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A model for the dynamics between paired and unpaired diffusion for a binary system
Volume 137, number 3
PHYSICS LETTERS A
8 May 1989
A MODEL FOR THE DYNAMICS BETWEEN PAIRED AND UNPAIRED DIFFUSION FOR A BINARY SYSTEM M. ORLOWSKI Siemens AG, CorporateResearch and Development,Otto-Hahn-Ring 6, 8000 Munich 83, FRG Received 3 January 1989; revised manuscript received 1 March 1989; accepted for publication 9 March 1989 Communicated by A.P. Fordy
A rigorous diffusion model is proposed for two species A and B which are allowed to diffuse partly by themselves and partly in pairs AB. The dynamics of this system is studied in terms of pairing probability and in terms of the relations among the diffusivities, DA, Da, DAa, for the unpaired and paired diffusion, respectively. Depending on the values of these quantities, the system displays a large variety of seemingly irreconcilable effects observed in experiments. It turns out that the highly non-linear character of these effects, for instance the uphill diffusion, is still based on a simple underlying Fickian (i.e. random walk ) mechanism.
Although binary systems of diffusing species have been occasionally considered in various fields [1 ], mostly in connection with chemical reactions, a rigorous model for the dynamics of two species diffusion partly in pairs and partly unpaired has been hitherto not available. The pair diffusion involving vacancies and interstitials gained some prominence in diffusing in metals [2]. More recently diffusion of dopants in semiconductors via impurity-point defect pairs [ 3-6 ] has been explicitly considered in the pertinent diffusion equations. However, the impurity-point defect pair dynamics has been considered only approximately, being appropriate for the kinetics of the point defects which is determined to a large extent through equilibrium values of the thermodynamic force of the crystal lattice itself. The present investigation has been motivated by the recent theoretical and experimental work by Aronowitz and collaborators [ 7- I 0 ]. Quantum chemical calculations on a model silicon lattice have predicted the interactions between p-type dopants of different atomic species to be attractive. For overlapping interstitial populations of aluminum and arsenic of aluminum and phosphorus a repulsive interaction has been predicted [7,8]. These predictions have been verified in diffusion experiments [ 9, l 0 ] indirectly resulting in the pull (attractive interaction)
and the push effect (repulsive interaction) between the pertinent dopant species. However, a careful analysis of the experimental resuits in ref. [9] and also of experiments performed by Orlowski and Mader [ 11 ] displays a much richer structure than expected suggesting a diffusion of each species being partly unpaired and partly in bound pairs as a result of the attractive interaction. In this Letter a rigorous model of a system of two diffusing species with allowance for a reaction to form a mobile compound of a bound pair is derived and its dynamics investigated. The model is applicable to various fields involving diffusion phenomena and is therefore of general character. In particular, the nature of effects observed in the experiment, such as uphill and dip modi, enhancement and retardation of the diffusion and the drag effect are for the first time clarified and the effects explained consistently in terms of a nonlinear diffusion dynamics generated by a simple pairing mechanism. Consider a medium (or phase) S, which may be, for example, a crystal, with a two-component system of species A and B. It is assumed that each of the species in S is partly bound in pairs AB and partly unpaired according to the pairing reaction: kl
A+B.
0375-9601/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )
k2
" AB. 115
Volume 137, number 3
PHYSICS LETTERS A
The diffusion dynamics of the binary system involving the above pairing reaction can be described by the following reaction-diffusion equations: o c ~ n p __1') Ot --aJA
02c~nP OX 2
klr~unpc-'unp.l_k2CAB, ~"A
"~B
OC~"p OzC~np O----{-= DB Ox----5--- kl C~,"p C~"p + k2 CAn,
OCAB I--'A ~"~B Ot --DAB~02CAB +ktr"nPr""P-k2CAB' where C~ rip, C~ np and CAB denote the concentration of the unpaired species A and B, and the concentration of pairs AB, respectively. Here, t is time, x is spatial position in the domain [0, L] and O2/Ox 2 is the Laplacian with respect to x. k~ denotes the pairing rate and k2 the rate of pair decay. DA (DB) and BAa denote diffusivities for the unpaired A (B) and for the pairs AB, respectively. It is reasonable in terms of physics and sufficient in terms of the completeness of the dynamics contained in the model to consider the pairing reaction in thermodynamic equilibrium. Quantum mechanical calculations of dopants in a crystal lattice on interstitial sites [ 12 ] and experimental perturbed ~ angular correlation studies [ 13 ] indicate that the pair binding energy is considerably smaller than the activation energy (potential barrier height in the Arrhenius law) of the dopant diffusivities DA and DB. This means that under such circumstances the processes for pairing and pair decay are much faster than the characteristic diffusion time z ~ d 2 / D (D denoting the diffusivity and d denoting a characteristic distance of an elementary displacement). Furthermore and more important, if the pairing and depairing mechanisms are slower than the diffusion mechanisms, then the diffusions of species A and B become decoupled from one another and determined predominantly by the diffusivities DA and DH. The dynamics of the system becomes not trivial only when the characteristic reaction time becomes at least comparable with the characteristic diffusion time. The closer the equilibrium limit is reached the more pronounced are the effects resulting from the interaction between the two species. Therefore departures from the equilibrium limit do not imply any new features of the dynamics considered. Hence it is appropriate to consider the ther116
8 May 1989
modynamic equilibrium to characterize completely the dynamics of the system. Thermodynamic equilibrium occurs when CAs=aCAunp CBunp ,
(1)
with a=k~/k2. The total concentrations CA and Ca for both species are given by CA = C~,np + CAB,
(2a)
Ca = C~np + CAn.
