In
solid-state physics, the electronic band structure (or simply band structure) of a
solid
Solid is one of the four fundamental states of matter (the others being liquid, gas, and plasma). The molecules in a solid are closely packed together and contain the least amount of kinetic energy. A solid is characterized by structural ...
describes the range of
energy levels
A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical particles, which can have any amount of energy. The t ...
that electrons may have within it, as well as the ranges of energy that they may not have (called ''
band gap
In solid-state physics, a band gap, also called an energy gap, is an energy range in a solid where no electronic states can exist. In graphs of the electronic band structure of solids, the band gap generally refers to the energy difference ( ...
s'' or ''forbidden bands'').
Band theory derives these bands and band gaps by examining the allowed quantum mechanical
wave function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
s for an electron in a large, periodic lattice of atoms or molecules. Band theory has been successfully used to explain many physical properties of solids, such as
electrical resistivity
Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...
and
optical absorption
In physics, absorption of electromagnetic radiation is how matter (typically electrons bound in atoms) takes up a photon's energy — and so transforms electromagnetic energy into internal energy of the absorber (for example, thermal energy). ...
, and forms the foundation of the understanding of all
solid-state devices (transistors, solar cells, etc.).
Why bands and band gaps occur
![Solid state electronic band structure](https://upload.wikimedia.org/wikipedia/commons/e/ef/Solid_state_electronic_band_structure.svg)
The electrons of a single, isolated atom occupy
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 an ...
s each of which has a discrete
energy level
A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical particles, which can have any amount of energy. The t ...
. When two or more atoms join together to form a
molecule
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioche ...
, their atomic orbitals overlap and
hybridize.
Similarly, if a large number ''N'' of identical atoms come together to form a solid, such as a
crystal lattice, the atoms' atomic orbitals overlap with the nearby orbitals.
Each discrete energy level splits into ''N'' levels, each with a different energy. Since the number of atoms in a macroscopic piece of solid is a very large number (N~10
22) the number of orbitals is very large and thus they are very closely spaced in energy (of the order of 10
−22 eV). The energy of the adjacent levels is so close together that they can be considered as a continuum, an energy band.
This formation of bands is mostly a feature of the outermost electrons (
valence electron
In chemistry and physics, a valence electron is an electron in the outer shell associated with an atom, and that can participate in the formation of a chemical bond if the outer shell is not closed. In a single covalent bond, a shared pair form ...
s) in the atom, which are the ones involved in chemical bonding and
electrical conductivity. The inner electron orbitals do not overlap to a significant degree, so their bands are very narrow.
Band gap
In solid-state physics, a band gap, also called an energy gap, is an energy range in a solid where no electronic states can exist. In graphs of the electronic band structure of solids, the band gap generally refers to the energy difference ( ...
s are essentially leftover ranges of energy not covered by any band, a result of the finite widths of the energy bands. The bands have different widths, with the widths depending upon the degree of overlap in 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 an ...
s from which they arise. Two adjacent bands may simply not be wide enough to fully cover the range of energy. For example, the bands associated with core orbitals (such as
1s electron
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 spe ...
s) are extremely narrow due to the small overlap between adjacent atoms. As a result, there tend to be large band gaps between the core bands. Higher bands involve comparatively larger orbitals with more overlap, becoming progressively wider at higher energies so that there are no band gaps at higher energies.
Basic concepts
Assumptions and limits of band structure theory
Band theory is only an approximation to the quantum state of a solid, which applies to solids consisting of many identical atoms or molecules bonded together. These are the assumptions necessary for band theory to be valid:
* ''Infinite-size system'': For the bands to be continuous, the piece of material must consist of a large number of atoms. Since a macroscopic piece of material contains on the order of 10
22 atoms, this is not a serious restriction; band theory even applies to microscopic-sized
transistor
upright=1.4, gate (G), body (B), source (S) and drain (D) terminals. The gate is separated from the body by an insulating layer (pink).
