3.5.2 Ionized Impurity Scattering

The influence of strain on the Fermi level and the screening parameters of the ionized impurity scattering model [44] is considered. The effects of strain on impurity centers [62] in doped layers, however, are neglected.

For an analytical band structure taking into account nonparabolicity and anisotropy the density of states of one valley is given by

$$g(\epsilon)=\frac{\sqrt{2}m_{d}^{\frac{3}{2}}\sqrt{\epsilon}}{\pi^{2}\hbar^{3}}\sqrt{1+\alpha\epsilon}\cdot(1+2\alpha\epsilon)$$ (3.67)

In order to calculate the Fermi energy in the strained material only terms up to the second order in the nonparabolicity coefficient are kept. A nonlinear equation for the Fermi energy $E_{f}$ is obtained:

$$n=\sum_{i}N_{c_{i}}^{(or)}\sum_{j}\biggl[\mathcal{F}_{1/2}(\eta_{... ... \frac{105}{32}\alpha^{2} k_{B}^{2}T_{0}^{2}\mathcal{F}_{5/2}(\eta_{ij})\biggr]$$ (3.68)

where $\eta_{ij}=(E_{f}-E_{c_{i}}-\Delta E_{c_{ij}})/k_{B}T_{0}$, $N_{c_{i}}^{(or)}$ stands for the effective density of states of valley $i$ with orientation $j$, $\Delta E_{c_{ij}}$ is the energy splitting of that valley and $T_{0}$ is the lattice temperature. The linear and quadratic terms in (3.68) play an important role as carriers can populate higher energy levels in highly degenerate semiconductors. (3.68) is solved by Newton iteration using as an initial guess the solution obtained for non-degenerate statistics and parabolic bands.

Including nonparabolicity up to the second order the contribution of valley $i$ with orientation $j$ to the inverse screening length takes the following form:

$$\beta_{s_{ij}}^2=\frac{e^{2}}{\varepsilon_{s}\varepsilon_{0}k_{B}... ...{105}{32}\alpha^{2} K_{B}^{2}T_{0}^{2}\cdot\mathcal{F}_{3/2}(\eta_{ij})\biggr],$$ (3.69)

It should be noted that in semiconductors with non-parabolic bands the inverse screening length increases which may weaken the ionized impurity scattering rate in particular for a high doping level when due to the Pauli exclusion principle the population of higher energies increases significantly. Thus there are two opposite factors which determine the strength of ionized impurity scattering. Another interesting effect occurs in strained doped materials. Due to strain some valleys shift up and do not contribute to the kinetics. However, this may change at high degeneracy when the Pauli principle causes the upper split bands to be populated, which then also give a contribution to the transport properties. The repopulation may be significant leading to a reduction of the valley splitting effect.

In case of momentum-dependent screening the dielectric function is modified to take into account the strain induced splitting of the conduction band minima for different valleys and orientations:

$$\varepsilon(q)=\varepsilon(0)\cdot\left(1+\frac{1}{q^{2}}\sum_{ij}\beta_{s_{ij}}^{2}G_{ij}(\xi,\eta_{ij})\right),$$ (3.70)

where $G_{ij}$ stands for the screening function in valley $i$ with orientation $j$. The momentum transfer $\vec{q} = \vec{p^{'}}-\vec{p}$ and the temperature dependence enters through $\xi$. S. Smirnov: