Semiclassical approximations beyond the leading order in h

Chapter 1: Abstract, contents and introduction

Mesoscopic systems, a rapidly progressing field of physical research in the last two decades, are of increasing technological and commercial interest. Semiclassical approximations i.e. expansions of quantum mechanical equations to leading order $\hbar$,are appropriate tools for the theoretical description of these systems in betweenthe microscopic and the macroscopic regime. The validity of these approximations requires higher-order $\hbar$ corrections to be negligible. The influence of higher-order corrections is studied theoretically using model systems, and their contributions are traced down in experimental data on magnetoconductance.

Semiclassics are usually defined as approximations of the quantum mechanical equations to leading order in $\hbar$. This definition is accurate, short, and self-contained --- but by no means self-explaining. This chapter first provides the necessary context by giving a short overview of the history of  semiclassical approximations before presenting the modern form used in the subsequent chapters.

This chapter is devoted to the inclusion of finite temperature and impurity scattering in semiclassical approximations. The common microscopic approach is outlined, and another, more mathematically oriented ansatz is presented. The comparison of the two procedures allows an extension of the smoothing formalism to higher-order contributions in $\hbar$. This section provides some of the technical details which will be important when considering $\hbar$ corrections in the subsequent chapters.

The disk billiard in homogeneous  magnetic fields is used as a model system for semiclassical approximations. Its quantum mechanical level density can be calculated analytically. Therefore, a precise comparison of the semiclassical approach to the exact result is possible. The influence of various $\hbar$-corrections to the trace formula is examined. With the help of the trace formula's close relation to classical dynamics it is possible to give a simple, intuitive picture explaining all features of the level density.

Transport properties are, in contrast to the level density considered above, readily accessible in experiment. This chapter gives a short introduction to the semiclassical approximation of electrical transport within the linear response formalism. The formulas presented will be used in the subsequent chapters.

The experimental realization of a freetwo dimensional electron gas (2DEG) is outlined. The Shubnikov-de-Haas oscillations (SdH) in its longitudinal resistivity are reproduced by the semiclassical Kubo formula, but the plateaus in the Hall resistivity, i.,e. the integer quantum Hall effect (QHE), are not. The description of the QHE succeeds by including a specific higher-order $\hbar$ term originating from the level density. The corresponding correction is derived for general systems.

This chapter studies the longitudinal magnetoconductance of a mesoscopic channel with a central antidot dimer. The experimentally observed conductance oscillates in dependence of both the magnetic field strengths and the antidot radius (regulated by the applied gate voltage). The period of the oscillations in B is approximately constant, and the maxima positions exhibit characteristic dislocations when varying the antidot diameter. This behavior was previously related to inherent quantum effects and believed not to be accessible by semiclassical methods. The semiclassical description developed in this chapter is able to reproduce qualitatively as well as quantitatively all observed features. Additionally, it allows an intuitive explanation of the origin of the maxima dislocations.

This work investigated the applicability of semiclassical approximations to mesoscopic systems. The different problems analyzed are grouped around three setups: two simple model systems and a more complicated structure realized in experiment. The theoretical studies, namely the calculation of the level density of the disk billiard and the conductivity tensor of the free 2DEG, analyzed the influence of various higher-order $\hbar$ contributions to the semiclassical description. It was shown that only a few of these corrections are relevant.  The magnetoconductance of the experimental system -- a mesoscopic channel with antidots -- was successfully described within leading order of $\hbar$. All observed features were quantitatively and qualitatively reproduced, and an intuitive picture of their origin could be given.

This technical appendix describes numerical techniques to deal with classical periodic orbits. Efficient methods for finding those orbits, calculating the relevant properties for the trace formula, and following them through parameter space are presented.

Bibliography and acknowledgments