A transistor is a semiconductor device used to Electronic amplifier, amplify or electronic switch, switch ...
s in
integrated circuits. With modifications, the concept of band structure can also be extended to systems which are only "large" along some dimensions, such as
two-dimensional electron systems.
* ''Homogeneous system'': Band structure is an intrinsic property of a material, which assumes that the material is homogeneous. Practically, this means that the chemical makeup of the material must be uniform throughout the piece.
* ''Non-interactivity'': The band structure describes "single electron states". The existence of these states assumes that the electrons travel in a static potential without dynamically interacting with
lattice vibration
In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phonon is an excited state in the quantum mechanical ...
s, other electrons,
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
s, etc.
The above assumptions are broken in a number of important practical situations, and the use of band structure requires one to keep a close check on the limitations of band theory:
* Inhomogeneities and interfaces: Near surfaces, junctions, and other inhomogeneities, the bulk band structure is disrupted. Not only are there local small-scale disruptions (e.g.,
surface states
Surface states are electronic states found at the surface of materials. They are formed due to the sharp transition from solid material that ends with a surface and are found only at the atom layers closest to the surface. The termination of a mate ...
or
dopant
A dopant, also called a doping agent, is a trace of impurity element that is introduced into a chemical material to alter its original electrical or optical properties. The amount of dopant necessary to cause changes is typically very low. When ...
states inside the band gap), but also local charge imbalances. These charge imbalances have electrostatic effects that extend deeply into semiconductors, insulators, and the vacuum (see
doping,
band bending
In solid-state physics, band bending refers to the process in which the electronic band structure in a material curves up or down near a junction or interface. It does not involve any physical (spatial) bending. When the electrochemical potential ...
).
* Along the same lines, most electronic effects (
capacitance
Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized ar ...
,
electrical conductance
The electrical resistance of an object is a measure of its opposition to the flow of electric current. Its reciprocal quantity is , measuring the ease with which an electric current passes. Electrical resistance shares some conceptual paralle ...
,
electric-field screening
In physics, screening is the damping of electric fields caused by the presence of mobile charge carriers. It is an important part of the behavior of charge-carrying fluids, such as ionized gases (classical plasmas), electrolytes, and charge ...
) involve the physics of electrons passing through surfaces and/or near interfaces. The full description of these effects, in a band structure picture, requires at least a rudimentary model of electron-electron interactions (see
space charge
Space charge is an interpretation of a collection of electric charges in which excess electric charge is treated as a continuum of charge distributed over a region of space (either a volume or an area) rather than distinct point-like charges. Thi ...
,
band bending
In solid-state physics, band bending refers to the process in which the electronic band structure in a material curves up or down near a junction or interface. It does not involve any physical (spatial) bending. When the electrochemical potential ...
).
* Small systems: For systems which are small along every dimension (e.g., a small
molecule
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioche ...
or a
quantum dot), there is no continuous band structure. The crossover between small and large dimensions is the realm of
mesoscopic physics
Mesoscopic physics is a subdiscipline of condensed matter physics that deals with materials of an intermediate size. These materials range in size between the nanoscale for a quantity of atoms (such as a molecule) and of materials measuring mic ...
.
*
Strongly correlated materials (for example,
Mott insulator
Mott insulators are a class of materials that are expected to conduct electricity according to conventional band theories, but turn out to be insulators (particularly at low temperatures). These insulators fail to be correctly described by band ...
s) simply cannot be understood in terms of single-electron states. The electronic band structures of these materials are poorly defined (or at least, not uniquely defined) and may not provide useful information about their physical state.
Crystalline symmetry and wavevectors
![Bulkbandstructure](https://upload.wikimedia.org/wikipedia/commons/1/1f/Bulkbandstructure.gif)
Band structure calculations take advantage of the periodic nature of a crystal lattice, exploiting its symmetry. The 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 solved for an electron in a lattice-periodic potential, giving
Bloch electron
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 d ...
s as solutions
:
,
where k is called the wavevector. For each value of k, there are multiple solutions to the Schrödinger equation labelled by ''n'', the band index, which simply numbers the energy bands.
Each of these energy levels evolves smoothly with changes in k, forming a smooth band of states. For each band we can define a function ''E''
''n''(k), which is the
dispersion relation
In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given t ...
for electrons in that band.
The wavevector takes on any value inside the
Brillouin zone
In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice ...
, which is a polyhedron in wavevector (
reciprocal lattice
In physics, the reciprocal lattice represents the Fourier transform of another lattice (group) (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is a periodic spatial fu ...
) space that is related to the crystal's lattice.
Wavevectors outside the Brillouin zone simply correspond to states that are physically identical to those states within the Brillouin zone.
Special high symmetry points/lines in the Brillouin zone are assigned labels like Γ, Δ, Λ, Σ (see Fig 1).
It is difficult to visualize the shape of a band as a function of wavevector, as it would require a plot in four-dimensional space, ''E'' vs. ''k
x'', ''k
y'', ''k
z''. In scientific literature it is common to see band structure plots which show the values of ''E''
''n''(k) for values of k along straight lines connecting symmetry points, often labelled Δ, Λ, Σ, or
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Another method for visualizing band structure is to plot a constant-energy isosurface in wavevector space, showing all of the states with energy equal to a particular value. The isosurface of states with energy equal to the Fermi level is known as the Fermi surface.
Energy band gaps can be classified using the wavevectors of the states surrounding the band gap:
*
Direct band gap
In semiconductor physics, the band gap of a semiconductor can be of two basic types, a direct band gap or an indirect band gap. The minimal-energy state in the conduction band and the maximal-energy state in the valence band are each characteriz ...
: the lowest-energy state above the band gap has the same k as the highest-energy state beneath the band gap.
*
Indirect band gap: the closest states above and beneath the band gap do not have the same k value.
Asymmetry: Band structures in non-crystalline solids
Although electronic band structures are usually associated with
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 ...
line materials,
quasi-crystalline and
amorphous solid
In condensed matter physics and materials science, an amorphous solid (or non-crystalline solid, glassy solid) is a solid that lacks the long-range order that is characteristic of a crystal.
Etymology
The term comes from the Greek ''a'' ("wi ...
s may also exhibit band gaps. These are somewhat more difficult to study theoretically since they lack the simple symmetry of a crystal, and it is not usually possible to determine a precise dispersion relation. 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
The density of states function ''g''(''E'') is defined as the number of electronic states per unit volume, per unit energy, for electron energies near ''E''.
The density of states function is important for calculations of effects based on band theory.
In
Fermi's Golden Rule
In quantum physics, Fermi's golden rule is a formula that describes the transition rate (the probability of a transition per unit time) from one energy eigenstate of a quantum system to a group of energy eigenstates in a continuum, as a result of a ...
, a calculation for the rate of
optical absorption
In physics, absorption of electromagnetic radiation is how matter (typically electrons bound in atoms) takes up a photon's energy — and so transforms electromagnetic energy into internal energy of the absorber (for example, thermal energy). ...
, it provides both the number of excitable electrons and the number of final states for an electron. It appears in calculations of
electrical conductivity where it provides the number of mobile states, and in computing electron scattering rates where it provides the number of final states after scattering.
For energies inside a band gap, ''g''(''E'') = 0.
Filling of bands
At
thermodynamic equilibrium
Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In the ...
, the likelihood of a state of energy ''E'' being filled with an electron is given by the
Fermi–Dirac distribution Fermi–Dirac may refer to:
* Fermi–Dirac statistics
Fermi–Dirac statistics (F–D statistics) is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pa ...
, a thermodynamic distribution that takes into account the
Pauli exclusion principle
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulat ...
:
:
where:
*''k''
B''T'' is the product of
Boltzmann's constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant ...
and
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, measured with a thermometer.
