In this book, we will introduce many concepts, some of them rather abstract, that are used to describe solids. Since most materials are ultimately used in some kind of application, it seems logical to investigate the link between the atomic structure of a solid, and its resulting macroscopic properties. After all, that is what the materials scientist or engineer is really interested in: how can we make a material useful for a certain task? What type of material do we need for a given application? And why can some materials not be used for particular applications? All these questions must be answered when a material is considered as part of a design. The main focus of this book is on the fundamental description of the positions and types of the atoms, the ultimate building blocks of solids, and on some of the experimental techniques used to determine how these atoms are arranged.

The subject of this chapter is solving the Schrödinger equation for electrons moving in a periodic mean field in crystals. The solution is called the Bloch orbital. We discuss Bloch's theorem, which applies to it. As for the energy eigenvalues, there are allowed and forbidden values. The allowed values are distributed in extended regions that are called bands. The forbidden values comprise the gaps. In order to calculate the Bloch orbital, we make use of an approximation based on the free electron model, and another approximation based on the linear combination of atomic orbitals. As a more realistic approach, we discuss the Wigner-Seitz method.

The periodic structure of crystals

In the previous chapter, we showed that many experimental facts can be explained by assuming that the conduction electrons in metals move in a uniform mean field. However, in real metals, there are collisions with ions. A question arises as to why these collisions do not cause a significant effect. Moreover, there are some problems that cannot be explained in the free electron model. One such problem is the quantitative deviations from the experimental results. Also, the signs associated with the Hall effect and the thermoelectric power are often opposite to that expected in the free electron model. Another problem is why this model does not apply at all to the case of insulators.

Solids are classified according to the qualitative nature of the interaction by which the atoms in solids attract each other. In metals, in particular, there is an important contribution due to the itinerant electrons. We then discuss the Sommerfeld theory, which describes such electrons. Almost all electrons are dormant because of the Fermi-Dirac statistics, and so they do not contribute to either specific heat or susceptibility. Such an electron system is said to be Fermi degenerate, and exhibits a number of singular properties. These are especially marked when the electrons interact with localized spin, and this point will be the main theme of this book in Chapters 5 and beyond.

Classification of solids

Thanks to recent quantum mechanical treatment, thorough understanding has been obtained not only on the problem of why molecular bonds appear but also on the problem of why it is more energetically favorable for atoms to get together and form solids than to be separated into individual atoms and molecules. While some aspects of these problems can be understood to some extent in terms of classical or phenomenological analysis, for the problem of the binding in metals, in particular, a quantum mechanical treatment is indispensable. In this section, let us classify solids on the basis of their condensation mechanism.

Molecular crystals

Since rare-gas atoms such as He, Ne and so on, which have closed-shell structures, and also molecules such as H2, Cl2 and so on, do not interact strongly with one another, it is difficult for them to form solids.

We consider the H2 molecule. We introduce the molecular orbitals from the viewpoint of mean-field approximation, and these are classified into bonding and antibonding orbitals. We discuss the molecular bonds also from the Heitler-London viewpoint, which is based on atomic orbitals. In both of these cases, the state with zero total spin, or the spin singlet state, is found to form a stable molecule. We discuss the relationship between the two viewpoints, and introduce the configuration interaction as an improvement to both. In order to facilitate the treatment of complex molecules, a model is proposed, and a second-quantization procedure which is convenient for its description is introduced.

The H+2 molecule

Let us, in this chapter, define the molecule as a system that consists of more than one nucleus and one or more electrons. Its wavefunction is a function of the coordinates of the nuclei and the electrons. An intuitive picture is as follows. Electrons are lighter by far than nuclei and are moving around fast. It is therefore reasonable to consider the nuclei as being instantaneously fixed, and to solve for the wavefunction of the electrons. The wavefunction and energy thus obtained are functions of the positions of the nuclei, and the energy can be considered to play the role of the potential energy with respect to the motion of the nuclei. This approximation is called the adiabatic approximation, or the Born-Oppenheimer approximation, and is valid when the ratio m/M of the masses of the electron and the nuclei is sufficiently small.

I wrote this book as an introduction to the theory of electrons in metals. There are a good many texts on this topic, and the emphasis of this present book is in discussing the physics of dilute magnetic alloys. The first half of this book is devoted to discussion of the topics that are necessary for the discussions in the later chapters. Recent activity in the theory of dilute magnetic alloys has made the field highly complex, so I have tried to describe this at a level that is suitable for those who are new to the subject area.

