09 Metals

jueves, 29 de julio de 2010

Metals and Insulators

Solids are divided into two major classes: metals and insulators. A metal – or a conductor – is a solid in which an electric current flows under the application of electric field. By contrast, application of an electric field produces no electric current in an insulator. There is a simple criterion for distinguishing between the two classes on the basis of the band structure. If the valence electrons exactly fill one or more bands, leaving others empty, the crystal will be an insulator. An external electric field will not cause current flow in an insulator. Provided that a filled band is separated by an energy gap from the next higher band, there is no continuous way to change the total momentum of the electrons if every accessible state is filled. Nothing changes when the field is applied.

Fig. 1 Occupied states and band structures giving (a) an insulator, (b) a metal or a semimetal because of band overlap, and (c) a metal because of electron concentration. In (b) the overlap need not occur along the same directions in the Brillouin zone. If the overlap is small, with relatively few states involved, we speak of a semimetal.

On the contrary if the valence band is not completely filled the solid is a metal. In a metal there are empty states available above the Fermi level like in a free electron gas. An application of an external electric field results in the current flow.

It is possible to determine whether a solid is a metal of an insulator by considering the number of valence electrons. A crystal can be an insulator only if the number of valence electrons in a primitive cell of the crystal is an even integer. This is because each band can accommodate only two electrons per primitive cell. For example, diamond has two atoms of valence four, so that there are eight valence electrons per primitive cell. The band gap in diamond is 7eV and this crystal is a good insulator

However, if a crystal has an even number of valence electrons per primitive cell, it is not necessarily an insulator. It may happen that the bands overlap in energy. If the bands overlap in energy, then instead of one filled band giving an insulator, we can have two partly filled bands giving a metal (Fig.1b). For example, the divalent metals, such as Mg or Zn, have two valence electrons per cell. However, they are metals, although a poor ones – their conductivity is small.

If this overlap is very small, we deal with semimetals. The best known example of a semimetal is bismuth (Bi).

If the number of valence electrons per cell is odd the solid is a metal. For example, the alkali metals and the noble metals have one valence electron per primitive cell, so that they have to be metals.

The alkaline earth metals have two valence electrons per primitive cell; they could be insulators, but the bands overlap in energy to give metals, but not very good metals. Diamond, silicon, and germanium each have two atoms of valence four, so that there are eight valence electrons per primitive cell; the bands do not overlap, and the pure crystals are insulators at absolute zero.

There are substances, which fall in an intermediate position between metals and insulators. If the gap between the valence band and the band immediately above it is small, then electrons are readily excitable thermally from the former to the latter band. Both bands become only partially filled and both contribute to the electric condition. Such a substance is known as a semiconductor. Examples are Si and Ge, in which the gaps are about 1 and 0.7 eV, respectively. Roughly speaking, a substance behaves as a semiconductor at room temperature whenever the gap is less than 2 eV. The conductivity of a typical semiconductor is very small compared to that of a metal, but it is still many orders of magnitude larger than that of an insulator. It is justifiable, therefore, to classify semiconductors as a new class of substance, although they are, strictly speaking, insulators at very low temperatures.

Semiconductors

A semiconductor is a material that has an electrical conductivity due to flowing electrons (as opposed to ionic conductivity) which is intermediate in magnitude between that of a conductor and an insulator. This means roughly in the range 103 to 10−8 siemens per centimeter. Devices made from semiconductor materials are the foundation of modern electronics, including radio, computers, telephones, and many other devices. Semiconductor devices include the various types of transistor, solar cells, many kinds of diodes including the light-emitting diode, the silicon controlled rectifier, and digital and analog integrated circuits. Similarly, semiconductor solar photovoltaic panels directly convert light energy into electrical energy. In a metallic conductor, current is carried by the flow of electrons. In semiconductors, current is often schematized as being carried either by the flow of electrons or by the flow of positively charged "holes" in the electron structure of the material. Actually, however, in both cases only electron movements are involved.

Common semiconducting materials are crystalline solids but amorphous and liquid semiconductors are known. These include hydrogenated amorphous silicon and mixtures of arsenic, selenium and tellurium in a variety of proportions. Such compounds share with better known semiconductors intermediate conductivity and a rapid variation of conductivity with temperature, as well as occasional negative resistance. Such disordered materials lack the rigid crystalline structure of conventional semiconductors such as silicon and are generally used in thin film structures, which are less demanding for as concerns the electronic quality of the material and thus are relatively insensitive to impurities and radiation damage. Organic semiconductors, that is, organic materials with properties resembling conventional semiconductors, are also known.

Silicon is used to create most semiconductors commercially. Dozens of other materials are used, including germanium, gallium arsenide, and silicon carbide. A pure semiconductor is often called an “intrinsic” semiconductor. The electronic properties and the conductivity of a semiconductor can be changed in a controlled manner by adding very small quantities of other elements, called “dopants”, to the intrinsic material. In crystalline silicon typically this is achieved by adding impurities of boron or phosphorus to the melt and then allowing the melt to solidify into the crystal. This process is called "doping".

Energy bands and electrical conduction

In classic crystalline semiconductors, the electrons can have energies only within certain bands (i.e. ranges of levels of energy). Energetically, these bands are located between the energy of the ground state, corresponding to electrons tightly bound to the atomic nuclei of the material, and the free electron energy. The latter is the energy required for an electron to escape entirely from the material. The energy bands each correspond to a large number of discrete quantum states of the electrons, and most of the states with low energy (closer to the nucleus) are full, up to a particular band called the valence band. Semiconductors and insulators are distinguished from metals because the valence band in the semiconductor materials is nearly filled under usual operating conditions, thus causing more electrons to be available in the "conduction band," the band immediately above the valence band.

The ease with which electrons in a semiconductor can be excited from the valence band to the conduction band depends on the band gap between the bands. The size of this energy bandgap serves as an arbitrary dividing line (roughly 4 eV) between semiconductors and insulators.

With covalent bonds, an electron moves by hopping to a neighboring bond. The Pauli exclusion principle requires the electron to be lifted into the higher anti-bonding state of that bond. For delocalized states, for example in one dimension – that is in a nanowire, for every energy there is a state with electrons flowing in one direction and another state for the electrons flowing in the other. For a net current to flow some more states for one direction than for the other direction have to be occupied and for this energy is needed, in the semiconductor the next higher states lie above the band gap. Often this is stated as: full bands do not contribute to the electrical conductivity. However, as the temperature of a semiconductor rises above absolute zero, there is more energy in the semiconductor to spend on lattice vibration and — more importantly for us — on lifting some electrons into an energy states of the conduction band. The current-carrying electrons in the conduction band are known as "free electrons", although they are often simply called "electrons" if context allows this usage to be clear.

Electrons excited to the conduction band also leave behind electron holes, or unoccupied states in the valence band. Both the conduction band electrons and the valence band holes contribute to electrical conductivity. The holes themselves don't actually move, but a neighboring electron can move to fill the hole, leaving a hole at the place it has just come from, and in this way the holes appear to move, and the holes behave as if they were actual positively charged particles.

One covalent bond between neighboring atoms in the solid is ten times stronger than the binding of the single electron to the atom, so freeing the electron does not imply destruction of the crystal structure.

Holes: electron absence as a charge carrier

The concept of holes can also be applied to metals, where the Fermi level lies within the conduction band. With most metals the Hall effect indicates electrons are the charge carriers. However, some metals have a mostly filled conduction band. In these, the Hall effect reveals positive charge carriers, which are not the ion-cores, but holes. In contrast, some conductors like solutions of salts, or plasma. In the case of a metal, only a small amount of energy is needed for the electrons to find other unoccupied states to move into, and hence for current to flow. Sometimes even in this case it may be said that a hole was left behind, to explain why the electron does not fall back to lower energies: It cannot find a hole. In the end in both materials electron-phonon scattering and defects are the dominant causes for resistance.

The energy distribution of the electrons determines which of the states are filled and which are empty. This distribution is described by Fermi-Dirac statistics. The distribution is characterized by the temperature of the electrons, and the Fermi energy or Fermi level. Under absolute zero conditions the Fermi energy can be thought of as the energy up to which available electron states are occupied. At higher temperatures, the Fermi energy is the energy at which the probability of a state being occupied has fallen to 0.5.

The dependence of the electron energy distribution on temperature also explains why the conductivity of a semiconductor has a strong temperature dependency, as a semiconductor operating at lower temperatures will have fewer available free electrons and holes able to do the work.

Fermi-Dirac distribution. States with energy ε below the Fermi energy, here µ, have higher probability n to be occupied, and those above are less likely to be occupied. Smearing of the distribution increases with temperature.

