High-temperature superconductivity

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A small sample of the high-temperature superconductor BSCCO-2223.

 

High-temperature superconductors (abbreviated high-T_c or HTS) are materials that behave as superconductors at unusually high temperatures. The first high-T_c superconductor was discovered in 1986 by IBM researchers Georg Bednorz and K. Alex Müller, who were awarded the 1987 Nobel Prize in Physics "for their important break-through in the discovery of superconductivity in ceramic materials".

Whereas "ordinary" or metallic superconductors usually have transition temperatures (temperatures below which they are superconductive) below 30 K (−243.2 °C) and must be cooled using liquid helium in order to achieve superconductivity, HTS have been observed with transition temperatures as high as 138 K (−135 °C), and can be cooled to superconductivity using liquid nitrogen. Until 2008, only certain compounds of copper and oxygen (so-called "cuprates") were known to have HTS properties, and the term high-temperature superconductor was used interchangeably with cuprate superconductor for compounds such as bismuth strontium calcium copper oxide (BSCCO) and yttrium barium copper oxide (YBCO). Several iron-based compounds (the iron pnictides) are now known to be superconducting at high temperatures.

In 2015, hydrogen sulfide (H₂S) under extremely high pressure (around 150 gigapascals) was found to undergo superconducting transition near 203 K (-70 °C), due the formation of H₃S, a new record high temperature superconductor.

History

The phenomenon of superconductivity was discovered by Kamerlingh Onnes in 1911, in metallic mercury below 4 K (−269.15 °C). Ever since, researchers have attempted to observe superconductivity at increasing temperatures with the goal of finding a room-temperature superconductor. By the late 1970s, superconductivity was observed in several metallic compounds (in particular Nb-based, such as NbTi, Nb₃Sn, and Nb₃Ge) at temperatures that were much higher than those for elemental metals and which could even exceed 20 K (−253.2 °C). In 1986, J. Georg Bednorz and K. Alex Müller, working at the IBM research lab near Zurich, Switzerland were exploring a new class of ceramics for superconductivity. Bednorz encountered a barium-doped compound of lanthanum and copper oxide whose resistance dropped to zero at a temperature around 35 K (−238.2 °C). Their results were soon confirmed by many groups, notably Paul Chu at the University of Houston and Shoji Tanaka at the University of Tokyo.

Shortly after, P. W. Anderson, at Princeton University came up with the first theoretical description of these materials, using the resonating valence bond theory, but a full understanding of these materials is still developing today. These superconductors are now known to possess a d-wave pair symmetry. The first proposal that high-temperature cuprate superconductivity involves d-wave pairing was made in 1987 by Bickers, Scalapino and Scalettar, followed by three subsequent theories in 1988 by Inui, Doniach, Hirschfeld and Ruckenstein, using spin-fluctuation theory, and by Gros, Poilblanc, Rice and Zhang, and by Kotliar and Liu identifying d-wave pairing as a natural consequence of the RVB theory. The confirmation of the d-wave nature of the cuprate superconductors was made by a variety of experiments, including the direct observation of the d-wave nodes in the excitation spectrum through Angle Resolved Photoemission Spectroscopy, the observation of a half-integer flux in tunneling experiments, and indirectly from the temperature dependence of the penetration depth, specific heat and thermal conductivity.

Until 2015 the superconductor with the highest transition temperature that had been confirmed by multiple independent research groups (a prerequisite to being called a discovery, verified by peer review) was mercury barium calcium copper oxide (HgBa₂Ca₂Cu₃O₈) at around 133 K.

After more than twenty years of intensive research, the origin of high-temperature superconductivity is still not clear, but it seems that instead of electron-phonon attraction mechanisms, as in conventional superconductivity, one is dealing with genuine electronic mechanisms (e.g. by antiferromagnetic correlations), and instead of conventional, purely s-wave pairing, more exotic pairing symmetries are thought to be involved (d-wave in the case of the cuprates; primarily extended s-wave, but occasionally d-wave, in the case of the iron-based superconductors). In 2014, evidence showing that fractional particles can happen in quasi two-dimensional magnetic materials, was found by EPFL scientists lending support for Anderson's theory of high-temperature superconductivity.

