It is argued that the Pauli master equation can be used to simulate electron transport in very small electronic devices under steady-state conditions. When written in a basis of suitable wave functions and with the appropriate open boundary conditions, this transport equation removes some of the approximations which render the Boltzmann equation unsatisfactory at small length scales, permitting the inclusion of tunneling, interference effects, arbitrary “steep” potentials, and intracollisional field effects. However, the master equation is based on the same weak-scattering and long-time limits on which also the Boltzmann equation rests and cannot provide the complete solution of time dependent quantum transport problems. The main problems consist in describing the interaction of the system with the reservoirs—here treated phenomenologically—and in assessing the range of validity of the equation: Only devices smaller than the size of the electron wave packets injected from the contacts can be handled, and this constitutes the interesting range of sub-50 nm devices. Three one-dimensional examples solved by a simple Monte Carlo technique are finally presented.

Theory of electron transport in small semiconductor devices using the Pauli master equation

We present a computational scheme which can be readily included in a standard Monte Carlo code to give quantitative estimates of the intracollisional field effect (ICFE) or of collisional broadening (CB). The central quantity in our theory is the joint spectral density K(e i , e f  ) which connects the initial and final kinetic energy of a carrier in a scattering event. Crucial approximations, connected with the algorithms which model the joint spectral densities obtained from the appropriate Dyson equation, are clearly stated and discussed. Numerical results for a simple modelsemiconductor are provided for illustrative purposes. These show that both ICFE and CB lead to a significant increase of carriers in the high‐energy tail of the distribution function.

Monte Carlo algorithms for collisional broadening and intracollisional field effect in semiconductor high‐field transport

A previously obtained generalised field-dependent screening function involving one-electron Wigner distributions and spectral functions is studied under various model approximations. The field dependence of the screening arises from the action of a (constant) electric field within duration of a collision process (estimated from the uncertainty relations) and has the effect of reducing the screening efficiency. This field-induced descreening effect is studied analytically and numerically for a range of electric field strengths and electron temperatures and densities in both three- and two-dimensional non-degenerate electron assemblies. The model assumed allows the influences of displaced Maxwellian distribution functions and finite electron quasiparticle lifetimes to be considered.

http://iopscience.iop.org/article/10.1088/0022-3719/18/12/013/pdf

A model study of field-dependent dynamical screening due to mobile electrons in submicron semiconductor devices

Transport in semiconductors has traditionally been described with the help of the Boltzmann equation (BE) (Conwell, 1967) (or by some of its simplifications, such as the drift-diffusion equation). Powerful Monte Carlo techniques have been developed to solve the BE (for a review see e.g. Jacoboni and Reggiani, 1983), and combined with Poisson equation solvers these methods form the basis for the theoretical analysis and design of modern semiconductor microdevices. As the characteristic length scales continue to decrease, however, quantum effects begin to dominate the transport, and consequently the semiclassical BE cannot be used as a starting point. Examples of such quantum effects are space quantization (the characteristic length scales in one or several directions are such that plane waves are no longer appropriate wave functions for describing the charge carriers), or ballistic transport (the charge carriers experience no, or only few, collisions within the active region of the device); here we point out the difference between quantum ballistic transport, such as underlies resistance quantization in quantum point contacts (van Wees et al., 1988; Wharam et al., 1988) and ballistic peaks observed in the distribution function in Boltzmann equation studies of microstructures (Baranger and Wilkins, 1984). Other effects beyond the conventional Boltzmann picture include the influence of field on collision processes, the so-called intra-collisional field effect (ICFE) (Barker, 1973; Levinson, 1970), and collisional broadening (CB), or, in other words, effects due to finite quasi-particle life-times.

Nonequilibrium green function techniques applied to hot-electron quantum transport

Models of hot-carrier-induced phenomena in the high-energy region for small-scale MOSFETs are presented. A free-flight optimization technique has also been developed to simulate the high-energy carrier dynamics by the Monte Carlo method with reasonable CPU time. The hot-carrier effect was investigated in detail from the carrier velocity and energy distribution viewpoint with rigorous treatment of carrier scattering processes, especially for photogeneration and impact ionization. The simulated results were in good agreement with experimental results. The analyses revealed that the hot-carrier effect is mainly dominated by carrier temperature, but is suppressed by impact ionization. >

Hot-carrier simulation for MOSFETs using a high-speed Monte Carlo method

First-order time-dependent perturbation theory is used to derive expression for scattering rates in the presence of high electric fields. Analysis in terms of Houston functions suggests that scattering rates are approximately field independent, but a more accurate treatment in terms of Airy functions shows that at sufficiently high fields, the resonant width of the scattering vertex is limited by the domain of integration, and a field dependent suppression of scattering occurs. This effect occurs first for small-wavevector phonons, and appreciable effects are expected for fields exceeding 104-105 V cm-1. It is also suggested that the complicated retarded transport equations recently introduced into hot electron theory are not necessary to treat the intra-collisional field effect in the weak phonon coupling limit at fields of interest in conventional semiconductor transport. At fields where suppression of scattering is significant, it is found that additional scattering into field-induced band tails is also important.

http://iopscience.iop.org/article/10.1088/0022-3719/15/26/020/pdf

D C Herbert

Papers

The intra-collisional field effect in semiconductors. I. Analytic results