Subsections

• 3.6.2.1 Elementary and Binary Semiconductors
• 3.6.2.2 Semiconductor Alloys
• 3.6.2.3 Model Application

3.6.2 The Relaxation Time Model

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

$$\tau_{\epsilon,n}=\tau_{\epsilon,0} + \tau_{\epsilon,1}\hspa... ...3mm}\left(\frac{T_{\mathrm{L}}}{\mathrm{300 K}}\right)\right)$$ (3.138)

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

$$\tau_{\epsilon,p}=\tau_{\epsilon,2}$$ (3.139)

The flexibility of this function allows its easy adaption to all considered materials. For Si, Ge, and III-V 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	$\tau_{\epsilon,0}$ [ps]	$\tau_{\epsilon,1}$ [ps]	$C_0$	$C_1$	$C_2$	$C_3$	$\tau_{\epsilon,2}$ [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 III-V semiconductor alloys the dependence of $\tau_{\epsilon,n}$ on the material composition $x$ is included. $\tau_{\epsilon,0}$ and $C_0$ are modeled as a quadratic function of $x$. The parameters are summarized in Table 3.34.

3.6.2.1 Elementary and Binary Semiconductors

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 $\tau_{\epsilon,n}$ in Si obtained from the model (lines) and Monte-Carlo 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, $\tau_{\epsilon,n}$ tends to saturate.

At very low electron temperature $\tau_{\epsilon,n}$ starts to increase. This effect is not reproduced by the model. When the electron temperature is close to the lattice temperature, the term $(T_n-T_{{\mathrm{L}}})/\tau_{\epsilon,n}$ appearing in the energy balance tends to zero [183], thus the influence of $\tau_{\epsilon,n}$ 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 $\tau_{\epsilon,n}$ is nearly independent of the lattice temperature, except for very low electron temperature. Therefore, any lattice temperature dependence is neglected ($C_3=0$ 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 $\tau_{\epsilon,n}$ 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 $\Gamma$ to the ${\mathrm{L}}$ valleys. The electron temperature for which $\tau_{\epsilon,n}$ reaches the peak value is independent of the lattice temperature. The associated average energy, $0.31$ eV, is close to the energy difference between the two valleys, $0.27$ 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

3.6.2.2 Semiconductor Alloys

The similar behavior of the energy relaxation time $\tau_{\epsilon,n}$ in In$_x$Ga$_{1-x}$As and Al$_x$Ga$_{1-x}$As to that in GaAs can be seen in Fig. 3.43 and Fig. 3.44. Thus, $\tau_{\epsilon,n}$ 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 $\tau_{\epsilon,0}$ ( ${\tau}_{\epsilon,\mathrm {sat}}$) and $C_0$ ( $T_{n,\mathrm {peak}}$).

For $\tau_{\epsilon,0}$ and $C_0$ of alloy materials (A$_{1-x}$B$_x$) 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 $x$.

$$\tau_{\epsilon,0}^\mathrm {AB} = \tau_{\epsilon,0}^\mathrm {A}\cdot(1-x) + \tau_{\epsilon,0}^\mathrm {B} \cdot x + \tau_C\cdot(1-x)\cdot x$$ (3.140)

$$C_{0}^\mathrm {AB} = C_{0}^\mathrm {A}\cdot(1-x) + C_{0}^\mathrm {B}\cdot x + C\cdot(1-x)\cdot x$$ (3.141)

$\tau_C$ and $C$ 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	$\tau_C$ [ps]	$\tau_{\epsilon,1}$ [ps]	$C$	$C_1$	$C_2$	$C_3$	$\tau_{\epsilon,2}$ [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 $\tau_{\epsilon,n}$ of GaAs is preserved for both semiconductor alloys considered, Al$_x$Ga$_{1-x}$As and In$_x$Ga$_{1-x}$As. This approximation is more accurate for low material composition, which is more frequently used ($x<0.3$).

In Fig. 3.43 the results of the model for $\mathrm {Al}_{x}\mathrm {Ga}_{1-x}\mathrm {As}$ at 300 K for different material compositions $x$ are shown. Note the shift of the electron temperature, at which $\tau_{\epsilon,n}$ reaches its maximum to lower values with the increase of $x$. For high values ($x>0.4$) no peak value of $\tau_{\epsilon,n}$ is observed. This behavior can be attributed to the composition dependence of the $\Gamma$, ${\mathrm{L}}$ and ${\mathrm{X}}$ valley minima. When the Al fraction changes from $0$ to $0.3$, the energy difference between the $\Gamma$ and ${\mathrm{L}}$ valleys varies between $0.27$ and $0.1$ eV. The corresponding change of the electron energy associated to the peak of $\tau_{\epsilon,n}$, varies between $0.31$ to $0.1$ eV. Furthermore, for Al contents $x>0.4$ the $X$ valleys are the lowest ones, and the bandgap becomes indirect. This explains the absence of a peak of $\tau_{\epsilon,n}$ for $x>0.4$.

For In$_x$Ga$_{1-x}$As similar results are obtained in Fig. 3.44. There is a shift of the maximum $\tau_{\epsilon,n}$ to higher values with increasing Indium composition up to $x=0.53$. This can be explained with the electron population change due to $\Gamma-{\mathrm{L}}$ transitions. For InAs a quick shift to lower values is observed, not explained by the dependence of the energy valleys on $x$. Monte-Carlo 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

3.6.2.3 Model Application

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 self-heating is included, the energy relaxation times are used in the relaxation terms of the lattice heat flow equation (3.17).

Using non-constant electron energy relaxation times in (3.129) allows proper modeling of the velocity overshoot in the velocity-field characteristics for III-V materials. In addition, it helps better accuracy for the bias dependence of small-signal parameters to be achieved. Good agreement with the MC simulation results and its simple analytical structure make it attractive for device simulation.