Subsections

• 3.6.1.1 The Direct Method
• 3.6.1.2 The Indirect Method

3.6.1 Methodology

Depending on the semiconductor material under investigation, different results are available from Monte-Carlo simulation. Two methods - direct and indirect - are used to obtain $\tau_{\epsilon,n}$.

3.6.1.1 The Direct Method

For Si, Ge and GaAs, the dependence of the electron energy relaxation time and the average electron energy, $w$, on the electric field are available in [94]. The average energy is approximated by the thermal energy, with the kinetic term being neglected:

$$w=\frac{1}{2}\hspace*{-0.2mm}\cdot\hspace*{-0.3mm}{m_n}\hspa... ...e*{-0.3mm}\mathrm{k_B}\hspace*{-0.2mm}\cdot\hspace*{-0.3mm}T_n$$ (3.136)

where $m_n$, $v_n$, and $T_n$ are the electron mass, velocity and temperature, respectively. This approximation, together with the interpolation of the Monte-Carlo simulation results for different electric fields allows to obtain directly $\tau_{\epsilon,n}$ as a function of the electron temperature at different lattice temperatures. This procedure is called direct method.

3.6.1.2 The Indirect Method

In the case of binary and ternary III-V compounds, such as InAs, AlAs, In$_x$Ga$_{1-x}$As, and Al$_x$Ga$_{1-x}$As, the dependence of $\tau_{\epsilon,n}$ on the electric field is not available. In this case $\tau_{\epsilon,n}$ is calculated in an indirect way, using the dependence of the electron velocity on the electric field from [94]. The local energy balance equation [86] is obtained by neglecting the energy flux:

$$\tau_{\epsilon,n}=\frac{3\cdot \mathrm{k_B}}{2\cdot q}\cdot\... ...n-T_{{\mathrm{L}}}}{v_n\hspace*{-0.2mm}\cdot\hspace*{-0.3mm}E}$$ (3.137)

where $\mathrm{q}$ is the electron charge, $T_{{\mathrm{L}}}$ the lattice temperature, and $E$ is the electric field. Using (3.136) and the dependences of the average electron energy and the electron velocity on the electric field, $\tau_{\epsilon,n}$ is extracted. This procedure is called indirect method.

Fig. 3.39 shows $\tau_{\epsilon,n}$ for GaAs as a function of the electron temperature at $T_{{\mathrm{L}}}$=300 K, as it results from both the direct and indirect methods. The correct values are extracted by the direct method as it is based on less approximations. It turns out that $\tau_{\epsilon,n}$ is overestimated by using (3.137) in the indirect method. Nevertheless, the saturation value of $\tau_{\epsilon,n}$ at high electron temperatures, ${\tau}_{\epsilon,\mathrm {sat}}$, and the location of the peak, $T_{n,\mathrm {peak}}$, are independent of the methodology used. This criteria are used for correct estimation of $\tau_{\epsilon,n}$ in materials where only the indirect method can be applied.

Figure 3.39: Energy relaxation time as a function of electron temperature: Results from the direct and indirect method for GaAs