Energy–momentum dispersion

In the preceding description an important fact is ignored for the sake of simplicity: the dispersion of the energy. The reason that the energies of the states are broadened into a band is that the energy depends on the value of the wave vector, or k-vector, of the electron. The k-vector, in quantum mechanics, is the representation of the momentum of a particle.

The dispersion relationship determines the effective mass, m^*, of electrons or holes in the semiconductor, according to the formula:

$m^{*} = \hbar^2 \cdot \left[ {{d^2 E(k)} \over {d k^2}} \right]^{-1}.$

The effective mass is important as it affects many of the electrical properties of the semiconductor, such as the electron or hole mobility, which in turn influences the diffusivity of the charge carriers and the electrical conductivity of the semiconductor.

Typically the effective mass of electrons and holes are different. This affects the relative performance of p-channel and n-channel IGFETs.

The top of the valence band and the bottom of the conduction band might not occur at that same value of k. Materials with this situation, such as silicon and germanium, are known as indirect bandgap materials. Materials in which the band extrema are aligned in k, for example gallium arsenide, are called direct bandgap semiconductors. Direct gap semiconductors are particularly important in optoelectronics because they are much more efficient as light emitters than indirect gap materials.

Electronic band structure

In solid-state physics, the electronic band structure (or simply band structure) of a solid describes ranges of energy that an electron is "forbidden" or "allowed" to have. It is due to the diffraction of the quantum mechanical electron waves in the periodic crystal lattice. The band structure of a material determines several characteristics, in particular its electronic and optical properties.

Why bands occur in materials

The electrons of a single isolated atom occupy atomic orbitals, which form a discrete set of energy levels. If several atoms are brought together into a molecule, their atomic orbitals split, as in a coupled oscillation. This produces a number of molecular orbitals proportional to the number of atoms. When a large number of atoms (of order × 10²⁰ or more) are brought together to form a solid, the number of orbitals becomes exceedingly large, and the difference in energy between them becomes very small, so the levels may be considered to form continuous bands of energy rather than the discrete energy levels of the atoms in isolation. However, some intervals of energy contain no orbitals, no matter how many atoms are aggregated, forming band gaps.

Within an energy band, energy levels are so numerous as to be a near continuum. First, the separation between energy levels in a solid is comparable with the energy that electrons constantly exchange with phonons (atomic vibrations). Second, it is comparable with the energy uncertainty due to the Heisenberg uncertainty principle, for reasonably long intervals of time. As a result, the separation between energy levels is of no consequence.

Basic concepts

Any solid has a large number of bands. In theory, it can be said to have infinitely many bands (just as an atom has infinitely many energy levels). However, all but a few lie at energies so high that any electron that reaches those energies escapes from the solid. These bands are usually disregarded.

Bands have different widths, based upon the properties of the atomic orbitals from which they arise. Also, allowed bands may overlap, producing (for practical purposes) a single large band.

Figure 1 shows a simplified picture of the bands in a solid that allows the three major types of materials to be identified: metals, semiconductors and insulators.

Metals contain a band that is partly empty and partly filled regardless of temperature. Therefore they have very high conductivity.

The lowermost, almost fully occupied band in an insulator or semiconductor, is called the valence band by analogy with the valence electrons of individual atoms. The uppermost, almost unoccupied band is called the conduction band because only when electrons are excited to the conduction band can current flow in these materials. The difference between insulators and semiconductors is only that the forbidden band gap between the valence band and conduction band is larger in an insulator, so that fewer electrons are found there and the electrical conductivity is lower. Because one of the main mechanisms for electrons to be excited to the conduction band is due to thermal energy, the conductivity of semiconductors is strongly dependent on the temperature of the material.

This band gap is one of the most useful aspects of the band structure, as it strongly influences the electrical and optical properties of the material. Electrons can transfer from one band to the other by means of carrier generation and recombination processes. The band gap and defect states created in the band gap by doping can be used to create semiconductor devices such as solar cells, diodes, transistors, laser diodes, and others.

Figure 1: Simplified diagram of the electronic band structure of metals, semiconductors, and insulators.

Symmetry

A more complete view of the band structure takes into account the periodic nature of a crystal lattice using the symmetry operations that form a space group. The Schrödinger equation is solved for the crystal, which has Bloch waves as solutions:

$\psi_{n\mathbf{k}}(\mathbf{r})=e^{i\mathbf{k}\cdot\mathbf{r}}u_{n\mathbf{k}}(\mathbf{r})$ ,

where k is called the wavevector, and is related to the direction of motion of the electron in the crystal, and n is the band index, which simply numbers the energy bands. The wavevector k takes on values within the Brillouin zone (BZ) corresponding to the crystal lattice, and particular directions/points in the BZ are assigned conventional names like Γ, Δ, Λ, Σ, etc. These directions are shown for the face-centered cubic lattice geometry in Figure 2.

The available energies for the electron also depend upon k, as shown in Figure 3 for silicon in the more complex energy band diagram at the right. In this diagram the topmost energy of the valence band is labeled E_v and the bottom energy in the conduction band is labeled E_c. The top of the valence band is not directly below the bottom of the conduction band (E_v is for an electron traveling in direction Γ, E_c in direction X), so silicon is called an indirect gap material. For an electron to be excited from the valence band to the conduction band, it needs something to give it energy E_c – E_v and a change in direction/momentum. In other semiconductors (for example GaAs) both are at Γ, and these materials are called direct gap materials (no momentum change required). Direct gap materials benefit the operation of semiconductor laser diodes.

Anderson's rule is used to align band diagrams between two different semiconductors in contact.

Figure 2: First Brillouin zone of FCC lattice showing symmetry labels

Figure 3. Bulk band structure for Si,Ge,GaAs and InAs generated with tight binding model. Note that Si and Ge are indirect while GaAs and InAs are direct band gap materials.

Band structures in different types of solids

Although electronic band structures are usually associated with crystalline materials, quasi-crystalline and amorphous solids may also exhibit band structures. However, the periodic nature and symmetrical properties of crystalline materials makes it much easier to examine the band structures of these materials theoretically. In addition, the well-defined symmetry axes of crystalline materials makes it possible to determine the dispersion relationship between the momentum (a 3-dimension vector quantity) and energy of a material. As a result, virtually all of the existing theoretical work on the electronic band structure of solids has focused on crystalline materials.

Density of states

While the density of energy states in a band could be very large for some materials, it may not be uniform. It approaches zero at the band boundaries, and is generally highest near the middle of a band. The density of states for the free electron model in three dimensions is given by,

$D(\epsilon)= \frac{V}{2\pi^2}\left(\frac {2m}{\hbar^2}\right)^{3/2} \epsilon^{1/2}$

Filling of bands

Although the number of states in all of the bands is effectively infinite, in an uncharged material the number of electrons is equal only to the number of protons in the atoms of the material. Therefore not all of the states are occupied by electrons ("filled") at any time. The likelihood of any particular state being filled at any temperature is given by Fermi-Dirac statistics. The probability is given by the following expression:

$f(E) = \frac{1}{1 + e^{\frac{E-\mu}{k_B T}}}$

where:

k_B is Boltzmann's constant,
T is the temperature,
µ is the chemical potential (in semiconductor physics, this quantity is more often called the "Fermi level" and denoted E_F).

The Fermi level naturally is the level at which the electrons and protons are balanced.

At T=0, the distribution is a simple step function:

$f(E) = \begin{cases} 1 & \mbox{if}\ 0 < E \le E_F \\ 0 & \mbox{if}\ E_F < E \end{cases}$

At nonzero temperatures, the step "smooths out", so that an appreciable number of states below the Fermi level are empty, and some states above the Fermi level are filled.

Band structure of crystals

Brillouin zone

Because electron momentum is the reciprocal of space, the dispersion relation between the energy and momentum of electrons can best be described in reciprocal space. It turns out that for crystalline structures, the dispersion relation of the electrons is periodic, and that the Brillouin zone is the smallest repeating space within this periodic structure. For an infinitely large crystal, if the dispersion relation for an electron is defined throughout the Brillouin zone, then it is defined throughout the entire reciprocal space.

Theory of band structures in crystals

The ansatz is the special case of electron waves in a periodic crystal lattice using Bloch waves as treated generally in the dynamical theory of diffraction. Every crystal is a periodic structure which can be characterized by a Bravais lattice, and for each Bravais lattice we can determine the reciprocal lattice, which encapsulates the periodicity in a set of three reciprocal lattice vectors (b₁,b₂,b₃). Now, any periodic potential V(r) which shares the same periodicity as the direct lattice can be expanded out as a Fourier series whose only non-vanishing components are those associated with the reciprocal lattice vectors. So the expansion can be written as:

$V(\mathbf{r}) = \sum_{\mathbf{K}}{V_{\mathbf{K}}e^{i \mathbf{K}\cdot\mathbf{r}}}$

where K = m₁b₁ + m₂b₂ + m₃b₃ for any set of integers (m₁,m₂,m₃).

From this theory, an attempt can be made to predict the band structure of a particular material, however most ab initio methods for electronic structure calculations fail to predict the observed band gap.

Tight binding

In solid-state physics, the tight binding model (or TB model) is an approach to the calculation of electronic band structure using an approximate set of wave functions based upon superposition of wave functions for isolated atoms located at each atomic site. The method is closely related to the linear combination of atomic orbitals molecular orbital method used for molecules. Tight binding calculates the ground state electronic energy and position of bandgaps for a molecule.

This approximation could be considered the analog of the LCAO-MO (Linear Combination of Atomic Orbital- Molecular Orbital) approach. The idea of a molecular orbital was advanced by Friedrich Hund and Robert Mulliken from 1926 through 1928. The LCAO method for approximating molecular orbitals was introduced in 1928 by B. N. Finklestein and G. E. Horowitz, while the LCAO for solids was developed by Felix Bloch, as part of his doctoral dissertation in 1928, concurrently with and independent of the LCAO-MO approach. A much simpler interpolation scheme for approximating the electronic band structure, the tight-binding method was originally conceived in 1954 by John Clarke Slater and George Fred Koster. Hence, it is sometimes referred to as the SK method. As in the LCAO-MO approach, the atomic locations can be specified arbitrarily (or additional calculations can be done to find the atomic positions), so the method can be applied to non-crystalline materials. However, the most common applications are to crystalline materials where the atomic positions are located on a periodic space lattice of sites. With the tight-binding method, the electronic band structure of a solid is interpolated over the entire Brillouin zone by fitting to first-principles calculations carried out at high-symmetry points.

In this approach, interactions between different atomic sites are considered as perturbations. There exists several kinds of interactions we must consider. The crystal Hamiltonian is only approximately a sum of atomic Hamiltonians located at different sites and atomic wave functions overlap adjacent atomic sites in the crystal, and so are not accurate representations of the exact wave function. There are further explanations in the next section with some mathematical expressions.

Recently, in the research about strongly correlated material, the tight binding approach is basic approximation because highly localized electrons like 3-d transition metal electrons sometimes indicate strongly correlated behaviors. In this case, the role of electron-electron interaction must be considered using the many-body physics description.

The tight-binding model is typically used for calculations of electronic band structure and band gaps in the static regime. However, in combination with other methods such as the random phase approximation (RPA) model, the dynamic response of systems may also be studied.

Formulation

We introduce the atomic orbitals φ_m( r ), which are eigenfunctions of the Hamiltonian H_at of a single isolated atom. When the atom is placed in a crystal, this atomic wave function overlaps adjacent atomic sites, and so are not true eigenfunctions of the crystal Hamiltonian. The overlap is less when electrons are tightly bound, which is the source of the descriptor "tight-binding". Any corrections to the atomic potential ΔU required to obtain the true Hamiltonian H of the system, are assumed small:

$H (\boldsymbol{r}) = \sum_{\boldsymbol{R_n}} H_{\mathrm{at}}(\boldsymbol{r - R_n}) +\Delta U (\boldsymbol{r}) \ .$

A solution ψ(r) to the time-independent single electron Schrödinger equation is then approximated as a linear combination of atomic orbitals φ_m( r − R_n ):

$\psi(\boldsymbol{r}) = \sum_{m,\boldsymbol{R_n}} b_m ( \boldsymbol{R_n}) \ \varphi_m (\boldsymbol{r-R_n})$ ,

where m refers to the m-th atomic energy level and R_n locates an atomic site in the crystal lattice.

