In
solid-state physics
Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how th ...
, the tight-binding model (or TB model) is an approach to the calculation of
electronic band structure
In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called '' band gaps'' or ...
using an approximate set of
wave functions based upon
superposition of wave functions for isolated
atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, a ...
s 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
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
describes the properties of tightly bound electrons in solids. The
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have n ...
s in this model should be tightly bound to the
atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, a ...
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
In atomic theory and quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in any ...
of the free atom to which it belongs. The energy of the electron will also be rather close to the
ionization energy
Ionization, or Ionisation is the process by which an atom or a molecule acquires a negative or positive charge by gaining or losing electrons, often in conjunction with other chemical changes. The resulting electrically charged atom or molecule ...
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
In chemistry, bond energy (''BE''), also called the mean bond enthalpy or average bond enthalpy is the measure of bond strength in a chemical bond. IUPAC defines bond energy as the average value of the gas-phase bond-dissociation energy (usually ...
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
A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macro ...
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
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
.
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
In solid-state physics, the nearly free electron model (or NFE model) or quasi-free electron model is a quantum mechanical model of physical properties of electrons that can move almost freely through the crystal lattice of a solid. The mode ...
. The model itself, or parts of it, can serve as the basis for other calculations.
[
] In the study of
conductive polymers,
organic semiconductor
Organic semiconductors are solids whose building blocks are pi-bonded molecules or polymers made up by carbon and hydrogen atoms and – at times – heteroatoms such as nitrogen, sulfur and oxygen. They exist in the form of molecular crystals or ...
s 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 orbital
In chemistry, a molecular orbital is a mathematical function describing the location and wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of find ...
s of
conjugated system
In theoretical chemistry, a conjugated system is a system of connected p-orbitals with delocalized electrons in a molecule, which in general lowers the overall energy of the molecule and increases stability. It is conventionally represented ...
s 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 metal
In chemistry, a transition metal (or transition element) is a chemical element in the d-block of the periodic table (groups 3 to 12), though the elements of group 12 (and less often group 3) are sometimes excluded. They are the elements that can ...
s, is the parameterized tight-binding method conceived in 1954 by
John Clarke Slater
John Clarke Slater (December 22, 1900 – July 25, 1976) was a noted American physicist who made major contributions to the theory of the electronic structure of atoms, molecules and solids. He also made major contributions to microwave electroni ...
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
perturbation
Perturbation or perturb may refer to:
* Perturbation theory, mathematical methods that give approximate solutions to problems that cannot be solved exactly
* Perturbation (geology), changes in the nature of alluvial deposits over time
* Perturbat ...
s. 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
Strongly correlated materials are a wide class of compounds that include insulators and electronic materials, and show unusual (often technologically useful) electronic and magnetic properties, such as metal-insulator transitions, heavy fermio ...
the tight binding approach is basic approximation because highly localized electrons like 3-d
transition metal
In chemistry, a transition metal (or transition element) is a chemical element in the d-block of the periodic table (groups 3 to 12), though the elements of group 12 (and less often group 3) are sometimes excluded. They are the elements that can ...
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
In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called '' band gaps'' or ...
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 orbital
In atomic theory and quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in any ...
s
, which are
eigenfunction
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
s of the
Hamiltonian 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
required to obtain the true Hamiltonian
of the system, are assumed small:
:
where
denotes the atomic potential of one atom located at site
in the
crystal lattice
In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by
: \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n ...
. A solution
to the time-independent single electron
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
is then approximated as a
linear combination of atomic orbitals :
:
,
where
refers to the m-th atomic energy level.
Translational symmetry and normalization
The
Bloch theorem
In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. The theorem is named after the physicist Felix Bloch, who di ...
states that the wave function in a crystal can change under translation only by a phase factor:
:
where
is the
wave vector of the wave function. Consequently, the coefficients satisfy
:
By substituting
, we find
:
(where in RHS we have replaced the dummy index
with
)
or
:
Normalizing the wave function to unity:
:
:::
:::
:::
:::
so the normalization sets ''
'' as
:
where ''α
m'' (''R''
p ) are the atomic overlap integrals, which frequently are neglected resulting in
[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 so-called Löwdin orbitals. See ]
:
and
::
The tight binding Hamiltonian
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
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 ''m''-th atomic level, and
,
and
are the tight binding matrix elements discussed below.
The tight binding matrix elements
The elements
are 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 class of terms
is the
interatomic 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 dominant term in the tight binding model.
The last class of terms
denote the
overlap integrals between the atomic orbitals ''m'' and ''l'' on adjacent atoms. These, too, are typically small; if not, then
Pauli repulsion has a non-negligible influence on the energy of the central atom.
Evaluation of the matrix elements
As mentioned before the values of the
-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
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 interatomic distances can be too small or the charges on the atoms or ions in the lattice is wrong for example.
