Using ensemble Monte Carlo methods to evaluate non-equilibrium Green’s functions

Abstract

The use of ensemble Monte Carlo (EMC) methods for the simulation of transport in semiconductor devices has become extensive over the past few decades. This method allows for simulation utilizing particles while addressing the full physics within the device, leaving the computational difficulties to the computer. More recently, the study of quantum mechanical effects within the devices, effects which also strongly affect the carrier transport itself, have become important. While particles have continued to be useful in quantum simulations using Wigner functions, interest in analytical solutions based upon the non-equilibrium Green's functions (NEGF) have become of greater interest in device simulation. While NEGF has been adopted by many commercial semiconductor, there remains considerable computational difficulty in this approach. Here, a particle approach to NEGF is discussed, and preliminary results presented illustrating the computational efficiency that remains with the use of particles. This approach adopts the natural basis functions for use in a high electric field and the preliminary results are obtained for quantum transport in Si at 300 K. This approach appears to offer significant advantages for the use of NEGF.

Using ensemble Monte Carlo methods to evaluate non-equilibrium Green’s functions, II. Polar-optical phonons

“…While the previous paper [10], and the present work have treated three-dimensional semiconductors, it should directly translate to reduced dimensionality that arises in modern materials and devices. In this work, it may be noticed that the transverse directions appear to become two-dimensional, as may be noticed in (16).…”

“…One important function is the less-than self-energy, or at least the imaginary part that will appear in the integral equation for the distribution function. This may be written, with the modifications due to the change in matrix elements, as [10] Figure 3. The imaginary part of the less-than self-energy for the case of wurtzite InN.…”

“…Naturally, the time ∆t and the distance ∆s are related to one another. This relationship in nonparabolic bands takes the form [10]…”

“…The computed drift velocity of the ensemble used here is shown in figure 8. This is determined by averaging over the 10 5 particles and the 200 iterations of the EMC procedure, as described in I [10]. Thus, this is a time and ensemble average.…”

“…As quantum effects have become important, simulation of transport has led to the use of the nonequilibrium Green's functions (NEGF) [8,9], which are analytically extremely difficult. However, recently an approach based upon particle simulation of the NEGF has been proposed (hereafter referred to as I [10]). In this approach, the NEGF are defined in terms of an Airy transform in the direction along the electric field.…”