The translational symmetry of the crystal implies the wave function under translation can change only by a phase factor:

$\psi(\boldsymbol{r+R_{\ell}}) = e^{i\boldsymbol{k \cdot R_{\ell}}}\psi(\boldsymbol{r}) \ ,$

where k is the wave vector of the wave function. Consequently, the coefficients satisfy

$\sum_{m,\boldsymbol{R_n}} b_m ( \boldsymbol{R_n}) \ \varphi_m (\boldsymbol{r-R_n+R_{\ell}})=e^{i\boldsymbol{k \cdot R_{\ell}}}\sum_{m,\boldsymbol{R_n}} b_m ( \boldsymbol{R_n}) \ \varphi_m (\boldsymbol{r-R_n})\ .$

By substituting R_p = R_n − R_ℓ, we find

$b_m ( \boldsymbol{R_p+R_{\ell}}) = e^{i\boldsymbol{k \cdot R_{\ell}}}b_m ( \boldsymbol{R_p}) \ ,$

$b_m (\boldsymbol{R_p}) = e^{i\boldsymbol{k \cdot R_{p}}} b_m ( \boldsymbol{0}) \ .$

Normalizing the wave function to unity:

$\int d^3 r \ \psi^* (\boldsymbol{r}) \psi (\boldsymbol{r}) = 1$

$= \sum_{\boldsymbol{R_n}} b^* ( \boldsymbol{R_n})\sum_{\boldsymbol{R_{\ell}}} b ( \boldsymbol{R_{\ell}})\int d^3 r \ \varphi^* (\boldsymbol{r-R_n}) \varphi (\boldsymbol{r-R_{\ell}})$

$= b^*(0)b(0)\sum_{\boldsymbol{R_n}} e^{-i \boldsymbol{k \cdot R_n}}\sum_{\boldsymbol{R_{\ell}}} e^ {i \boldsymbol{k \cdot R_{\ell}}}\ \int d^3 r \ \varphi^* (\boldsymbol{r-R_n}) \varphi (\boldsymbol{r-R_{\ell}})$

$=N b^*(0)b(0)\sum_{\boldsymbol{R_p}} e^{-i \boldsymbol{k \cdot R_p}}\ \int d^3 r \ \varphi^* (\boldsymbol{r-R_p}) \varphi (\boldsymbol{r})\$

$=N b^*(0)b(0)\sum_{\boldsymbol{R_p}} e^{i \boldsymbol{k \cdot R_p}}\ \int d^3 r \ \varphi^* (\boldsymbol{r}) \varphi (\boldsymbol{r-R_p})\ ,$

so the normalization sets b(0) as

$b^*(0)b(0) = \frac {1} {N}\ \cdot \ \frac {1}{1 + \sum_{\boldsymbol{R_p \neq 0}} e^{-i \boldsymbol{k \cdot R_p}} \alpha (\boldsymbol{R_p})} \ ,$

where α (R_p ) are the atomic overlap integrals, which frequently are neglected resulting in^[1]

$b_n (0) \approx \frac {1} {\sqrt{N}} \ ,$

and

$\psi (\boldsymbol{r}) \approx \frac {1} {\sqrt{N}} \sum_{m,\boldsymbol{R_n}} e^{i \boldsymbol{k \cdot R_n}} \ \varphi_m (\boldsymbol{r-R_n}) \ .$

Using the tight binding form for the wave function, and assuming only the m-th atomic energy level is important for the m-th energy band, the Bloch energies $\varepsilon_m$ are of the form

$\varepsilon_m = \int d^3 r \ \psi^* (\boldsymbol{r})H(\boldsymbol{r}) \psi (\boldsymbol{r})$

$=\sum_{\boldsymbol{R_n}} b^* ( \boldsymbol{R_n})\ \int d^3 r \ \varphi^* (\boldsymbol{r-R_n})H(\boldsymbol{r}) \psi (\boldsymbol{r}) \$

$=\sum_{\boldsymbol{R_{\ell}}} \ \sum_{\boldsymbol{R_n}} b^* ( \boldsymbol{R_n})\ \int d^3 r \ \varphi^* (\boldsymbol{r-R_n})H_{\mathrm{at}}(\boldsymbol{r-R_{\ell}}) \psi (\boldsymbol{r}) \ + \sum_{\boldsymbol{R_n}} b^*( \boldsymbol{R_n})\ \int d^3 r \ \varphi^* (\boldsymbol{r-R_n})\Delta U (\boldsymbol{r}) \psi (\boldsymbol{r}) \ .$

$\approx E_m + b^*(0)\sum_{\boldsymbol{R_n}} e^{-i \boldsymbol{k \cdot R_n}}\ \int d^3 r \ \varphi^* (\boldsymbol{r-R_n})\Delta U (\boldsymbol{r}) \psi (\boldsymbol{r}) \ .$

