 Tight binding

In solidstate 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 LCAO method used in chemistry. Tight binding models are applied to a wide variety of solids. The model gives good qualitative results in many cases and can be combined with other models that give better results where the tight binding model fails. Though the tight binding model is a oneelectron model, the model also provides a basis for more advanced calculations like the calculation of surface states and application to various kinds of manybody problem and quasiparticle calculations.
Contents
Introduction
The name "tight binding" of this electronic band structure model suggests that this quantum mechanical model describes the properties of tightly bound electrons in solids. The electrons in this model should be tightly bound to the atom to which they belong and they should have limited interaction with states and potentials on surrounding atoms of the solid. As a result the wave function of the electron will be rather similar to the atomic orbital of the free atom it belongs to. The energy of the electron will also be rather close to the ionization energy of the electron in the free atom or ion because the interaction with potentials and states on neighboring atoms is limited.
Though the mathematical formulation^{[1]} of the oneparticle tight binding Hamiltonian may look complicated at first glance, the model is not complicated at all and can be understood intuitively quite easily. There are only three kinds of elements that play a significant role in the theory. Two of those three kinds of elements should be close to zero and can often be neglected. The most important elements in the model are the inter atomic matrix elements, which would simply be called the bond energies by a chemist.
In general there are a number of atomic energy levels and atomic orbitals involved in the model. This can lead to complicated band structures because the orbitals belong to different point group representations. The reciprocal lattice and the Brillouin zone often belong to a different space group than the crystal of the solid. High symmetry points in the Brillouin zone belong to different point group representations. When simple systems are studied, like the lattices of elements or simple compounds, it is often not very difficult to calculate eigenstates in high symmetry points analytically. So the tight binding model can provide nice examples for those who want to learn more about group theory.
What is known as the "tight binding model" has a long history and has been applied in many ways and with many different purposes and different outcomes. The model doesn't stand on its own. Parts of the model can be filled in or extended by other kinds of calculations and models like the nearlyfree electron model. The model itself, or parts of it, can serve as the basis for other calculations.^{[2]} In the study of conductive polymers, organic semiconductors and molecular electronics for example, tight binding like models are applied in which the role of the atoms in the original concept is replaced by the molecular orbitals of conjugated systems and where the inter atomic matrix elements are replaced by inter or intra molecular hopping and tunneling parameters. These conductors nearly all have very anisotropic properties and sometimes are almost perfectly onedimensional.
Historical background
By 1928, the idea of a molecular orbital had been advanced by Robert Mulliken, who was influenced considerably by the work of Friedrich Hund. The LCAO method for approximating molecular orbitals was introduced in 1928 by B. N. Finklestein and G. E. Horowitz, while the LCAO method for solids was developed by Felix Bloch, as part of his doctoral dissertation in 1928, concurrently with and independent of the LCAOMO approach. A much simpler interpolation scheme for approximating the electronic band structure, especially for the dbands of transition metals, is the parameterized tightbinding method conceived in 1954 by John Clarke Slater and George Fred Koster,^{[1]} the absolute climax in the history of this model. Hence, it is sometimes referred to as the SK tightbinding method. With the SK tightbinding method, electronic band structure calculations on a solid need not be carried out with full rigor as in the original Bloch's theorem but, rather, firstprinciples calculations are carried out only at highsymmetry points and the band structure is interpolated over the remainder of the Brillouin zone between these points.
In this approach, interactions between different atomic sites are considered as perturbations. There exist 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 3d transition metal electrons sometimes display strongly correlated behaviors. In this case, the role of electronelectron interaction must be considered using the manybody physics description.
The tightbinding 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.
Mathematical formulation
We introduce the atomic orbitals , 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 "tightbinding". Any corrections to the atomic potential ΔU required to obtain the true Hamiltonian H of the system, are assumed small:
A solution ψ(r) to the timeindependent single electron Schrödinger equation is then approximated as a linear combination of atomic orbitals :
 ,
where m refers to the mth atomic energy level and R_{n} locates an atomic site in the crystal lattice.
Translational symmetry and normalization
The translational symmetry of the crystal implies the wave function under translation can change only by a phase factor:
where k is the wave vector of the wave function. Consequently, the coefficients satisfy
By substituting R_{p} = R_{n} − R_{ℓ}, we find
or
Normalizing the wave function to unity:
so the normalization sets b(0) as
where α (R_{p} ) are the atomic overlap integrals, which frequently are neglected resulting in^{[3]}
and
The tight binding Hamiltonian
Using the tight binding form for the wave function, and assuming only the mth atomic energy level is important for the mth energy band, the Bloch energies ε_{m} are of the form
Here terms involving the atomic Hamiltonian at sites other than where it is centered are neglected. The energy then becomes
where E_{m} is the energy of the mth atomic level, and α_{m,l}, β_{m} and γ_{m,l} are the tight binding matrix elements.