Thermometers are calibrated in various Conversion of units of temperature, temp ...
, and
*''µ'' is the
total chemical potential
In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...
of electrons, or ''Fermi level'' (in
semiconductor physics
A semiconductor is a material which has an electrical conductivity value falling between that of a conductor, such as copper, and an insulator, such as glass. Its resistivity falls as its temperature rises; metals behave in the opposite way. ...
, this quantity is more often denoted ''E''
F). The Fermi level of a solid is directly related to the voltage on that solid, as measured with a voltmeter. Conventionally, in band structure plots the Fermi level is taken to be the zero of energy (an arbitrary choice).
The density of electrons in the material is simply the integral of the Fermi–Dirac distribution times the density of states:
:
Although there are an infinite number of bands and thus an infinite number of states, there are only a finite number of electrons to place in these bands.
The preferred value for the number of electrons is a consequence of electrostatics: even though the surface of a material can be charged, the internal bulk of a material prefers to be charge neutral.
The condition of charge neutrality means that ''N''/''V'' must match the density of protons in the material. For this to occur, the material electrostatically adjusts itself, shifting its band structure up or down in energy (thereby shifting ''g''(''E'')), until it is at the correct equilibrium with respect to the Fermi level.
Names of bands near the Fermi level (conduction band, valence band)
A solid has an infinite number of allowed bands, just as an atom has infinitely many energy levels. However, most of the bands simply have too high energy, and are usually disregarded under ordinary circumstances.
Conversely, there are very low energy bands associated with the core orbitals (such as
1s electron
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 spe ...
s). These low-energy ''core band''s are also usually disregarded since they remain filled with electrons at all times, and are therefore inert.
Likewise, materials have several band gaps throughout their band structure.
The most important bands and band gaps—those relevant for electronics and optoelectronics—are those with energies near the Fermi level.
The bands and band gaps near the Fermi level are given special names, depending on the material:
* In a
semiconductor
A semiconductor is a material which has an electrical conductivity value falling between that of a conductor, such as copper, and an insulator, such as glass. Its resistivity falls as its temperature rises; metals behave in the opposite way. ...
or
band insulator, the Fermi level is surrounded by a band gap, referred to as ''the'' band gap (to distinguish it from the other band gaps in the band structure). The closest band above the band gap is called ''the
conduction band'', and the closest band beneath the band gap is called ''the
valence band
In solid-state physics, the valence band and conduction band are the bands closest to the Fermi level, and thus determine the electrical conductivity of the solid. In nonmetals, the valence band is the highest range of electron energies in w ...
''. The name "valence band" was coined by analogy to chemistry, since in semiconductors (and insulators) the valence band is built out of the
valence orbital
In chemistry and physics, a valence electron is an electron in the outer shell associated with an atom, and that can participate in the formation of a chemical bond if the outer shell is not closed. In a single covalent bond, a shared pair forms ...
s.
* In a metal or
semimetal
A semimetal is a material with a very small overlap between the bottom of the conduction band and the top of the valence band.
According to electronic band theory, solids can be classified as insulators, semiconductors, semimetals, or metals ...
, the Fermi level is inside of one or more allowed bands. In semimetals the bands are usually referred to as "conduction band" or "valence band" depending on whether the charge transport is more electron-like or hole-like, by analogy to semiconductors. In many metals, however, the bands are neither electron-like nor hole-like, and often just called "valence band" as they are made of valence orbitals. The band gaps in a metal's band structure are not important for low energy physics, since they are too far from the Fermi level.
Theory in crystals
The
ansatz
In physics and mathematics, an ansatz (; , meaning: "initial placement of a tool at a work piece", plural Ansätze ; ) is an educated guess or an additional assumption made to help solve a problem, and which may later be verified to be part of the ...
is the special case of electron waves in a periodic crystal lattice using
Bloch's 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 d ...
as treated generally in the
dynamical theory of diffraction
The dynamical theory of diffraction describes the interaction of waves with a regular lattice. The wave fields traditionally described are X-rays, neutrons or electrons and the regular lattice are atomic crystal structures or nanometer-scale mul ...