Crystal structures of high-temperature ceramic superconductors

The structure of high-T_c copper oxide or cuprate superconductors are often closely related to perovskite structure, and the structure of these compounds has been described as a distorted, oxygen deficient multi-layered perovskite structure. One of the properties of the crystal structure of oxide superconductors is an alternating multi-layer of CuO₂ planes with superconductivity taking place between these layers. The more layers of CuO₂, the higher T_c. This structure causes a large anisotropy in normal conducting and superconducting properties, since electrical currents are carried by holes induced in the oxygen sites of the CuO₂ sheets. The electrical conduction is highly anisotropic, with a much higher conductivity parallel to the CuO₂ plane than in the perpendicular direction. Generally, critical temperatures depend on the chemical compositions, cations substitutions and oxygen content. They can be classified as superstripes; i.e., particular realizations of superlattices at atomic limit made of superconducting atomic layers, wires, dots separated by spacer layers, that gives multiband and multigap superconductivity.

YBaCuO superconductors

YBCO unit cell

 

The first superconductor found with T_c > 77 K (liquid nitrogen boiling point) is yttrium barium copper oxide (YBa₂Cu₃O_7−x); the proportions of the three different metals in the YBa₂Cu₃O₇ superconductor are in the mole ratio of 1 to 2 to 3 for yttrium to barium to copper, respectively. Thus, this particular superconductor is often referred to as the 123 superconductor. 

The unit cell of YBa₂Cu₃O₇ consists of three pseudocubic elementary perovskite unit cells. Each perovskite unit cell contains a Y or Ba atom at the center: Ba in the bottom unit cell, Y in the middle one, and Ba in the top unit cell. Thus, Y and Ba are stacked in the sequence [Ba–Y–Ba] along the c-axis. All corner sites of the unit cell are occupied by Cu, which has two different coordinations, Cu(1) and Cu(2), with respect to oxygen. There are four possible crystallographic sites for oxygen: O(1), O(2), O(3) and O(4). The coordination polyhedra of Y and Ba with respect to oxygen are different. The tripling of the perovskite unit cell leads to nine oxygen atoms, whereas YBa₂Cu₃O₇ has seven oxygen atoms and, therefore, is referred to as an oxygen-deficient perovskite structure. The structure has a stacking of different layers: (CuO)(BaO)(CuO₂)(Y)(CuO₂)(BaO)(CuO). One of the key feature of the unit cell of YBa₂Cu₃O_7−x (YBCO) is the presence of two layers of CuO₂. The role of the Y plane is to serve as a spacer between two CuO₂ planes. In YBCO, the Cu–O chains are known to play an important role for superconductivity. T_c is maximal near 92 K when x ≈ 0.15 and the structure is orthorhombic. Superconductivity disappears at x ≈ 0.6, where the structural transformation of YBCO occurs from orthorhombic to tetragonal.

Bi-, Tl- and Hg-based high-T_c superconductors

The crystal structure of Bi-, Tl- and Hg-based high-T_c superconductors are very similar. Like YBCO, the perovskite-type feature and the presence of CuO₂ layers also exist in these superconductors. However, unlike YBCO, Cu–O chains are not present in these superconductors. The YBCO superconductor has an orthorhombic structure, whereas the other high-T_c superconductors have a tetragonal structure. 

The Bi–Sr–Ca–Cu–O system has three superconducting phases forming a homologous series as Bi₂Sr₂Ca_n−1Cu_nO_4+2n+x (n=1, 2 and 3). These three phases are Bi-2201, Bi-2212 and Bi-2223, having transition temperatures of 20, 85 and 110 K, respectively, where the numbering system represent number of atoms for Bi, Sr, Ca and Cu respectively. The two phases have a tetragonal structure which consists of two sheared crystallographic unit cells. The unit cell of these phases has double Bi–O planes which are stacked in a way that the Bi atom of one plane sits below the oxygen atom of the next consecutive plane. The Ca atom forms a layer within the interior of the CuO₂ layers in both Bi-2212 and Bi-2223; there is no Ca layer in the Bi-2201 phase. The three phases differ with each other in the number of CuO₂ planes; Bi-2201, Bi-2212 and Bi-2223 phases have one, two and three CuO₂ planes, respectively. The c axis lattice constants of these phases increases with the number of CuO₂ planes (see table below). The coordination of the Cu atom is different in the three phases. The Cu atom forms an octahedral coordination with respect to oxygen atoms in the 2201 phase, whereas in 2212, the Cu atom is surrounded by five oxygen atoms in a pyramidal arrangement. In the 2223 structure, Cu has two coordinations with respect to oxygen: one Cu atom is bonded with four oxygen atoms in square planar configuration and another Cu atom is coordinated with five oxygen atoms in a pyramidal arrangement.

