Insolid-state physics, theelectronic band structure (or simplyband structure) of asolid describes the range ofenergy levels that electrons may have within it, as well as the ranges of energy that they may not have (calledband gaps orforbidden bands).
Band theory derives these bands and band gaps by examining the allowed quantum mechanicalwave 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 aselectrical resistivity andoptical absorption, and forms the foundation of the understanding of allsolid-state devices (transistors, solar cells, etc.).
The formation of electronic bands and band gaps can be illustrated with two complementary models for electrons in solids.[1]: 161 The first one is thenearly free electron model, in which the electrons are assumed to move almost freely within the material. In this model, the electronic states resemblefree 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.[2]: 121
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 occupyatomic orbitals with discreteenergy levels. If two atoms come close enough so that their atomic orbitals overlap, the electrons cantunnel between the atoms. This tunneling splits (hybridizes) the atomic orbitals intomolecular orbitals with different energies.[2]: 117–122
Similarly, if a large numberN of identical atoms come together to form a solid, such as acrystal lattice, the atoms' atomic orbitals overlap with the nearby orbitals.[3] Each discrete energy level splits intoN levels, each with a different energy. Since the number of atoms in a macroscopic piece of solid is a very large number (N ≈ 1022), 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 of10−22 eV),[4][5][6] 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 andelectrical 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 theatomic 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 as1s 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.
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:
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Band structure calculations take advantage of the periodic nature of a crystal lattice, exploiting its symmetry. The single-electronSchrödinger equation is solved for an electron in a lattice-periodic potential, givingBloch electrons as solutionswherek is called the wavevector. For each value ofk, there are multiple solutions to the Schrödinger equation labelled byn, the band index, which simply numbers the energy bands.Each of these energy levels evolves smoothly with changes ink, forming a smooth band of states. For each band we can define a functionEn(k), which is thedispersion relation for electrons in that band.
The wavevector takes on any value inside theBrillouin 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 seeband structure plots which show the values ofEn(k) for values ofk along straight lines connecting symmetry points, often labelled Δ, Λ, Σ, or[100], [111], and [110], respectively.[7][8] Another method for visualizing band structure is to plot a constant-energyisosurface in wavevector space, showing all of the states with energy equal to a particular value. The isosurface of states with energy equal to theFermi level is known as theFermi surface.
Energy band gaps can be classified using the wavevectors of the states surrounding the band gap:
Although electronic band structures are usually associated withcrystalline materials,quasi-crystalline andamorphous 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.
The density of states functiong(E) is defined as the number of electronic states per unit volume, per unit energy, for electron energies nearE.
The density of states function is important for calculations of effects based on band theory.InFermi's Golden Rule, a calculation for the rate ofoptical absorption, it provides both the number of excitable electrons and the number of final states for an electron. It appears in calculations ofelectrical 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.[citation needed]
For energies inside a band gap,g(E) = 0.
Atthermodynamic equilibrium, the likelihood of a state of energyE being filled with an electron is given by theFermi–Dirac distribution, a thermodynamic distribution that takes into account thePauli exclusion principle:where:
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 thatN/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 shiftingg(E)), until it is at the correct equilibrium with respect to the Fermi level.
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.[9]Conversely, there are very low energy bands associated with the core orbitals (such as1s electrons). These low-energycore bands are also usually disregarded since they remain filled with electrons at all times, and are therefore inert.[10]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:
Theansatz is the special case of electron waves in a periodic crystal lattice usingBloch's theorem as treated generally in thedynamical theory of diffraction. Every crystal is a periodic structure which can be characterized by aBravais lattice, and for eachBravais lattice we can determine thereciprocal lattice, which encapsulates the periodicity in a set of three reciprocal lattice vectors(b1,b2,b3). Now, any periodic potentialV(r) which shares the same periodicity as the direct lattice can be expanded out as aFourier series whose only non-vanishing components are those associated with the reciprocal lattice vectors. So the expansion can be written as:whereK =m1b1 +m2b2 +m3b3 for any set of integers(m1,m2,m3).
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.
