Electronic Transport in Metallic Systems and Generalized Kinetic EquationsInternational Journal of Modern Physics B (IJMPB), Volume: 25, Issue: 23-24 (2011) p.3071-3183

This paper reviews some selected approaches to the description of transport properties, mainly electroconductivity, in crystalline and disordered metallic systems. A detailed qualitative theoretical formulation of the electron transport processes in metallic systems within a model approach is given. Generalized kinetic equations which were derived by the method of the nonequilibrium statistical operator are used. Tight-binding picture and modified tight-binding approximation (MTBA) were used for describing the electron subsystem and the electron-lattice interaction correspondingly. The low- and high-temperature behavior of the resistivity was discussed in detail. The main objects of discussion are nonmagnetic (or paramagnetic) transition metals and their disordered alloys. The choice of topics and the emphasis on concepts and model approach makes it a good method for a better understanding of the electrical conductivity of the transition metals and their disordered binary substitutional alloys, but the formalism developed can be applied (with suitable modification), in principle, to other systems. The approach we used and the results obtained complements the existent theories of the electrical conductivity in metallic systems. The present study extends the standard theoretical format and calculation procedures in the theories of electron transport in solids.

Keywords: Transport phenomena in solids; electrical conductivity in metals and alloys; transition metals and their disordered alloys; tight-binding and modified tight-binding approximation; method of the nonequilibrium statistical operator; generalized kinetic equations.

Contents:

1 Introduction

Transport properties of matter constitute the transport of charge, mass, spin, energy and momentum [1, 2, 3, 4, 5, 6, 7, 8]. It has not been our aim to discuss all the aspects of the charge and thermal transport in metals. We are concerned in the present work mainly with some selected approaches to the problem of electric charge transport (mainly electroconductivity) in crystalline and disordered metallic systems. Only the fundamentals of the subject are treated. In the present work we aim to obtain a better understanding of the electrical conductivity of the transition metals and their disordered binary substitutional alloys both by themselves and in relationship to each other within the statistical mechanical approach. Thus our consideration will concentrate on the derivation of generalized kinetic equations suited for the relevant models of metallic systems.
The problem of the electronic transport in solids is an interesting and actual part of the physics of condensed matter [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. It includes the transport of charge and heat in crystalline and disordered metallic conductors of various nature. Transport of charge is connected with an electric current. Transport of heat has many aspects, main of which is the heat conduction. Other important aspects are the thermoelectric effects. The effect, termed Seebeck effect, consists of the occurrence of a potential difference in a circuit composed of two distinct metals at different temperatures. Since the earlier seminal attempts to construct the quantum theory of the electrical, thermal [27, 28, 29, 30] and thermoelectric and thermomagnetic transport phenomena [31], there is a great interest in the calculation of transport coefficients in solids in order to explain the experimental results as well as to get information on the microscopic structure of materials [32, 33, 34, 35].
A number of physical effects enter the theory of quantum transport processes in solids at various density of carriers and temperature regions. A variety of theoretical models has been proposed to describe these effects [1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 25, 36, 37, 38, 39, 40, 41]. Theory of the electrical and heat conductivities of crystalline and disordered metals and semiconductors have been developed by many authors during last decades [1, 2, 3, 4, 5, 6, 20, 36, 37, 38, 39, 40, 41]. There exist a lot of theoretical methods for the calculation of transport coefficients [18, 20, 36, 37, 38, 42, 43, 44, 45, 46], as a rule having a fairly restricted range of validity and applicability. In the present work the description of the electronic and some aspects of heat transport in metallic systems are briefly reviewed, and the theoretical approaches to the calculation of the resistance at low and high temperature are surveyed. As a basic tool we use the method of the nonequilibrium statistical operator [42, 43](NSO). It provides a useful and compact description of the transport processes. Calculation of transport coefficients within NSO approach [42] was presented and discussed in the author’s work [45]. The present paper can be considered as the second part of the review article [45]. The close related works on the study of electronic transport in metals are briefly summarized in the present work. It should be emphasized that the choice of generalized kinetic equations among all other methods of the theory of transport in metals is related with its efficiency and compact form. They are an alternative (or complementary) tool for studying of transport processes, which complement other existing methods.
Due to the lack of space many interesting and actual topics must be omitted. An important and extensive problem of thermoelectricity was mentioned very briefly; thus it has not been possible to do justice to all the available theoretical and experimental results of great interest. The thermoelectric and transport properties of the layered high- cuprates were reviewed by us already in the extended review article [47].
Another interesting aspect of transport in solids which we did not touched is the spin transport [7, 8]. The spin degree of freedom of charged carriers in metals and semiconductors has attracted in last decades big attention and continues to play a key role in the development of many applications, establishing a field that is now known as spintronics. Spin transport and manipulation in not only ferromagnets but also nonmagnetic materials are currently being studied actively in a variety of artificial structures and designed new materials. This enables the fabrication of spintronic properties on intention. A study on spintronic device structures was reported as early as in late sixties. Studies of spin-polarized internal field emission using the magnetic semiconductor sandwiched between two metal electrodes opened a new epoch in electronics. Since then, many discoveries have been made using spintronic structures [7, 8]. Among them is giant magnetoresistance in magnetic multilayers. Giant magnetoresistance has enabled the realization of sensitive sensors for hard-disk drives, which has facilitated successful use of spintronic devices in everyday life. There is big literature on this subject and any reasonable discussion of the spin transport deserves a separate extended review. We should mention here that some aspects of the spin transport in solids were discussed by us in Refs. [45, 48].
In the present study a qualitative theory for conductivity in metallic systems is developed and applied to systems like transition metals and their disordered alloys. The nature of transition metals is discussed in details and the tight-binding approximation and method of model Hamiltonians are described. For the interaction of the electron with the lattice vibrations we use the modified tight-binding approximation (MTBA). Thus this approach can not be considered as the first-principle method and has the same shortcomings and limitations as describing a transition metal within the Hubbard model. In the following pages, we shall present a formulation of the theory of the electrical transport in the approach of the nonequilibrium statistical operator. Because several other sections in this review require a certain background in the use of statistical-mechanical methods, physics of metals, etc., it was felt that some space should be devoted to this background. Sections 2 to 8 serves as an extended introduction to the core sections 9-12 of the present paper. Thus those sections are intended as a brief summary and short survey of the most important notions and concepts of charge transport (mainly electroconductivity) for the sake of a self-contained formulation. We wish to describe those concepts which have proven to be of value, and those notions which will be of use in clarifying subtle points.
First, in order to fix the domain of study, we must briefly consider the various formulation of the subject and introduce the basic notions of the physics of metals and alloys.

2 Metals and Nonmetals. Band Structure

The problem of the fundamental nature of the metallic state is of long standing [1, 2, 3, 13]. It is well known that materials are conveniently divided into two broad classes: insulators (nonconducting) and metals (conducting) [13, 49, 50, 51]. More specific classification divided materials into three classes: metals, insulators, and semiconductors. The most characteristic property of a metal is its ability to conduct electricity. If we classify crystals in terms of the type of bonding between atoms, they may be divided into the following five categories (see Table 1).

Table1. Five Categories of Crystals

Table 1: Five Categories of Crystals
Type of Crystal substances
ionic alkali halides, alkaline oxides, etc.
homopolar bounded (covalent) diamond, silicon, etc.
metallic various metals and alloys
molecular , , , , , etc.
hydrogen bonded ice, , fluorides, etc.

Ultimately we are interested in studying all of the properties of metals [1]. At the outset it is natural to approach this problem through studies of the electrical conductivity and closely related problem of the energy band structure [32, 33, 34, 35].
The energy bands in solids [13, 33, 35] represent the fundamental electronic structure of a crystal just as the atomic term values represent the fundamental electronic structure of the free atom. The behavior of an electron in one-dimensional periodic lattice is described by Schrödinger equation

(2.1)

where is periodic with the period of the lattice . The variation of energy as a function of quasi-momentum within the Brillouin zones, and the variation of the density of states with energy, are of considerable importance for the understanding of real metals. The assumption that the potential is small compared with the total kinetic energy of the electrons (approximation of nearly free electrons) is not necessarily true for all metals. The theory may also be applied to cases where the atoms are well separated, so that the interaction between them is small. This treatment is usually known as the approximation of ”tight binding” [13]. In this approximation the behavior of an electron in the region of any one atom being only slightly influenced by the field of the other atoms [33, 52]. Considering a simple cubic structure, it is found that the energy of an electron may be written as

(2.2)

where is an integral depending on the difference between the potentials in which the electron moves in the lattice and in the free atom, and has a similar significance [33, 52] (details will be given below). Thus in the tight-binding limit, when electrons remain to be tightly bound to their original atoms, the valence electron moves mainly about individual ion core, with rare hopping from ion to ion. This is the case for the -electrons of transition metals. In the typical transition metal the radius of the outermost -shell is less than half the separation between the atoms. As a result, in the transition metals the -bands are relatively narrow. In the nearly free-electron limit the bands are derived from the - and -shells which radii are significantly larger than half the separation between the atoms. Thus, according this simplified picture simple metals have nearly-free-electron energy bands (see Fig.1).

Figure 1: Schematic form of the band structure of various metals

Fortunately in the case of simple metals the combined results of the energy band calculation and experiment have indicated that the effects of the interaction between the electrons and ions which make up the metallic lattice is extremely weak. It is not the case for transition metals and their disordered alloys [53, 54].
An obvious characterization of a metal is that it is a good electrical and thermal conductor [1, 2, 13, 55, 56]. Without considering details it is possible to see how the simple Bloch picture outlined above accounts for the existence of metallic properties, insulators, and semiconductors. When an electric current is carried, electrons are accelerated, that is promoted to higher energy levels. In order that this may occur, there must be vacant energy levels, above that occupied by the most energetic electron in the absence of an electric field, into which the electron may be excited. At some conditions there exist many vacant levels within the first zone into which electrons may be excited. Conduction is therefore possible. This case corresponds with the noble metals. It may happen that the lowest energy in the second zone is lower than the highest energy in the first zone. It is then possible for electrons to begin to occupy energy contained within the second zone, as well as to continue to fill up the vacant levels in the first zone and a certain number of levels in the second zone will be occupied. In this case the metallic conduction is possible as well. The polyvalent metals are materials of this class.
If, however, all the available energy levels within the first Brillouin zone are full and the lowest possible electronic energy at the bottom of the second zone is higher than the highest energy in the first zone by an amount , there exist no vacant levels into which electrons may be excited. Under these conditions no current can be carried by the material and an insulating crystal results.
For another class of crystals, the zone structure is analogous to that of insulators but with a very small value of . In such cases, at low temperatures the material behaves as an insulator with a higher specific resistance. When the temperature increases a small number of electrons will be thermally excited across the small gap and enter the second zone, where they may produce metallic conduction. These substances are termed semiconductors [13, 55, 56], and their resistance decreases with rise in temperature in marked contrast to the behavior of real metals (for a detailed review of semiconductors see Refs. [57, 58]).
The differentiation between metal and insulator can be made by measurement of the low frequency electrical conductivity near K. For the substance which we can refer as an ideal insulator the electrical conductivity should be zero, and for metal it remains finite or even becomes infinite. Typical values for the conductivity of metals and insulators differ by a factor of the order . So big difference in the electrical conductivity is related directly to a basic difference in the structural and quantum chemical organization of the electron and ion subsystems of solids. In an insulator the position of all the electrons are highly connected with each other and with the crystal lattice and a weak direct current field cannot move them. In a metal this connection is not so effective and the electrons can be easily displaced by the applied electric field. Semiconductors occupy an intermediate position due to the presence of the gap in the electronic spectra.
An attempt to give a comprehensive empirical classification of solids types was carried out by Zeitz [55] and Kittel [56]. Zeitz reanalyzed the generally accepted classification of materials into three broad classes: insulators, metals and semiconductors and divided materials into five categories: metals, ionic crystals, valence or covalent crystals, molecular crystals, and semiconductors. Kittel added one more category: hydrogen-bonded crystals. Zeitz also divided metals further into two major classes, namely, monoatomic metals and alloys.
Alloys constitute an important class of the metallic systems [25, 55, 56, 49, 59, 60, 61]. This class of substances is very numerous [49, 59, 60, 61]. A metal alloy is a mixed material that has metal properties and is made by melting at least one pure metal along with another pure chemical or metal. Examples of metal alloys , and an alloy of carbon and iron, or copper, antimony and lead. Brass is an alloy of copper and zinc, and bronze is an alloy of copper and tin. Alloys of titanium, vanadium, chromium and other metals are used in many applications. The titanium alloys (interstitial solid solutions) form a big variety of equilibrium phases. Alloy metals are usually formed to combine properties of metals and the exact proportion of metals in an alloy will change the characteristic properties of the alloy. We confine ourselves to those alloys which may be regarded essentially as very close to pure metal with the properties intermediate to those of the constituents.
There are different types of monoatomic metals within the Bloch model for the electronic structure of a crystal: simple metals, alkali metals, noble metals, transition metals, rare-earth metals, divalent metals, trivalent metals, tetravalent metals, pentavalent semimetals, lantanides, actinides and their alloys. The classes of metals according to crude Bloch model provide us with a simple qualitative picture of variety of metals. This simplified classification takes into account the state of valence atomic electrons when we decrease the interatomic separation towards its bulk metallic value. Transition metals have narrow -bands in addition to the nearly-free-electron energy bands of the simple metals [53, 54]. In addition, the correlation of electrons plays an essential role [53, 54, 62]. The Fermi energy lies within the -band so that the -band is only partially occupied. Moreover the Fermi surface have much more complicated form and topology. The concrete calculations of the band structure of many transition metals (, , , , , etc.) can be found in Refs. [13, 32, 33, 34, 35, 53, 63, 64, 65, 66] and in Landolt-Bornstein reference books [60, 67].
The noble metal atoms have one -electron outside of a just completed -shell. The -bands of the noble metals lie below the Fermi energy but not too deeply. Thus they influence many of the physical properties of these metals. It is, in principle, possible to test the predictions of the single-electron band structure picture by comparison with experiment. In semiconductors it has been performed with the measurements of the optical absorption, which gives the values of various energy differences within the semiconductor bands. In metals the most direct approach is related to the experiments which studied the shape and size of the Fermi surfaces. In spite of their value, these data represent only a rather limited scope in comparison to the many properties of metals which are not so directly related to the energy band structure. Moreover, in such a picture there are many weak points: there is no sharp boundary between insulator and semiconductor, the theoretical values of have discrepancies with experiment, the metal-insulator transition [68] cannot be described correctly, and the notion ”simple” metal have no single meaning [69]. The crude Bloch model even met more serious difficulties when it was applied to insulators. The improved theory of insulating state was developed by Kohn [70] within a many-body approach. He proposed a new and more comprehensive characterization of the insulating state of matter. This line of reasoning was continued further in Refs. [68, 71, 72] on a more precise and firm theoretical and experimental basis.
Anderson [50] gave a critical analysis of the Zeitz and Kittel classification schemes. He concluded that ”in every real sense the distinction between semiconductors and metals or valence crystals as to type of binding, and between semiconductor and any other type of insulator as to conductivity, is entirely artificial; semiconductors do not represent in any real sense a distinct class of crystal” [50] (see, however Refs. [13, 55, 56, 23, 38]). Anderson has pointed also the extent to which the standard classification falls. His conclusions were confirmed by further development of solid state physics. During the last decades a lot of new substances and materials were synthesized and tested. Their conduction properties and temperature behavior of the resistivity are differed substantially and constitute a difficult task for consistent classification [73] (see Fig.1).