Tl–Ba–Ca–Cu–O superconductor: The first series of the Tl-based superconductor containing one Tl–O layer has the general formula TlBa₂Ca_n-1Cu_nO_2n+3, whereas the second series containing two Tl–O layers has a formula of Tl₂Ba₂Ca_n-1Cu_nO_2n+4 with n =1, 2 and 3. In the structure of Tl₂Ba₂CuO₆ (Tl-2201), there is one CuO₂ layer with the stacking sequence (Tl–O) (Tl–O) (Ba–O) (Cu–O) (Ba–O) (Tl–O) (Tl–O). In Tl₂Ba₂CaCu₂O₈ (Tl-2212), there are two Cu–O layers with a Ca layer in between. Similar to the Tl₂Ba₂CuO₆ structure, Tl–O layers are present outside the Ba–O layers. In Tl₂Ba₂Ca₂Cu₃O₁₀ (Tl-2223), there are three CuO2 layers enclosing Ca layers between each of these. In Tl-based superconductors, T_c is found to increase with the increase in CuO₂ layers. However, the value of T_c decreases after four CuO₂ layers in TlBa₂Ca_n-1Cu_nO_2n+3, and in the Tl₂Ba₂Ca_n-1Cu_nO_2n+4 compound, it decreases after three CuO₂ layers.

Hg–Ba–Ca–Cu–O superconductor: The crystal structure of HgBa₂CuO₄ (Hg-1201), HgBa₂CaCu₂O₆ (Hg-1212) and HgBa₂Ca₂Cu₃O₈ (Hg-1223) is similar to that of Tl-1201, Tl-1212 and Tl-1223, with Hg in place of Tl. It is noteworthy that the T_c of the Hg compound (Hg-1201) containing one CuO₂ layer is much larger as compared to the one-CuO₂-layer compound of thallium (Tl-1201). In the Hg-based superconductor, T_c is also found to increase as the CuO₂ layer increases. For Hg-1201, Hg-1212 and Hg-1223, the values of T_c are 94, 128 and the record value at ambient pressure 134 K, respectively, as shown in table below. The observation that the T_c of Hg-1223 increases to 153 K under high pressure indicates that the T_c of this compound is very sensitive to the structure of the compound.

Critical temperature (T_c), crystal structure and lattice constants of some high-T_c superconductors

Formula	Notation	Tc (K)	No. of Cu-O planes in unit cell	Crystal structure
YBa2Cu3O7	123	92	2	Orthorhombic
Bi2Sr2CuO6	Bi-2201	20	1	Tetragonal
Bi2Sr2CaCu2O8	Bi-2212	85	2	Tetragonal
Bi2Sr2Ca2Cu3O10	Bi-2223	110	3	Tetragonal
Tl2Ba2CuO6	Tl-2201	80	1	Tetragonal
Tl2Ba2CaCu2O8	Tl-2212	108	2	Tetragonal
Tl2Ba2Ca2Cu3O10	Tl-2223	125	3	Tetragonal
TlBa2Ca3Cu4O11	Tl-1234	122	4	Tetragonal
HgBa2CuO4	Hg-1201	94	1	Tetragonal
HgBa2CaCu2O6	Hg-1212	128	2	Tetragonal
HgBa2Ca2Cu3O8	Hg-1223	134	3	Tetragonal