(2b)
It is important to describe the entire dynamics in terms of total concentrations, because only total chemical (and less reliably the total electrically active) concentrations are accessible to experimental evaluation [ 14 ]. Besides the equilibrium constant a strongly depends on the temperature Timplying that the number of pairs varies strongly with T, i.e. it is different at elevated temperatures where the diffusion occurs from the number of pairs at low temperatures during the measurement. This means that even with an experimental technique at hand allowing a measurement of CAB, one would have to perform the measurements at elevated temperatures (in case of silicon crystal 600-1200°C), which is not feasible for the experimental techniques nowadays. Inserting eq. ( 1 ) in eq. (2) one obtains algebraic equations for C~"p and C~np, which can be solved analytically. The result for species A is given by C~,.p = (2a)-1{ [ 1 +2a(CA+CB) +aZ(CA - Ca) 2] 1/2
--[ I +a(CB--CA) ]} .
(3)
The solution for the species B is obtained by interchanging simply the indices A and B. Thus the unpaired concentration of each species depends on the total concentration of both species and on the reaction rate a. Assume now that the respective species are transported partly unpaired and partly in pairs AB. Thus the total fluxes JA and JB are given by JA =j~np +JAB,
JB =j~np +JAB.
For both fractional fluxes contributing to the total flux a simple Fickian diffusion mechanism is assumed in accordance with the model introduced above, i.e.
Volume 137, number 3
j~np = D A OC~nO/OX,
PHYSICS LETTERSA
J~np = D B O C ~ n p / o x ,
JAB = D A a O C A B / O X "
The rate of change of the total concentration CA is then given by 0CA - -
Ot
-
O
Ox
(JI"P
+JAa)
0 - Ox (DAOC~"P/OX+DABOCAa/OX) "
(4)
An analogous equation holds for species B. Using eqs. ( 1 )- (3) the concentrations of the unpaired species and the concentration of the pairs AB in eq. (4) can be eliminated and the r.h.s, ofeq. (4) can be expressed in terms of total concentrations CA and Ca. After some manipulations one obtains the final form of diffusion equations for a binary diffusion system including the diffusion by pairs:
OCA O( l Ot - ~x [~(DA + DAa)+ (DAa--DA)fAA] OCAox OCB )
+ (Oha--Og)(½+faa) OX 1 '
ot
-
[½(Da+DAaI+(DAa--DaIA~I
ocA)
+ (Dna - Da) ( ½+fkA) --~--X] '
(5a)
OC,~ox ( 5b )
where the dynamic factor fna is given by
a C a - ½ [ l +a(CA +Ca) l fgA = { [ 1 +or(CA + Ca) ]z--4azcA Ca } ,/z" The dynamic factorfaa can be obtained from the last equation by interchanging the indices A and B. Eq. (5) reduces to a simple second Fickian law equation (independent of the extant species) if we assume that the diffusivity of the species under consideration, say A, is the same as the diffusivity for pairs AB, i.e. DA =DAa=D; as expected from the general considerations one obtains
OCA =D 0 2CA Ot OX 2
"
It should be noted that now the diffusion of species
8 May 1989
A is independent of the diffusion of species B; firstly the diffusivity of the direct term is independent of any concentrations, and secondly the cross term proportional to the gradient of species B vanishes now, because DAB- DB = 0. Thus although species A is still partly unpaired and partly bound in pairs, this difference degenerates on the level of fluxes and rates, for it cannot be distinguished whether the transport is effectuated by pairs or by unpaired species, since the efficacy of both transport mechanisms is the same. In order to solve eq. (5) it is, in general, necessary to prescribe both initial distributions and boundary conditions. For convenience, we shall adopt Neumann boundary conditions OCi/OXJx=O.x=L=O for i=A, B. These boundary conditions imply automatically the conservation of the total species A and B, as realized in most experimental situations (introduction of the dopants by implantation). Adoption of Dirichlet boundary conditions has no impact on the basic mechanisms contained in the model; it leads merely to quantitative modifications of the final profiles particularly close to the boundaries. Eq. (5) represents the first rigorous prototype model to study pairing dynamics. The extant models for impurity-point defect pairs in silicon were marred by two circumstances: Firstly the diffusivities of the point defects are orders of magnitude faster than the diffusivities of the dopants. Secondly and more important, the thermodynamics force, which introduces some degree of entropy into the crystal lattice in order to minimize the free energy above 0 K, drives the concentrations of the point defects very quickly (on the characteristic time scale of impurity diffusion) into equilibrium values. Nevertheless some aspects of the dynamics contained in eq. (5) are displayed by systems involving point defects, if one is able to produce highly inhomogeneous distributions of these species by the presence of crystal defects [ 15 ] or by irradiation [ 16-19 ]. Eq. (5) is also a paradigm of general structure for the analysis of diffusion equations given by Hu [ 20 ] almost twenty years ago, who pointed out, that even a simple system needs in general to be formulated using a complete set of equations of irreversible thermodynamics, including cross terms from all constituents. In general, a simple Fick law is inapplicable, even for random walk basic mechanisms. The diffusion flux is not just given 117