Figure 2: Resistivity of various conducting materials (from Ref. [73])

Bokij [74] carried out an interesting analysis of notions ”metals” and ”nonmetals” for chemical elements. According to him, there are typical metals (, , ) and typical nonmetals (, , halogens), but the boundary between them and properties determined by them are still an open question. The notion ”metal” is defined by a number of specific properties of the corresponding elemental substances, e.g. by high electrical conductivity and thermal capacity, the ability to reflect light waves(luster), plasticity, and ductility. Bokij emphasizes [74], that when defining the notion of a metal, one has also to take into account the crystal structure. As a rule, the structure of metals under normal conditions are characterized by rather high symmetries and high coordination numbers (c.n.) of atoms equal to or higher than eight, whereas the structures of crystalline nonmetals under normal conditions are characterized by lower symmetries and coordination numbers of atoms (2-4).
It is worth noting that such topics like studies of the strongly correlated electronic systems [62], high- superconductivity [75], colossal magnetoresistance [5] and multiferroicity [5] have led to a new development of solid state physics during the last decades. Many transition-metal oxides show very large (”colossal”) magnitudes of the dielectric constant and thus have immense potential for applications in modern microelectronics and for the development of new capacitance-based energy-storage devices. These and other interesting phenomena to a large extend first have been revealed and intensely investigated in transition-metal oxides. The complexity of the ground states of these materials arises from strong electronic correlations, enhanced by the interplay of spin, orbital, charge and lattice degrees of freedom [62]. These phenomena are a challenge for basic research and also bear big potentials for future applications as the related ground states are often accompanied by so-called ”colossal” effects, which are possible building blocks for tomorrow’s correlated electronics. The measurement of the response of transition-metal oxides to electric fields is one of the most powerful techniques to provide detailed insight into the underlying physics that may comprise very different phenomena, e.g., charge order, molecular or polaronic relaxations, magnetocapacitance, hopping charge transport, ferroelectricity or density-wave formation. In the recent work [76], authors thoroughly discussed the mechanisms that can lead to colossal values of the dielectric constant, especially emphasizing effects generated by external and internal interfaces, including electronic phase separation. The authors of the work [76] studied the materials showing so-called colossal dielectric constants (CDC), i.e. values of the real part of the permittivity exceeding 1000. Since long, materials with high dielectric constants are in the focus of interest, not only for purely academic reasons but also because new high- materials are urgently sought after for the further development of modern electronics. In addition, authors of the work [76] provided a detailed overview and discussion of the dielectric properties of and related systems, which is today’s most investigated material with colossal dielectric constant. Also a variety of further transition-metal oxides with large dielectric constants were treated in detail, among them the system where electronic phase separation may play a role in the generation of a colossal dielectric constant. In general, for the miniaturization of capacitive electronic elements materials with high- are prerequisite. This is true not only for the common silicon-based integrated-circuit technique but also for stand-alone capacitors.
Nevertheless, as regards to metals, the workable practical definition of Kittel can be adopted: metals are characterized by high electrical conductivity, so that a portion of electrons in metal must be free to move about. The electrons available to participate in the conductivity are called conduction electrons. Our picture of a metal, therefore, must be that it contains electrons which are free to move, and which may, when under the influence of an electric field, carry a current through the material.
In summary, the 68 naturally occurring metallic and semimetallic elements [49] can be classified as it is shown in Table 2.