While metals are characterized by the presence of electrons that move about freely in them, it is most important that, unlike in semi-conductors, there is huge electron density. As a result, quantum effects become predominant and, because electrons are fermions, the phenomenon of degeneracy takes place. We may even go so far as to say that almost all of the characteristic behavior of metals is due to this phenomenon of degeneracy.

Concerning the quantum theory of electrons in metals, there have been five major developments. The first is the Sommerfeld theory, which introduced the concept of degeneracy to explain the behavior of the electronic specific heat. The second is the Bohm-Pines theory, which discusses the effect of the inter electronic Coulomb interaction, together with the many-body treatment of the problem which emerged from this theory.

We present here an overview of the electronic structure of atoms. We begin with the mean-field approximation. This scheme is sometimes also called the Hartree approximation, and is the most basic starting point when discussing many-electron systems. In this approach, the atomic states are distinguished from one another by their electronic configuration. An electronic configuration is, in general, degenerate with a number of other configurations. However, when we take into account the corrections due to the deviation of the Coulomb interaction away from the mean field, the energy levels are split into a number of distinct levels, and each of these split energy levels is called a multiplet. In order to demonstrate this point, we introduce the Slater determinant. After this, we discuss the Coulomb integral and the exchange integral. In particular, because of the Pauli principle, the exchange integral exists only between electrons with the same spin orientations. This allows us to explain Hund's rule, that is, the multiplet that has the largest value of composite spin has the lowest energy.

Mean-field approximation and electronic configurations

The usual starting point for discussing the electronic structure of atoms is the mean-field approximation.

The motion of an electron is affected by attractive Coulomb interaction due to the positive charge Ze of the nucleus and repulsive Coulomb interaction due to the other electrons. The latter is time dependent owing to the motion of the other electrons, but we may, as an approximation, replace these electrons by an appropriate charge distribution and consider the Coulomb force due to it.

Wilson (1975) introduced the following method, which is distinct from the perturbative calculation where an expansion in terms of J is used. In his method, one starts from a two-body system composed of a localized spin and a conduction electron located at the same point. One then proceeds to expand the system by gradually taking into account electrons that are in the nearest neighborhood. As one goes further away from the localized spin, the states that are far away from the Fermi surface are taken out of consideration. That is, when using this method, the states around the Fermi surface are described more and more precisely as the system grows larger in size, and this corresponds to lowering the temperature.

As the system grows larger, the number of basis states increases rapidly. However, consideration of around a thousand excited states near the ground state is sufficient, and therefore the size of the matrix that needs to be diagonalized remains small enough for numerical simulation on the computer. In this way, for a reasonable value of J, when the system size is increased in a series of about 100 steps, Wilson obtained a state which was considered to be at sufficiently low temperature.

Let us investigate in more detail the origin of the logarithmic term that we found in the previous chapter. We examine the s–d Hamiltonian from this point of view. We first consider the two total wavefunctions of conduction electrons moving in two different local potentials. The overlap integral and matrix elements of these two wavefunctions contain logarithmic terms. In particular, the overlap integral vanishes at zero temperature in general. Second, we investigate the development of the wavefunction under a time-dependent perturbation where one potential changes into the other one abruptly at a particular time. This problem is closely related to the physics of the s-d interaction, since the potential changes quite abruptly as the spin-flips occur due to the s-d interaction. This problem is solved using the Nozières-de Dominicis method, and we apply these general considerations to the s–d Hamiltonian to obtain the perturbation expansion of the partition function and other physical quantities. The behavior of the localized spin at low temperatures is clarified using the scaling method applied to the s–d Hamiltonian.

The Anderson orthogonality theorem

This section is based mainly on Anderson (1967a,b).

In the previous chapter, we showed that the logarithmic singularity arises as a function of temperature in several physical quantities for systems governed by the s–d Hamiltonian due to flips of the localized spin. It is essential in this phenomenon that the localized spin has an internal degree of freedom.

There have been a number of notable developments in this field after the publication of the Japanese-language version of this book. The purpose of this chapter is to discuss two such topics.

The first topic involves a series of intermetallic compounds, which usually contain Ce. These have been found to have electronic heat capacity that is several hundred to thousand times greater than the corresponding heat capacity of ordinary metals. Here, although a Ce atom is magnetic, it does not interact with the neighboring Ce atoms, and it behaves as if it is an isolated magnetic atom. This fact explains the large heat capacity.