Preparation of high-T_c superconductors

The simplest method for preparing high-T_c superconductors is a solid-state thermochemical reaction involving mixing, calcination and sintering. The appropriate amounts of precursor powders, usually oxides and carbonates, are mixed thoroughly using a Ball mill. Solution chemistry processes such as coprecipitation, freeze-drying and sol-gel methods are alternative ways for preparing a homogeneous mixture. These powders are calcined in the temperature range from 800 °C to 950 °C for several hours. The powders are cooled, reground and calcined again. This process is repeated several times to get homogeneous material. The powders are subsequently compacted to pellets and sintered. The sintering environment such as temperature, annealing time, atmosphere and cooling rate play a very important role in getting good high-T_c superconducting materials. The YBa₂Cu₃O_7−x compound is prepared by calcination and sintering of a homogeneous mixture of Y₂O₃, BaCO₃ and CuO in the appropriate atomic ratio. Calcination is done at 900–950 °C, whereas sintering is done at 950 °C in an oxygen atmosphere. The oxygen stoichiometry in this material is very crucial for obtaining a superconducting YBa₂Cu₃O_7−x compound. At the time of sintering, the semiconducting tetragonal YBa₂Cu₃O₆ compound is formed, which, on slow cooling in oxygen atmosphere, turns into superconducting YBa₂Cu₃O_7−x. The uptake and loss of oxygen are reversible in YBa₂Cu₃O_7−x. A fully oxygenated orthorhombic YBa₂Cu₃O_7−x sample can be transformed into tetragonal YBa₂Cu₃O₆ by heating in a vacuum at temperature above 700 °C.

The preparation of Bi-, Tl- and Hg-based high-T_c superconductors is difficult compared to YBCO. Problems in these superconductors arise because of the existence of three or more phases having a similar layered structure. Thus, syntactic intergrowth and defects such as stacking faults occur during synthesis and it becomes difficult to isolate a single superconducting phase. For Bi–Sr–Ca–Cu–O, it is relatively simple to prepare the Bi-2212 (T_c ≈ 85 K) phase, whereas it is very difficult to prepare a single phase of Bi-2223 (T_c ≈ 110 K). The Bi-2212 phase appears only after few hours of sintering at 860–870 °C, but the larger fraction of the Bi-2223 phase is formed after a long reaction time of more than a week at 870 °C. Although the substitution of Pb in the Bi–Sr–Ca–Cu–O compound has been found to promote the growth of the high-T_c phase, a long sintering time is still required.

Properties

"High-temperature" has two common definitions in the context of superconductivity:

1. Above the temperature of 30 K that had historically been taken as the upper limit allowed by BCS theory (1957). This is also above the 1973 record of 23 K that had lasted until copper-oxide materials were discovered in 1986.
2. Having a transition temperature that is a larger fraction of the Fermi temperature than for conventional superconductors such as elemental mercury or lead. This definition encompasses a wider variety of unconventional superconductors and is used in the context of theoretical models.

The label high-T_c may be reserved by some authors for materials with critical temperature greater than the boiling point of liquid nitrogen (77 K or −196 °C). However, a number of materials – including the original discovery and recently discovered pnictide superconductors – had critical temperatures below 77 K but are commonly referred to in publication as being in the high-T_c class.

Technological applications could benefit from both the higher critical temperature being above the boiling point of liquid nitrogen and also the higher critical magnetic field (and critical current density) at which superconductivity is destroyed. In magnet applications, the high critical magnetic field may prove more valuable than the high T_c itself. Some cuprates have an upper critical field of about 100 tesla. However, cuprate materials are brittle ceramics which are expensive to manufacture and not easily turned into wires or other useful shapes. Also, high-temperature superconductors do not form large, continuous superconducting domains, but only clusters of microdomains within which superconductivity occurs. They are therefore unsuitable for applications requiring actual superconducted currents, such as magnets for magnetic resonance spectrometers.

After two decades of intense experimental and theoretical research, with over 100,000 published papers on the subject, several common features in the properties of high-temperature superconductors have been identified. As of 2011, no widely accepted theory explains their properties. Relative to conventional superconductors, such as elemental mercury or lead that are adequately explained by the BCS theory, cuprate superconductors (and other unconventional superconductors) remain distinctive. There also has been much debate as to high-temperature superconductivity coexisting with magnetic ordering in YBCO, iron-based superconductors, several ruthenocuprates and other exotic superconductors, and the search continues for other families of materials. HTS are Type-II superconductors, which allow magnetic fields to penetrate their interior in quantized units of flux, meaning that much higher magnetic fields are required to suppress superconductivity. The layered structure also gives a directional dependence to the magnetic field response.