Volume 137, number 3
PHYSICS LETTERS A
by one term associated with the gradient of the respective species, but is also contributed by cross terms which associated with gradients of the extant species, may in some cases actually dominate. In the following we discuss the dynamics of the system eq. (5) by means of a few examples, which illustrate such effects as the uphill and dip modi, retardation and enhancement of the diffusion. These effects and the drag effect can now be explained in a unifying way by the non-linear diffusion dynamics proposed above. Consider the first example with the initial distribution for species A and B given as in fig. 1 by solid lines. Further assume that DB < DAB< DA. The distribution of species A and B after the diffusion for some time is indicated by dashed lines. It can be clearly seen that the species A exhibits an uphill diffusion effectuated by the cross term (second term) in eq. (5a). This result is easily explained: In the vicinity of Xo (interval I) a large percentage of species A will be bound in pairs (depending on the reaction constant ct). Outside of I there will be almost no AB pairs because the concentration Ca is negligible. Outside of the interval I the transport of A is accomplished mainly via the diffusion of unpaired species A, whereas within the interval I the transport of species A by pairs will be of increasing importance. It should be noted that pairs AB move much more slowly than the unpaired species A. This means that initially there is much larger flux of the species A en-
I CA,CB DAB
8 May 1989
tering the region I than leaving it, until a sufficient gradient in the distribution of the species A builds up. This leads to an accumulation of species A in I and depletion of species A outside of the interval I. An opposite effect - the dip modus of diffusion can be obtained by assuming DB < DA < DAB. Here the pairs AB move faster that the species A leading to a depletion of the species A in the region I and to accumulation in the vicinity of I which diminishes with increasing distance from the region I. This is illustrated in fig. 2. For both cases it is clear that initially the redistribution of the species A is provided solely by the cross term (OCa/Ox), since the contribution of the direct term vanishes due to the initial uniform distribution of A. As the diffusion proceeds the direct term comes into existence and opposes the action of the cross term leading to a temporarily quasistable distribution for species A, due to the slow diffusion of species B. The degree of order for the distribution A increases. Of course, after sufficiently long time the inhomogeneity of the species A will gradually smear out as the initially highly inhomogeneous distribution of species B also smears out. In fig. 3 the effect of the retardation of the species A in presence of species B is shown. We assume DAB< DA. In case of codiffusion of species A and B a uniform initial distribution of species B has been assumed. The retardation effect for species A in presence of species B is clearly recognized. Conversely the diffusion of the species A will be enhanced in presence of species B if DAn > DA. A related effect to the uphill diffusion effect and
eA, CB
/~--~
T
. .~. . . ' 7
DAB >DA
f~ I ~B
I
1
I ?" . . . . .
I
~-
I
I
I
!1 1"
IB
Xo
Bt
Distance Fig. 1. Uphill modus of the diffusion. Initial distribution of species A and B is given by solid lines. Final distributions after a diffusion step are indicated by dashed lines. Concentrations are given on a logarithmic scale and distance on a linear scale in arbritrary units (applies to all figures ).
118
B// ,/
/
.
A .
.
.
.
.
x.% x
\,
Distance Fig. 2. Dip modus of the diffusion. Final distribution of species A and B (dashed lines). The initial distributions are the same as in fig. 1.
Volume 137, number 3
PHYSICS LETTERS A
TCA DAB
~-_ .~.':-...-..~ "~,,~
\
..no B
-,?-,.
with B ....-.xXN\ ~
Distance
p
Fig. 3. Retardation effect: Initial distribution of species A is given by solid line. Final distribution of species A in presence and in absence of species B are given by dashed lines.
