Describes the range of energies of an electron within the solid
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 "forbidden bands").
Band theory derives these bands and band gaps by examining the allowed quantum mechanical wave functions 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 and optical absorption, and forms the foundation of the understanding of all solid-state devices (transistors, solar cells, etc.).
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Why bands and band gaps occur.
The formation of electronic bands and band gaps can be illustrated with two complementary models for electrons in solids. The first one is the nearly free electron model, in which the electrons are assumed to move almost freely within the material. In this model, the electronic states resemble free electron plane waves, and are only slightly perturbed by the crystal lattice. This model explains the origin of the electronic dispersion relation, but the explanation for band gaps is subtle in this model.
The second model starts from the opposite limit, in which the electrons are tightly bound to individual atoms. The electrons of a single, isolated atom occupy atomic orbitals with discrete energy levels. If two atoms come close enough so that their atomic orbitals overlap, the electrons can tunnel between the atoms. This tunneling splits (hybridizes) the atomic orbitals into molecular orbitals with different energies.
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 (), the number of orbitals that hybridize with each other is very large. For this reason, the adjacent levels are very closely spaced in energy (of the order of ), and can be considered to form a continuum, an energy band.
This formation of bands is mostly a feature of the outermost electrons (valence electrons) 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 gaps 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 orbitals 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 electrons) 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:
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:
Crystalline symmetry and wavevectors.
Band structure calculations take advantage of the periodic nature of a crystal lattice, exploiting its symmetry. The single-electron Schrödinger equation is solved for an electron in a lattice-periodic potential, giving Bloch electrons as solutions
formula_1
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 for electrons in that band.
The wavevector takes on any value inside the Brillouin zone, which is a polyhedron in wavevector (reciprocal lattice) 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. "kx", "ky", "kz". 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 [100], [111], and [110], respectively. 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:
Asymmetry: Band structures in non-crystalline solids.
Although electronic band structures are usually associated with crystalline materials, quasi-crystalline and amorphous solids 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, a calculation for the rate of optical absorption, 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, the likelihood of a state of energy "E" being filled with an electron is given by the Fermi–Dirac distribution, a thermodynamic distribution that takes into account the Pauli exclusion principle:
formula_2
where:
The density of electrons in the material is simply the integral of the Fermi–Dirac distribution times the density of states:
formula_3
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 electrons). 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:
Theory in crystals.
The ansatz is the special case of electron waves in a periodic crystal lattice using Bloch's theorem as treated generally in the dynamical theory of diffraction. Every crystal is a periodic structure which can be characterized by a Bravais lattice, and for each Bravais lattice we can determine the reciprocal lattice, which encapsulates the periodicity in a set of three reciprocal lattice vectors (b1, b2, b3). Now, any periodic potential "V"(r) which shares the same periodicity as the direct lattice can be expanded out as a Fourier series whose only non-vanishing components are those associated with the reciprocal lattice vectors. So the expansion can be written as:
formula_4
where K = "m"1b1 + "m"2b2 + "m"3b3 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 which states that electrons in a periodic potential have wavefunctions and energies which are periodic in wavevector up to a constant phase shift between neighboring reciprocal lattice vectors. The consequences of periodicity are described mathematically by the Bloch's theorem, which states that the eigenstate wavefunctions have the form
formula_5
where the Bloch function formula_6 is periodic over the crystal lattice, that is,
formula_7
Here index n refers to the nth 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.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 orbitals and potentials on neighbouring atoms is relatively large. In that case the wave function of the electron can be approximated by a (modified) plane wave. The band structure of a metal like aluminium even gets close to the empty lattice approximation.
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 formula_8 is well approximated by a linear combination of atomic orbitals formula_9.
formula_10
where the coefficients formula_11 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:Eq. 42 p. 267
formula_12
in which formula_13 is the periodic part of the Bloch's theorem and the integral is over the Brillouin zone. 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:
formula_14
The TB model works well in materials with limited overlap between atomic orbitals and potentials on neighbouring atoms. Band structures of materials like Si, GaAs, SiO2 and diamond for instance are well described by TB-Hamiltonians on the basis of atomic sp3 orbitals. In transition metals 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 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 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