Table 2. Metallic and Semimetallic Elements

Table 2: Metallic and Semimetallic Elements
item number elements
alkali metals 5 , , , ,
noble metals 3 , ,
polyvalent simple metals 11 , , , , , , , , , ,
alkali-earth metals 4 , , ,
semi-metals 4 , , ,
transition metals 23 , , , etc.
rare earths 14
actinides 4

3 Many-Particle Interacting Systems and Current operator

Let us now consider a general system of interacting electrons in a volume described by the Hamiltonian

(3.1)

Here is a one-body potential, e.g. an externally applied potential like that due to the field of the ions in a solid, and is a two-body potential like the Coulomb potential between electrons. It is essential that and do not depend on the velocities of the particles.
It is convenient to introduce a ”quantization” in a continuous space [77, 78, 79] via the operators and which create and destroy a particle at . In terms of and we have

(3.2)

Studies of flow problems lead to the continuity equation [20, 42]

(3.3)

This equation based on the concept of conservation of certain extensive variable. In nonequilibrium thermodynamics [42] the fundamental flow equations are obtained using successively mass, momentum, and energy as the relevant extensive variables. The analogous equation are known from electromagnetism. The central role plays a global conservation law of charge, , for it refers to the total charge in a system. Charge is also conserved locally [80]. This is described by Eq.(3.3), where and are the charge and current densities, respectively.
In quantum mechanics there is the connection of the wavefunction to the particle mass-probability current distribution

(3.4)

where satisfy the time-dependent Schrödinger equation [81, 79]

(3.5)

Consider the motion of a particle under the action of a time-independent force determined by a real potential Equation (3.5) becomes

(3.6)

It can be shown that for the probability density we have

(3.7)

This is the equation of continuity and it is quite general for real potentials. The equation of continuity mathematically states the local conservation of particle mass probability in space.
A thorough consideration of a current carried by a quasi-particle for a uniform gas of fermions, containing particles in a volume , which was assumed to be very large, was performed within a semi-phenomenological theory of Fermi liquid [82]. This theory describes the macroscopic properties of a system at zero temperature and requires knowledge of the ground state and the low-lying excited states. The current carried by the quasi-particle is the sum of two terms: the current which is equal to the velocity of the quasi-particle and the backflow of the medium [82]. The precise definition of the current in an arbitrary state within the Fermi liquid theory is given by

(3.8)

where is the momentum of the th particle and its bare mass. To measure it is necessary to use a reference frame moving with respect to the system with the uniform velocity . The Hamiltonian in the rest frame can be written

(3.9)

It was assumed that depends only on the positions and the relative velocities of the particles; it is not modified by a translation. In the moving system only the kinetic energy changes; the apparent Hamiltonian becomes

(3.10)

Taking the average value of in the state , and let be the energy of the system as seen from the moving reference frame, one find in the

(3.11)

where refers to one of the three coordinates. This expression gives the definition of current in the framework of the Fermi liquid theory. For the particular case of a translationally invariant system the total current is a constant of the motion, which commutes with the interaction and which, as a consequence, does not change when is switched on adiabatically. For the particular state containing one quasi-particle the total current is the same as for the ideal system

(3.12)

This result is a direct consequence of Galilean invariance.
Let us consider now the many-particle Hamiltonian (3.2)

(3.13)

It will also be convenient to consider density of the particles in the following form [20]

The Fourier transform of the particle density operator becomes

(3.14)

The particle mass-probability current distribution in this ”lattice” representation will take the form

(3.15)

Here is the velocity operator. The direct calculation shows that

(3.16)

Thus the equation of motion for the particle density operator becomes

(3.17)

or in another form

(3.18)

which is the continuity equation considered above. Note, that

These relations holds in general for any periodic potential and interaction potential of the electrons which depend only on the coordinates of the electrons.
It is easy to check the validity of the following relation

(3.19)

This formulae is the known f-sum rule [82] which is a consequence from the continuity equation (for a more general point of view see Ref. [83]).
Now consider the second-quantized Hamiltonian (3.2). The particle density operator has the form [84, 77, 85]

(3.20)

Then we define

(3.21)

Here is the probability current density, i.e. the probability flow per unit time per unit area perpendicular to . The continuity equation will persist for this case too. Let us consider the equation motion

(3.22)

Note, that .
We find

(3.23)

Thus the continuity equation have the same form in both the ”particle” and ”field” versions.