I CA, CB B
DA = DB << DAB
---.X
\'\ B
\
\"-,
\ . . . . \B \ Distance
Fig. 4. Drag effect: Initial distributions of species A and B are given by solid lines, the final distributions by dashed lines.
retardation effect is the drag effect. To demonstrate this effect we suppose that the following relation holds between the diffusivities DA=DB<< DAB. Thus the transport of both species is accomplished almost entirely by the pair diffusion. Comparison between the initial and final distribution in fig. 4 shows the drag effect exerted by species B on species A. The distribution CA has been shifted much more strongly to the right than distribution CB. The numerical results in figs. 1-4 have been obtained using the general purpose differential equation solver ZOMBIE [21 ]. Finally it is appropriate to note that all the effects mentioned above have been observed in the experiment for various systems. An uphill diffusion, retarded diffusion and the drag ef-
8 May 1989
feet have been observed for systems in silicon such as aluminum-boron-aluminum-gallium, and boron-gallium [ 9,11 ]. In summary, we have presented a prototype model for the diffusion of two species coupled to each other by the pairing mechanism. This model can be easily extended to more species in which the species can diffuse by themselves and in compounds. The system described in this Letter displays an astonishing variety of effects of highly non-linear behavior being based nevertheless on a simple underlying Fickian ansatz. The author wishes to acknowledge fruitful discussions with Dr. Ch. Werner. He also wishes to thank Dr. H~insch for discussions of the current drag effect [ 22 ] due to Coulomb interaction between holes and electrons described correctly by the present model. The author acknowledges useful suggestions by the referee.
References
[ 1 ] W. Jost, Diffusion in solids, liquids, gases (Academic Press, New York, 1969) pp. 200, 392; E.L. Cussler, Diffusion: mass transfer in fluid systems (Cambridge Univ. Press, Cambridge, 1984) pp. 176, 402; D.A. Frak-Kementskij, Diffusion and heat transfer in chemical kinetics (Plenum, New York, 1969 ) p.201; A. Seeger and D. Schumacher, eds., Vacancies and interstitials in metals (North-HoUand, Amsterdam, 1970). [2] W.K. Warburton and D. Turnbull, in: Diffusion in solids: recent developments, eds. A.S. Nowick and J.J. Burton (Academic Press, New York, 1975 ) p. 171. [ 3 ] F.F. Morehead and R.F. Lever, Appl. Phys. Lett. 48 ( 1986 ) 151. [4] B.I. Mulvaney and W.B. Richardson, Appl. Phys. Lett. 51 (1987) 139. [5] M. Orlowski, Appl. Phys. Lett. 53 (1988) 1323. [6 ] M. Orlowski, in: Proc. SISDEP Conferences, Bologna, 1988. [7] S. Aronowitz, Appl. Phys. 54 (1983) 3930. [ 8 ] S. Aronowitz, L. Coyne, J. Lawless and J. Rishpon, J. Inorg. Chem. 21 (1962) 3589. [9] S. Aronowitz, Appl. Phys. 61 (1986) 2495. [ 10 ] S. Aronowitz and G. Risks, J. Electrochem. Soc. 134 (1987) 702. [ 11 ] M. Orlowski and L. Mader, to be published. [ 12 ] S. Aronowitz et al., J. Inorg. Chem. 21 ( 1982 ) 3589. [ 13 ] Th. Wichert, M.L. Swanson and A.F. Quenneville, Phys. Rev. Lett. 57 (1986) 1757; M.L. Swanson, Th. Wichert and A.F. Quenneville, Appl. Phys. Lett. 49 (1986) 265.
119
Volume 137, number 3
PHYSICS LETTERS A
[14]A. Benninghoven, F.G. Riidenauer and H.W. Werner, Secondary ion mass spectroscopy (Wiley, New York, 1987 ); S.M. Sze, VLSI technology (McGraw-Hill, New York, 1983). [15] M. Orlowski, C. Mazure and L. Mader, J. Phys. (Paris) Colloq. CH. Suppl. 9 (1988) 557. [ 16 ] W. Akutagawa, H.L. Dunlap, R. Hart and O.J. Marsh, Appl. Phys. 50 (1979) 777. [ 17 ] Y. Morikawa, K. Yamamoto and K. Nagami, Appl. Phys. Lett. 36 (1980) 997.
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[ 18 ] P. Baruch, J. Monnier, B. Blanchard and C. Castaing, Appl. Phys. Lett. 26 (1975) 77. [ 19] B.J. Masters and E.F. Gorey, Appl. Phys. 49 (1978) 2717. [20] S.M. Hu, Phys. Rev. 180 (1969) 773. [21 ] W. Jiingling, P. Pichler, S. Selberherr, E. Guerrero and H.W. P~Stzl, IEEE Trans. ED 32 (1985) 156. [22] W. H~insch and G.D. Maham, J. Phys. Chem. Solids 44 (1983) 663.
