|Electronic structure methods|
|Valence bond theory|
| Coulson–Fischer theory |
Generalized valence bond
Modern valence bond
|Molecular orbital theory|
| Hartree–Fock method |
Semi-empirical quantum chemistry methods
Møller–Plesset perturbation theory
Multi-configurational self-consistent field
Quantum chemistry composite methods
Quantum Monte Carlo
|Density functional theory|
| Time-dependent density functional theory |
Orbital-free density functional theory
Linearized augmented-plane-wave method
Projector augmented wave method
|Electronic band structure|
| Nearly free electron model |
k·p perturbation theory
Empty lattice approximation
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.).
The electrons of a single, isolated atom occupy atomic orbitals each of which has a discrete energy level. When two or more atoms join together to form a molecule, 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. eV). The energy of the adjacent levels is so close together that they can be considered as a continuum, an energy band.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~1022) the number of orbitals is very large and thus they are very closely spaced in energy (of the order of 10−22
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.
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:
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
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 En(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 En(k) for values of k along straight lines connecting symmetry points, often labelled Δ, Λ, Σ, or , , and , 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:
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.
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.[ citation needed ]
For energies inside a band gap, g(E) = 0.
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:
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.
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 bands 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:
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:
where K = 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 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
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.
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.
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 is well approximated by a linear combination of atomic orbitals .
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:
in which 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:
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.
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 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.
In recent physics literature, a large majority of the electronic structures and band plots are calculated using density-functional theory (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. DFT-calculated bands are in many cases found to be in agreement with experimentally measured bands, for example by angle-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.
It is commonly believed that DFT is a theory to predict ground state properties of a system only (e.g. the total energy, the atomic structure, 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 electronic structure of a system, and there is no Koopmans' theorem 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 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.
To calculate the bands including electron-electron interaction many-body effects, one can resort to so-called Green'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 the Dyson equation 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, 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.
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, 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, 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 theoretical solid state physics. In addition to the models mentioned above, other models include the following:
The band structure has been generalised to wavevectors that are complex numbers, 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 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 a band 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.
In solid state physics, a particle's effective mass is the mass that it seems to have when responding to forces, or the mass that it seems to have when interacting with other identical particles in a thermal distribution. One of the results from the band theory of solids is that the movement of particles in a periodic potential, over long distances larger than the lattice spacing, can be very different from their motion in a vacuum. The effective mass is a quantity that is used to simplify band structures by modeling the behavior of a free particle with that mass. For some purposes and some materials, the effective mass can be considered to be a simple constant of a material. In general, however, the value of effective mass depends on the purpose for which it is used, and can vary depending on a number of factors.
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, electrolytes, and charge carriers in electronic conductors . In a fluid, with a given permittivity ε, composed of electrically charged constituent particles, each pair of particles interact through the Coulomb force as
Density-functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure of many-body systems, in particular atoms, molecules, and the condensed phases. Using this theory, the properties of a many-electron system can be determined by using functionals, i.e. functions of another function. In the case of DFT, these are functionals of the spatially dependent electron density. DFT is among the most popular and versatile methods available in condensed-matter physics, computational physics, and computational chemistry.
In solid state physics and condensed matter physics, the density of states (DOS) of a system describes the number of modes per unit frequency range. The density of states is defined as , where is the number of states in the system of volume whose energies lie in the range from to . It is mathematically represented as a distribution by a probability density function, and it is generally an average over the space and time domains of the various states occupied by the system. The density of states is directly related to the dispersion relations of the properties of the system. High DOS at a specific energy level means that many states are available for occupation.
In condensed matter physics, the Fermi surface is the surface in reciprocal space which separates occupied from unoccupied electron states at zero temperature. The shape of the Fermi surface is derived from the periodicity and symmetry of the crystalline lattice and from the occupation of electronic energy bands. The existence of a Fermi surface is a direct consequence of the Pauli exclusion principle, which allows a maximum of one electron per quantum state. The study of the Fermi surfaces of materials is called fermiology.
In quantum physics, the spin–orbit interaction is a relativistic interaction of a particle's spin with its motion inside a potential. A key example of this phenomenon is the spin–orbit interaction leading to shifts in an electron's atomic energy levels, due to electromagnetic interaction between the electron's magnetic dipole, its orbital motion, and the electrostatic field of the positively charged nucleus. This phenomenon is detectable as a splitting of spectral lines, which can be thought of as a Zeeman effect product of two relativistic effects: the apparent magnetic field seen from the electron perspective and the magnetic moment of the electron associated with its intrinsic spin. A similar effect, due to the relationship between angular momentum and the strong nuclear force, occurs for protons and neutrons moving inside the nucleus, leading to a shift in their energy levels in the nucleus shell model. In the field of spintronics, spin–orbit effects for electrons in semiconductors and other materials are explored for technological applications. The spin–orbit interaction is one cause of magnetocrystalline anisotropy and the spin Hall effect.
Electronic correlation is the interaction between electrons in the electronic structure of a quantum system. The correlation energy is a measure of how much the movement of one electron is influenced by the presence of all other electrons.
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 by Hans Hellmann in 1934.
In solid-state physics, the tight-binding model is an approach to the calculation of electronic band structure using an approximate set of wave functions based upon superposition of wave functions for isolated atoms located at each atomic site. The method is closely related to the LCAO method 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.
Local-density approximations (LDA) are a class of approximations to the exchange–correlation (XC) energy functional in density functional theory (DFT) that depend solely upon the value of the electronic density at each point in space. Many approaches can yield local approximations to the XC energy. However, overwhelmingly successful local approximations are those that have been derived from the homogeneous electron gas (HEG) model. In this regard, LDA is generally synonymous with functionals based on the HEG approximation, which are then applied to realistic systems.
In solid-state physics, the nearly free electron 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 model is closely related to the more conceptual empty lattice approximation. The model enables understanding and calculating the electronic band structure of especially metals.
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 material with a surface leads to a change of the electronic band structure from the bulk material to the vacuum. In the weakened potential at the surface, new electronic states can be formed, so called surface states.
In the field of physics concerning condensed matter, a Kohn anomaly is an anomaly in the dispersion relation of a phonon branch in a metal. It is named for Walter Kohn. For a specific wavevector, the frequency of the associated phonon is considerably lowered, and there is a discontinuity in its derivative. They have been first proposed by Walter Kohn in 1959. In extreme cases, the energy of this phonon is zero, meaning that a static distortion of the lattice appears. This is one explanation for charge density waves in solids. The wavevectors at which a Kohn anomaly is possible are the nesting vectors of the Fermi surface, that is vectors that connect a lot of points of the Fermi surface. The electron phonon interaction causes a rigid shift of the Fermi sphere and a failure of the Born-Oppenheimer approximation since the electrons do not follow any more the ionic motion adiabatically.
In solid-state physics, the k·p perturbation theory is an approximated semi-empirical approach for calculating the band structure and optical properties of crystalline solids. It is pronounced "k dot p", and is also called the "k·p method". This theory has been applied specifically in the framework of the Luttinger–Kohn model, and of the Kane model.
The Thomas–Fermi (TF) model, named after Llewellyn Thomas and Enrico Fermi, is a quantum mechanical theory for the electronic structure of many-body systems developed semiclassically shortly after the introduction of the Schrödinger equation. It stands separate from wave function theory as being formulated in terms of the electronic density alone and as such is viewed as a precursor to modern density functional theory. The Thomas–Fermi model is correct only in the limit of an infinite nuclear charge. Using the approximation for realistic systems yields poor quantitative predictions, even failing to reproduce some general features of the density such as shell structure in atoms and Friedel oscillations in solids. It has, however, found modern applications in many fields through the ability to extract qualitative trends analytically and with the ease at which the model can be solved. The kinetic energy expression of Thomas–Fermi theory is also used as a component in more sophisticated density approximation to the kinetic energy within modern orbital-free density functional theory.
The empty lattice approximation is a theoretical electronic band structure model in which the potential is periodic and weak. One may also consider an empty irregular lattice, in which the potential is not even periodic. The empty lattice approximation describes a number of properties of energy dispersion relations of non-interacting free electrons that move through a crystal lattice. The energy of the electrons in the "empty lattice" is the same as the energy of free electrons. The model is useful because it clearly illustrates a number of the sometimes very complex features of energy dispersion relations in solids which are fundamental to all electronic band structures.
In quantum mechanics, orbital magnetization, Morb, refers to the magnetization induced by orbital motion of charged particles, usually electrons in solids. The term "orbital" distinguishes it from the contribution of spin degrees of freedom, Mspin, to the total magnetization. A nonzero orbital magnetization requires broken time-reversal symmetry, which can occur spontaneously in ferromagnetic and ferrimagnetic materials, or can be induced in a non-magnetic material by an applied magnetic field.
Heat transfer physics describes the kinetics of energy storage, transport, and energy transformation by principal energy carriers: phonons, electrons, fluid particles, and photons. Heat is energy stored in temperature-dependent motion of particles including electrons, atomic nuclei, individual atoms, and molecules. Heat is transferred to and from matter by the principal energy carriers. The state of energy stored within matter, or transported by the carriers, is described by a combination of classical and quantum statistical mechanics. The energy is different made (converted) among various carriers. The heat transfer processes are governed by the rates at which various related physical phenomena occur, such as the rate of particle collisions in classical mechanics. These various states and kinetics determine the heat transfer, i.e., the net rate of energy storage or transport. Governing these process from the atomic level to macroscale are the laws of thermodynamics, including conservation of energy.
The linearized augmented-plane-wave method (LAPW) is an implementation of Kohn-Sham density functional theory (DFT) adapted to periodic materials. It typically goes along with the treatment of both valence and core electrons on the same footing in the context of DFT and the treatment of the full potential and charge density without any shape approximation. This is often referred to as the all-electron full-potential linearized augmented-plane-wave method (FLAPW). It does not rely on the pseudopotential approximation and employs a systematically extendable basis set. These features make it one of the most precise implementations of DFT, applicable to all crystalline materials, regardless of their chemical composition. It can be used as a reference for evaluating other approaches.
The FLEUR code is an open-source scientific software package for the simulation of material properties of crystalline solids, thin films, and surfaces. It implements Kohn-Sham density functional theory (DFT) in terms of the all-electron full-potential linearized augmented-plane-wave method. With this, it is a realization of one of the most precise DFT methodologies. The code has the common features of a modern DFT simulation package. In the past, major applications have been in the field of magnetism, spintronics, quantum materials, e.g. in ultrathin films, complex magnetism like in spin spirals or magnetic Skyrmion lattices, and in spin-orbit related physics, e.g. in graphene and topological insulators.