A Steady-State Model for Coupled Defect Impurity Diffusion in Silicon
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A STEADY-STATE MODEL FOR COUPLED DEFECT IMPURITY DIFFUSION IN SILICON F.
F. MOREHEAD AND R. F. LEVER IBM East Fishkill Facility, Hopewell Junction,
New York 12533
ABSTRACT We extend our earlier model which was proposed to explain tails in the diffusion profiles of high concentration boron and phosphorus in silicon. Our quasi-steady-state approach is generalized here to include both vacancies (V) and interstitials (I) at equivalent levels. I-V recombination is regarded as near local equilibrium, occurring through reactions of the defects with defect-impurity pairs. This approach leads to the wellknown plateau, kink and tail in high surface concentration P diffusions in Si and to the less well recognized tails in B as well. Our extended model, in its simplest form, allows a more complete and less restrictive treatment of Au diffusion in Si. An important advantage is the direct inclusion of these defectimpurity interactions and the resulting gradients in the defect concentrations. Recently models for impurity diffusion in silicon have been proposed, by us [1] and subsequently by others [2,3), which attempt directly to include the effects of counter-fluxes of impurity-defect pairs and the unpaired defects, such as tails in high concentrations of boron and phosphorus. In Ref. (1), we used the equality of the oppositely directed fluxes of interstitials (e.g.,l and I+) and the interstitial-boron pairs [(BI) and (BI)-] to yield a quasi-steady-state distribution of I's in the bulk Si crystal, rising from an equilibrium value 10 at the surface to a higher level in the interior. The V's were determined from IV = I 0 V0 . Here we write a complete set of coupled diffusion equations for B, I,V, Bland BV-pairs and find a steady-state solution by setting to 0 a particular sum containing the time derivatives of all these entities except B (in comparison with which they are all very small). The sum is formed so that all of the kinetic terms for interactions of these diffusing species cancel out. To illustrate, consider a simple system with no concenration enhanced diffusion, no clustering, no field effect for a substitutional dopant P (not necessarily phosphorus) which difuses via E-centers (PV) and F-centers (PI) with four reactions: P
+
I
->
F;
Ri
=
Kj'PI
P
+
V
-
E;
R2
=
K2 'PV
E
+
I
-
P;
R3
=
K3 'EI
F
+
V
-,
P;
R4
=
K4 *FV
-
Ki"F
(1)
K2 ''E
(2)
-
K3 "P
(3)
-
K4 "P
(4)
Here Ki' is the kinetic constant for the forward ith reaction, For simplicity (following closely the conreverse. Ki" its Mat. Res. Soc. Symp. Proc. Vol. 163. ©1990 Materials Research Society
562
ventions of Ref. [4]), we have used the same symbol both for the entity and its concentration, e.g., I is the symbol both for the interstitial Si and its concentration. The complete set of coupled differential equations is dP/dt =
- R1 - R2 + R3 + R4
dF/dt =
(d/dx)DF(dF/dx)
+ Ri-
dE/dt =
(d/dx)DE(dE/dx)
+ R-
R3,
(7)
- RI-
R3 ,
(8)
R4.
(9)
dI/dt
(d/dx)Di(dI/dx)
dV/dt =
(d/dx)Dv(dV/dx)
- R2
(5) R4,
-
(6)
H
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