PHYSICS LETTERS A
8 May 1989
A MODEL FOR THE DYNAMICS BETWEEN PAIRED AND UNPAIRED DIFFUSION FOR A BINARY SYSTEM M. ORLOWSKI Siemens AG, CorporateResearch and Development,Otto-Hahn-Ring 6, 8000 Munich 83, FRG Received 3 January 1989; revised manuscript received 1 March 1989; accepted for publication 9 March 1989 Communicated by A.P. Fordy
A rigorous diffusion model is proposed for two species A and B which are allowed to diffuse partly by themselves and partly in pairs AB. The dynamics of this system is studied in terms of pairing probability and in terms of the relations among the diffusivities, DA, Da, DAa, for the unpaired and paired diffusion, respectively. Depending on the values of these quantities, the system displays a large variety of seemingly irreconcilable effects observed in experiments. It turns out that the highly non-linear character of these effects, for instance the uphill diffusion, is still based on a simple underlying Fickian (i.e. random walk ) mechanism.
Although binary systems of diffusing species have been occasionally considered in various fields [1 ], mostly in connection with chemical reactions, a rigorous model for the dynamics of two species diffusion partly in pairs and partly unpaired has been hitherto not available. The pair diffusion involving vacancies and interstitials gained some prominence in diffusing in metals [2]. More recently diffusion of dopants in semiconductors via impurity-point defect pairs [ 3-6 ] has been explicitly considered in the pertinent diffusion equations. However, the impurity-point defect pair dynamics has been considered only approximately, being appropriate for the kinetics of the point defects which is determined to a large extent through equilibrium values of the thermodynamic force of the crystal lattice itself. The present investigation has been motivated by the recent theoretical and experimental work by Aronowitz and collaborators [ 7- I 0 ]. Quantum chemical calculations on a model silicon lattice have predicted the interactions between p-type dopants of different atomic species to be attractive. For overlapping interstitial populations of aluminum and arsenic of aluminum and phosphorus a repulsive interaction has been predicted [7,8]. These predictions have been verified in diffusion experiments [ 9, l 0 ] indirectly resulting in the pull (attractive interaction)
and the push effect (repulsive interaction) between the pertinent dopant species. However, a careful analysis of the experimental resuits in ref. [9] and also of experiments performed by Orlowski and Mader [ 11 ] displays a much richer structure than expected suggesting a diffusion of each species being partly unpaired and partly in bound pairs as a result of the attractive interaction. In this Letter a rigorous model of a system of two diffusing species with allowance for a reaction to form a mobile compound of a bound pair is derived and its dynamics investigated. The model is applicable to various fields involving diffusion phenomena and is therefore of general character. In particular, the nature of effects observed in the experiment, such as uphill and dip modi, enhancement and retardation of the diffusion and the drag effect are for the first time clarified and the effects explained consistently in terms of a nonlinear diffusion dynamics generated by a simple pairing mechanism. Consider a medium (or phase) S, which may be, for example, a crystal, with a two-component system of species A and B. It is assumed that each of the species in S is partly bound in pairs AB and partly unpaired according to the pairing reaction: kl
A+B.
0375-9601/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )
k2
" AB. 115
Volume 137, number 3
PHYSICS LETTERS A
The diffusion dynamics of the binary system involving the above pairing reaction can be described by the following reaction-diffusion equations: o c ~ n p __1') Ot --aJA
02c~nP OX 2
klr~unpc-'unp.l_k2CAB, ~"A
"~B
OC~"p OzC~np O----{-= DB Ox----5--- kl C~,"p C~"p + k2 CAn,
OCAB I--'A ~"~B Ot --DAB~02CAB +ktr"nPr""P-k2CAB' where C~ rip, C~ np and CAB denote the concentration of the unpaired species A and B, and the concentration of pairs AB, respectively. Here, t is time, x is spatial position in the domain [0, L] and O2/Ox 2 is the Laplacian with respect to x. k~ denotes the pairing rate and k2 the rate of pair decay. DA (DB) and BAa denote diffusivities for the unpaired A (B) and for the pairs AB, respectively. It is reasonable in terms of physics and sufficient in terms of the completeness of the dynamics contained in the model to consider the pairing reaction in thermodynamic equilibrium. Quantum mechanical calculations of dopants in a crystal lattice on interstitial sites [ 12 ] and experimental perturbed ~ angular correlation studies [ 13 ] indicate that the pair binding energy is considerably smaller than the activation energy (potential barrier height in the Arrhenius law) of the dopant diffusivities DA and DB. This means that under such circumstances the processes for pairing and pair decay are much faster than the characteristic diffusion time z ~ d 2 / D (D denoting the diffusivity and d denoting a characteristic distance of an elementary displacement). Furthermore and more important, if the pairing and depairing mechanisms are slower than the diffusion mechanisms, then the diffusions of species A and B become decoupled from one another and determined predominantly by the diffusivities DA and DH. The dynamics of the system becomes not trivial only when the characteristic reaction time becomes at least comparable with the characteristic diffusion time. The closer the equilibrium limit is reached the more pronounced are the effects resulting from the interaction between the two species. Therefore departures from the equilibrium limit do not imply any new features of the dynamics considered. Hence it is appropriate to consider the ther116
8 May 1989
modynamic equilibrium to characterize completely the dynamics of the system. Thermodynamic equilibrium occurs when CAs=aCAunp CBunp ,
(1)
with a=k~/k2. The total concentrations CA and Ca for both species are given by CA = C~,np + CAB,
(2a)
Ca = C~np + CAn.
(2b)
It is important to describe the entire dynamics in terms of total concentrations, because only total chemical (and less reliably the total electrically active) concentrations are accessible to experimental evaluation [ 14 ]. Besides the equilibrium constant a strongly depends on the temperature Timplying that the number of pairs varies strongly with T, i.e. it is different at elevated temperatures where the diffusion occurs from the number of pairs at low temperatures during the measurement. This means that even with an experimental technique at hand allowing a measurement of CAB, one would have to perform the measurements at elevated temperatures (in case of silicon crystal 600-1200°C), which is not feasible for the experimental techniques nowadays. Inserting eq. ( 1 ) in eq. (2) one obtains algebraic equations for C~"p and C~np, which can be solved analytically. The result for species A is given by C~,.p = (2a)-1{ [ 1 +2a(CA+CB) +aZ(CA - Ca) 2] 1/2
--[ I +a(CB--CA) ]} .
(3)
The solution for the species B is obtained by interchanging simply the indices A and B. Thus the unpaired concentration of each species depends on the total concentration of both species and on the reaction rate a. Assume now that the respective species are transported partly unpaired and partly in pairs AB. Thus the total fluxes JA and JB are given by JA =j~np +JAB,
JB =j~np +JAB.
For both fractional fluxes contributing to the total flux a simple Fickian diffusion mechanism is assumed in accordance with the model introduced above, i.e.
Volume 137, number 3
j~np = D A OC~nO/OX,
PHYSICS LETTERSA
J~np = D B O C ~ n p / o x ,
JAB = D A a O C A B / O X "
The rate of change of the total concentration CA is then given by 0CA - -
Ot
-
O
Ox
(JI"P
+JAa)
0 - Ox (DAOC~"P/OX+DABOCAa/OX) "
(4)
An analogous equation holds for species B. Using eqs. ( 1 )- (3) the concentrations of the unpaired species and the concentration of the pairs AB in eq. (4) can be eliminated and the r.h.s, ofeq. (4) can be expressed in terms of total concentrations CA and Ca. After some manipulations one obtains the final form of diffusion equations for a binary diffusion system including the diffusion by pairs:
OCA O( l Ot - ~x [~(DA + DAa)+ (DAa--DA)fAA] OCAox OCB )
+ (Oha--Og)(½+faa) OX 1 '
ot
-
[½(Da+DAaI+(DAa--DaIA~I
ocA)
+ (Dna - Da) ( ½+fkA) --~--X] '
(5a)
OC,~ox ( 5b )
where the dynamic factor fna is given by
a C a - ½ [ l +a(CA +Ca) l fgA = { [ 1 +or(CA + Ca) ]z--4azcA Ca } ,/z" The dynamic factorfaa can be obtained from the last equation by interchanging the indices A and B. Eq. (5) reduces to a simple second Fickian law equation (independent of the extant species) if we assume that the diffusivity of the species under consideration, say A, is the same as the diffusivity for pairs AB, i.e. DA =DAa=D; as expected from the general considerations one obtains
OCA =D 0 2CA Ot OX 2
"
It should be noted that now the diffusion of species
8 May 1989
A is independent of the diffusion of species B; firstly the diffusivity of the direct term is independent of any concentrations, and secondly the cross term proportional to the gradient of species B vanishes now, because DAB- DB = 0. Thus although species A is still partly unpaired and partly bound in pairs, this difference degenerates on the level of fluxes and rates, for it cannot be distinguished whether the transport is effectuated by pairs or by unpaired species, since the efficacy of both transport mechanisms is the same. In order to solve eq. (5) it is, in general, necessary to prescribe both initial distributions and boundary conditions. For convenience, we shall adopt Neumann boundary conditions OCi/OXJx=O.x=L=O for i=A, B. These boundary conditions imply automatically the conservation of the total species A and B, as realized in most experimental situations (introduction of the dopants by implantation). Adoption of Dirichlet boundary conditions has no impact on the basic mechanisms contained in the model; it leads merely to quantitative modifications of the final profiles particularly close to the boundaries. Eq. (5) represents the first rigorous prototype model to study pairing dynamics. The extant models for impurity-point defect pairs in silicon were marred by two circumstances: Firstly the diffusivities of the point defects are orders of magnitude faster than the diffusivities of the dopants. Secondly and more important, the thermodynamics force, which introduces some degree of entropy into the crystal lattice in order to minimize the free energy above 0 K, drives the concentrations of the point defects very quickly (on the characteristic time scale of impurity diffusion) into equilibrium values. Nevertheless some aspects of the dynamics contained in eq. (5) are displayed by systems involving point defects, if one is able to produce highly inhomogeneous distributions of these species by the presence of crystal defects [ 15 ] or by irradiation [ 16-19 ]. Eq. (5) is also a paradigm of general structure for the analysis of diffusion equations given by Hu [ 20 ] almost twenty years ago, who pointed out, that even a simple system needs in general to be formulated using a complete set of equations of irreversible thermodynamics, including cross terms from all constituents. In general, a simple Fick law is inapplicable, even for random walk basic mechanisms. The diffusion flux is not just given 117