Here terms involving the atomic Hamiltonian at sites other than where it is centered are neglected. The energy then becomes

$\varepsilon_m(\boldsymbol{k}) = E_m - N\ |b (0)|^2 \left(\beta_m + \sum_{\boldsymbol{R_n}\neq 0} \gamma_m(\boldsymbol{R_n}) e^{i \boldsymbol{k} \cdot \boldsymbol{R_n}}\right) \ ,$

$= E_m - \ \frac {\beta_m + \sum_{\boldsymbol{R_n}\neq 0} \gamma_m(\boldsymbol{R_n}) e^{i \boldsymbol{k} \cdot \boldsymbol{R_n}}}{1 + \sum_{\boldsymbol{R_n \neq 0}} e^{i \boldsymbol{k \cdot R_n}} \alpha (\boldsymbol{R_n})} \ ,$

where E_m is the energy of the m-th atomic level,

$\beta_m = -\int \varphi_m^*(\boldsymbol{r})\Delta U(\boldsymbol{r}) \varphi_m(\boldsymbol{r}) \, d^3 r \$ ,

$\gamma_m(\boldsymbol{R_n}) = -\int \varphi_m^*(\boldsymbol{r}) \Delta U(\boldsymbol{r}) \varphi_m(\boldsymbol{r - R_n}) \, d^3 r \ ,$

and

$\alpha_m(\boldsymbol{R_n}) = \int \varphi_m^*(\boldsymbol{r}) \varphi_m(\boldsymbol{r - R_n}) \, d^3 r \$ ,

are the overlap integrals.

One-dimensional example

Here the tight binding model is illustrated for a string of atoms in a straight line with spacing a between atomic sites. Denote the translation operator $τ$ , which satisfies the property:

$\tau(a)|n\rangle =|n+1\rangle$

Here, the state ket $|n\rangle$ represents a particular choice of atomic orbital (for example, an s- or p- orbital from some shell of orbitals) located at the site R_n = n a in the lattice with lattice constant a. Because the Hamiltonian $H$ is invariant under the operation $τ(a)$ , we have commutation relation

$[H,\ \tau{(a)}]=0$ .

This commutation relation implies the Hamiltonian operator $H$ and translation operator $τ(a)$ can be simultaneously diagonalized.

To find approximate eigenstates of the Hamiltonian, we can use a linear combination of the atomic orbitals

$|k\rangle =\frac{1}{\sqrt{N}}\sum_{n=1}^N e^{inka} |n\rangle$

where N = total number of sites and $k$ is a real parameter with $-\frac{\pi}{a}\leqq k\leqq\frac{\pi}{a}$ . (This wave function is normalized to unity by the leading factor 1/√N provided overlap of atomic wave functions is ignored.) If we apply the lattice translation operator $τ(a)$ to this state $|k\rangle$ , this state is found to be an eigenstate of this operator:

$\tau(a)|k\rangle=\frac{1}{\sqrt{N}}\sum_n e^{inka}|n+1\rangle=\frac{1}{\sqrt{N}}\sum_n e^{i(n-1)ka}|n\rangle = e^{ika}|k\rangle$

Assuming only nearest neighbor overlap (that is, tight binding), the only non-zero matrix elements of the Hamiltonian can be expressed as

$\langle n\pm 1|H|n\rangle=-\Delta \ ;$ $\langle n|H|n\rangle=E_0 \ .$

The energy E₀ is approximately the atomic energy level corresponding to the chosen atomic orbital if H at site R_n = n a is approximately H_at at that site. We can derive the energy of the state $|k\rangle$ using the above equation:

$H|k\rangle=\frac{1}{\sqrt{N}}\sum_n e^{inka} H |n\rangle$

$\langle k| H|k\rangle =\frac{1}{N}\sum_{n,\ m} e^{i(n-m)ka} \langle m|H|n\rangle$ $=\frac{1}{N}\sum_n \langle n|H|n\rangle+\frac{1}{N}\sum_n \langle n-1|H|n\rangle e^{+ika}+\frac{1}{N}\sum_n\langle n+1|H|n\rangle e^{-ika}$ $= E_0 -2\Delta\,\cos(ka)\ ,$

where, for example,

$\frac{1}{N}\sum_n \langle n|H|n\rangle = E_0 \frac{1}{N}\sum_n 1 = E_0 \ ,$

and

$\frac{1}{N}\sum_n \langle n-1|H|n\rangle e^{+ika}=-\Delta e^{ika}\frac{1}{N}\sum_n 1 = -\Delta e^{ika} \ .$

Thus the energy of this state $|k\rangle$ can be represented in the familiar form of the energy dispersion:

$E(k)=E_0-2\Delta\,\cos(ka)$ .

