Editing Electronic band structure (section)

== 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 {{math|('''b'''<sub>1</sub>, '''b'''<sub>2</sub>, '''b'''<sub>3</sub>)}}. Now, any periodic potential {{math|''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:
<math display="block">V(\mathbf{r}) = \sum_{\mathbf{K}} {V_{\mathbf{K}} e^{i \mathbf{K}\cdot\mathbf{r}}}</math>
where {{math|1='''K''' = ''m''<sub>1</sub>'''b'''<sub>1</sub> + ''m''<sub>2</sub>'''b'''<sub>2</sub> + ''m''<sub>3</sub>'''b'''<sub>3</sub>}} for any set of integers {{math|(''m''<sub>1</sub>, ''m''<sub>2</sub>, ''m''<sub>3</sub>)}}.

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 ===
{{Main|Nearly free electron model|Free electron model|pseudopotential}}
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 [[wavefunction]]s 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
<math display="block">\Psi_{n,\mathbf{k}} (\mathbf{r}) = e^{i \mathbf{k}\cdot\mathbf{r}} u_n(\mathbf{r}) </math>
where the Bloch function <math>u_n(\mathbf{r})</math> is periodic over the crystal lattice, that is,
<math display="block">u_n(\mathbf{r}) = u_n(\mathbf{r}-\mathbf{R}) .</math>

Here index {{mvar|n}} refers to the {{mvar|n}}th energy band, wavevector {{math|'''k'''}} is related to the direction of motion of the electron, {{math|'''r'''}} is the position in the crystal, and {{math|'''R'''}} is the location of an atomic site.<ref name=Kittel>
{{cite book
 |author=Charles Kittel
 |title=[[Introduction to Solid State Physics (Kittel book)|Introduction to Solid State Physics]]
 |year= 1996
 |edition=Seventh
 |publisher=Wiley
 |location=New York
 |isbn=978-0-471-11181-8
}}</ref>{{rp|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]]s and potentials on neighbouring [[atom]]s 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 ===
{{Main|Tight binding}}
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]] <math>\Psi</math> is well approximated by a [[Linear combination of atomic orbitals molecular orbital method|linear combination]] of [[atomic orbitals]] <math>\psi_n(\mathbf{r})</math>.<ref name=Kittel/>{{rp|pp=245-248}}
<math display="block">\Psi(\mathbf{r}) = \sum_{n,\mathbf{R}} b_{n,\mathbf{R}} \psi_n(\mathbf{r}-\mathbf{R}),</math>
where the coefficients <math> b_{n,\mathbf{R}}</math> are selected to give the best approximate solution of this form. Index {{mvar|n}} refers to an atomic energy level and {{math|'''R'''}} refers to an atomic site. A more accurate approach using this idea employs [[Wannier functions]], defined by:<ref name=Kittel/>{{rp|at=Eq. 42 p. 267}}<ref name=Mattis>{{cite book | author=Daniel Charles Mattis | title=The Many-Body Problem: Encyclopaedia of Exactly Solved Models in One Dimension | year = 1994 | publisher=World Scientific | page=340 | isbn=978-981-02-1476-0 | url=https://books.google.com/books?id=BGdHpCAMiLgC&q=wannier+functions&pg=PA332}}</ref>
<math display="block">a_n(\mathbf{r}-\mathbf{R}) = \frac{V_{C}}{(2\pi)^{3}} \int_\text{BZ} d\mathbf{k} e^{-i\mathbf{k}\cdot(\mathbf{R} -\mathbf{r})}u_{n\mathbf{k}};</math>
in which <math>u_{n\mathbf{k}}</math> is the periodic part of the Bloch's theorem and the integral is over the [[Brillouin zone]]. Here index {{math|''n''}} refers to the {{math|''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 {{math|'''R'''}} are orthogonal. The Wannier functions can be used to form the Schrödinger solution for the {{math|''n''}}-th energy band as:
<math display="block">\Psi_{n,\mathbf{k}} (\mathbf{r}) = \sum_{\mathbf{R}} e^{-i\mathbf{k}\cdot(\mathbf{R}-\mathbf{r})}a_n(\mathbf{r} - \mathbf{R}).</math>

The TB model works well in materials with limited overlap between [[atomic orbital]]s and potentials on neighbouring atoms. Band structures of materials like [[Silicon|Si]], [[GaAs]], SiO<sub>2</sub> and [[diamond]] for instance are well described by TB-Hamiltonians on the basis of atomic sp<sup>3</sup> 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,<ref name=Harrison>{{cite book
|author=Walter Ashley Harrison
|title=Electronic Structure and the Properties of Solids
|year= 1989
|publisher=Dover Publications
|url=https://books.google.com/books?id=R2VqQgAACAAJ
|isbn=978-0-486-66021-9
}}</ref>
sometimes extended with wave function approximations based on pseudopotential methods, are often used as an economic starting point for further calculations.

=== KKR model ===
{{Main|Multiple scattering theory}}
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 [[Jan Korringa|Korringa]], [[Walter Kohn|Kohn]] and Rostocker, and is often referred to as the ''[[Korringa–Kohn–Rostoker method]]''.<ref name=Galsin>{{cite book |title=Impurity Scattering in Metal Alloys |author=Joginder Singh Galsin |page=Appendix C |url=https://books.google.com/books?id=kmcLT63iX_EC&q=KKR+method+band+structure&pg=PA498 |isbn=978-0-306-46574-1 |year=2001 |publisher=Springer |no-pp=true}}</ref><ref name=Ohtaka>{{cite book |title=Photonic Crystals |author=Kuon Inoue, Kazuo Ohtaka |page=66 |url=https://books.google.com/books?id=GIa3HRgPYhAC&q=KKR+method+band+structure&pg=PA66 |isbn=978-3-540-20559-3 |year=2004 |publisher=Springer}}</ref> 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 [[Screening effect|screened potential]] is approximated as a constant. Continuity of the potential between the atom-centered spheres and interstitial region is enforced.

=== Density-functional theory ===
{{Main|Density functional theory}}
{{See also|Kohn–Sham equations}}
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 interaction|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.<ref>{{Cite journal|last1=Assadi|first1=M. Hussein. N.|last2=Hanaor|first2=Dorian A. H.| date=2013-06-21|title=Theoretical study on copper's energetics and magnetism in TiO<sub>2</sub> polymorphs|journal=Journal of Applied Physics|volume=113|issue=23| pages=233913–233913–5 | arxiv=1304.1854 |doi=10.1063/1.4811539|bibcode=2013JAP...113w3913A|s2cid=94599250|issn=0021-8979}}</ref>

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.<ref>{{cite journal | last=Hohenberg|first=P | author2=Kohn, W.|title=Inhomogeneous Electron Gas|journal=Phys. Rev.|date=Nov 1964| volume=136 | issue=3B | pages=B864–B871 | doi=10.1103/PhysRev.136.B864 | bibcode = 1964PhRv..136..864H |doi-access=free}}</ref> 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 equations|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 density functional theory|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]]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.<ref name=Paier>{{Cite journal | last1 = Paier | first1 = J. | last2 = Marsman | first2 = M. | last3 = Hummer | first3 = K. | last4 = Kresse | first4 = G. | last5 = Gerber | first5 = I. C. | last6 = Angyán | first6 = J. G. | title = Screened hybrid density functionals applied to solids | journal = J Chem Phys | volume = 124 | issue = 15 | pages = 154709 |date=2006 | doi = 10.1063/1.2187006 | pmid = 16674253 |bibcode = 2006JChPh.124o4709P }}</ref>

=== Green's function methods and the ''ab initio'' GW approximation ===
{{Main|Green's function (many-body theory)|Green–Kubo relations}}
To calculate the bands including electron-electron interaction [[Many-body problem|many-body effects]], one can resort to so-called [[Green's function (many-body theory)|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.

=== Dynamical mean-field theory ===
{{Main|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 [[Cobalt(II) oxide|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.

=== 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 "band structure" of a region of free space that has been divided into a lattice.
* [[k·p perturbation theory]] 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 [[Particle in a one-dimensional lattice (periodic potential)#Kronig–Penney model|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]]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]] salts (e.g. [[Sodium chloride|NaCl]]).