Walter A. Harrison - Solid State Theory

This manuscript has been designed to serve as a text for a one-year graduate course in solid state theory. A year of quantum mechanics is prerequisite (more advanced methods are introduced as needed) but introductory solid state is not. Thus the course attempts to span the subject matter which in many universities is covered in an introductory solid state course and in a quantum theory of solids course. As an alternative to such a two-year program, Stanford offers this one general course and a number of shorter courses on specific topics in solid state.

In this one-year program I attempt to develop a sufficient familiarity with the concepts and methods of solid state physics that a reader will feel moderately at home in any solid state seminar or when reading any article in solid state physics. I attempt also to eliminate the threshold which may prevent a specialist in another field from attacking a solid state problem which arises in the context of his own work. Both these objectives, in my view, can be accomplished in a single year of study.

In this context there are very few topics in traditional or modern solid state physics which can be omitted altogether. Yet the subject is much too broad for any kind of complete coverage in a single course. An attempt to resolve this dilemma has been made by developing the fundamental ideas and approaches in many areas but this necessitates giving only a very small number of detailed applications. The first chapter develops the fundamentals of group and representation theory and gives only three illustrative applications (though use is made of group theory at many points later in the text). In discussing band structure only one prototype in each category is considered. The transport equation is derived, but is applied to only a few properties. Included, when possible, are discussions of currently active problems such as the Mott transition, the electronic structure of disordered systems, tunneling, the Kondo effect, and fluctuations near critical points. Which topics are most active vary from year to year and few remain of central interest indefinitely. Their inclusion, however, adds an air of relevance to the subject matter and brings the reader to the frontiers of knowledge in a number of areas.

Because virtually all of the properties of solids are determined by the valence electrons, I have focused on electron states in much of the first third of the text. This is predominantly a formulation of the approximate descriptions in different systems which allow a treatment of properties rather than a discussion of the energy bands themselves; the latter provide the basis for the treatment of very few properties. The remaining portion of the text is devoted to discussions of solids and liquids using these approximate descriptions. This organization of the material allows parallel treatments of each property for metals, semiconductors, and insulators, though in some cases it separates the discussion of related properties in one material. Throughout the text emphasis is placed on self-consistent-field methods for the treatment of interaction among electrons and pseudopotential methods for the treatment of the interaction among electrons and ions.

The timing of this course in the graduate curriculum and the breadth of its coverage can make it a vehicle for a major clarification of quantum mechanics for the student. Such clarification comes in the comparison of the semiclassical (Boltzmann equation) treatment of screening and the corresponding quantum (Liouville equation) treatment. It comes in understanding of the breakdown of the Franck-Condon principle and it comes with the understanding of off-diagonal long-range order in vibrating lattices and in superconductivity.

A course at this level also provides an opportunity to present two impressive phenomenologies, the Landau theory of Fermi liquids and the Ginsburg-Landau theory of superconductivity. Both of these are presented in the spirit in which they were generated rather than as consequences of microscopic theories. This form of presentation seemed much more in tune with an effort to enable students to generate good physics rather than to simply use the physics of others.

Throughout, an attempt is made to maintain a balance between the physics and the mathematics of our understanding of solids. Where the theory provides general methods for treating solids, the attempt has been to make the derivations careful and complete. Where the physical concepts are in my belief central, the tendency is only to outline the mathematical analysis. In some cases the details are shifted to the problems following each chapter; in some cases the reader may go to the more complete discussions referred to. Sufficient completeness in most arguments is presented, however, (frequently more complete than the original source) so that the references are more likely to provide additional applications than clarification of the methods.

In total, I view this as a text in physics rather than in mathematical methods. The emphasis is on current problems and modern concepts, but the traditional background which provides the foundation for modern theory is also included. My desire is to provide a framework that the reader can fill in with his subsequent reading and work, and upon which he can build understanding where it does not exist today.

Walter A. Harrison