Properties of Vacancies in Silicon Determined by Out-Diffusion of Zinc from Silicon
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ns + I
(1)
k-2 In this model an interstitially dissolved Zn atom (Zni) changes over to the substitutional position (Zns) thereby kicking a Si atom from a regular lattice site (Sis) into the interstice. The reaction rates k+2 and k-2 apply to forward and backward reactions, respectively. Since diffusion of Zns into dislocation-free Si is governed by the transport capacity CTqDr of Si self-interstitials I-related properties are now rather well established in literature. At lower temperatures it was observed that V contributions become more important, which may be rationalized by additional diffusion via the dissociative mechanism Zni + V
,-k-I
Zns + E
(2)
where E represents an empty interstitial site. In contrast to the I-related data, the V-related data so-obtained are questionable because they were deduced from a minor contribution 219 Mat. Res. Soc. Syrup. Proc. Voi. 532 ©1998 Materials Research Society
of the dissociative mechanism to overall in-diffusion processes dominated by the kick-out mechanism. Out-diffusion experiments of Zn from Si offer an alternative approach to determine properties of V. THEORY In accordance with previous reports we assume that Zn diffuses in Si simultaneously via the kick-out and the dissociative mechanism. Taking into account that the concentrations of the point defects involved may change with time either by diffusion or due to interstitialsubstitutional exchanges a full set of four coupled partial differential equations (PDE) can be derived [3]:
Cz
= (k+ 1Cz.,Cv - k- 1CoCz.C ) + (k+2Cz.iC0 - k-2CZ.C,)
at
aczni
t
--
at
V__ 02CZni".
_ (' 2ý&L! a.X
Dzni
at = D
-
9C
(I • zi aC
ns
a
t J~n
at
(4)
Z
(5)
2CV _ (aCZn) aCv at =-D a-O-X•2v
(6)
The preceding equations hold for diffusion in one dimensional geometry with diffusion time 3 t and penetration depth x. Co is the volume density of Si atoms (5 x 1022 c-m ). The purely substitutional diffusion of Zns either via direct site exchange or via the normal vacancy mechanism was neglected (Dz,, - 0) since this process is assumed to be much slower than diffusion via interstitial-substitutional exchange. With respect to the quasi-chemical reactions (1) and (2) the mass law of action connects the equilibrium concentrations with the reaction rates as follows
kk+1
-
CC Ceq Cý 0q
-- (eq GJni
,
Ceqceq
+
eq _i~
C'Vk-
CZ
Co nsI
(7)
After sufficient long diffusion time local equilibrium among the reacting species is reached and characterized by Czn, CI
-
Czns
Czn
n
C; cz• n
C' q n Ofq
Czni CV
c eq nC eq(
(8)
For these conditions and provided that the relationships c6qnDzni >> C» qD, CveqDv and Cq, " hold diffusion in dislocation-free Si can be described by the n and Czn. equation
a
ac z,(X) at
(9)
aCZ( X) ax a
with a concentration-dependent effective diffusion coefficient eff
D ef
D D,
eff -
CD
220
(
zn()
+ CV(10)
CZeq
In the case of Zn in-diffusion Deff is mainly determined by D' 1 for x far enough from the surface even when C"D, and C•q Dv are assumed to be the same order of magnitud
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