6.3.2 Degeneracy Effects on Inversion Layer Mobility

The MC algorithm presented in Section 5.4 taking into account the Pauli principle has been compared to two other methods found in the literature [Bosi76,Yamakawa96]. The first algorithm to include the Pauli principle in the MC technique [Bosi76] is based on a self-consistent iterative algorithm that uses a rejection technique to account for the occupation probability of the final state after each scattering event. Since this auxiliary self-scattering mechanism is proportional to the occupation of the final states, the algorithm prevents a large number of classically allowed transitions.

A different approach to include degeneracy in MC simulations was given in [Yamakawa96]. Inelastic scattering rates are multiplied with a factor of $(1-f_{\mathrm{FD}}(\epsilon_f))/(1-f_{\mathrm{FD}}(\epsilon_i))$ , where $\epsilon_i$ and $\epsilon_f$ denote the initial and final electron energy, respectively. This additional factor stems from the use of Fermi-Dirac statistics within the relaxation time approximation [Roychoudhury80].

**Figure 6.23:** The simulated effective mobility using a new algorithm (solid line) is compared to results of a non-self-consistent version of Bosi's algorithm [Bosi76] (dotted line), the algorithm proposed in [Yamakawa96], and to the mobility calculated with the Kubo-Greenwood formalism (open circles).
$\includegraphics[width=11.0cm]{xcrv-scipts/compZfldLiterature.eps}$

In the limit of vanishing field the mobility can also be calculated based on the relaxation time approximation using the Kubo-Greenwood formula [Fischetti03]. Figure 6.23 shows that the new method yields the closest agreement, whereas a non-self-consistent implementation of the algorithm proposed in [Bosi76], where $f(\bf {k})$ has been approximated by the equilibrium distribution function $f_{\mathrm{FD}}(\bf {k})$ , and the algorithm proposed by [Yamakawa96] overestimate the effective mobility.

Using the proposed MC algorithm the mean electron velocity as a function of total electron energy was extracted in the small driving field limit. For this purpose the particle's energy domain was divided into a set of intervals $\Delta E$ . The mean velocity component of an electron in direction of the driving field in a particular interval $E \in [E_0,E_0+\Delta E]$ can be obtained during a MC simulation from a history of duration

**Figure 6.24:** Velocity distribution functions (left) and the corresponding electron mobilities limited by phonon scattering (right) for various effective fields.
$\includegraphics[width=7.cm]{xmgrace-files/velDistrPh-0.1.eps}$ $\includegraphics[width=7.cm]{xmgrace-files/mobPh-0.1.eps}$ $\includegraphics[width=7.cm]{xmgrace-files/velDistrPh0.4.eps}$ $\includegraphics[width=7.cm]{xmgrace-files/mobPh0.4.eps}$ $\includegraphics[width=7.cm]{xmgrace-files/velDistrPh1.0.eps}$ $\includegraphics[width=7.cm]{xmgrace-files/mobPh1.0.eps}$ $\includegraphics[width=7.cm]{xmgrace-files/velDistrPh2.0.eps}$ $\includegraphics[width=7.cm]{xmgrace-files/mobPh2.0.eps}$

**Figure 6.25:** Velocity distribution functions (left) and the corresponding electron mobilities limited by phonon- and surface scattering (right) for various effective fields.
$\includegraphics[width=7.cm]{xmgrace-files/velDistrSR-0.1.eps}$ $\includegraphics[width=7.cm]{xmgrace-files/mobSR-0.1.eps}$ $\includegraphics[width=7.cm]{xmgrace-files/velDistrSR0.4.eps}$ $\includegraphics[width=7.cm]{xmgrace-files/mobSR0.4.eps}$ $\includegraphics[width=7.cm]{xmgrace-files/velDistrSR1.0.eps}$ $\includegraphics[width=7.cm]{xmgrace-files/mobSR1.0.eps}$ $\includegraphics[width=7.cm]{xmgrace-files/velDistrSR2.0.eps}$ $\includegraphics[width=7.cm]{xmgrace-files/mobSR2.0.eps}$

A interesting behavior can be observed when comparing the mean velocities resulting from simulations with classical and Fermi-Dirac statistics from Figure 6.24 and 6.25. As shown in Figure 6.24 the mean velocities coincide for both simulation modes in the nondegenerate regime (small $\ensuremath {E_\mathrm{eff}}$ ) when only phonon scattering is considered in the MC simulation. At high $\ensuremath {E_\mathrm{eff}}$ , where the 2DEG is highly degenerate, a shift of the mean velocity distribution toward higher energies and a decrease of its peak can be observed as compared to the mean velocity resulting from nondegenerate simulations without the Pauli principle. The coincidence of the mean velocities in the nondegenerate regime is a check of consistency that the algorithm with the Pauli principle included converges to the classical algorithm for the nondegenerate 2DEG. At high $\ensuremath {E_\mathrm{eff}}$ the different mean velocities can be interpreted as follows: In simulations neglecting the Pauli principle electrons have an equilibrium energy of $\ensuremath {{\mathrm{k_B}}}T$ whereas the mean energy resulting from simulations with the Pauli principle can be more than twice as much. Since phonon scattering is proportional to the density of states, which is an increasing step-like function for the 2DEG, electrons at higher energies - as it is the case in simulations with the Pauli principle - experience more scattering and thus the phonon-limited mobility is strongly decreased (see right plots of Figure 6.24).

The plots of Figure 6.25 show the mean velocities and the effective mobilities at various effective fields when surface roughness scattering is included in MC simulations. At low $\ensuremath {E_\mathrm{eff}}$ surface roughness scattering does not play an important role, and the mean velocities compare well with the simulation results for the phonon-limited mobility in Figure 6.24. However, now even at high $\ensuremath {E_\mathrm{eff}}$ the mean velocities stemming from simulations with and without the Pauli principle do not differ as much. The large peak that was observed in the nondegenerate phonon-limited mean velocity at small energies is suppressed. This is a direct consequence of surface roughness scattering, which is at small energies more effective than phonon scattering. Thus, the effective mobility resulting from simulations with degenerate statistics are incidentally in close agreement to those using classical statistics even though the phonon-limited mobility experiences a noticeable reduction when using degenerate statistics. As previously discussed, this close agreement can only be understood from the cancellation of two effects: Degeneracy leads to an increase of the mean kinetic energy. This leads to an increase in phonon scattering and a decrease in the mobility. At the same time electrons with larger kinetic energies experience less effective surface-roughness scattering, thus the surface roughness limited mobility is increased. By coincidence, in Si inversion layers at room temperature these two effects cancel each other at all effective fields, and the difference between a simulation with nondegenerate and degenerate statistics is very small.

As a final observation from Figure 6.24 and 6.25 one can see that due to degeneracy effects electrons at energies below the Fermi level have smaller mean velocities, which corresponds well to the general picture that highly occupied states have little contribution to transport.


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