- Email: [email protected]
Theory of enhanced diffusion in predoped and implanted silicon
MATERIALS SCIENCE & ENGIWEERIMG ELSEVIER
B
Materials Science and Engineering B37 (1996) 205-207
Theory of enhanced diffusion in predoped and implanted
silicon
E. Antoncik Institute
of Physics
and Astronomy,
University
of Aahrs,
DK-8000
Aarhs
C, Denmark
Abstract A new physicalmodelis utilized in computersimulationsto calculatedopant diffusion profilesin silicon. It is basedon a system of reaction-diffusion equationstaking into account all the defectsinvolved and their interaction; moreover,at high concentrations the predominant influence of the limited dopant solubility is included. This makesit possibleto interpret both the enhanced transient and the steady-statedopant diffusion phenomenaand to calculatethe effective diffusion coefficient. Keywords:Silicon; Dopant diffusion profiles;Theory of diffusion; Ion implantion
1. What is wrong with calculated diffusion profiles? Until recently the calculation of the diffusion profiles of the dopants in semiconductors has not been very successful.This situation has been best characterized in a recent review article by Fahey et al. [I] saying that “all quantitative models proposed thus far for phosphorus have claimed to be able to fit profile shapes, even though they have subsequently been found to be (i) directly at odds with other experimental results, (ii) incapable of predicting profiles under only slightly different experimental conditions, or (iii) physically untenable”. In a series of papers the present author has shown that the above mentioned failure of the diffusion theory is mainly owing to two drawbacks: (i) the neglect of the limited dopant solubility at high concentrations, and (ii) the use of the diffusion equation outside its validity domain (e.g. with transient effects).
to the creation of donors/acceptors a probability factor (1 - u/u,,,) reflecting the fact that the velocity of this reaction is proportional to the number of accessible sites (u,,, - 21)in the neighbourhood of the dopants, while it is equal to zero once the dopant concentration zdreaches or transcends its solubility limit u,,,. In this way the solutions of the system depend explicitly on u,,, and thus correctly reflect the formation of a dopant plateau both for highly predoped or implanted samples.For example, if one assumesthat the diffusion of donors is vacancy assistedaccording to the reaction D + Jf+=P
(1)
then the following system can be written [2,3] aud -
at
=
%LD
at
kc,
’ ~(%~~ax
&l -4-kd
p ax2
-
ud)(l
It turns out that the equilibration processesunder rapid thermal and furnace (diffusion) annealing are best described using a system of reaction-diffusion equations giving the time dependence of individual defect profiles [l]. To respect the limited solubility of the dopants one has to introduce with terms corresponding 0921-5107/96/$15.00 0 1996 - Elsevier Science S.A. All rights reserved ““r.rh,,e., r
Ud/Udmax)~p
’ 0 (z~dmmax -
x (1 - ~&&,&p 2. Reaction-diffusion equations
-
-
k&Pd
ud)
+ kpU,Ud
(2)
Here LI,(x,~) for i = d, u, p represent the concentration profiles of donors, vacancies and pairs, respectively. B(x) = 1 for x 3 0 and zero otherwise. ki’s are the coefficients corresponding to reaction (l), and D, is the diffusion coefficient of pairs while the donors are immobile. A similar equation for vacancies has been omitted in Eq. (2) since the flat distribution of u,, owing to very high migration of vacancies, can be approximately re-
206
E. Antoncik / Materials Science and Engineering B37 (1996) 205-207
placed by a constant uGO.For completeness, let us recall that the kd term on the RHS of Eq. (2) represents the formation of donors below the solubility limit udmaxby the decay of donor-vacancy pairs while the k, term corresponds to the formation of pairs, according to reaction (1). It is this reaction that determines the number of mobile pairs and, consequently, the course of the diffusion process. Further details can be found in Refs. [2] and [3]. Reaction-diffusion equations have two kinds of physically interesting solutions: (i) transient solutions, and (ii) steady-state solutions.
‘.’ /
-gO.6 %
3. Transient solutions 0.4
Transient solutions fulfil the initial conditions and dominate at sufficiently short times. Their character does not allow the reduction of system (2) to an effective diffusion equation; in other words, no effective diffusion coefficient can be calculated in this case since both the reaction(s) and the diffusion determine the time development of the profiles, (i.e., it cannot be ascribed to diffusion only). A typical example of this kind is the enhanced transient diffusion of implanted boron with peak concentration below the solubility limit [4] which has been explained in great detail using two parameters only. A similar effect, however, above the solubility limit shows, for example, arsenic implanted with peak concentration transcending udmax(see Figs. 1 and 2) which is owing to the &,-term in (2) while the k, term is equal to zero for ud > udmax [2]. The same situation arises with b-doped silicon with similar time dependence of the peak concentration as given in Fig. 2. Unfortunately, very often the change of the profiles with time in the transient regime is erroneously assigned to a time dependent
Fig. 1. The change of the initial LSS donor profile at the end of the transient regime (qimax = 0.6 cmm3). (From Ref. [2]).
0.b
II 2.0
II 4.0
anneal
I/ 6:O
t/me
II,
“‘“(4
8
‘O*O
0
’
Fig. 2. Time dependence of the peak concentration of the donor profiles shown in Fig. 1. Note the transient and steady-state periods. (From Ref. [2]).
diffusion coefficient which, however, mathematically does not exist [5].
4. Steady-state solutions These solutions are valid in the limit of long times when the defect reactions are in dynamic equilibrium, i.e., when the sum of the reaction terms in Eq. (2) is equal to zero. As can be seen from the first equation in Eq. (2) this means that in this case the donor/acceptor concentration is nearly constant. The dopant profiles both in diffused and implanted samples show a box-like shape with a typical plateau slightly decreasing toward smaller concentrations [2,3]; however, at the end of the
Fig. 3. Calculated concentration profiles of phosphorus in silicon for two diffusion times; (---) electrically active dopants, (----) total concentration (from Ref. [3]).
E. Antoncik
/ Materials
Science
profiles the concentration drops to zero very quickly (see Fig. 3). Actually Fig. 3 shows two more processes not included in system (2): (i) precipitation, and (ii) additional diffusion mechanism responsible for the tail (see Ref. [3] for further details). It can be shown mathematically that in this case the system of reaction-diffusion equations may be reduced to a single diffusion equation; at the same time the effective diffusion coefficient can be calculated. For example, in the case of the model considered in the previous section one obtains after trivial calculations
[61. D,= DJU + KU - ~&~max)21
(3)
Here K is a (large) constant including the reaction coefficients. It is evident that D, is concentration dependent being spike-shaped at udmmax[6]. The explanation of this form is rather simple: the condition of steady-state diffusion requires high D, at places with a small concentration gradient in the plateau region and small D, in the tail region. Let us add that D, and Dp /(K-I- 1) can be directly determined from isoconcentration experiments [7] while the concentration dependence of D, can be obtained using the Boltzmann-Matano procedure [6]. As concerns dopant diffusion in predoped silicon the same formalism of reaction-diffusion equations can be
and Engineering
B37 (1996)
205-207
207
used, however, in this case the probability factor has to include both the predoped and the diffusing dopants, i.e., [I - (u~,.~~+ u)/zI,,,,] . B(u,,,, - u~,.~~- u). It has to be noted that except in some special cases it is not always possible to calculate De,+ consequently, in this case the profiles have to be calculated using the original system of reaction-diffusion equations [7]. A similar procedure can also be used with silicon samples predoped with different but physically related dopants (e.g. P and As) [2]. Actually, according to this model, the predoping of the sample decreases the reaction rate of the donor/acceptor formation through the probability factor and thus enhances the diffusion process by increasing the lifetime of pairs (with a high diffusion coefficient 0,). In other words, the predoping effectively decreases the number of accessible sites and thus reduces the formation of immobile donors (acceptors).
References [l] P.M. Fahey, (1989) 289. [2] E. Antoncik, [3] E. Antoncik, [4] E. Antoncik, [5] E. Antoncik, [6] E. Antoncik, [7] E. Antoncik,
P.B. Griffin and J.D. Plummer, Reu. Mod. Phys., 61 Appl. Phys. A, 56 (1993) 291. Appl. Phys. A, 58 (1994) 117. Rnd. Efi DeJ Solids, 116 (1991) 375. Appl. Phys. Lett., 65 (1994) 1320. J. Electrochem. Sot., 141 (1994) 3593.
J. Electrochem. Sot., 142 (1995) 3170.
B
Materials Science and Engineering B37 (1996) 205-207
Theory of enhanced diffusion in predoped and implanted
silicon
E. Antoncik Institute
of Physics
and Astronomy,
University
of Aahrs,
DK-8000
Aarhs
C, Denmark
Abstract A new physicalmodelis utilized in computersimulationsto calculatedopant diffusion profilesin silicon. It is basedon a system of reaction-diffusion equationstaking into account all the defectsinvolved and their interaction; moreover,at high concentrations the predominant influence of the limited dopant solubility is included. This makesit possibleto interpret both the enhanced transient and the steady-statedopant diffusion phenomenaand to calculatethe effective diffusion coefficient. Keywords:Silicon; Dopant diffusion profiles;Theory of diffusion; Ion implantion
1. What is wrong with calculated diffusion profiles? Until recently the calculation of the diffusion profiles of the dopants in semiconductors has not been very successful.This situation has been best characterized in a recent review article by Fahey et al. [I] saying that “all quantitative models proposed thus far for phosphorus have claimed to be able to fit profile shapes, even though they have subsequently been found to be (i) directly at odds with other experimental results, (ii) incapable of predicting profiles under only slightly different experimental conditions, or (iii) physically untenable”. In a series of papers the present author has shown that the above mentioned failure of the diffusion theory is mainly owing to two drawbacks: (i) the neglect of the limited dopant solubility at high concentrations, and (ii) the use of the diffusion equation outside its validity domain (e.g. with transient effects).
to the creation of donors/acceptors a probability factor (1 - u/u,,,) reflecting the fact that the velocity of this reaction is proportional to the number of accessible sites (u,,, - 21)in the neighbourhood of the dopants, while it is equal to zero once the dopant concentration zdreaches or transcends its solubility limit u,,,. In this way the solutions of the system depend explicitly on u,,, and thus correctly reflect the formation of a dopant plateau both for highly predoped or implanted samples.For example, if one assumesthat the diffusion of donors is vacancy assistedaccording to the reaction D + Jf+=P
(1)
then the following system can be written [2,3] aud -
at
=
%LD
at
kc,
’ ~(%~~ax
&l -4-kd
p ax2
-
ud)(l
It turns out that the equilibration processesunder rapid thermal and furnace (diffusion) annealing are best described using a system of reaction-diffusion equations giving the time dependence of individual defect profiles [l]. To respect the limited solubility of the dopants one has to introduce with terms corresponding 0921-5107/96/$15.00 0 1996 - Elsevier Science S.A. All rights reserved ““r.rh,,e., r
Ud/Udmax)~p
’ 0 (z~dmmax -
x (1 - ~&&,&p 2. Reaction-diffusion equations
-
-
k&Pd
ud)
+ kpU,Ud
(2)
Here LI,(x,~) for i = d, u, p represent the concentration profiles of donors, vacancies and pairs, respectively. B(x) = 1 for x 3 0 and zero otherwise. ki’s are the coefficients corresponding to reaction (l), and D, is the diffusion coefficient of pairs while the donors are immobile. A similar equation for vacancies has been omitted in Eq. (2) since the flat distribution of u,, owing to very high migration of vacancies, can be approximately re-
206
E. Antoncik / Materials Science and Engineering B37 (1996) 205-207
placed by a constant uGO.For completeness, let us recall that the kd term on the RHS of Eq. (2) represents the formation of donors below the solubility limit udmaxby the decay of donor-vacancy pairs while the k, term corresponds to the formation of pairs, according to reaction (1). It is this reaction that determines the number of mobile pairs and, consequently, the course of the diffusion process. Further details can be found in Refs. [2] and [3]. Reaction-diffusion equations have two kinds of physically interesting solutions: (i) transient solutions, and (ii) steady-state solutions.
‘.’ /
-gO.6 %
3. Transient solutions 0.4
Transient solutions fulfil the initial conditions and dominate at sufficiently short times. Their character does not allow the reduction of system (2) to an effective diffusion equation; in other words, no effective diffusion coefficient can be calculated in this case since both the reaction(s) and the diffusion determine the time development of the profiles, (i.e., it cannot be ascribed to diffusion only). A typical example of this kind is the enhanced transient diffusion of implanted boron with peak concentration below the solubility limit [4] which has been explained in great detail using two parameters only. A similar effect, however, above the solubility limit shows, for example, arsenic implanted with peak concentration transcending udmax(see Figs. 1 and 2) which is owing to the &,-term in (2) while the k, term is equal to zero for ud > udmax [2]. The same situation arises with b-doped silicon with similar time dependence of the peak concentration as given in Fig. 2. Unfortunately, very often the change of the profiles with time in the transient regime is erroneously assigned to a time dependent
Fig. 1. The change of the initial LSS donor profile at the end of the transient regime (qimax = 0.6 cmm3). (From Ref. [2]).
0.b
II 2.0
II 4.0
anneal
I/ 6:O
t/me
II,
“‘“(4
8
‘O*O
0
’
Fig. 2. Time dependence of the peak concentration of the donor profiles shown in Fig. 1. Note the transient and steady-state periods. (From Ref. [2]).
diffusion coefficient which, however, mathematically does not exist [5].
4. Steady-state solutions These solutions are valid in the limit of long times when the defect reactions are in dynamic equilibrium, i.e., when the sum of the reaction terms in Eq. (2) is equal to zero. As can be seen from the first equation in Eq. (2) this means that in this case the donor/acceptor concentration is nearly constant. The dopant profiles both in diffused and implanted samples show a box-like shape with a typical plateau slightly decreasing toward smaller concentrations [2,3]; however, at the end of the
Fig. 3. Calculated concentration profiles of phosphorus in silicon for two diffusion times; (---) electrically active dopants, (----) total concentration (from Ref. [3]).
E. Antoncik
/ Materials
Science
profiles the concentration drops to zero very quickly (see Fig. 3). Actually Fig. 3 shows two more processes not included in system (2): (i) precipitation, and (ii) additional diffusion mechanism responsible for the tail (see Ref. [3] for further details). It can be shown mathematically that in this case the system of reaction-diffusion equations may be reduced to a single diffusion equation; at the same time the effective diffusion coefficient can be calculated. For example, in the case of the model considered in the previous section one obtains after trivial calculations
[61. D,= DJU + KU - ~&~max)21
(3)
Here K is a (large) constant including the reaction coefficients. It is evident that D, is concentration dependent being spike-shaped at udmmax[6]. The explanation of this form is rather simple: the condition of steady-state diffusion requires high D, at places with a small concentration gradient in the plateau region and small D, in the tail region. Let us add that D, and Dp /(K-I- 1) can be directly determined from isoconcentration experiments [7] while the concentration dependence of D, can be obtained using the Boltzmann-Matano procedure [6]. As concerns dopant diffusion in predoped silicon the same formalism of reaction-diffusion equations can be
and Engineering
B37 (1996)
205-207
207
used, however, in this case the probability factor has to include both the predoped and the diffusing dopants, i.e., [I - (u~,.~~+ u)/zI,,,,] . B(u,,,, - u~,.~~- u). It has to be noted that except in some special cases it is not always possible to calculate De,+ consequently, in this case the profiles have to be calculated using the original system of reaction-diffusion equations [7]. A similar procedure can also be used with silicon samples predoped with different but physically related dopants (e.g. P and As) [2]. Actually, according to this model, the predoping of the sample decreases the reaction rate of the donor/acceptor formation through the probability factor and thus enhances the diffusion process by increasing the lifetime of pairs (with a high diffusion coefficient 0,). In other words, the predoping effectively decreases the number of accessible sites and thus reduces the formation of immobile donors (acceptors).
References [l] P.M. Fahey, (1989) 289. [2] E. Antoncik, [3] E. Antoncik, [4] E. Antoncik, [5] E. Antoncik, [6] E. Antoncik, [7] E. Antoncik,
P.B. Griffin and J.D. Plummer, Reu. Mod. Phys., 61 Appl. Phys. A, 56 (1993) 291. Appl. Phys. A, 58 (1994) 117. Rnd. Efi DeJ Solids, 116 (1991) 375. Appl. Phys. Lett., 65 (1994) 1320. J. Electrochem. Sot., 141 (1994) 3593.
J. Electrochem. Sot., 142 (1995) 3170.