The interatomic matrix elements
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 interatomic overlap matrix elements
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 interatomic distance for example. In metals and transition metals the broad s-band or sp-band can be fitted better to an existing band structure calculation by the introduction of next-nearest-neighbor 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 d-bands and f-bands. 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 NFE-TB model.
Connection to Wannier functions
Bloch functions describe the electronic states in a periodic
crystal lattice
In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by
: \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n ...
. Bloch functions can be represented as a
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
[Orfried Madelung, ''Introduction to Solid-State Theory'' (Springer-Verlag, Berlin Heidelberg, 1978).]
:
where ''R''
n denotes an atomic site in a periodic crystal lattice, ''k'' is the
wave vector of the Bloch's function, ''r'' is the electron position, ''m'' is the band index, and the sum is over all ''N'' atomic sites. The Bloch's function 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
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
analysis, a spatially localized wave function for the ''m''-th energy band can be constructed from multiple Bloch's functions:
:
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 so-called tight binding approximation.
Second quantization
Modern explanations of electronic structure like
t-J model
In solid-state physics, the ''t''-''J'' model is a model first derived in 1977 from the Hubbard model by Józef Spałek to explain antiferromagnetic properties of the Mott insulators and taking into account experimental results about the strengt ...
and
Hubbard model
The Hubbard model is an approximate model used to describe the transition between conducting and insulating systems.
It is particularly useful in solid-state physics. The model is named for John Hubbard.
The Hubbard model states that each el ...
are based on tight binding model.
Tight binding can be understood by working under a
second quantization
Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as t ...
formalism.
Using the atomic orbital as a basis state, the second quantization Hamiltonian operator in the tight binding framework can be written as:
:
,
:
- creation and annihilation operators
:
- spin polarization
:
- hopping integral
:
- nearest neighbor index
:
- the hermitian conjugate of the other term(s)
Here, hopping integral
corresponds to the transfer integral
in tight binding model. Considering extreme cases of
, it is impossible for an 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 physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acce ...
.
In the strongly correlated electron system, it is necessary to consider the electron-electron interaction. This term can be written in
:
This interaction Hamiltonian includes direct
Coulomb
The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI).
In the present version of the SI it is equal to the electric charge delivered by a 1 ampere constant current in 1 second and to elementary char ...
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 transition
In physics, a quantum phase transition (QPT) is a phase transition between different quantum phases ( phases of matter at zero temperature). Contrary to classical phase transitions, quantum phase transitions can only be accessed by varying a phy ...
s.
Example: one-dimensional s-band
Here the tight binding model is illustrated with a s-band model for a string of atoms with a single
s-orbital 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
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 non-zero 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
In chemistry, bond energy (''BE''), also called the mean bond enthalpy or average bond enthalpy is the measure of bond strength in a chemical bond. IUPAC defines bond energy as the average value of the gas-phase bond-dissociation energy (usually ...
. In this one dimensional s-band model we only have
-bonds between the s-orbitals with bond energy
. 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
the energy is
and the state consists of a sum of all atomic orbitals. This state can be viewed as a chain of
bonding orbitals.
*For
the energy is
and the state consists of a sum of atomic orbitals which are a factor
out of phase. This state can be viewed as a chain of
non-bonding orbitals.
*Finally for
the energy is
and the state consists of an alternating sum of atomic orbitals. This state can be viewed as a chain of
anti-bonding orbitals.
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.
Table of interatomic matrix elements
In 1954 J.C. Slater and G.F. Koster published, mainly for the calculation of
transition metal
In chemistry, a transition metal (or transition element) is a chemical element in the d-block of the periodic table (groups 3 to 12), though the elements of group 12 (and less often group 3) are sometimes excluded. They are the elements that can ...
d-bands, a table of interatomic matrix elements
:
which can also be derived from the
cubic harmonic orbitals straightforwardly. The table expresses the matrix elements as functions of
LCAO
A linear combination of atomic orbitals or LCAO is a quantum superposition of atomic orbitals and a technique for calculating molecular orbitals in quantum chemistry. In quantum mechanics, electron configurations of atoms are described as wavef ...
two-centre
bond integrals between two
cubic harmonic orbitals, ''i'' and ''j'', on adjacent atoms. The bond integrals are for example the
,
and
for
sigma
Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used a ...
,
pi and
delta bonds (Notice that these integrals should also depend on the distance between the atoms, i.e. are a function of
, even though it is not explicitly stated every time.).
The interatomic vector is expressed as
:
where ''d'' is the distance between the atoms and ''l'', ''m'' and ''n'' are the
direction cosine
In analytic geometry, the direction cosines (or directional cosines) of a vector are the cosines of the angles between the vector and the three positive coordinate axes. Equivalently, they are the contributions of each component of the basis to ...
s to the neighboring atom.
:
:
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