evolution of the parameter 4 D t with the fit on Eq.₁ Ž . Ž .6 . a FD results same as in Fig. 1a ; b MC results same as inŽ . Ž . Ž Fig. 1b ..

convenience. The following values of C

_s⁰

s 10

²²

y3 0 19 y3

Ž

^{0 0 y3}

.

^{) 20}

cm , C

_i

s 10 cm C

_i

r C

_s

s 10 , C

_v

s 10 cm

^y3

, and L s 50 m m are used, while D , D and

_{i s}

D

_v

satisfy the condition D

_i

% D

_v

% D and k

_{s iv}

is

Ž

^X

.

related to k

_s

via the Eq. 1 . On the other hand, w x

previous study 15 showed that, in FD simulation, the variation of the width to the half height D of the Gaussian distribution do not affect the diffusion pro- files for values lower than 3 A. So, we use this value ˚ in all FD simulations. Furthermore, we use space step D x s 10 A and time step ˚ D t s 2 s to get sufficient precision.

4.1. Obtained profiles

We draw, in Fig. 1a, the profiles obtained by FD simulation, and in Fig. 1b, those relative to MC

results, for various times. In these profiles we re- trieve the two stages of diffusion obtained experi- mentally in NbCo 1 where the deposit at the sur- w x face corresponds to a finite source. Note that the two

Ž . w x stages have been also found in GaAs Zn 2 with an infinite source condition. For more details, we pre- sent, in Fig. 2, the penetration profiles relative to

Ž .

interstitial atoms C x, t and those relative to sub-

_i

Ž . Ž .

stitutional atoms C x, t

_s

FD-2a; MC-2b . One can observe an inversion between the C and C concen-

_{i s}

tration curves, when time increases, which occurs at the second stage. When decreasing C

_i⁰

r C

_s⁰

and r or C

_v⁾

the inversion is biased toward shorter times.

Otherwise, C profiles shape is similar to that of total

_s

Ž .

solute atoms C x, t , whereas the C concentration

_i

remains constant in the first stage depths for a fixed time and undergoes a drop in the second stage.

Ž .

Fig. 5. Time evolution of surface concentration C 0, t fitted on1

Ž . Ž . Ž . Ž . Ž

Eq. 7 . a FD results same as in Fig. 1a ; b MC results same as in Fig. 1b ..

4.2. Profiles analysis

w x

In an analytical study, Stolwijk 10 showed that the shape of diffusion profiles depends on boundary

conditions and on the mechanism controlling the solute atoms penetration. The parameter values adopted, in our simulations, correspond to intermedi- ate cases. Hence, it is more suitable to try the

Ž . Ž .

Fig. 6. Time evolution of the extrapolated surface concentration of the second stage C 0, t for various values₂ g or k_iv. The solid lines

Ž . Ž . Ž . Ž . Ž .

correspond to fit on Eq. 15 . a FD results same as in Fig. 1a ; b MC results same as in Fig. 1b .

simulated penetration profiles fit on a function which is a sum of the two stolwijk’s solutions. Various tests

w x showed that the best fit is obtained for 15,16 :

x

²

C x ,t Ž . s C 0,t exp

1

Ž . ž y ^{4 D t}

¹

/

x

q C 0,t erfc

2

Ž . ž ^{2 D t (}

²

/ Ž 14 . where D

₁

and D

₂

are respectively the effective diffusion coefficients of the first and the second

Ž .

stage. The curve fits Fig. 1a and b lead to 4 D t

₂

values very larger than those of 4 D t. For example,

₁

some values of these parameters are reported in Table 1 for FD and MC methods.

The first stage being well approached by a Gauss- ian function except for short times where the fit curve falls below the numerical points, the condition used in both methods correspond to a finite source, while the second stage is well fitted by an erfc type function, suggesting that the first stage is acting as an infinite source for the second one in the consid- ered time domain.

Otherwise, to identify the mechanisms controlling each stage, various simulations have been done for different diffusion coefficient D and D values and

_{i v}

Ž .

for different g or k

_iv

values. In FD simulation, the reaction rate k

_iv

is calculated using the mass action law and taking the crystal density C as 6.25

₀

= 10

²²

cm

^y3

.

4.3. First stage

As shown in Fig. 3, the parameter 4 D t increases

₁

linearly with increasing D

_v

for both methods. The straight line slope p t Ž . being time-dependent, the effective diffusion coefficient D can be expressed

₁

Ž . Ž .

by D

₁

s p t D as predicted by Eq. 6 . Note also

_v

that the values of 4 D t are not affected by the D

_{2 v}

variation.

Furthermore, the time evolution of the parameter 4 D t, plotted in Fig. 4, is also well fitted by Eq. 6 .