4 Tight-Binding and Modified Tight-Binding Approximation

Electrons and phonons are the basic elementary excitations of a metallic solid. Their mutual interactions [2, 52, 86, 87, 88, 89] manifest themselves in such observations as the temperature dependent resistivity and low-temperature superconductivity. In the quasiparticle picture, at the basis of this interaction is the individual electron-phonon scattering event, in which an electron is deflected in the dynamically distorted lattice. We consider here the scheme which is called the modified tight-binding interaction (MTBA). But firstly, we remind shortly the essence of the tight-binding approximation. The main purpose in using the tight-binding method is to simplify the theory sufficiently to make workable. The tight-binding approximation considers solid as a giant molecule.

4.1 Tight-binding approximation

The main problem of the electron theory of solids is to calculate the energy level spectrum of electrons moving in an ion lattice [52, 90]. The tight binding method [52, 91, 92, 93, 94] for energy band calculations has generally been regarded as suitable primarily for obtaining a simple first approximation to a complex band structure. It was shown that the method should also be quite powerful in quantitative calculations from first principles for a wide variety of materials. An approximate treatment requires to obtain energy levels and electron wave functions for some suitable chosen one-particle potential (or pseudopotential), which is usually local. The standard molecular orbital theories of band structure are founded on an independent particle model.
As atoms are brought together to form a crystal lattice the sharp atomic levels broaden into bands. Provided there is no overlap between the bands, one expects to describe the crystal state by a Bloch function of the type,

(4.1)

where is a free atom single electron wave function, for example such as and is the position of the atom in a rigid lattice. If the bands overlap or approach each other one should use instead of a combination of the wave functions corresponding to the levels in question, e.g. , etc. In the other words, this approach, first introduced to crystal calculation by F.Bloch, expresses the eigenstates of an electron in a perfect crystal in a linear combination of atomic orbitals and termed LCAO method [52, 91, 92, 93, 94].
Atomic orbitals are not the most suitable basis set due to the nonorthogonality problem. It was shown by many authors [52, 95, 96, 97] that the very efficient basis set for the expansion (4.1) is the atomic-like Wannier functions  [52, 95, 96, 97]. These are the Fourier transforms of the extended Bloch functions and are defined as

(4.2)

Wannier functions form a complete set of mutually orthogonal functions localized around each lattice site within any band or group of bands. They permit one to formulate an effective Hamiltonian for electrons in periodic potentials and span the space of a singly energy band. However, the real computation of Wannier functions in terms of sums over Bloch states is a complicated task [33, 97].
To define the Wannier functions more precisely let us consider the eigenfunctions belonging to a particular simple band in a lattice with the one type of atom at a center of inversion. Let it satisfy the following equations with one-electron Hamiltonian

(4.3)

and the orthonormality relation where the integration is performed over the unit cells in the crystal. The property of periodicity together with the property of the orthonormality lead to the orthonormality condition of the Wannier functions

(4.4)

The set of the Wannier functions is complete, i.e.

(4.5)

Thus it is possible to find the inversion of the Eq.(4.2) which has the form

(4.6)

These conditions are not sufficient to define the functions uniquely since the Bloch states are determined only within a multiplicative phase factor according to

(4.7)

where is any real function of , and are Bloch functions [98]. The phases are usually chosen so as to localize about the origin. The usual choice of phase makes real and positive. This lead to the maximum possible value in and decaying exponentially away from . In addition, function with this choice will satisfy the symmetry properties

It follows from the above consideration that the Wannier functions are real and symmetric,

Analytical, three dimensional Wannier functions have been constructed from Bloch states formed from a lattice gaussians. Thus, in the condensed matter theory, the Wannier functions play an important role in the theoretical description of transition metals, their compounds and disordered alloys, impurities and imperfections, surfaces, etc.

