Theory of Electron Transport in Small Semiconductor Devices Using the Pauli Master Equation

M V Fischetti

doi:10.1155/1998/67427

Abstract

It is argued that the Pauli master equation can be used to simulate electron transport in very small electronic devices under steady-state conditions. Written in a basis of suitable wavefunctions and with the appropriate open boundary conditions, this equation removes some of the approximations which render the Boltzmann equation unsatisfactory at small length-scales. The main problems consist in describing the interaction of the system with the reservoirs and in assessing the range of validity of the equation: Only devices smaller than the size of the electron wavepackets injected from the contacts can be handled. Two one-dimensional examples solved by a simple Monte Carlo technique are presented.

Highlights

As noted by Frensley discussing quantum transport in open systems [1], the Pauli master equation [2], (PME) could constitute an intuitive description of electron transport in semiconductor devices of size comparable to the electron wavelength: One could capture the wavelike nature of transport, lost in the Boltzmann-transport-equation (BTE) picture, so bypassing the weak-field, long devices limitations
Arguments have been raised against the correctness of the PME [1], it is applicable to very small devices and to steady-state phenomena: Small devices, because the PME rests on the absence of off-diagonal elements of the density matrix p
The main conclusion of this paper is that there is an interesting class of problems for which the Pauli master provides a solution of the electron transport problem overcoming some of the strongest limitations of the semiclassical Boltzmann equation: For devices with active regions smaller than a few hundreds of nm, the master equation makes it possible to account for the wave nature of the electrons, only in the weak scattering limit and in steady state or slow time transients

Summary

INTRODUCTION

As noted by Frensley discussing quantum transport in open systems [1], the Pauli master equation [2], (PME) could constitute an intuitive description of electron transport in semiconductor devices of size comparable to the electron wavelength: One could capture the wavelike nature of transport, lost in the Boltzmann-transport-equation (BTE) picture, so bypassing the weak-field, long devices limitations. Arguments have been raised against the correctness of the PME [1], it is applicable to very small devices and to steady-state phenomena: Small devices, because the PME rests on the absence of off-diagonal elements of the density matrix p. Their absence implies that the contacts inject ’plane waves’, which corresponds physically to the injection of spatially highly delocalized wavepackets. As note by Frensley [1], the PME is inconsistent with current continuity in this case

MASTER EQUATION AND CONTACTS

MONTE CARLO SOLUTION

SUMMARY AND CONCLUSIONS