₁

Ž . The curve fits lead to the following:

t

0

s Ž 3317 " 195 mcs and 2C .

v⁾

D

v

r C 0,0

s

Ž .

s 1.235 = 10

^y5

a

²

r mcs MC results Ž . t

0

s Ž 16,730 " 1907 s and 2C .

v⁾

D

v

r C 0,0

s

Ž .

s Ž 2.3 " 0.1 . = 10

^{y1 5}

r cm

^y1

s

^y1

Ž FD result . .

Ž .

On the other hand, the C 0, t variations with time

₁

are drawn in Fig. 5 together with fit curves on Eq.

Ž . 7 . The values of the fit parameters are:

t

₀

s Ž 3378 " 65 mcs and C 0,0 .

_s

Ž . s Ž 135.9 " 2.5 at . r plan MC results Ž . t

0

s Ž 14,906 " 22 s and C 0,0 .

s

Ž .

s Ž 2.85 " 0.01 . = 10

²¹

cm

^y3

Ž FD results . . We notice the good agreement between the values of

Ž .

t

₀

obtained from C 0, t and 4 D t time evolutions.

_{1 1}

Ž .

In the same time, the obtained values of C 0, 0 are

_s

Table 2

Ž . Ž . Ž . Ž . Ž .

The parameter values resulting from fit of C 0, t on Eq. 15 for various2 g or kiv values. a FD results, b MC results

y2 y1 y7 y1 19 y3 y5 y1 20 y3 1r2

Ž .a kivŽ10 s . ksŽ10 s . p1Ž10 cm . p2Ž10 s . p3Ž10 cm s .

6.25 1 1.61"0.01 6.00"0.18 0.91"0.01

18.75 3 2.95"0.01 8.00"0.10 0.93"0.01

37.50 6 4.12"0.01 11.00"0.08 0.95"0.01

62.50 10 5.20"0.01 15.00"0.28 0.96"0.01

y5 y5 y1 1r2

Ž .b gsk_ivrk₀ k_sŽ10 mcs. p₁Žatrplan. p₂Ž10 mcs . p₃Žatrplan mcs .

0.5 2.11 0.57"0.03 7.59"0.55 14.65"0.12

0.7 2.95 0.74"0.03 8.07"0.53 14.96"0.15

1 4.22 0.89"0.03 10.74"0.58 15.47"0.23

2 8.44 1.28"0.01 13.69"0.34 15.99"0.22

10 42.2 1.78"0.01 37.67"1.31 16.43"0.78

close to those initially distributed at the surface, namely 135 at r plan and 10

²²

cm

^y3

for MC and FD, respectively.

However, we note, for both methods, a discrep- ancy between the numerical C

_v⁾

values deduced

)

Ž .

from 2C D

_{v v}

r C 0,0 and those initially considered.

_s

This leads to some ambiguity in the C

_v⁾

value to be used in analytical calculations. We conclude that the first stage is vacancy-controlled one, and the laws predicted by the Stolwijk’s analysis are followed even when the conditions are not strictly those of the vacancy-limited case.

Ž . Ž . Ž . Ž .

Fig. 7. Time evolution of the parameter 4 D t for various values of₂ gor k_iv. The fits are taken from Eq. 17 continuous lines . a FD

Ž . Ž . Ž .

results same as in Fig. 1a ; b MC results same as in Fig. 1b .

4.4. Second stage

Ž .

For this stage, the parameters C 0, t and 4 D t

_{2 2}

Ž .

are studied for various D and

_i

g or k

_iv

values. We obtain here as in the first one, a linear variation of 4 D t vs. D , suggesting that this stage is due to an

_{2 i}

interstitial diffusion. As it can be observed in Fig. 1,

Ž . Ž

C

₂

x , t

₀

where x

₀

is the depth corresponding to the second stage beginning . could be well ap-

Ž .

proached by C 0, t , hence we use, in the follow-

₂

Ž .

ing, C 0, t as an approached value of the second

₂

stage beginning concentration.

Otherwise, the extrapolated surface concentration

Ž .

C 0, t drawn in Fig. 6 presents a decreasing for

₂

short times, an increasing for intermediate times and a saturation for long times. This saturation is reached

Ž .

at shorter times when g or k

_iv

is higher.

Ž .

The evolution of C 0, t concentration is well

₂

Ž .

analysed by Fig. 6 :

C 0,t

₂

Ž . s C

_s2

Ž 0,t . q C

_i2

Ž 0,t . p

₃

s p 1

1

Ž y exp Ž y p t

2

. . q . Ž 15 .

' t

The first part of the right hand is the solution of the equation:

E C

_s

s k C C

iv i v

y k C

s s

Ž 16 . E t

which describe the B atoms penetration for depths

_s

larger than x , by neglecting D and assuming, in a

_{0 s}

first approximation, C C

_{i v}

as constant. The second

Ž .

part takes into account the decreasing of C 0, t

_i

when the diffusion term is predominant in Eq. 2 for Ž . which the solution could be a Gaussian shape in the

Ž .

short time and low g or k

_iv

value domains. Hence,

Ž .

the decrease observed in the beginning of C 0, t

₂

Ž .

curves is due to C 0, t diminution. The values of

_i

the fit parameters are reported in Table 2.