Cuprates

Simplified doping dependent phase diagram of cuprate superconductors for both electron (n) and hole (p) doping. The phases shown are the antiferromagnetic (AF) phase close to zero doping, the superconducting phase around optimal doping, and the pseudogap phase. Doping ranges possible for some common compounds are also shown. After.

 

Cuprate superconductors are generally considered to be quasi-two-dimensional materials with their superconducting properties determined by electrons moving within weakly coupled copper-oxide (CuO₂) layers. Neighbouring layers containing ions such as lanthanum, barium, strontium, or other atoms act to stabilize the structure and dope electrons or holes onto the copper-oxide layers. The undoped "parent" or "mother" compounds are Mott insulators with long-range antiferromagnetic order at low enough temperature. Single band models are generally considered to be sufficient to describe the electronic properties. 

The cuprate superconductors adopt a perovskite structure. The copper-oxide planes are checkerboard lattices with squares of O²⁻ ions with a Cu²⁺ ion at the centre of each square. The unit cell is rotated by 45° from these squares. Chemical formulae of superconducting materials generally contain fractional numbers to describe the doping required for superconductivity. There are several families of cuprate superconductors and they can be categorized by the elements they contain and the number of adjacent copper-oxide layers in each superconducting block. For example, YBCO and BSCCO can alternatively be referred to as Y123 and Bi2201/Bi2212/Bi2223 depending on the number of layers in each superconducting block (n). The superconducting transition temperature has been found to peak at an optimal doping value (p=0.16) and an optimal number of layers in each superconducting block, typically n=3. 

Possible mechanisms for superconductivity in the cuprates are still the subject of considerable debate and further research. Certain aspects common to all materials have been identified. Similarities between the antiferromagnetic low-temperature state of the undoped materials and the superconducting state that emerges upon doping, primarily the d_x²-y² orbital state of the Cu²⁺ ions, suggest that electron-electron interactions are more significant than electron-phonon interactions in cuprates – making the superconductivity unconventional. Recent work on the Fermi surface has shown that nesting occurs at four points in the antiferromagnetic Brillouin zone where spin waves exist and that the superconducting energy gap is larger at these points. The weak isotope effects observed for most cuprates contrast with conventional superconductors that are well described by BCS theory.

Similarities and differences in the properties of hole-doped and electron doped cuprates:

• Presence of a pseudogap phase up to at least optimal doping.
• Different trends in the Uemura plot relating transition temperature to the superfluid density. The inverse square of the London penetration depth appears to be proportional to the critical temperature for a large number of underdoped cuprate superconductors, but the constant of proportionality is different for hole- and electron-doped cuprates. The linear trend implies that the physics of these materials is strongly two-dimensional.
• Universal hourglass-shaped feature in the spin excitations of cuprates measured using inelastic neutron diffraction.
• Nernst effect evident in both the superconducting and pseudogap phases.

Fig. 1. The Fermi surface of bi-layer BSCCO, calculated (left) and measured by ARPES (right). The dashed rectangle represents the first Brillouin zone.

 

The electronic structure of superconducting cuprates is highly anisotropic (see the crystal structure of YBCO or BSCCO). Therefore, the Fermi surface of HTSC is very close to the Fermi surface of the doped CuO₂ plane (or multi-planes, in case of multi-layer cuprates) and can be presented on the 2D reciprocal space (or momentum space) of the CuO₂ lattice. The typical Fermi surface within the first CuO₂ Brillouin zone is sketched in Fig. 1 (left). It can be derived from the band structure calculations or measured by angle resolved photoemission spectroscopy (ARPES). Fig. 1 (right) shows the Fermi surface of BSCCO measured by ARPES. In a wide range of charge carrier concentration (doping level), in which the hole-doped HTSC are superconducting, the Fermi surface is hole-like (i.e. open, as shown in Fig. 1). This results in an inherent in-plane anisotropy of the electronic properties of HTSC.