Volume 137, number 3
PHYSICS LETTERS A
by one term associated with the gradient of the respective species, but is also contributed by cross terms which associated with gradients of the extant species, may in some cases actually dominate. In the following we discuss the dynamics of the system eq. (5) by means of a few examples, which illustrate such effects as the uphill and dip modi, retardation and enhancement of the diffusion. These effects and the drag effect can now be explained in a unifying way by the non-linear diffusion dynamics proposed above. Consider the first example with the initial distribution for species A and B given as in fig. 1 by solid lines. Further assume that DB < DAB< DA. The distribution of species A and B after the diffusion for some time is indicated by dashed lines. It can be clearly seen that the species A exhibits an uphill diffusion effectuated by the cross term (second term) in eq. (5a). This result is easily explained: In the vicinity of Xo (interval I) a large percentage of species A will be bound in pairs (depending on the reaction constant ct). Outside of I there will be almost no AB pairs because the concentration Ca is negligible. Outside of the interval I the transport of A is accomplished mainly via the diffusion of unpaired species A, whereas within the interval I the transport of species A by pairs will be of increasing importance. It should be noted that pairs AB move much more slowly than the unpaired species A. This means that initially there is much larger flux of the species A en-
I CA,CB DAB
8 May 1989
tering the region I than leaving it, until a sufficient gradient in the distribution of the species A builds up. This leads to an accumulation of species A in I and depletion of species A outside of the interval I. An opposite effect - the dip modus of diffusion can be obtained by assuming DB < DA < DAB. Here the pairs AB move faster that the species A leading to a depletion of the species A in the region I and to accumulation in the vicinity of I which diminishes with increasing distance from the region I. This is illustrated in fig. 2. For both cases it is clear that initially the redistribution of the species A is provided solely by the cross term (OCa/Ox), since the contribution of the direct term vanishes due to the initial uniform distribution of A. As the diffusion proceeds the direct term comes into existence and opposes the action of the cross term leading to a temporarily quasistable distribution for species A, due to the slow diffusion of species B. The degree of order for the distribution A increases. Of course, after sufficiently long time the inhomogeneity of the species A will gradually smear out as the initially highly inhomogeneous distribution of species B also smears out. In fig. 3 the effect of the retardation of the species A in presence of species B is shown. We assume DAB< DA. In case of codiffusion of species A and B a uniform initial distribution of species B has been assumed. The retardation effect for species A in presence of species B is clearly recognized. Conversely the diffusion of the species A will be enhanced in presence of species B if DAn > DA. A related effect to the uphill diffusion effect and
eA, CB
/~--~
T
. .~. . . ' 7
DAB >DA
f~ I ~B
I
1
I ?" . . . . .
I
~-
I
I
I
!1 1"
IB
Xo
Bt
Distance Fig. 1. Uphill modus of the diffusion. Initial distribution of species A and B is given by solid lines. Final distributions after a diffusion step are indicated by dashed lines. Concentrations are given on a logarithmic scale and distance on a linear scale in arbritrary units (applies to all figures ).
118
B// ,/
/
.
A .
.
.
.
.
x.% x
\,
Distance Fig. 2. Dip modus of the diffusion. Final distribution of species A and B (dashed lines). The initial distributions are the same as in fig. 1.
Volume 137, number 3
PHYSICS LETTERS A
TCA DAB
~-_ .~.':-...-..~ "~,,~
\
..no B
-,?-,.
with B ....-.xXN\ ~
Distance
p
Fig. 3. Retardation effect: Initial distribution of species A is given by solid line. Final distribution of species A in presence and in absence of species B are given by dashed lines.