. Every crystal is a periodic structure which can be characterized by a
Bravais 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 ...
, and for each
Bravais 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 ...
we can determine the
reciprocal lattice
In physics, the reciprocal lattice represents the Fourier transform of another lattice (group) (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is a periodic spatial fu ...
, which encapsulates the periodicity in a set of three reciprocal lattice vectors (b
1, b
2, b
3). 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:
:
where K = ''m''
1b
1 + ''m''
2b
2 + ''m''
3b
3 for any set of integers (''m''
1, ''m''
2, ''m''
3).
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.
Nearly free electron approximation
In the nearly free electron approximation, interactions between electrons are completely ignored. This approximation allows use of
Bloch's 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 d ...
which states that electrons in a periodic potential have
wavefunction
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
s and energies which are periodic in wavevector up to a constant phase shift between neighboring
reciprocal lattice
In physics, the reciprocal lattice represents the Fourier transform of another lattice (group) (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is a periodic spatial fu ...
vectors. The consequences of periodicity are described mathematically by the Bloch's theorem, which states that the eigenstate wavefunctions have the form
:
where the Bloch function
is periodic over the crystal lattice, that is,
:
.
Here index ''n'' refers to the ''n-th'' energy band, wavevector k is related to the direction of motion of the electron, r is the position in the crystal, and R is the location of an atomic site.
[Kittel, p. 179]
The NFE model works particularly well in materials like metals where distances between neighbouring atoms are small. In such materials
the overlap of
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 an ...
s and potentials on neighbouring
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, ...
s is relatively large. In that case the
wave function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
of the electron can be approximated by a (modified) plane wave. The band structure of a metal like
aluminium
Aluminium (aluminum in American and Canadian English) is a chemical element with the symbol Al and atomic number 13. Aluminium has a density lower than those of other common metals, at approximately one third that of steel. I ...
even gets close to the
empty lattice approximation
The empty lattice approximation is a theoretical electronic band structure model in which the potential is ''periodic'' and ''weak'' (close to constant). One may also consider an empty irregular lattice, in which the potential is not even periodi ...
.
Tight binding model
The opposite extreme to the nearly free electron approximation assumes the electrons in the crystal behave much like an assembly of constituent atoms. This
tight binding model assumes the 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 well approximated by a
linear combination of
atomic orbitals
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 sp ...
.
[Kittel, pp. 245-248]
:
,
where the coefficients
are selected to give the best approximate solution of this form. Index ''n'' refers to an atomic energy level and R refers to an atomic site. A more accurate approach using this idea employs
Wannier functions, defined by:
[Kittel, Eq. 42 p. 267]
:
;
in which
is the periodic part of the Bloch's theorem and the integral is over the
Brillouin zone
In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice ...
. Here index ''n'' refers to the ''n''-th energy band in the crystal. The Wannier functions are localized near atomic sites, like atomic orbitals, but being defined in terms of Bloch functions they are accurately related to solutions based upon the crystal potential. Wannier functions on different atomic sites R are orthogonal. The Wannier functions can be used to form the Schrödinger solution for the ''n''-th energy band as:
:
.
The TB model works well in materials with limited overlap between
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 an ...
s and potentials on neighbouring atoms. Band structures of materials like
Si,
GaAs
Gallium arsenide (GaAs) is a III-V direct band gap semiconductor with a zinc blende crystal structure.
Gallium arsenide is used in the manufacture of devices such as microwave frequency integrated circuits, monolithic microwave integrated circ ...
