4. Monte Carlo Methods for the Solution of the Boltzmann Equation

To analyze the semiclassical kinetics in semiconductors described in Chapter 2 in a comprehensive manner it is necessary to solve the Boltzmann kinetic equation (2.60). This equation mathematically represents an integro-differential equation. In general the exact solution cannot be obtained analytically. Thus approximate solution techniques have been developed. They can be divided into three classes which are analytical techniques, deterministic numerical methods, and Monte Carlo approaches.

Analytical techniques were the first attempt to understand the semiclassical transport in semiconductors. Some of these techniques assume an analytical form of the distribution function which contains parameters. These parameters are obtained from the Boltzmann kinetic equation. For example, for a heated and drifted normalized Maxwellian distribution

$$f(\vec{k})=A\exp\biggl\{-\frac{\hbar^{2}(\vec{k}-\vec{k}_{d})^{2}}{2mk_{B}T_{e}}\biggr\},$$ (4.1)

where $A$ is the normalization constant, two parameters $\vec{k}_{d}$ and $T_{e}$ are introduced. The first parameter, $\hbar\vec{k}_{d}$ is the average quasi-momentum of the distribution. The second one, $T_{e}$, is the electron temperature. These two parameters are obtained from the equations which represent the first three moments of the Boltzmann transport equation which are coupled. Despite this approach is restricted to simple cases it turned out to be very useful for the hot-electron problem [63]. Another analytical approach is related to an expansion of the distribution function in spherical harmonics [64,65,66,67]. If a problem has a cylindrical symmetry around the direction of the electric field, the Legendre polynomial expansion can be written as:

$$f(\vec{k})=f_{0}(\epsilon)+f_{1}(\epsilon)P_{1}(\cos\theta)+\cdots,$$ (4.2)

where $\epsilon$ is the electron energy, $\theta$ stands for the angle between the electron quasi-momentum $\hbar\vec{k}$ and electric field^4.1, $f_{0}$ denotes the symmetrical part of the distribution function and $f_{1}$ is the anti-symmetrical contribution to the distribution. Expansion (4.2) is substituted into the kinetic equation. Making use of the orthogonality of the Legendre polynomials a coupled system of equations for the coefficients $f_{0}$, $f_{1}$, $\ldots$ is derived. This system can then be solved numerically. However, the treatment of general band structures is quite difficult within this approach.

Iterative techniques [68,69,70] are based on the following equality for the distribution function for a homogenous system^4.2:

$$f(\vec{k},t)=\int_{0}^{\infty}\,dt^{'}\int\,d\vec{k}^{'}\exp\bigg... ...da[\vec{K}(y)]\,dy\biggr\} f(\vec{k}^{'},t-t^{'})S(\vec{K}(t^{'}),\vec{k}^{'}).$$ (4.3)

The iterative method consists of substituting an initial distribution function $f_{0}(\vec{k},t)$ into the right-hand side of (4.3) and evaluating $f(\vec{k},t)$. The new distribution function is again substituted into the right-hand side of (4.3) and this procedure is repeated until $f(\vec{k},t)$ has converged to its solution with a given accuracy.

The Monte Carlo method simulates the electron's motion in a crystal under external forces. Within this technique a particle trajectory is constructed as a sequence of free flights and scattering events^4.3. The free flight times between collisions and the parameters of the scattering events are generated stochastically using probabilities of microscopic processes. Thus the main advantage of this approach is the direct description of the microscopic particle dynamics. This allows to incorporate within the same technique very complicated kinetic phenomena. General band structures of different semiconductors can be taken into account. In addition the implementation is simpler compared to other numerical methods.

In this work preference is given to the Monte Carlo approach. In the following new Monte Carlo algorithms are developed for modeling of strained bulk SiGe, and known Monte Carlo techniques and their aspects are also described.

Subsections

• 4.1 Perturbation Approach to the Boltzmann Equation
  Including the Pauli Principle
  • 4.1.1 The Zeroth Order Equation
  • 4.1.2 The First Order Equation
  
  
• 4.2 Integral Form of the First Order Equation
  • 4.2.1 New Differential and Total Scattering Rates
  • 4.2.2 Integral Form for the Distribution Function Perturbation
  • 4.2.3 Free Term and Initial Distribution
  • 4.2.4 The Resolvent Series
  • 4.2.5 The Second Iteration of the Forward Resolvent Series
  
  
• 4.3 Zero Field Monte Carlo Algorithm Including the
  Pauli Principle
  • 4.3.1 Low Field Carrier Mobility and Monte Carlo Techniques
  • 4.3.2 Specific of the Equilibrium Distribution
  • 4.3.3 Total Scattering Rate
  • 4.3.4 Expression for the Initial Distribution
  • 4.3.5 Monte Carlo Algorithm for the Mobility Tensor
  
  
• 4.4 The High Field Small Signal Monte Carlo Algorithm
  • 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: