78
4 Structural Defects
Fig. 4.8 Simulated mean
path length as a function of
implantation direction
(azimuthal angle φ and
polar angle θ) near [001]
for 5 keV boron in silicon.
The [001] channeling peak
appears as a ridge at the
left side of the plot (θ = 0,
any value of φ). Adapted
from [296]
40
30
20
10
0
50
(°)
30
20
10
0 0
10
20
30
40
50
60
40
(°)
d (nm)
m
[114]
[112]
[111]
[101]
B in Si
4.2.5 Large Concentration Effects
4.2.5.1 Lattice Constant
At high doping concentration, a noticeable effect on the lattice constant a 0 is found. For silicon the atom
density
7 is N Si = 5×10
22 cm
−3 . A doping level of N = 10
19 cm
−3 corresponds thus to a dopant fraction
of 0.02%. Such crystal could also be considered a very dilute alloy. About each (N Si /N )
1/3
≈ 17-th
atom in a given direction is a dopant.
The effect of high doping on the lattice constant is due to different ionic radius of the dopant and
the hydrostatic deformation potential of the band edge occupied by the free carriers [301]. In a linear
approach, the effect is summarized in the coefficient β via
β =
1
N
0
a 0
.
(4.21)
The effect due to charge carriers on β is negative (positive) for p-doping (n-doping). Experimental
data for Si, Ge, GaAs and GaP are compiled in [302, 303] and theoretically discussed. The effect
is in the order of β = ±(1–10)×10
−24 cm
3 . For example, in the case of Si:B, the shrinkage of the
lattice constant is mostly due to the charge carrier effect, for Si:P both effects almost cancel. In [304]
it is shown that boron incorporation in silicon changes the lattice constant in various directions quite
differently, e.g. d 333 is shrunk by 0.4% for a doping level of 10
19 cm
−3 while the {620} lattice constant
remains constant.
4.2.5.2 Clustering
Point defects can cluster, i.e. several point defects aggregate at neighboring sites. An example the
configuration of five nearby vacancies in silicon, the so-called V 5 cluster is shown in Fig. 4.9a. In
[305] the ring-like hexavacancy in silicon is predicted a very stable defect. A large number of clustered
vacancies is equivalent to a void. An example is depicted in Fig. 4.9b for an In 2 O 3 crystal which has
locally ’decomposed’ into an indium particle and a void as revealed by TEM [306]. Also impurities
can exhibit clustering.
7 eight atoms per cubic unit cell of length a 0 = 0.543 nm.
4 Structural Defects
Fig. 4.8 Simulated mean
path length as a function of
implantation direction
(azimuthal angle φ and
polar angle θ) near [001]
for 5 keV boron in silicon.
The [001] channeling peak
appears as a ridge at the
left side of the plot (θ = 0,
any value of φ). Adapted
from [296]
40
30
20
10
0
50
(°)
30
20
10
0 0
10
20
30
40
50
60
40
(°)
d (nm)
m
[114]
[112]
[111]
[101]
B in Si
4.2.5 Large Concentration Effects
4.2.5.1 Lattice Constant
At high doping concentration, a noticeable effect on the lattice constant a 0 is found. For silicon the atom
density
7 is N Si = 5×10
22 cm
−3 . A doping level of N = 10
19 cm
−3 corresponds thus to a dopant fraction
of 0.02%. Such crystal could also be considered a very dilute alloy. About each (N Si /N )
1/3
≈ 17-th
atom in a given direction is a dopant.
The effect of high doping on the lattice constant is due to different ionic radius of the dopant and
the hydrostatic deformation potential of the band edge occupied by the free carriers [301]. In a linear
approach, the effect is summarized in the coefficient β via
β =
1
N
0
a 0
.
(4.21)
The effect due to charge carriers on β is negative (positive) for p-doping (n-doping). Experimental
data for Si, Ge, GaAs and GaP are compiled in [302, 303] and theoretically discussed. The effect
is in the order of β = ±(1–10)×10
−24 cm
3 . For example, in the case of Si:B, the shrinkage of the
lattice constant is mostly due to the charge carrier effect, for Si:P both effects almost cancel. In [304]
it is shown that boron incorporation in silicon changes the lattice constant in various directions quite
differently, e.g. d 333 is shrunk by 0.4% for a doping level of 10
19 cm
−3 while the {620} lattice constant
remains constant.
4.2.5.2 Clustering
Point defects can cluster, i.e. several point defects aggregate at neighboring sites. An example the
configuration of five nearby vacancies in silicon, the so-called V 5 cluster is shown in Fig. 4.9a. In
[305] the ring-like hexavacancy in silicon is predicted a very stable defect. A large number of clustered
vacancies is equivalent to a void. An example is depicted in Fig. 4.9b for an In 2 O 3 crystal which has
locally ’decomposed’ into an indium particle and a void as revealed by TEM [306]. Also impurities
can exhibit clustering.
7 eight atoms per cubic unit cell of length a 0 = 0.543 nm.