Tight binding

For other uses, see Tight binding (disambiguation).

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 LCAO method (linear combination of atomic orbitals 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 one-electron model, the model also provides a basis for more advanced calculations like the calculation of surface states and application to various kinds of many-body problem and quasiparticle calculations.

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 to which it belongs. 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 of the one-particle 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 matrix 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 interatomic 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 like the lattices of elements or simple compounds are studied 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.

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 nearly-free electron model. The model itself, or parts of it, can serve as the basis for other calculations. 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 interatomic matrix elements are replaced by inter- or intramolecular hopping and tunneling parameters. These conductors nearly all have very anisotropic properties and sometimes are almost perfectly one-dimensional.

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 LCAO-MO approach. A much simpler interpolation scheme for approximating the electronic band structure, especially for the d-bands of transition metals, is the parameterized tight-binding method conceived in 1954 by John Clarke Slater and George Fred Koster, sometimes referred to as the SK tight-binding method. With the SK tight-binding 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, first-principles calculations are carried out only at high-symmetry 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.

In the recent research about strongly correlated material the tight binding approach is basic approximation because highly localized electrons like 3-d transition metal electrons sometimes display 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.

Mathematical formulation

We introduce the atomic orbitals $\varphi _{m}(\mathbf {r} )$, which are eigenfunctions of the Hamiltonian $H_{\rm {at}}$ 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 $\Delta U$ required to obtain the true Hamiltonian $H$ of the system, are assumed small:

$H(\mathbf {r} )=H_{\mathrm {at} }(\mathbf {r} )+\sum _{\mathbf {R_{n}} \neq \mathbf {0} }V(\mathbf {r} -\mathbf {R_{n}} )=H_{\mathrm {at} }(\mathbf {r} )+\Delta U(\mathbf {r} )\ ,$

where $V(\mathbf {r} -\mathbf {R_{n}} )$ denotes the atomic potential of one atom located at site $\mathbf {R} _{n}$ in the crystal lattice. A solution $\psi _{m}$ to the time-independent single electron Schrödinger equation is then approximated as a linear combination of atomic orbitals $\varphi _{m}(\mathbf {r-R_{n}} )$:

$\psi _{m}(\mathbf {r} )=\sum _{\mathbf {R_{n}} }b_{m}(\mathbf {R_{n}} )\ \varphi _{m}(\mathbf {r-R_{n}} )$,

where $m$ refers to the m-th atomic energy level.

Translational symmetry and normalization

The Bloch theorem states that the wave function in a crystal can change under translation only by a phase factor:

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

where $\mathbf {k}$ is the wave vector of the wave function. Consequently, the coefficients satisfy

$\sum _{\mathbf {R_{n}} }b_{m}(\mathbf {R_{n}} )\ \varphi _{m}(\mathbf {r-R_{n}+R_{\ell }} )=e^{i\mathbf {k\cdot R_{\ell }} }\sum _{\mathbf {R_{n}} }b_{m}(\mathbf {R_{n}} )\ \varphi _{m}(\mathbf {r-R_{n}} )\ .$

By substituting $\mathbf {R_{p}} =\mathbf {R_{n}} -\mathbf {R_{\ell }}$, we find

$b_{m}(\mathbf {R_{p}+R_{\ell }} )=e^{i\mathbf {k\cdot R_{\ell }} }b_{m}(\mathbf {R_{p}} )\ ,$ (where in RHS we have replaced the dummy index $\mathbf {R_{n}}$ with $\mathbf {R_{p}}$)

or

$b_{m}(\mathbf {R_{l}} )=e^{i\mathbf {k\cdot R_{l}} }b_{m}(\mathbf {0} )\ .$

Normalizing the wave function to unity:

$\int d^{3}r\ \psi _{m}^{*}(\mathbf {r} )\psi _{m}(\mathbf {r} )=1$

$=\sum _{\mathbf {R_{n}} }b_{m}^{*}(\mathbf {R_{n}} )\sum _{\mathbf {R_{\ell }} }b_{m}(\mathbf {R_{\ell }} )\int d^{3}r\ \varphi _{m}^{*}(\mathbf {r-R_{n}} )\varphi _{m}(\mathbf {r-R_{\ell }} )$

$=b_{m}^{*}(0)b_{m}(0)\sum _{\mathbf {R_{n}} }e^{-i\mathbf {k\cdot R_{n}} }\sum _{\mathbf {R_{\ell }} }e^{i\mathbf {k\cdot R_{\ell }} }\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r-R_{n}} )\varphi _{m}(\mathbf {r-R_{\ell }} )$

$=Nb_{m}^{*}(0)b_{m}(0)\sum _{\mathbf {R_{p}} }e^{-i\mathbf {k\cdot R_{p}} }\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r-R_{p}} )\varphi _{m}(\mathbf {r} )\$

$=Nb_{m}^{*}(0)b_{m}(0)\sum _{\mathbf {R_{p}} }e^{i\mathbf {k\cdot R_{p}} }\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} )\varphi _{m}(\mathbf {r-R_{p}} )\ ,$

so the normalization sets $b_{m}(0)$ as

$b_{m}^{*}(0)b_{m}(0)={\frac {1}{N}}\ \cdot \ {\frac {1}{1+\sum _{\mathbf {R_{p}\neq 0} }e^{i\mathbf {k\cdot R_{p}} }\alpha _{m}(\mathbf {R_{p}} )}}\ ,$

Example 1: Dimer

The simplest example of a tight binding system are two atoms with a single S-orbital with hopping t. The Hamiltonian in second quantization of this system is

$H=t(c_{1}^{\dagger }c_{2}+c_{2}^{\dagger }c_{1})\ ,$

with $c_{i}$ and $c_{i}^{\dagger }$ creation and annihilation operators respectively. We write the Hamiltonian martially in the following way $H_{m,n}=c_{m}^{\dagger }Hc_{n}$ and the Hamiltonian in matrix form is

$H={\begin{pmatrix}0&t\\t&0\end{pmatrix}}\ .$

We find the solutions of this system finding the eigenvalues (energy) and eigenstates of the Hamiltonian. We write the eigenvalue $E_{i}$ with its respective eigenstate $|\psi _{i}\rangle$

$E_{1}=t\ ,$

$|\psi _{1}\rangle ={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\1\end{pmatrix}}\ ,$

$E_{2}=-t\ ,$

$|\psi _{2}\rangle ={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\-1\end{pmatrix}}\ .$

Table of interatomic matrix elements

In 1954 J.C. Slater and G.F. Koster published, mainly for the calculation of transition metal d-bands, a table of interatomic matrix elements

$E_{i,j}({\vec {\mathbf {r} }}_{n,n'})=\langle n,i|H|n',j\rangle$

which can also be derived from the cubic harmonic orbitals straightforwardly. The table expresses the matrix elements as functions of LCAO two-centre bond integrals between two cubic harmonic orbitals, i and j, on adjacent atoms. The bond integrals are for example the $V_{ss\sigma }$, $V_{pp\pi }$ and $V_{dd\delta }$ for sigma, pi and delta bonds (Notice that these integrals should also depend on the distance between the atoms, i.e. are a function of $(l,m,n)$, even though it is not explicitly stated every time.).

The interatomic vector is expressed as

${\vec {\mathbf {r} }}_{n,n'}=(r_{x},r_{y},r_{z})=d(l,m,n)$

where d is the distance between the atoms and l, m and n are the direction cosines to the neighboring atom.

$E_{s,s}=V_{ss\sigma }$

$E_{s,x}=lV_{sp\sigma }$

$E_{x,x}=l^{2}V_{pp\sigma }+(1-l^{2})V_{pp\pi }$

$E_{x,y}=lmV_{pp\sigma }-lmV_{pp\pi }$

$E_{x,z}=lnV_{pp\sigma }-lnV_{pp\pi }$

$E_{s,xy}={\sqrt {3}}lmV_{sd\sigma }$

$E_{s,x^{2}-y^{2}}={\frac {\sqrt {3}}{2}}(l^{2}-m^{2})V_{sd\sigma }$

$E_{s,3z^{2}-r^{2}}=[n^{2}-(l^{2}+m^{2})/2]V_{sd\sigma }$

$E_{x,xy}={\sqrt {3}}l^{2}mV_{pd\sigma }+m(1-2l^{2})V_{pd\pi }$

$E_{x,yz}={\sqrt {3}}lmnV_{pd\sigma }-2lmnV_{pd\pi }$

$E_{x,zx}={\sqrt {3}}l^{2}nV_{pd\sigma }+n(1-2l^{2})V_{pd\pi }$

$E_{x,x^{2}-y^{2}}={\frac {\sqrt {3}}{2}}l(l^{2}-m^{2})V_{pd\sigma }+l(1-l^{2}+m^{2})V_{pd\pi }$

$E_{y,x^{2}-y^{2}}={\frac {\sqrt {3}}{2}}m(l^{2}-m^{2})V_{pd\sigma }-m(1+l^{2}-m^{2})V_{pd\pi }$

$E_{z,x^{2}-y^{2}}={\frac {\sqrt {3}}{2}}n(l^{2}-m^{2})V_{pd\sigma }-n(l^{2}-m^{2})V_{pd\pi }$

$E_{x,3z^{2}-r^{2}}=l[n^{2}-(l^{2}+m^{2})/2]V_{pd\sigma }-{\sqrt {3}}ln^{2}V_{pd\pi }$

$E_{y,3z^{2}-r^{2}}=m[n^{2}-(l^{2}+m^{2})/2]V_{pd\sigma }-{\sqrt {3}}mn^{2}V_{pd\pi }$

$E_{z,3z^{2}-r^{2}}=n[n^{2}-(l^{2}+m^{2})/2]V_{pd\sigma }+{\sqrt {3}}n(l^{2}+m^{2})V_{pd\pi }$

$E_{xy,xy}=3l^{2}m^{2}V_{dd\sigma }+(l^{2}+m^{2}-4l^{2}m^{2})V_{dd\pi }+(n^{2}+l^{2}m^{2})V_{dd\delta }$