I CA, CB B
DA = DB << DAB
---.X
\'\ B
\
\"-,
\ . . . . \B \ Distance
Fig. 4. Drag effect: Initial distributions of species A and B are given by solid lines, the final distributions by dashed lines.
retardation effect is the drag effect. To demonstrate this effect we suppose that the following relation holds between the diffusivities DA=DB<< DAB. Thus the transport of both species is accomplished almost entirely by the pair diffusion. Comparison between the initial and final distribution in fig. 4 shows the drag effect exerted by species B on species A. The distribution CA has been shifted much more strongly to the right than distribution CB. The numerical results in figs. 1-4 have been obtained using the general purpose differential equation solver ZOMBIE [21 ]. Finally it is appropriate to note that all the effects mentioned above have been observed in the experiment for various systems. An uphill diffusion, retarded diffusion and the drag ef-
8 May 1989
feet have been observed for systems in silicon such as aluminum-boron-aluminum-gallium, and boron-gallium [ 9,11 ]. In summary, we have presented a prototype model for the diffusion of two species coupled to each other by the pairing mechanism. This model can be easily extended to more species in which the species can diffuse by themselves and in compounds. The system described in this Letter displays an astonishing variety of effects of highly non-linear behavior being based nevertheless on a simple underlying Fickian ansatz. The author wishes to acknowledge fruitful discussions with Dr. Ch. Werner. He also wishes to thank Dr. H~insch for discussions of the current drag effect [ 22 ] due to Coulomb interaction between holes and electrons described correctly by the present model. The author acknowledges useful suggestions by the referee.
References
[ 1 ] W. Jost, Diffusion in solids, liquids, gases (Academic Press, New York, 1969) pp. 200, 392; E.L. Cussler, Diffusion: mass transfer in fluid systems (Cambridge Univ. Press, Cambridge, 1984) pp. 176, 402; D.A. Frak-Kementskij, Diffusion and heat transfer in chemical kinetics (Plenum, New York, 1969 ) p.201; A. Seeger and D. Schumacher, eds., Vacancies and interstitials in metals (North-HoUand, Amsterdam, 1970). [2] W.K. Warburton and D. Turnbull, in: Diffusion in solids: recent developments, eds. A.S. Nowick and J.J. Burton (Academic Press, New York, 1975 ) p. 171. [ 3 ] F.F. Morehead and R.F. Lever, Appl. Phys. Lett. 48 ( 1986 ) 151. [4] B.I. Mulvaney and W.B. Richardson, Appl. Phys. Lett. 51 (1987) 139. [5] M. Orlowski, Appl. Phys. Lett. 53 (1988) 1323. [6 ] M. Orlowski, in: Proc. SISDEP Conferences, Bologna, 1988. [7] S. Aronowitz, Appl. Phys. 54 (1983) 3930. [ 8 ] S. Aronowitz, L. Coyne, J. Lawless and J. Rishpon, J. Inorg. Chem. 21 (1962) 3589. [9] S. Aronowitz, Appl. Phys. 61 (1986) 2495. [ 10 ] S. Aronowitz and G. Risks, J. Electrochem. Soc. 134 (1987) 702. [ 11 ] M. Orlowski and L. Mader, to be published. [ 12 ] S. Aronowitz et al., J. Inorg. Chem. 21 ( 1982 ) 3589. [ 13 ] Th. Wichert, M.L. Swanson and A.F. Quenneville, Phys. Rev. Lett. 57 (1986) 1757; M.L. Swanson, Th. Wichert and A.F. Quenneville, Appl. Phys. Lett. 49 (1986) 265.
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Volume 137, number 3
PHYSICS LETTERS A
[14]A. Benninghoven, F.G. Riidenauer and H.W. Werner, Secondary ion mass spectroscopy (Wiley, New York, 1987 ); S.M. Sze, VLSI technology (McGraw-Hill, New York, 1983). [15] M. Orlowski, C. Mazure and L. Mader, J. Phys. (Paris) Colloq. CH. Suppl. 9 (1988) 557. [ 16 ] W. Akutagawa, H.L. Dunlap, R. Hart and O.J. Marsh, Appl. Phys. 50 (1979) 777. [ 17 ] Y. Morikawa, K. Yamamoto and K. Nagami, Appl. Phys. Lett. 36 (1980) 997.
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[ 18 ] P. Baruch, J. Monnier, B. Blanchard and C. Castaing, Appl. Phys. Lett. 26 (1975) 77. [ 19] B.J. Masters and E.F. Gorey, Appl. Phys. 49 (1978) 2717. [20] S.M. Hu, Phys. Rev. 180 (1969) 773. [21 ] W. Jiingling, P. Pichler, S. Selberherr, E. Guerrero and H.W. P~Stzl, IEEE Trans. ED 32 (1985) 156. [22] W. H~insch and G.D. Maham, J. Phys. Chem. Solids 44 (1983) 663.