The tight binding matrix elements
The element
 ,
is the atomic energy shift due to the potential on neighboring atoms. This term is relatively small in most cases. If it is large it means that potentials on neighboring atoms have a large influence on the energy of the central atom.
The next term
is the inter atomic matrix element between the atomic orbitals m and l on adjacent atoms. It is also called the bond energy or two center integral and it is the most important element in the tight binding model.
The last terms
 ,
denote the overlap integrals between the atomic orbitals m and l on adjacent atoms.
Evaluation of the matrix elements
As mentioned before the values of the β_{m}matrix elements are not so large in comparison with the ionization energy because the potentials of neighboring atoms on the central atom are limited. If β_{m} is not relatively small it means that the potential of the neighboring atom on the central atom is not small either. In that case it is an indication that the tight binding model is not a very good model for the description of the band structure for some reason. The inter atomic distances can be too small or the charges on the atoms or ions in the lattice is wrong for example.
The inter atomic matrix elements γ_{m,l} can be calculated directly if the atomic wave functions and the potentials are known in detail. Most often this is not the case. There are numerous ways to get parameters for these matrix elements. Parameters can be obtained from chemical bond energy data. Energies and eigenstates on some high symmetry points in the Brillouin zone can be evaluated and values integrals in the matrix elements can be matched with band structure data from other sources.
The inter atomic overlap matrix elements α_{m,l} should be rather small or neglectable. If they are large it is again an indication that the tight binding model is of limited value for some purposes. Large overlap is an indication for too short inter atomic distance for example. In metals and transition metals the broad sband or spband can be fitted better to an existing band structure calculation by the introduction of nextnearestneighbor matrix elements and overlap integrals but fits like that don't yield a very useful model for the electronic wave function of a metal. Broad bands in dense materials are better described by a nearly free electron model.
The tight binding model works particularly well in cases where the band width is small and the electrons are strongly localized, like in the case of dbands and fbands. The model also gives good results in the case of open crystal structures, like diamond or silicon, where the number of neighbors is small. The model can easily be combined with a nearly free electron model in a hybrid NFETB model.^{[2]}
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^{[4]}
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 mth energy band can be derived from this Bloch wave:
These real space wave functions 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 socalled tight binding approximation.
Second quantization
Modern explanations of electronic structure like tJ model and Hubbard model are based on tight binding model.^{[5]} 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.
 ,
  creation and annihilation operators
  spin polarization
  hopping integral
 nearest neighbor index
Here, hopping integral corresponds to the transfer integral in tight binding model. Considering extreme cases of , it is impossible for electron to hop into neighboring sites. This case is the isolated atomic system. If the hopping term is turned on () electrons can stay in both sites lowering their kinetic energy.
In the strongly correlated electron system, it is necessary to consider the electronelectron interaction. This term can be written in
This interaction Hamiltonian includes direct Coulomb interaction energy and exchange interaction energy between electrons. There are several novel physics induced from this electronelectron interaction energy, such as metalinsulator transitions (MIT), hightemperature superconductivity, and several quantum phase transitions.
Example: onedimensional sband
Here the tight binding model is illustrated with a sband model for a string of atoms with a single sorbital in a straight line with spacing a and σ bonds between atomic sites.
To find approximate eigenstates of the Hamiltonian, we can use a linear combination of the atomic orbitals
where N = total number of sites and k is a real parameter with . (This wave function is normalized to unity by the leading factor 1/√N provided overlap of atomic wave functions is ignored.) Assuming only nearest neighbor overlap, the only nonzero matrix elements of the Hamiltonian can be expressed as
The energy E_{i} is the ionization energy corresponding to the chosen atomic orbital and U is the energy shift of the orbital as a result of the potential of neighboring atoms. The elements, which are the Slater and Koster interatomic matrix elements, are the bond energies E_{i,j}. In this one dimensional sband model we only have σbonds between the sorbitals with bond energy E_{s,s} = V_{ssσ}. The overlap between states on neighboring atoms is S. We can derive the energy of the state using the above equation:
where, for example,
and
Thus the energy of this state can be represented in the familiar form of the energy dispersion:
 .
 For k = 0 the energy is E = (E_{0} − 2Δ) / (1 + 2S) and the state consists of a sum of all atomic orbitals. This state can be viewed as a chain of bonding orbitals.
 For k = π / (2a) the energy is E = E_{0} and the state consists of a sum of atomic orbitals which are a factor e^{iπ / 2} out of phase. This state can be viewed as a chain of nonbonding orbitals.
 Finally for k = π / a the energy is E = (E_{0} + 2Δ) / (1 − 2S) and the state consists of an alternating sum of atomic orbitals. This state can be viewed as a chain of antibonding orbitals.
This example is readily extended to three dimensions, for example, to a bodycentered cubic or facecentered cubic lattice by introducing the nearest neighbor vector locations in place of simply n a.^{[6]} 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.
Table of interatomic matrix elements
In 1954 J.C. Slater and F.G. Koster published, mainly for the calculation of transition metal dbands, a table of inter atomic matrix elements^{[1]}
which, with a little patience and effort, can also be derived from the cubic harmonic orbitals straightforwardly. The table expresses the matrix elements as functions of LCAO twocentre bond integrals between two cubic harmonic orbitals, i and j, on adjacent atoms. The bond integrals are for example the V_{ssσ}, V_{ppπ} and V_{ddδ} for sigma, pi and delta bonds.
The inter atomic vector is expressed as
where d is the distance between the atoms and l, m and n are the direction cosines to the neighboring atom.
 E_{s,s} = V_{ssσ}
 E_{s,x} = lV_{spσ}
 E_{x,x} = l^{2}V_{ppσ} + (1 − l^{2})V_{ppπ}
 E_{x,y} = lmV_{ppσ} − lmV_{ppπ}
 E_{x,z} = lnV_{ppσ} − lnV_{ppπ}
 E_{xy,xy} = 3l^{2}m^{2}V_{ddσ} + (l^{2} + m^{2} − 4l^{2}m^{2})V_{ddπ} + (n^{2} + l^{2}m^{2})V_{ddδ}
 E_{xy,yz} = 3lm^{2}nV_{ddσ} + ln(1 − 4m^{2})V_{ddπ} + ln(m^{2} − 1)V_{ddδ}
 E_{xy,zx} = 3l^{2}mnV_{ddσ} + mn(1 − 4l^{2})V_{ddπ} + mn(l^{2} − 1)V_{ddδ}
Not all inter atomic matrix elements are listed explicitly. Matrix elements that are not listed in this table can be constructed by permutation of indices and cosine directions of other matrix elements in the table.
See also
References
 ^ ^{a} ^{b} ^{c} J. C. Slater, G. F. Koster (1954). "Simplified LCAO method for the Periodic Potential Problem". Physical Review 94 (6): 1498–1524. Bibcode 1954PhRv...94.1498S. doi:10.1103/PhysRev.94.1498.
 ^ ^{a} ^{b} Walter Ashley Harrison (1989). Electronic Structure and the Properties of Solids. Dover Publications. ISBN 0486660214. http://books.google.com/books?id=R2VqQgAACAAJ.
 ^ As an alternative to neglecting overlap, one may choose as a basis instead of atomic orbitals a set of orbitals based upon atomic orbitals but arranged to be orthogonal to orbitals on other atomic sites, the socalled Löwdin orbitals. See PY Yu & M Cardona (2005). "Tightbinding or LCAO approach to the band structure of semiconductors". Fundamentals of Semiconductors (3 ed.). Springrer. p. 87. ISBN 3540254706. http://books.google.com/books?id=W9pdJZoAeyEC&pg=PA87.
 ^ Orfried Madelung, Introduction to SolidState Theory (SpringerVerlag, Berlin Heidelberg, 1978).
 ^ Alexander Altland and Ben Simons (2006). "Interaction effects in the tightbinding system". Condensed Matter Field Theory. Cambridge University Press. pp. 58 ff. ISBN 9780521845083. http://books.google.com/books?id=0KMkfAMe3JkC&pg=RA4PA58.
 ^ Sir Nevill F Mott & H Jones (1958). "II §4 Motion of electrons in a periodic field". The theory of the properties of metals and alloys (Reprint of Clarendon Press (1936) ed.). Courier Dover Publications. pp. 56 ff. ISBN 048660456X. http://books.google.com/?id=LIPsUaTqUXUC&printsec=frontcover#PPA58,M1.
 N. W. Ashcroft and N. D. Mermin, Solid State Physics (Thomson Learning, Toronto, 1976).
 Stephen Blundell Magnetism in Condensed Matter(Oxford, 2001).
 S.Maekawa et al. Physics of Transition Metal Oxides (SpingerVerlag Berlin Heidelberg, 2004).
 John Singleton Band Theory and Electronic Properties of Solids (Oxford, 2001).
Further reading
 Walter Ashley Harrison (1989). Electronic Structure and the Properties of Solids. Dover Publications. ISBN 0486660214. http://books.google.com/books?id=R2VqQgAACAAJ.
 N. W. Ashcroft and N. D. Mermin (1976). Solid State Physics. Toronto: Thomson Learning.
 Davies, John H. (1998). The physics of lowdimensional semiconductors: An introduction. Cambridge, United Kingdom: Cambridge University Press. ISBN 052148491X.
 Goringe, C M; Bowler, D R; Hernández, E (1997). "Tightbinding modelling of materials". Reports on Progress in Physics 60 (12): 1447–1512. Bibcode 1997RPPh...60.1447G. doi:10.1088/00344885/60/12/001.
 Slater, J. C.; Koster, G. F. (1954). "Simplified LCAO Method for the Periodic Potential Problem". Physical Review 94 (6): 1498–1524. Bibcode 1954PhRv...94.1498S. doi:10.1103/PhysRev.94.1498.
Categories: Electronic structure methods
 Electronic band structures
Wikimedia Foundation. 2010.