4.2 Interacting electrons on a lattice and the Hubbard model

There are big difficulties in description of the complicated problems of electronic and magnetic properties of a metal with the band electrons which are really neither ”local” nor ”itinerant” in a full sense. A better understanding of the electronic correlation effects in metallic systems can be achieved by the formulating of the suitable flexible model that could be used to analyze major aspects of both the insulating and metallic states of solids in which electronic correlations are important.
The Hamiltonian of the interacting electrons with pair interaction in the second-quantized form is given by Eq.(3.2). Consider this Hamiltonian in the Bloch representation. We have

(4.8)

Here is the Bloch function satisfying the equation

(4.9)

The functions form a complete orthonormal set of functions

(4.10)

We find

(4.11)

Since the method of second quantization is based on the choice of suitable complete set of orthogonal normalized wave functions, we take now the set of the Wannier functions. Here is the band index. The field operators in the Wannier-function representation are given by

(4.12)

Thus we have

(4.13)

Many of treatment of the correlation effects are effectively restricted to a non-degenerate band. The Wannier functions basis set is the background of the widely used Hubbard model. The Hubbard model[99, 100] is, in a certain sense, an intermediate model (the narrow-band model) and takes into account the specific features of transition metals and their compounds by assuming that the electrons form a band, but are subject to a strong Coulomb repulsion at one lattice site. The single-band Hubbard Hamiltonian is of the form [99, 62]

(4.14)

Here and are the second-quantized operators of the creation and annihilation of the electrons in the lattice state with spin . The Hamiltonian includes the intra-atomic Coulomb repulsion and the one-electron hopping energy . The corresponding parameters of the Hubbard Hamiltonian are given by

(4.15)
(4.16)

The electron correlation forces electrons to localize in the atomic-like orbitals which are modelled here by a complete and orthogonal set of the Wannier wave functions . On the other hand, the kinetic energy is increased when electrons are delocalized. The band energy of Bloch electrons is defined as follows:

(4.17)

where is the number of lattice sites. The Pauli exclusion principle which does not allow two electrons of common spin to be at the same site, , plays a crucial role. Note, that the standard derivation of the Hubbard model presumes the rigid ion lattice with the rigidly fixed ion positions. We note that -electrons are not explicitly taken into account in our model Hamiltonian. They can be, however, implicitly taken into account by screening effects and effective -band occupation.

4.3 Current operator for the tight-binding electrons

Let us consider again a many-particle interacting systems on a lattice with the Hamiltonian (4.11). At this point, it is important to realize the fundamental difference between many-particle system which is uniform in space and many-particle system on a lattice. For the many-particle systems on a lattice the proper definition of current operator is a subtle problem. It was shown above that a physically satisfactory definition of the current operator in the quantum many-body theory is given based upon the continuity equation. However, this point should be reconsidered carefully for the lattice fermions which are described by the Wannier functions.
Let us remind once again that the Bloch and Wannier wave functions are related to each other by the unitary transformation of the form

(4.18)

The number occupation representation for a single-band case lead to

(4.19)

In this representation the particle density operator and current density take the form

(4.20)

The equation of the motion for the particle density operator will consists of two contributions

(4.21)

The first contribution is

(4.22)

Here the notation was introduced

(4.23)

In the Bloch representation for the particle density operator one finds

(4.24)

where

(4.25)

For the second contribution we find

(4.26)

For the single-band Hubbard Hamiltonian the last equation will take the form

(4.27)

The direct calculations give for the case of electrons on a lattice ( is a charge of an electron)

(4.28)

Taking into account that

(4.29)

we find

(4.30)

This unusual result was analyzed critically by many authors. The proper definition of the current operator for the Hubbard model has been the subject of intensive discussions [101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111]. To clarify the situation let us consider the ”total position operator” for our system of the electrons on a lattice

(4.31)

In the ”quantized” picture it has the form

(4.32)

where we took into account the relation

(4.33)

We find that

(4.34)

Let us consider the local particle density operator

(4.35)

It is clear that the current operator should be defined on the basis of the equation