We can see from this table, that p is increasing

₁

Ž .

with increasing g or k

_iv

and tends toward its equi- librium value C

_v⁾

. This could be explained by the

Ž .

fact that when g or k

_iv

increases, the annihilation reactions of Eq. 1 are more frequent. Furthermore, Ž .

Ž .

the p

₂

value increases linearly with g or k

_iv

but presents some departure from analytical value which

Ž .

must be equal to k , particularly for lower

_s

g or k

_iv

values. The p parameter presents a weak increasing

₃

Ž .

when g or k

_iv

increases which reveals a slowdown in the B atom diffusion due to the reaction jumps.

_i

Thus, the reaction terms could not be neglected when

Ž . Ž .

solving the Eq. 2 for g or k

_iv

high values.

Otherwise, the time evolution of 4 D t for various

₂

Ž .

g or k

_iv

are presented in Fig. 7. The obtained curves are not linear and 4 D t decreases with in-

₂

Ž .

creasing g or k

_iv

. This evolution is due to the ratio

Ž . Ž .

C 0, t

_i

r C 0, t which is not constant in contrast to

₂

Eq. 8 . Hence, the effective diffusion coefficient D Ž .

₂

must be written as:

C 0,t

_i

Ž .

D

₂

s D

eff_i

s D .

_i

Ž 17 .

C 0,t

₂

Ž .

Indeed, the numerical results relative to 4 D t are

₂

Ž . Ž .

well fitted to Eq. 17 where C 0, t is expressed by

₂

Ž . Ž .

^y1r2

Eq. 15 and C 0, t

_i

is proportional to t as already noted. For long times, we retrieve a constant effective diffusion coefficient D as expected from

₂

Eq. 8 . Ž .

5. Conclusion

The simulation of the dissociative mechanism by

MC and FD methods for intermediate cases, where

analytical solutions are not available in literature,

leads to double-stage diffusion profiles found experi-

mentally in some metallic alloys and semi-conduc-

tors. An inversion between interstitial concentration

and substitutional one has been observed at the

second stage depths when increasing time. The anal-

ysis of the two stages suggests to attribute the first

one to a vacancy-controlled-diffusion, whereas the

second is due to a fast interstitial atom diffusion. The

time evolution of surface concentration and diffusion

coefficient of the first stage are in good agreement

with the analytical study of Stolwijk, in the case of

the vacancy-controlled regime. The extrapolated sec-

ond stage surface concentration has been found to

show a decreasing, an increasing and a saturation

when time increases. However, some discrepancies

between numerical and analytical values have been

noted which are due to the simultaneous presence of the two limited cases in all our simulations. In spite of this departure, the Stolwijk analysis based on two limited cases seems to be qualitatively valid for intermediate cases.

References

w x1 F. Wenwer, N.A. Stolwijk, H. Mehrer, Z. Metallkd. 80 Ž1989 205..

w x2 B. Tuck, M.A. Kadhim, J. Mater. Sci. 7 1972 585.Ž . w x3 N.A. Stolwijk, W. Frank, J. Holzl, S.J. Pearton, E.E. Haller,¨

Ž .

J. Appl. Phys. 57 1985 5211.

w x4 N.A. Stolwijk, F. Wenwer, H. Bracht, H. Mehrer, in: Inter.

Ž .

Conf. Diff. in Mat., Aussois France , 1989.

w x5 W.K. Warburton, D. Turnbull, in: A.S. Nowick, J.J. Burton

ŽEds. , Diffusion in Solids, Academic Press, New York,. 1975, p. 171.

w x6 Chr. Herzig, Dimeta-82, Diffus. Defect Monogr. Ser. 7 1983Ž . 23.

w x7 A. Menai, H. Amenzou, G. Moya, J. Bernardini, Acta Metall.

Ž .

Mater. 42 1994 4049.

w x8 G. Moya, X.H. Li, A. Menai, E.L.M. Kherraz, H. Amenzou,

Ž .

J. Bernardini, P. Moser, Mater. Sci. Forum 175–178 1995 457.

w x9 F.C. Frank, D. Turnbull, Phys. Rev. 104 1956 617.Ž . w10 N.A. Stolwijk, Phys. Status Solidi B 157 1990 107.x Ž . w11 G. Edelin, Phys. Status Solidi B 98 1980 699.x Ž . w12 M.D. Sturge, Proc. Phys. Soc., London 73 1959 297.x Ž . w13 M.D. Zahari, B. Tuck, J. Phys. D: Appl. Phys. 18 1985x Ž .

1595.

w14 G.E. Murch, Am. J. Phys. 47 1979 78.x Ž .

w15 A. Benmakhlouf, A. Menai, Alg. J. Adv. Mater. 3 1999 inx Ž . press.

w16 A. Hasnaoui, A. Hoummada, A. Menai, Alg. J. Adv. Mater.x

Ž .

3 1999 in press.