1. Introduction

The application of strained layers in semiconductor devices took on another point of view when it was recognized that strained-layer structures might display new electronic and optical properties not seen in the unstrained-constituent materials. Before this conceptually new point of view, strained-layer growth was considered a compromise between a desire to produce semiconductor heterostructures and simultaneously to avoid misfit dislocations in these structures. This was changed later by noting that strain could be a tool for modifying the band structure of semiconductors in a useful and predictable fashion. As the bigger part of modern technology is silicon based, much research interest on strained layers is devoted to this material. Before silicon became dominant in semiconductor industry, germanium was the main material, and its properties are also well developed. As there is a difference in the lattice constants of these two materials, it is quite natural to strain silicon based devices by introducing germanium or its alloy with silicon into both active layers and substrates.

First data on SiGe were published in 1955 based on measurements of magneto-resistance [1]. Later in 1975 first $\textrm{Si}_{1-x}\textrm{Ge}_x$ layers with $x\leq 0.15$ were grown on Si substrates using ultra high vacuum epitaxy [2]. As the lattice constants of Si and Ge differ by 4.2%, the epitaxial layers were strained when sufficiently thin. At the beginning the strain was essentially considered a drawback which destroys the perfect crystal structure and cannot be avoided. But later in 1982 it was realized that this provided an additional option for band structure engineering [3]. Soon after that the strained-layer modulation-doped field effect transistor was fabricated [4]. Two years later the hetero-junction bipolar transistor was reimplemented using strained-layer $\textrm{Si}_{1-x}\textrm{Ge}_{x}/\textrm{Si}$ heterostructures [5,6,7,8]. In the early 1990s short-period SiGe superlattices became popular as promising results were obtained in this field. The most important results were enhanced optical absorption at the bandgap of short-period superlattice structures [9] and band to band photoluminescence and electroluminescence [10,11]. At the end of the 1990s the idea of cascade lasers based on Si and SiGe heterostructures was discussed [12]. This idea turned into a large modern research area. Another present-day topic concerns quantum computing using SiGe heterostructures [13]. The idea is based on the difference of $g$ factors in Si and Ge and the possibility to shift the electron wave function into layers with different Ge composition. This changes the spin Zeeman energy and in this way produces single-qubit operations.

This short historical review demonstrates undiminished interest in electronic devices based on strained SiGe and that this area is quite large and diversified. Thus the investigation of SiGe properties and their changes under strain conditions plays an important role for future applications of strained SiGe layers in electronic devices. However, the complicated physics of SiGe systems gets even more sophisticated when strain comes into play. This leads to the need for physical modeling of strained SiGe layers.

In particular, the necessity of physical modeling stems from the fact that the kinetic processes in semiconductors have a complex behavior which cannot be described analytically. The strain makes the situation even worse as it affects the kinetic properties of the material. One of the main tasks for the modeling of strained SiGe layers is thus to examine the modifications introduced by strain into the kinetic properties of the semiconductor.

One possible way to describe the kinetic properties of the material is based on the kinetic Boltzmann equation. This is an integral-differential equation which can be nonlinear when the quantum mechanical Pauli exclusion principle is taken into account. There are very many approaches for the solution of the kinetic equation, both analytical and numerical ones. However, only the Monte Carlo approach allows a comprehensive physical model to be included without further approximations. Additionally, strain effects can be included in a natural way in the formalism provided by the Boltzmann equation.

In this thesis electron transport in strained $\textrm {Si}_{1-x}\textrm {Ge}_{x}$ layers is studied using an analytical anisotropic and nonparabolic band structure model. The influence of strain on low field as well as high field kinetics including the small signal response is studied using Monte Carlo methods. Undoped and doped layers are considered and the quantum mechanical Pauli exclusion principle is taken into account. New Monte Carlo methods which are equally applicable to any level of degeneracy are developed.

The thesis is organized as follows:

Chapter 2 provides the description of the semiclassical transport model used in this work. The equations of motion, the distribution function, the Boltzmann equation and its validity, band structure and scattering mechanisms are discussed.

Chapter 3 treats the strain effects in SiGe within the deformation-potential theory. The stress and the strain tensor are introduced. Their transformations is then given and applied to obtain the elements of the strain tensor in the principle axes using Hooke's law. Finally, the influence of the strain on the band structure and the scattering mechanisms is considered.

Chapter 4 develops the zero field Monte Carlo algorithm used to obtain the low field electron mobility tensor in strained SiGe. The Pauli exclusion principle is included into the scattering operator. The Monte Carlo algorithm is then derived using an integral representation of the Boltzmann equation. The role of the Pauli exclusion principle on the scattering processes is explained and reversing of the inelastic processes is found at high degeneracy. Finally, a small signal Monte Carlo algorithm accounting for the Pauli exclusion principle is developed.

In Chapter 5 applications are presented. The dependence of the low field mobility on the transport direction, Ge compositions of the substrate and the layer, substrate orientation and the doping level is investigated. To explain the mobility behavior the valley populations and the role of repoplution effects is studied. The small signal response including degeneracy effects is also explained.

Finally, Chapter 6 presents a summary of the thesis.

S. Smirnov: