4.4 The High Field Small Signal Monte Carlo Algorithm

To investigate the small signal response of the carriers in semiconductors different Monte Carlo techniques are widely applied to solve the time-dependent Boltzmann equation [74,73,77,78,79]. There are also small signal approaches based on the velocity and energy balance equations [80,81]. However, a significant advantage of Monte Carlo methods based on the Boltzmann kinetic equation is that they allow a comprehensive treatment of kinetic phenomena within the quasi-classical approach and to account for accurate band structures. Additionally, the quantum mechanical Pauli exclusion principle can be taken into consideration to study the small signal response of carriers in degenerate semiconductors.

When the carrier density is very high the Pauli exclusion principle becomes important and may have a strong influence on various differential response functions which relate a small perturbation of the electric field and a mean value of some physical quantity. The influence is expected to be strong and it has been pointed out [74] that the behavior of impulse response functions is determined by the overlap of the distributions of two carrier ensembles introduced in the formalism. This overlap is much stronger when the Pauli exclusion principle is included due to the additional statistical broadening. When degenerate statistics is taken into account, the Boltzmann equation is nonlinear which makes its solution more difficult. One of the possible solution methods is based on a Legendre polynomial expansion [65]. In this work however, the Monte Carlo method is employed.

In this section the approach presented in [74] is extended and a Monte Carlo algorithm for small signal analysis of degenerate electron gases in homogeneous bulk semiconductors is constructed.
Subsections

• 4.4.1 Solution of the Zeroth Order Equation
  • 4.4.1.1 Initial Distributions of the Two Ensembles
  • 4.4.1.2 Normalization of the Stationary Distribution Function
  • 4.4.1.3 Integral Form of the Nonlinear Boltzmann Equation
  
  
• 4.4.2 Solution of the First Order Equation
• 4.4.3 Monte Carlo Algorithm for the Impulse Response

S. Smirnov: