Investigating Hot-Carrier Effects using the Backward Monte Carlo Method

6.3 Conclusion

This thesis covers the basics of the semi-classic transport theory in semiconductors. It covers different types of band-structure models and how to calculate them, it shows the limitation of the semi-classical transport theory and introduces the BTE, which is most commonly used to describe semi-classical transport in semiconductors.

The result of the BTE is a distribution function (DF). With this DF defined in -space, -space and time, every other quantity of interest can be calculated. There are three different methods shown, which are used to obtain approximate solutions of the BTE. One method is based on the moments of the BTE. Taking only two moments into account, the method of moments yield to the drift-diffusion (DD) model. The second method discussed in this work is the spherical harmonics expansion (SHE) method. The solution of the BTE is obtained deterministically. The third method discussed is the Monte Carlo (MC) method. Based on the stochastic nature of the physical processes the solution of the BTE is obtained by MC estimates. Advantages and drawbacks of all three methods have been discussed.

The backward MC (BMC) method was proposed to overcome some drawbacks of the forward MC method [45, 62, 73]. Here, we established a stable BMC estimator based on [62] and developed several novel techniques such as symmetric injection to further reduce variance. All derived estimators were implemented in a pre-existing MC device simulator. We show that the implementations could fulfill all proposed predictions and lead to tremendous variance reduction for rare events in MC simulations. Thus, it is possible to only simulate the DF in one specific region of interest or in other words: one can choose the energy and the location in the device and estimate the DF for this point only. Thus, only the trajectories passing this specific point have to be considered and no other trajectories have to be calculated. The advantage of using full-bands, physical scattering mechanisms, and the possibility to only estimate the results of interest make the BMC method a powerful tool for investigating hot-carrier effects.

We took some effort to establish a formalism to describe EES for two particles as well as for single particles. This formalism was derived in general, and implemented with full-bands and with the parabolic band approximation. Furthermore, it was shown that the transition rates obey the principle of detailed balance. Thus, no altering of the DF’s high energy tail is expected. In order to compare it to results from an SHE simulator in the drain region, the single-particle scattering model assumes that the hot-electrons get mixed in the drain with an ensemble of electrons in thermal equilibrium. With partner electrons in equilibrium and parabolic bands for both, the sample and the partner electrons, the non-self-consistent results show no altering of the distribution function’s high energy tail.

The good compatibility with the standard algorithms for scattering and free-flight makes the BMC method very smooth to implement in existing MC simulators. The implemented version of the BMC method appears to be a powerful tool for investigating rare events in general and hot-carrier effects in particular. Transport across high energy barriers or physical processes with high energy thresholds can be investigated more easily than with other methods. This work shows that the BMC algorithm has no problems with high energy barriers when it comes to the calculation of the current. Moreover, the results even show a slight decrease in simulation time by increasing barrier heights. The algorithm proves to be stable, no matter of the injection distribution that can be chosen freely. Of course, the more realistic the injection distribution, the smaller the statistical error of the estimation. Furthermore, the BMC method is able to estimate quantities with a high energy threshold such as the acceleration integral used in hot-carrier degradation modeling. With our EES method, we showed that the investigation of effects affecting hot-carrier distribution including full-bands is possible with the BMC method.