The other topic is the quantum dot. This tiny ‘artificial atom’ can be connected to leads to give rise to a system which is analogous to a system of metals with impurities. When there are an odd number of electrons in the quantum dot, the quantum dot acquires a spin, and its behavior becomes similar to that of a magnetic atom in metals. In particular, the transmission probability of an electron through the leads becomes 1 at absolute zero, and this corresponds to the unitary limit of electrical resistivity in systems with magnetic impurity.

The spin-flip rate

A spin placed inside a metal undergoes an interaction with the conduction electrons of the form eq. (5.90), which causes the inversion of the direction of the spin.

This present volume is a translation of “kinzoku denshi-ron – jisei-gokin wo chushin to shite” (“Theory of electrons in metals – with emphasis on magnetic alloys”), written by Professor J. Kondo in Japanese and published by Shokabo in 1983. The translation contains an additional chapter which discusses some of the developments that have taken place since the original publication of the book. The title of the book could very well have been “The Kondo effect”, had the author not been Professor Kondo himself. The discussion of the Kondo effect takes the prime position in this book, though the author never refers to it as such!

The author and his theory need no introduction. Suffice it to say that his work has been a milestone in condensed matter physics, with far-reaching consequences such as those in the study of many-body problems in general. But not only that. The Kondo effect also marked the beginning of the concept of asymptotic freedom, where the relevant coupling strength increases logarithmically with decreasing energy/temperature scale. This phenomenon is of central importance in the physics of strong interaction, in particle and nuclear physics, which is now believed to be described by quantum chromo-dynamics (QCD). We should add that there has been renewed interest in the study of the Kondo effect in the context of heavy electron systems and quantum dots.

Professor Kondo's famous work was carried out in the 1960s, in what has now become a central block of the National Institute of Advanced Industrial Science and Technology (AIST) in Tsukuba, Japan, and was then called the Electro-technical Laboratory or ETL and situated in Tanashi, Japan.

Andrei and Wiegmann derived the exact solution to the s-d problem independently. If the s-d interaction is localized, only the radial degree of freedom needs to be considered, and the problem may be reduced to the Schrödinger equation in one-dimensional real space. When the solution is assumed to be in accord with Bethe's ansatz, the problem can be treated in exactly the same manner as in the one-dimensional Hubbard model, and the exact solution method used therein can be adapted as it is to the s–d problem. In this way, each of the various physical quantities can be represented by a single function all the way from high to low temperatures. This is a function of T/TK, and its functional form is found to be consistent with the result of Wilson, insofar as they can be compared. This approach is mathematically powerful, and we may apply it to the case with a magnetic field, the case with S >1/2, and the Anderson model. However, the focus of our attention in this chapter will be to discuss the initial analysis of Andrei (Andrei, 1980; Andrei and Lowenstein, 1981; Andrei et al., 1983) and Wiegmann (Weigmann, 1981; Filyov et al., 1981).

A one-dimensional model

At first, let us consider the movement of conduction electrons on a onedimensional line. Even in the three-dimensional case, if the interaction is δ function-like, only s-wave scattering arises.

In this chapter, we discuss the electronic states of a single impurity atom in metals. In particular, when the impurity atom is a 3d transition metal, its 3d orbital tends to assume the character of an isolated atom, and has a non-zero spin due to a similar mechanism to Hund's rule in atoms. We first describe the Friedel-Anderson theory regarding the emergence of this localized spin.

In alloys with a small amount of 3d transition metals, we have a long-standing problem which is known as the resistance minimum phenomenon. This is the phenomenon that the electrical resistance starts to rise as the temperature falls to around the boiling point of helium. We explain that this phenomenon is due to the exchange interaction between a localized spin and conduction electrons. This result suggests that we need to refine our discussion of the emergence of localized spin further, and many theoretical studies have been done. These theoretical works will be discussed later in Chapters 6 to 8. In this chapter, we compare the speed of the fluctuation of localized spin against the timescale of observation. We emphasize that when the latter is greater than the former, localized spin appears to vanish.

Local charge neutrality

In the discussion of the electronic states of an impurity atom in metals, the overall electric charge neutrality becomes an important issue. The potential due to a single impurity needs to fall to zero sufficiently fast as the distance r from the impurity becomes large.