, SiO
2 and
diamond
Diamond is a solid form of the element carbon with its atoms arranged in a crystal structure called diamond cubic. Another solid form of carbon known as graphite is the chemically stable form of carbon at room temperature and pressure, ...
for instance are well described by TB-Hamiltonians on the basis of atomic sp
3 orbitals. In
transition metals
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 ca ...
a mixed TB-NFE model is used to describe the broad NFE
conduction band and the narrow embedded TB d-bands. The radial functions of
the atomic orbital part
of the Wannier functions are most easily calculated by the use of
pseudopotential
In physics, a pseudopotential or effective potential is used as an approximation for the simplified description of complex systems. Applications include atomic physics and neutron scattering. The pseudopotential approximation was first introduced ...
methods. NFE, TB or combined NFE-TB band structure
calculations,
sometimes extended with wave function approximations based on pseudopotential methods, are often used as an economic starting point for further calculations.
KKR model
The KKR method, also called "multiple scattering theory" or Green's function method, finds the stationary values of the inverse transition matrix T rather than the Hamiltonian. A variational implementation was suggested by
Korringa,
Kohn Kohn is both a first name and a surname. Kohn means cook in Yiddish. It may also be related to Cohen. Notable people with the surname include:
* Angela Kohn ( Jacki-O), rapper
* Arnold Kohn, Croatian Zionist and longtime president of the Jewish c ...
and Rostocker, and is often referred to as the ''
Korringa–Kohn–Rostoker method''.
The most important features of the KKR or Green's function formulation are (1) it separates the two aspects of the problem: structure (positions of the atoms) from the scattering (chemical identity of the atoms); and (2) Green's functions provide a natural approach to a localized description of electronic properties that can be adapted to alloys and other disordered system. The simplest form of this approximation centers non-overlapping spheres (referred to as ''muffin tins'') on the atomic positions. Within these regions, the potential experienced by an electron is approximated to be spherically symmetric about the given nucleus. In the remaining interstitial region, the
screened potential is approximated as a constant. Continuity of the potential between the atom-centered spheres and interstitial region is enforced.
Density-functional theory
In recent physics literature, a large majority of the electronic structures and band plots are calculated using
density-functional theory
Density-functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body ...
(DFT), which is not a model but rather a theory, i.e., a microscopic first-principles theory of
condensed matter physics that tries to cope with the electron-electron many-body problem via the introduction of an
exchange-correlation term in the functional of the
electronic density
In quantum chemistry, electron density or electronic density is the measure of the probability of an electron being present at an infinitesimal element of space surrounding any given point. It is a scalar quantity depending upon three spatial va ...
. DFT-calculated bands are in many cases found to be in agreement with experimentally measured bands, for example by
angle-resolved photoemission spectroscopy
Angle-resolved photoemission spectroscopy (ARPES) is an experimental technique used in condensed matter physics to probe the allowed energies and momenta of the electrons in a material, usually a crystalline solid. It is based on the photoel ...
(ARPES). In particular, the band shape is typically well reproduced by DFT. But there are also systematic errors in DFT bands when compared to experiment results. In particular, DFT seems to systematically underestimate by about 30-40% the band gap in insulators and semiconductors.
It is commonly believed that DFT is a theory to predict
ground state properties of a system only (e.g. the
total energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat ...
, the
atomic structure
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, and ...
, etc.), and that
excited state properties cannot be determined by DFT. This is a misconception. In principle, DFT can determine any property (ground state or excited state) of a system given a functional that maps the ground state density to that property. This is the essence of the Hohenberg–Kohn theorem.
In practice, however, no known functional exists that maps the ground state density to excitation energies of electrons within a material. Thus, what in the literature is quoted as a DFT band plot is a representation of the DFT
Kohn–Sham energies, i.e., the energies of a fictive non-interacting system, the Kohn–Sham system, which has no physical interpretation at all. The Kohn–Sham electronic structure must not be confused with the real,
quasiparticle
In physics, quasiparticles and collective excitations are closely related emergent phenomena arising when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in vacuum.