$E_{xy,yz}=3lm^{2}nV_{dd\sigma }+ln(1-4m^{2})V_{dd\pi }+ln(m^{2}-1)V_{dd\delta }$

$E_{xy,zx}=3l^{2}mnV_{dd\sigma }+mn(1-4l^{2})V_{dd\pi }+mn(l^{2}-1)V_{dd\delta }$

$E_{xy,x^{2}-y^{2}}={\frac {3}{2}}lm(l^{2}-m^{2})V_{dd\sigma }+2lm(m^{2}-l^{2})V_{dd\pi }+[lm(l^{2}-m^{2})/2]V_{dd\delta }$

$E_{yz,x^{2}-y^{2}}={\frac {3}{2}}mn(l^{2}-m^{2})V_{dd\sigma }-mn[1+2(l^{2}-m^{2})]V_{dd\pi }+mn[1+(l^{2}-m^{2})/2]V_{dd\delta }$

$E_{zx,x^{2}-y^{2}}={\frac {3}{2}}nl(l^{2}-m^{2})V_{dd\sigma }+nl[1-2(l^{2}-m^{2})]V_{dd\pi }-nl[1-(l^{2}-m^{2})/2]V_{dd\delta }$

$E_{xy,3z^{2}-r^{2}}={\sqrt {3}}\left[lm(n^{2}-(l^{2}+m^{2})/2)V_{dd\sigma }-2lmn^{2}V_{dd\pi }+[lm(1+n^{2})/2]V_{dd\delta }\right]$

$E_{yz,3z^{2}-r^{2}}={\sqrt {3}}\left[mn(n^{2}-(l^{2}+m^{2})/2)V_{dd\sigma }+mn(l^{2}+m^{2}-n^{2})V_{dd\pi }-[mn(l^{2}+m^{2})/2]V_{dd\delta }\right]$

$E_{zx,3z^{2}-r^{2}}={\sqrt {3}}\left[ln(n^{2}-(l^{2}+m^{2})/2)V_{dd\sigma }+ln(l^{2}+m^{2}-n^{2})V_{dd\pi }-[ln(l^{2}+m^{2})/2]V_{dd\delta }\right]$

$E_{x^{2}-y^{2},x^{2}-y^{2}}={\frac {3}{4}}(l^{2}-m^{2})^{2}V_{dd\sigma }+[l^{2}+m^{2}-(l^{2}-m^{2})^{2}]V_{dd\pi }+[n^{2}+(l^{2}-m^{2})^{2}/4]V_{dd\delta }$

$E_{x^{2}-y^{2},3z^{2}-r^{2}}={\sqrt {3}}\left[(l^{2}-m^{2})[n^{2}-(l^{2}+m^{2})/2]V_{dd\sigma }/2+n^{2}(m^{2}-l^{2})V_{dd\pi }+[(1+n^{2})(l^{2}-m^{2})/4]V_{dd\delta }\right]$

$E_{3z^{2}-r^{2},3z^{2}-r^{2}}=[n^{2}-(l^{2}+m^{2})/2]^{2}V_{dd\sigma }+3n^{2}(l^{2}+m^{2})V_{dd\pi }+{\frac {3}{4}}(l^{2}+m^{2})^{2}V_{dd\delta }$

Not all interatomic 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. Note that swapping orbital indices amounts to taking $(l,m,n)\rightarrow (-l,-m,-n)$, i.e. $E_{\alpha ,\beta }(l,m,n)=E_{\beta ,\alpha }(-l,-m,-n)$. For example, $E_{x,s}=-lV_{sp\sigma }$.

See also

• Electronic band structure
• Nearly-free electron model
• Bloch's theorems
• Kronig-Penney model
• Fermi surface
• Wannier function
• Hubbard model
• t-J model
• Effective mass
• Anderson's rule
• Dynamical theory of diffraction
• Solid state physics
• Linear combination of atomic orbitals molecular orbital method (LCAO)
• Holstein–Herring method
• Peierls substitution
• Hückel method

References

• 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 (Springer-Verlag 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 0-486-66021-4.
• N. W. Ashcroft and N. D. Mermin (1976). Solid State Physics. Toronto: Thomson Learning.
• Davies, John H. (1998). The physics of low-dimensional semiconductors: An introduction. Cambridge, United Kingdom: Cambridge University Press. ISBN 0-521-48491-X.
• Goringe, C M; Bowler, D R; Hernández, E (1997). "Tight-binding modelling of materials". Reports on Progress in Physics. 60 (12): 1447–1512. Bibcode:1997RPPh...60.1447G. doi:10.1088/0034-4885/60/12/001.
• 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.

External links

• Crystal-field Theory, Tight-binding Method, and Jahn-Teller Effect in E. Pavarini, E. Koch, F. Anders, and M. Jarrell (eds.): Correlated Electrons: From Models to Materials, Jülich 2012, ISBN 978-3-89336-796-2
• Tight-Binding Studio: A Technical Software Package to Find the Parameters of Tight-Binding Hamiltonian