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Subsections
The following expression is used to model the electron relaxation time as a
function of the carrier and lattice temperatures:

(3.138) 
In case of holes a constant energy relaxation time is assumed.

(3.139) 
The flexibility of this function allows its easy adaption to all considered
materials. For Si, Ge, and IIIV binary materials, all parameters in
(3.138) are summarized in Table 3.33.
Table 3.33:
Parameter values for the energy relaxation time model for basic materials
Material 
[ps] 
[ps] 




[ps] 
Si 
1.0 
0.538 
0 
0.0015 
0.09 
0.17 
0.4 
Ge 
0.26 
1.49 
0 
0.434 
1.322 
0 
0.4 
GaAs 
0.48 
0.025 
0 
0.053 
0.853 
0.5 
1.0 
AlAs 
0.17 
0.025 
61 
0.053 
0.853 
0.5 
1.0 
InAs 
0.08 
0.025 
3 
0.053 
0.853 
0.5 
1.0 
InP 
0.5 
0.21 
18 
0.04 
0 
0 
1.0 
GaP 
0.04 
0.21 
10 
0.04 
0 
0 
1.0 

In the case of IIIV semiconductor alloys the dependence of
on the
material composition is included.
and are modeled as a
quadratic function of . The parameters are summarized in Table 3.34.
The direct method is used for Si, Ge and GaAs, and the indirect one for AlAs
and InAs, depending on the MC data available. Fig. 3.40 shows the values for
in Si obtained from the model (lines) and MonteCarlo results (circles
and triangles) at different lattice temperatures. The energy relaxation time
slightly decreases with increasing lattice temperature. It is also observed
that for high electron temperatures,
tends to saturate.
At very low electron temperature
starts to increase. This effect is
not reproduced by the model. When the electron temperature is close to the
lattice temperature, the term
appearing in the energy
balance tends to zero [183], thus the influence of
is
negligible, and therefore its increase can be neglected.
In GaAs and Ge similar behavior was observed at very low electron temperature,
and the same assumptions as for Si are made. In the case of Ge Fig. 3.41
shows that
is nearly independent of the lattice temperature, except
for very low electron temperature. Therefore, any lattice temperature
dependence is neglected ( in (3.138)). The sharp initial
fall is attributed to the increase of optical and inter valley scattering as
the electrons are heated by the field [186].
The results for GaAs are shown in Fig. 3.42. At high electron
temperatures
tends to some saturated value and becomes independent of
the lattice temperature. For low and intermediate electron temperatures, the
behavior can be attributed to the transition of electrons from the to
the valleys. The electron temperature for which
reaches the peak
value is independent of the lattice temperature. The associated average energy,
eV, is close to the energy difference between the two valleys, eV.
Figure 3.40:
Energy relaxation time as a function of electron temperature:
Comparison of the model and MC data for Si at several lattice temperatures

Figure 3.41:
Energy relaxation time as a function of electron temperature:
Comparison of the model and MC data for Ge

Figure 3.42:
Energy relaxation time as a function of electron temperature:
Comparison of the model and MC data GaAs at several lattice temperatures

The similar behavior of the energy relaxation time
in
InGaAs and AlGaAs to that in GaAs can be seen in
Fig. 3.43 and Fig. 3.44. Thus,
of alloy materials is modeled
by preserving the standard deviation and the amplitude of the function obtained
for GaAs with the direct method and by adjusting then the position with the
material composition dependence of
(
) and (
).
For
and of alloy materials (AB) the model
suggests a quadratic interpolation between the the values from Table 3.33
for the basic materials (A and B) depending on the material composition .
and are referred to as bowing parameters. The values used in this
model are summarized in Table 3.34.
Table 3.34:
Parameter values for energy relaxation times in alloy materials
Material 
[ps] 
[ps] 




[ps] 
AlGaAs 
0.35 
0.025 
61 
0.053 
0.853 
0.5 
1.0 
InGaAs 
1.8 
0.025 
34 
0.053 
0.853 
0.5 
1.0 
InGaP 
0.4 
0.21 
5.2 
0.04 
0 
0 
1.0 

The indirect method is applicable for all semiconductor alloys as explained in
Section 2.2. The lattice temperature dependence of
of GaAs is
preserved for both semiconductor alloys considered, AlGaAs and
InGaAs. This approximation is more accurate for low material
composition, which is more frequently used ().
In Fig. 3.43 the results of the model for
at 300 K for different material compositions are shown. Note the shift of
the electron temperature, at which
reaches its maximum to lower values
with the increase of . For high values () no peak value of
is observed. This behavior can be attributed to the composition dependence of the
, and valley minima. When the Al fraction changes from to
, the energy difference between the and valleys varies
between and eV. The corresponding change of the electron energy
associated to the peak of
, varies between to
eV. Furthermore, for Al contents the valleys are the lowest ones,
and the bandgap becomes indirect. This explains the absence of a peak of
for .
For InGaAs similar results are obtained in Fig. 3.44. There is
a shift of the maximum
to higher values with increasing
Indium composition up to . This can be explained with the electron
population change due to
transitions. For InAs a quick shift to
lower values is observed, not explained by the dependence of the energy valleys
on . MonteCarlo simulation results show that at very high Indium contents
the average electron energy decreases and the saturation drift velocity
increases very much, but no clear results are available in this case.
Figure 3.43:
Energy relaxation time as a function of electron temperature
for different Al contents in AlGaAs at room temperature

Figure 3.44:
Energy relaxation time as a function of electron temperature
for different In contents in InGaAs at room temperature

The energy relaxation times are used in the HD mobility model
(3.129) and in the relaxation terms of the energy balance
equations (3.8) and (3.9). Additionally,
if selfheating is included, the energy relaxation times are used in the
relaxation terms of the lattice heat flow equation (3.17).
Using nonconstant electron energy relaxation times in
(3.129) allows proper modeling of the velocity overshoot in
the velocityfield characteristics for IIIV materials. In addition, it helps
better accuracy for the bias dependence of smallsignal parameters to be achieved.
Good agreement with the MC simulation results and its simple
analytical structure make it attractive for device simulation.
Next: 3.7 Generation and Recombination
Up: 3.6 Energy Relaxation Time
Previous: 3.6.1 Methodology
Vassil Palankovski
20010228