For exa ...
electronic structure of a system, and there is no
Koopmans' theorem
Koopmans' theorem states that in closed-shell Hartree–Fock theory (HF), the first ionization energy of a molecular system is equal to the negative of the orbital energy of the highest occupied molecular orbital (HOMO). This theorem is named aft ...
holding for Kohn–Sham energies, as there is for Hartree–Fock energies, which can be truly considered as an approximation for
quasiparticle energies. Hence, in principle, Kohn–Sham based DFT is not a band theory, i.e., not a theory suitable for calculating bands and band-plots. In principle
time-dependent DFT can be used to calculate the true band structure although in practice this is often difficult. A popular approach is the use of
hybrid functional Hybrid functionals are a class of approximations to the exchange–correlation energy functional in density functional theory (DFT) that incorporate a portion of exact exchange from Hartree–Fock theory with the rest of the exchange–correla ...
s, which incorporate a portion of Hartree–Fock exact exchange; this produces a substantial improvement in predicted bandgaps of semiconductors, but is less reliable for metals and wide-bandgap materials.
Green's function methods and the ''ab initio'' GW approximation
To calculate the bands including electron-electron interaction
many-body effects, one can resort to so-called
Green's function
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if \operatorname is the linear differenti ...
methods. Indeed, knowledge of the Green's function of a system provides both ground (the total energy) and also excited state observables of the system. The poles of the Green's function are the quasiparticle energies, the bands of a solid. The Green's function can be calculated by solving the
Dyson equation
In quantum field theory, the energy that a particle has as a result of changes that it causes in its environment defines self-energy \Sigma, and represents the contribution to the particle's energy, or Effective mass (solid-state physics), effecti ...
once the
self-energy of the system is known. For real systems like solids, the self-energy is a very complex quantity and usually approximations are needed to solve the problem. One such approximation is the
GW approximation
The ''GW'' approximation (GWA) is an approximation made in order to calculate the self-energy of a many-body system of electrons. The approximation is that the expansion of the self-energy ''Σ'' in terms of the single particle Green's function ...
, so called from the mathematical form the self-energy takes as the product Σ = ''GW'' of the Green's function ''G'' and the dynamically screened interaction ''W''. This approach is more pertinent when addressing the calculation of band plots (and also quantities beyond, such as the spectral function) and can also be formulated in a completely ''ab initio'' way. The GW approximation seems to provide band gaps of insulators and semiconductors in agreement with experiment, and hence to correct the systematic DFT underestimation.
Dynamical mean-field theory
Although the nearly free electron approximation is able to describe many properties of electron band structures, one consequence of this theory is that it predicts the same number of electrons in each unit cell. If the number of electrons is odd, we would then expect that there is an unpaired electron in each unit cell, and thus that the valence band is not fully occupied, making the material a conductor. However, materials such as
CoO that have an odd number of electrons per unit cell are insulators, in direct conflict with this result. This kind of material is known as a
Mott insulator
Mott insulators are a class of materials that are expected to conduct electricity according to conventional band theories, but turn out to be insulators (particularly at low temperatures). These insulators fail to be correctly described by band ...
, and requires inclusion of detailed electron-electron interactions (treated only as an averaged effect on the crystal potential in band theory) to explain the discrepancy. The
Hubbard model is an approximate theory that can include these interactions. It can be treated non-perturbatively within the so-called
dynamical mean-field theory Dynamical mean-field theory (DMFT) is a method to determine the electronic structure of strongly correlated materials. In such materials, the approximation of independent electrons, which is used in density functional theory and usual band structure ...
, which attempts to bridge the gap between the nearly free electron approximation and the atomic limit. Formally, however, the states are not non-interacting in this case and the concept of a band structure is not adequate to describe these cases.
Others
Calculating band structures is an important topic in theoretical
solid state physics. In addition to the models mentioned above, other models include the following:
*
Empty lattice approximation
The empty lattice approximation is a theoretical electronic band structure model in which the potential is ''periodic'' and ''weak'' (close to constant). One may also consider an empty irregular lattice, in which the potential is not even periodi ...