In the nearly free electron approximation, interactions between electrons are completely ignored. This approximation allows use ofBloch's Theorem which states that electrons in a periodic potential havewavefunctions and energies which are periodic in wavevector up to a constant phase shift between neighboringreciprocal lattice vectors. The consequences of periodicity are described mathematically by the Bloch's theorem, which states that the eigenstate wavefunctions have the formwhere the Bloch function is periodic over the crystal lattice, that is,
Here indexn refers to thenth energy band, wavevectork is related to the direction of motion of the electron,r is the position in the crystal, andR is the location of an atomic site.[12]: 179
The NFE model works particularly well in materials like metals where distances between neighbouring atoms are small. In such materials the overlap ofatomic orbitals and potentials on neighbouringatoms is relatively large. In that case thewave function of the electron can be approximated by a (modified) plane wave. The band structure of a metal likealuminium even gets close to theempty lattice approximation.
The opposite extreme to the nearly free electron approximation assumes the electrons in the crystal behave much like an assembly of constituent atoms. Thistight binding model assumes the solution to the time-independent single electronSchrödinger equation is well approximated by alinear combination ofatomic orbitals.[12]: 245–248 where the coefficients are selected to give the best approximate solution of this form. Indexn refers to an atomic energy level andR refers to an atomic site. A more accurate approach using this idea employsWannier functions, defined by:[12]: Eq. 42 p. 267 [13]in which is the periodic part of the Bloch's theorem and the integral is over theBrillouin zone. Here indexn refers to then-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 sitesR are orthogonal. The Wannier functions can be used to form the Schrödinger solution for then-th energy band as:
The TB model works well in materials with limited overlap betweenatomic orbitals and potentials on neighbouring atoms. Band structures of materials likeSi,GaAs, SiO2 anddiamond for instance are well described by TB-Hamiltonians on the basis of atomic sp3 orbitals. Intransition metals a mixed TB-NFE model is used to describe the broad NFEconduction 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 ofpseudopotential methods. NFE, TB or combined NFE-TB band structure calculations,[14]sometimes extended with wave function approximations based on pseudopotential methods, are often used as an economic starting point for further calculations.
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 byKorringa,Kohn and Rostocker, and is often referred to as theKorringa–Kohn–Rostoker method.[15][16] 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 asmuffin 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, thescreened potential is approximated as a constant. Continuity of the potential between the atom-centered spheres and interstitial region is enforced.
In recent physics literature, a large majority of the electronic structures and band plots are calculated usingdensity-functional theory (DFT), which is not a model but rather a theory, i.e., a microscopic first-principles theory ofcondensed matter physics that tries to cope with the electron-electron many-body problem via the introduction of anexchange-correlation term in the functional of theelectronic density. DFT-calculated bands are in many cases found to be in agreement with experimentally measured bands, for example byangle-resolved photoemission spectroscopy (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.[17]
It is commonly believed that DFT is a theory to predictground state properties of a system only (e.g. thetotal energy, theatomic structure, etc.), and thatexcited 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.[18] 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 DFTKohn–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 electronic structure of a system, and there is noKoopmans' theorem holding for Kohn–Sham energies, as there is for Hartree–Fock energies, which can be truly considered as an approximation forquasiparticle 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 principletime-dependent DFT can be used to calculate the true band structure although in practice this is often difficult. A popular approach is the use ofhybrid functionals, 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.[19]
To calculate the bands including electron-electron interactionmany-body effects, one can resort to so-calledGreen's function 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 theDyson equation once theself-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 theGW approximation, so called from the mathematical form the self-energy takes as the product Σ =GW of the Green's functionG and the dynamically screened interactionW. 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 completelyab 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.
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 asCoO 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 aMott insulator, 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. TheHubbard model is an approximate theory that can include these interactions. It can be treated non-perturbatively within the so-calleddynamical mean-field theory, 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.
Calculating band structures is an important topic in theoreticalsolid state physics. In addition to the models mentioned above, other models include the following:
The band structure has been generalised to wavevectors that arecomplex numbers, resulting in what is called acomplex 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 asmetal halide salts (e.g.NaCl).
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 aband diagram. 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.
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