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Investigating Hot-Carrier Effects using the Backward Monte Carlo Method

5.6 Results and Discussion

The following results and discussion are taken from a previous work [P5].

5.6.1 Screened Coulomb Scattering Mechanisms

Since ionized-impurity scattering (IIS) and e-e scattering (EES) are caused by the very same screened Coulomb interaction, we compare the rates of these two scattering mechanisms in Fig. 5.2 and Fig. 5.3. For this comparison we used the IIS model of Brooks and Herring (BH)

$(5.106) \{begin}{align} \Gamma _\mathrm {BH}(\kv ) = \frac {N_Ie^4m}{4\pi \hbar ^3\epsilon ^2\beta _s^2}\frac {4\,k}{\beta _s^2 + 4\,k^2} = \frac {B}{2} \frac {4\,k}{\beta _s^2 + 4\,k^2} \{end}{align}$

Here, $B$ is the already defined pre-factor (5.49) and $N_I$ is the ionized impurity concentration. In both cases, the scattering rate becomes smaller with increasing concentration. However, since the scattering potential gets more localized with stronger screening, its distribution in momentum space gets wider, and hence the momentum transferred per scattering event gets larger. At high energies, the rates become concentration-independent in both cases. The main differences can be observed at low energies. While an electron at rest is strongly affected by the moving partner electrons, it will not be affected by the static impurities. At low energies, the EES rate assumes a constant value determined by $F(0,x)$ , whereas the IIS rate vanishes for a non-zero screening parameter $\beta _s$ . For weak screening ( $\beta _s \to 0)$ the EES rate converges to a finite value determined by $F(0,0) = 1$ , whereas the maximum of the IIS rate grows indefinitely, see Fig. 5.3.

5.6.2 Results for Bulk Silicon

As a first test, the equilibrium distribution function is simulated. We asume a parabolic dispersion, which is consistent with the integration of the transition rate (5.19). This results in an equilibrium distribution function, represented by a Maxwellian, also in the presence of EES. This is expected since EES satisfies the principle of detailed balance. The numbers of energy gain and loss processes are perfectly balanced for each scattering mechanism. Using the transition rate (5.58) in an MC simulation with a non-parabolic dispersion ( $\alpha = 0.5$  eV $^{-1}$ ), however, results in an imbalance of energy gain and loss processes. An excess of phonon emissions over absorptions indicates that the inconsistently used EES model provides net energy to the electron system.

This example indicates that the analytical formula derived for a parabolic dispersion should not be used in a transport model with any other dispersion.

5.6.3 Results for an $n^+n^-n^+$ Diode

The EES model has been implemented in the Monte Carlo device simulator VMC [109] for both analytical and numerical band structures. The first device investigated is an $n^+n^-n^+$ diode with abrupt junctions. The doping levels are $10^{19}$ cm $^{-3}$ and $10^{15}$ cm $^{-3}$ , respectively. Fig. 5.4 shows the conduction band edge for an applied voltage of $\SI {2}{V}$ and the electron densities. The plot distinguishes between the total electron density (S+D) and the density of electrons originating from the source contact only (S). Fig. 5.5 shows the energy relaxation process due to phonon scattering only (doted lines) and due to phonon and EES (solid lines). In the drain region ( $x \geq \SI {400}{\nano \meter }$ ), the difference between the decay of the mean energy due to scattering only (magenta lines) and due to mixing with the cold carriers from the drain contact (cyan lines) can be observed.

5.6.4 Results for a n-channel MOSFET

The second device we consider is a planar n-channel MOSFET with $L_G = \SI {65}{nm}$ , $t_\mathrm {ox} =$ 2.5 nm, and a channel width of $W=\SI {1}{\mu m}$ . Device geometry and doping profiles have been obtained by process simulation [104]. The first simulation uses the parabolic band approximation. Fig. 5.6 shows the EDF at three interface points in the channel at $V_{GS}=\SI {2.2}{\volt }$ and $V_{DS}=\SI {2.2}{\volt }$ . Fig. 5.6 indicates that EES has virtually no influence on the non-equilibrium EDF. The reason is that the EES transition rate (5.19) asumes the interaction with an equilibrium electron system and thus does not alter the Maxwellian high energy tail.

The full-band implementation of the EES model in the MC code requires a numerical integration over the Brillouin zone. Fermi-Dirac statistics for the initial state and the Pauli blocking factor for the final state of the partner electron are taken into account. The state after scattering is selected randomly using pre-calculated lists. To resolve the high energy tail accurately, we employ the backward MC method, described in Chapter 4.

In Fig. 5.7 results from full-band transport calculations are compared to the results of ViennaSHE, a deterministic solver for the BTE based on a spherical harmonics expansion of the distribution function [4, 63, 85, 108]. ViennaSHE accounts for an isotropic, multi-valley band-structure that captures some features of the full-band density of states. It is also able to take EES into account. In the EES model of ViennaSHE, additional approximations are introduced. For instance, the energy of the partner electron before scattering is treated as a constant $(\Epsilon ^*)$ which is set equal to the average energy. As shown in Fig. 5.7, the MC model predicts a Maxwellian tail at high energies by the assumption that the hot carriers interact with an equilibrium system of cold carriers, whereas the EES model of ViennaSHE predicts a significant deviation from the Maxwellian tail. We believe that an EES model that adequately fulfills energy and momentum conservation simultaneously would not be able to yield such strong enhancements of the high energy tail as reported in [5, 15, 104].