72
4 Structural Defects
Table 4.1 Formation enthalpy H f and entropy S f of the interstitial (I ) and vacancy (V ) in Si and the Ga vacancy in
GaAs. Data for Si from [268, 269], for GaAs from [270]
Material
Defect
H f (eV)
S f (k B )
Si
I
3.2
4.1
Si
V
2.8
∼ 1
GaAs
V Ga
3.2
9.6
G = G − G 0 = n (H
f
− T S
f
) − T S
d
= n G
f
− T S
d
,
(4.3)
where G
f
= H
f
− T S
f denotes the free enthalpy of formation of a single isolated defect. In Table 4.1
experimental values for the formation entropy and enthalpy are given for several defects. Surprisingly,
despite their fundamental importance in semiconductor defect physics, these numbers are not very well
known and disputed in the literature.
The defect concentration is obtained by minimizing G, i.e.
∂∂G
∂n
= G
f
− T
∂ S
d
∂n
= 0 .
(4.4)
The entropy S
d due to disorder is given as
S
d
= k B ln W ,
(4.5)
where W is the complexion number, usually the number of distinguishable ways to distribute n defects
on N lattice sites
W =
N
n
=
N !
n! (N − n)!
.
(4.6)
With Stirling’s formula ln x! ≈ x(ln x − 1) for large x we obtain
∂ S
d
∂n
= k B
N
n
ln
N
N − n
+ ln
N − n
n
.
(4.7)
If n N , ∂ N /∂n = 0 and the right side of (4.7) reduces to k B ln(N /n). The condition (4.4) reads
G
f
+ k B T ln(n/N ), or
n
N
= exp
−
G
f
kT
.
(4.8)
In the case of several different defects i with a degeneracy Z i , e.g. a spin degree of freedom or several
equivalent configurations, (4.8) can be generalized to
n i
Z i N
= exp
−
G
f
i
kT
.
(4.9)
In [271] the equilibrium concentration of interstitials C
eq
I in silicon has been given as
C
eq
I =
1.0 × 10
27 cm
−3
exp
−
3.8 eV
kT
,
(4.10)
4 Structural Defects
Table 4.1 Formation enthalpy H f and entropy S f of the interstitial (I ) and vacancy (V ) in Si and the Ga vacancy in
GaAs. Data for Si from [268, 269], for GaAs from [270]
Material
Defect
H f (eV)
S f (k B )
Si
I
3.2
4.1
Si
V
2.8
∼ 1
GaAs
V Ga
3.2
9.6
G = G − G 0 = n (H
f
− T S
f
) − T S
d
= n G
f
− T S
d
,
(4.3)
where G
f
= H
f
− T S
f denotes the free enthalpy of formation of a single isolated defect. In Table 4.1
experimental values for the formation entropy and enthalpy are given for several defects. Surprisingly,
despite their fundamental importance in semiconductor defect physics, these numbers are not very well
known and disputed in the literature.
The defect concentration is obtained by minimizing G, i.e.
∂∂G
∂n
= G
f
− T
∂ S
d
∂n
= 0 .
(4.4)
The entropy S
d due to disorder is given as
S
d
= k B ln W ,
(4.5)
where W is the complexion number, usually the number of distinguishable ways to distribute n defects
on N lattice sites
W =
N
n
=
N !
n! (N − n)!
.
(4.6)
With Stirling’s formula ln x! ≈ x(ln x − 1) for large x we obtain
∂ S
d
∂n
= k B
N
n
ln
N
N − n
+ ln
N − n
n
.
(4.7)
If n N , ∂ N /∂n = 0 and the right side of (4.7) reduces to k B ln(N /n). The condition (4.4) reads
G
f
+ k B T ln(n/N ), or
n
N
= exp
−
G
f
kT
.
(4.8)
In the case of several different defects i with a degeneracy Z i , e.g. a spin degree of freedom or several
equivalent configurations, (4.8) can be generalized to
n i
Z i N
= exp
−
G
f
i
kT
.
(4.9)
In [271] the equilibrium concentration of interstitials C
eq
I in silicon has been given as
C
eq
I =
1.0 × 10
27 cm
−3
exp
−
3.8 eV
kT
,
(4.10)