4.2 Point Defects
73
about 10
14 cm
−3 at 1200
◦ C. The vacancy concentration has been investigated in [272]. Around a
temperature of 1200
◦ C it is in the 10
14 –10
15 cm
−3 range. Due to the reaction
0 I + V ,
(4.11)
a mass action law holds for the concentrations of interstitials and vacancies
C I C V = C
eq
I C
eq
V .
(4.12)
4.2.3 Diffusion
The diffusion of point defects is technologically very important, in particular for silicon as host material.
Typically a dopant profile should be stable under following technological processing steps and also
during device performance. Also defect annihilation is crucial after implantation processes. Diffusion
of an interstitial I and a vacancy V to the same site is prerequisite for recombination of defects (so
called bulk process) according to the scheme I + V → 0. We note that the process 0 → I + V is
called Frenkel pair process.
1 Also the self-diffusion of silicon has been studied, e.g. using radioactively
marked isotopes [271]. The diffusion of point defects including dopants in silicon has been reviewed in
[273, 274]. Usually Fick’s law is applied, stating how the flux J depends on the concentration gradient,
for an interstitial it reads:
J I = −D I ∇C I ,
(4.13)
D I being the interstitial diffusion coefficient. For interstitials in Si it was found [271] that
D I = 0.2 exp
−
1.2 eV
kT
cm
2
/s .
(4.14)
The diffusion of neutral vacancies occurs with [275]
D V = 0.0012 exp
−
0.45 eV
kT
cm
2
/s .
(4.15)
The temperature dependent diffusion coefficients of point defects and dopants in silicon are shown in
Fig. 4.4.
The self-diffusion coefficient of silicon has been determined from the annealing of isotope superlattices (Sect. 12.5) of sequence
28 Si n /
30 Si n , n = 20 to be [276]
D
SD
Si =
2175.4 exp
−
4.95 eV
kT
+ 0.0023 exp
−
3.6 eV
kT
cm
2
/s ,
(4.16)
the first (second) term being due to interstitial (vacancy) mechanism, dominant for temperatures larger
(smaller) than 900
◦ C. The enthalpy in the exponent, e.g. H V = 3.6
+0.3
−0.1 eV [276], consists of the
formation and migration enthalpies,
H V = H
f
V + H
m
V .
(4.17)
1 At higher temperatures a silicon atom can occasionally acquire sufficient energy from lattice vibrations to leave its
lattice site and thus an interstitial and a vacancy are generated.
73
about 10
14 cm
−3 at 1200
◦ C. The vacancy concentration has been investigated in [272]. Around a
temperature of 1200
◦ C it is in the 10
14 –10
15 cm
−3 range. Due to the reaction
0 I + V ,
(4.11)
a mass action law holds for the concentrations of interstitials and vacancies
C I C V = C
eq
I C
eq
V .
(4.12)
4.2.3 Diffusion
The diffusion of point defects is technologically very important, in particular for silicon as host material.
Typically a dopant profile should be stable under following technological processing steps and also
during device performance. Also defect annihilation is crucial after implantation processes. Diffusion
of an interstitial I and a vacancy V to the same site is prerequisite for recombination of defects (so
called bulk process) according to the scheme I + V → 0. We note that the process 0 → I + V is
called Frenkel pair process.
1 Also the self-diffusion of silicon has been studied, e.g. using radioactively
marked isotopes [271]. The diffusion of point defects including dopants in silicon has been reviewed in
[273, 274]. Usually Fick’s law is applied, stating how the flux J depends on the concentration gradient,
for an interstitial it reads:
J I = −D I ∇C I ,
(4.13)
D I being the interstitial diffusion coefficient. For interstitials in Si it was found [271] that
D I = 0.2 exp
−
1.2 eV
kT
cm
2
/s .
(4.14)
The diffusion of neutral vacancies occurs with [275]
D V = 0.0012 exp
−
0.45 eV
kT
cm
2
/s .
(4.15)
The temperature dependent diffusion coefficients of point defects and dopants in silicon are shown in
Fig. 4.4.
The self-diffusion coefficient of silicon has been determined from the annealing of isotope superlattices (Sect. 12.5) of sequence
28 Si n /
30 Si n , n = 20 to be [276]
D
SD
Si =
2175.4 exp
−
4.95 eV
kT
+ 0.0023 exp
−
3.6 eV
kT
cm
2
/s ,
(4.16)
the first (second) term being due to interstitial (vacancy) mechanism, dominant for temperatures larger
(smaller) than 900
◦ C. The enthalpy in the exponent, e.g. H V = 3.6
+0.3
−0.1 eV [276], consists of the
formation and migration enthalpies,
H V = H
f
V + H
m
V .
(4.17)
1 At higher temperatures a silicon atom can occasionally acquire sufficient energy from lattice vibrations to leave its
lattice site and thus an interstitial and a vacancy are generated.