4.2 Point Defects
75
of B in Si has found great interest [277–279]. The diffusion depends on the charge state of boron. The
diffusion of positively charged boron has been suggested [279] to occur via the following route: The
boron leaves its substitutional site and goes to the hexagonal site (‘H’) (Fig. 4.5b) with an activation
energy of about 1 eV (Fig. 4.5d). It can then relax (∼ 0.1 eV) without barrier to the tetrahedral ‘T’
position (Fig. 4.5c). The direct migration B s –Si
T+
i
→B
T+
i
has a higher activation energy of 1.12 eV
and is thus unlikely. The boron atom can then diffuse through the crystal by going from ‘H’ to ‘T’ to ‘H’
and so on (Fig. 4.5e). However, long-range diffusion seems to be not possible in this way because the
kick-in mechanism will bring back the boron to its stable configuration. The pair diffusion mechanism
for neutral boron B s –Si
T
i →B
H
i →B s –Si
T
i via the hexagonal site has an activation energy of about
0.5 eV (Fig. 4.5d) while the path via B
T
i has a larger 0.9 eV barrier. The concentration dependence of
the diffusion mechanism has been discussed in [280].
Similarly, indium diffusion in silicon has been investigated suggesting a minimum energy In s –
Si
T
i →In
T
i →In s –Si
T
i diffusion pathway via the tetrahedral site with 0.8 eV activation energy [281].
Microscopic modeling has been reported also for diffusion of phosphorus [282].
4.2.4 Dopant Distribution
The introduction of impurities into a semiconductor (or other materials such as glasses) is termed doping. The unavoidable incorporation of impurities in the nominally pure (nominally undoped) material
is called unintentional doping and leads to a residual or background impurity concentration. Several
methods are used for doping and the creation of particular doping profiles (in depth or lateral). All
doping profiles underly subsequent diffusion of dopants (Sect. 4.2.3).
Various methods of doping are used. A straightforward method of doping is the incorporation during
crystal growth or epitaxy. For semiconductor wafers a homogeneous doping concentration is targeted,
both laterally and along the rod from which the wafer is cut (Sect. 12.2.2). When a crystal is grown
from melt, containing a concentration c 0 of the impurity, the concentration in the solid is given by
(‘normal freezing’ case [283–285])
3
c(x) = c 0 k (1 − x)
k−1
,
(4.18)
where c(x) is the impurity concentration in the crystal at the freezing interface, x is the frozen melt
fraction (ratio of solid mass to total mass, 0 ≤ x ≤ 1). k is the distribution coefficient (or segregation
coefficient) which is the fraction of impurities that is built into the crystal at the liquid–solid interface.
Since the melt volume reduces during the solidification, the impurity concentration rises over time.
For small distribution coefficients (4.18) can be approximated to
c(x) ≈ c 0
k
1 − x
,
(4.19)
An experimental example for Ge:In is shown in Fig. 4.6a.
In Table 4.2 the distribution coefficients for various impurities in Si, Ge and GaAs is given. The
modification of distribution coefficients in SiGe alloys is discussed in [286]. Equilibrium values (k eq )
are obtained for ‘slow’ crystal growth. For finite growth rates, k becomes a function of the growth rate
3 Mass preservation of the impurities can be written at any time c m (1−x)+
x
0 c(x ) dx = c 0 , where c m is the (remaining)
concentration in the melt. At the beginning c m (0) = c 0 . At the interface c(x) = k c m (x). Putting this into the mass
preservation, building c (x) and solving the resulting differential equation c = c(1 − k)/(1 − x) with c(0) = k c 0 leads
to (4.18).
75
of B in Si has found great interest [277–279]. The diffusion depends on the charge state of boron. The
diffusion of positively charged boron has been suggested [279] to occur via the following route: The
boron leaves its substitutional site and goes to the hexagonal site (‘H’) (Fig. 4.5b) with an activation
energy of about 1 eV (Fig. 4.5d). It can then relax (∼ 0.1 eV) without barrier to the tetrahedral ‘T’
position (Fig. 4.5c). The direct migration B s –Si
T+
i
→B
T+
i
has a higher activation energy of 1.12 eV
and is thus unlikely. The boron atom can then diffuse through the crystal by going from ‘H’ to ‘T’ to ‘H’
and so on (Fig. 4.5e). However, long-range diffusion seems to be not possible in this way because the
kick-in mechanism will bring back the boron to its stable configuration. The pair diffusion mechanism
for neutral boron B s –Si
T
i →B
H
i →B s –Si
T
i via the hexagonal site has an activation energy of about
0.5 eV (Fig. 4.5d) while the path via B
T
i has a larger 0.9 eV barrier. The concentration dependence of
the diffusion mechanism has been discussed in [280].
Similarly, indium diffusion in silicon has been investigated suggesting a minimum energy In s –
Si
T
i →In
T
i →In s –Si
T
i diffusion pathway via the tetrahedral site with 0.8 eV activation energy [281].
Microscopic modeling has been reported also for diffusion of phosphorus [282].
4.2.4 Dopant Distribution
The introduction of impurities into a semiconductor (or other materials such as glasses) is termed doping. The unavoidable incorporation of impurities in the nominally pure (nominally undoped) material
is called unintentional doping and leads to a residual or background impurity concentration. Several
methods are used for doping and the creation of particular doping profiles (in depth or lateral). All
doping profiles underly subsequent diffusion of dopants (Sect. 4.2.3).
Various methods of doping are used. A straightforward method of doping is the incorporation during
crystal growth or epitaxy. For semiconductor wafers a homogeneous doping concentration is targeted,
both laterally and along the rod from which the wafer is cut (Sect. 12.2.2). When a crystal is grown
from melt, containing a concentration c 0 of the impurity, the concentration in the solid is given by
(‘normal freezing’ case [283–285])
3
c(x) = c 0 k (1 − x)
k−1
,
(4.18)
where c(x) is the impurity concentration in the crystal at the freezing interface, x is the frozen melt
fraction (ratio of solid mass to total mass, 0 ≤ x ≤ 1). k is the distribution coefficient (or segregation
coefficient) which is the fraction of impurities that is built into the crystal at the liquid–solid interface.
Since the melt volume reduces during the solidification, the impurity concentration rises over time.
For small distribution coefficients (4.18) can be approximated to
c(x) ≈ c 0
k
1 − x
,
(4.19)
An experimental example for Ge:In is shown in Fig. 4.6a.
In Table 4.2 the distribution coefficients for various impurities in Si, Ge and GaAs is given. The
modification of distribution coefficients in SiGe alloys is discussed in [286]. Equilibrium values (k eq )
are obtained for ‘slow’ crystal growth. For finite growth rates, k becomes a function of the growth rate
3 Mass preservation of the impurities can be written at any time c m (1−x)+
x
0 c(x ) dx = c 0 , where c m is the (remaining)
concentration in the melt. At the beginning c m (0) = c 0 . At the interface c(x) = k c m (x). Putting this into the mass
preservation, building c (x) and solving the resulting differential equation c = c(1 − k)/(1 − x) with c(0) = k c 0 leads
to (4.18).