: the "band structure" of a region of free space that has been divided into a lattice.
*
k·p perturbation theory
In solid-state physics, the k·p perturbation theory is an approximated semi-empirical approach for calculating the band structure (particularly effective mass) and optical properties of crystalline solids.
It is pronounced "k dot p", and is ...
is a technique that allows a band structure to be approximately described in terms of just a few parameters. The technique is commonly used for
semiconductors, and the parameters in the model are often determined by experiment.
*The
Kronig–Penney model, a one-dimensional rectangular well model useful for illustration of band formation. While simple, it predicts many important phenomena, but is not quantitative.
*
Hubbard model
The band structure has been generalised to wavevectors that are
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, resulting in what is called a ''complex band structure'', which is of interest at surfaces and interfaces.
Each model describes some types of solids very well, and others poorly. The nearly free electron model works well for metals, but poorly for non-metals. The tight binding model is extremely accurate for ionic insulators, such as
metal halide
Metal halides are compounds between metals and halogens. Some, such as sodium chloride are ionic, while others are covalently bonded. A few metal halides are discrete molecules, such as uranium hexafluoride, but most adopt polymeric structures, ...
salts (e.g.
NaCl
Sodium chloride , commonly known as salt (although sea salt also contains other chemical salts), is an ionic compound with the chemical formula NaCl, representing a 1:1 ratio of sodium and chloride ions. With molar masses of 22.99 and 35.45 g/ ...
).
Band diagrams
To understand how band structure changes relative to the Fermi level in real space, a band structure plot is often first simplified in the form of a
band diagram
In solid-state physics of semiconductors, a band diagram is a diagram plotting various key electron energy levels (Fermi level and nearby energy band edges) as a function of some spatial dimension, which is often denoted ''x''. These diagrams ...
. In a band diagram the vertical axis is energy while the horizontal axis represents real space. Horizontal lines represent energy levels, while blocks represent energy bands. When the horizontal lines in these diagram are slanted then the energy of the level or band changes with distance. Diagrammatically, this depicts the presence of an electric field within the crystal system. Band diagrams are useful in relating the general band structure properties of different materials to one another when placed in contact with each other.
See also
*
Band-gap engineering
Band-gap engineering is the process of controlling or altering the band gap of a material. This is typically done to semiconductors by controlling the composition of alloys, constructing layered materials with alternating compositions, or by in ...
—the process of altering a material's band structure
*
Felix Bloch
Felix Bloch (23 October 1905 – 10 September 1983) was a Swiss-American physicist and Nobel physics laureate who worked mainly in the U.S. He and Edward Mills Purcell were awarded the 1952 Nobel Prize for Physics for "their development of new ...
—pioneer in the theory of band structure
*
Alan Herries Wilson—pioneer in the theory of band structure
Citations
General bibliography
*
Further reading
* Ashcroft, Neil and N. David Mermin, ''Solid State Physics'',
* Harrison, Walter A., ''Elementary Electronic Structure'',
* Harrison, Walter A.; W. A. Benjamin ''Pseudopotentials in the theory of metals'', (New York) 1966
* Marder, Michael P., ''Condensed Matter Physics'',
* Martin, Richard, ''Electronic Structure: Basic Theory and Practical Methods'',
* Millman, Jacob; Arvin Gabriel, ''Microelectronics'', , Tata McGraw-Hill Edition.
* Nemoshkalenko, V. V., and N. V. Antonov, ''Computational Methods in Solid State Physics'',
* Omar, M. Ali, ''Elementary Solid State Physics: Principles and Applications'',
* Singh, Jasprit, ''Electronic and Optoelectronic Properties of Semiconductor Structures'' Chapters 2 and 3,
* Vasileska, Dragica
Tutorial on Bandstructure Methods(2008)
External links
Animation, applications and research about quantum physics and band theory(Université Paris Sud)
{{Condensed matter physics topics
Solid state engineering
Articles containing video clips