This example is readily extended to three dimensions, for example, to a body-centered cubic or face-centered cubic lattice by introducing the nearest neighbor vector locations in place of simply n a.Likewise, the method can be extended to multiple bands using multiple different atomic orbitals at each site. The general formulation above shows how these extensions can be accomplished.

Connection to Wannier functions

Bloch wave functions describe the electronic states in a periodic crystal lattice. Bloch functions can be represented as a Fourier series

$\psi_m\mathbf{(k,r)}=\frac{1}{\sqrt{N}}\sum_{n}{a_m\mathbf{(R_n,r)}} e^{\mathbf{ik\cdot R_n}}\ ,$

where R_n denotes an atomic site in a periodic crystal lattice, k is the wave vector of the Bloch wave, r is the electron position, m is the band index, and the sum is over all N atomic sites. The Bloch wave is an exact eigensolution for the wave function of an electron in a periodic crystal potential corresponding to an energy E_m (k), and is spread over the entire crystal volume.

Using the Fourier transform analysis, a spatially localized wave function for the m-th energy band can be derived from this Bloch wave:

$a_m\mathbf{(R_n,r)}=\frac{1}{\sqrt{N}}\sum_{\mathbf{k}}{e^{\mathbf{-ik\cdot R_n}}\psi_m\mathbf{(k,r)}}=\frac{1}{\sqrt{N}}\sum_{\mathbf{k}}{e^{\mathbf{ik\cdot (r-R_n)}}u_m\mathbf{(k,r)}}.$

These real space wave functions ${a_m\mathbf{(R_n,r)}}$ are called Wannier functions, and are fairly closely localized to the atomic site R_n. Of course, if we have exact Wannier functions, the exact Bloch functions can be derived using the inverse Fourier transform.

However it is not easy to calculate directly either Bloch functions or Wannier functions. An approximate approach is necessary in the calculation of electronic structures of solids. If we consider the extreme case of isolated atoms, the Wannier function would become an isolated atomic orbital. That limit suggests the choice of an atomic wave function as an approximate form for the Wannier function, the so-called tight binding approximation.

Second quantization

Modern explanations of electronic structure like t-J model and Hubbard model are based on tight binding model. If we introduce second quantization formalism, it is clear to understand the concept of tight binding model.

Using the atomic orbital as a basis state, we can establish the second quantization Hamiltonian operator in tight binding model.

$H = -t \sum_{\langle i,j \rangle,\sigma}( c^{\dagger}_{i,\sigma} c^{}_{j,\sigma}+ h.c.)$ ,

$c^\dagger_{i\sigma} , c_{j\sigma}$ - creation and annihilation operators

$\displaystyle\sigma$ - spin polarization

$\displaystyle t$ - hopping integral

$\displaystyle \langle i,j \rangle$ -nearest neighbor index

Here, hopping integral $\displaystyle t$ corresponds to the transfer integral $\displaystyle\gamma$ in tight binding model. Considering extreme cases of $t\rightarrow 0$ , it is impossible for electron to hop into neighboring sites. This case is the isolated atomic system. If the hopping term is turned on ( $\text{[math]}$ 0" src="http://upload.wikimedia.org/math/c/4/2/c42a280003a64774acf91ad359d199bc.png">) electrons can stay in both sites lowering their kinetic energy.

In the strongly correlated electron system, it is necessary to consider the electron-electron interaction. This term can be written in

$\displaystyle H_{ee}=\frac{1}{2}\sum_{n,m,\sigma}\langle n_1 m_1, n_2 m_2|\frac{e^2}{|r_1-r_2|}|n_3 m_3, n_4 m_4\rangle c^\dagger_{n_1 m_1 \sigma_1}c^\dagger_{n_2 m_2 \sigma_2}c_{n_4 m_4 \sigma_2} c_{n_3 m_3 \sigma_1}$

This interaction Hamiltonian includes direct Coulomb interaction energy and exchange interaction energy between electrons. There are several novel physics induced from this electron-electron interaction energy, such as metal-insulator transitions (MIT), high-temperature superconductivity, and several quantum phase transitions.