Iron-based superconductors

Simplified doping dependent phase diagrams of iron-based superconductors for both Ln-1111 and Ba-122 materials. The phases shown are the antiferromagnetic/spin density wave (AF/SDW) phase close to zero doping and the superconducting phase around optimal doping. The Ln-1111 phase diagrams for La and Sm were determined using muon spin spectroscopy, the phase diagram for Ce was determined using neutron diffraction. The Ba-122 phase diagram is based on.

 

Iron-based superconductors contain layers of iron and a pnictogen—such as arsenic or phosphorus—or a chalcogen. This is currently the family with the second highest critical temperature, behind the cuprates. Interest in their superconducting properties began in 2006 with the discovery of superconductivity in LaFePO at 4 K and gained much greater attention in 2008 after the analogous material LaFeAs(O,F) was found to superconduct at up to 43 K under pressure. The highest critical temperatures in the iron-based superconductor family exist in thin films of FeSe, where a critical temperature in excess of 100 K was reported in 2014.

Since the original discoveries several families of iron-based superconductors have emerged:

• LnFeAs(O,F) or LnFeAsO_1−x (Ln=lanthanide) with T_c up to 56 K, referred to as 1111 materials. A fluoride variant of these materials was subsequently found with similar T_c values.
• (Ba,K)Fe₂As₂ and related materials with pairs of iron-arsenide layers, referred to as 122 compounds. T_c values range up to 38 K. These materials also superconduct when iron is replaced with cobalt.
• LiFeAs and NaFeAs with T_c up to around 20 K. These materials superconduct close to stoichiometric composition and are referred to as 111 compounds.
• FeSe with small off-stoichiometry or tellurium doping.

Most undoped iron-based superconductors show a tetragonal-orthorhombic structural phase transition followed at lower temperature by magnetic ordering, similar to the cuprate superconductors. However, they are poor metals rather than Mott insulators and have five bands at the Fermi surface rather than one. The phase diagram emerging as the iron-arsenide layers are doped is remarkably similar, with the superconducting phase close to or overlapping the magnetic phase. Strong evidence that the T_c value varies with the As-Fe-As bond angles has already emerged and shows that the optimal T_c value is obtained with undistorted FeAs₄ tetrahedra. The symmetry of the pairing wavefunction is still widely debated, but an extended s-wave scenario is currently favored.

Hydrogen sulfide

At pressures above 90 GPa (Gigapascals), hydrogen sulfide becomes a metallic conductor of electricity. When cooled below a critical temperature this high-pressure phase exhibits superconductivity. The critical temperature increases with pressure, ranging from 23 K at 100 GPa to 150 K at 200 GPa. If hydrogen sulfide is pressurized at higher temperatures, then cooled, the critical temperature reaches 203 K (−70 °C), the highest accepted superconducting critical temperature as of 2015. It has been predicted that by substituting a small part of sulfur with phosphorus and using even higher pressures it may be possible to raise the critical temperature to above 273 K (0 °C) and achieve room-temperature superconductivity.

Other materials sometimes referred to as high-temperature superconductors

Magnesium diboride is occasionally referred to as a high-temperature superconductor because its T_c value of 39 K is above that historically expected for BCS superconductors. However, it is more generally regarded as the highest-T_c conventional superconductor, the increased T_c resulting from two separate bands being present at the Fermi level. 

Fulleride superconductors where alkali-metal atoms are intercalated into C₆₀ molecules show superconductivity at temperatures of up to 38 K for Cs₃C₆₀.

Some organic superconductors and heavy fermion compounds are considered to be high-temperature superconductors because of their high T_c values relative to their Fermi energy, despite the T_c values being lower than for many conventional superconductors. This description may relate better to common aspects of the superconducting mechanism than the superconducting properties.

Metallic hydrogen

Theoretical work by Neil Ashcroft in 1968 predicted that solid metallic hydrogen at extremely high pressure should become superconducting at approximately room-temperature because of its extremely high speed of sound and expected strong coupling between the conduction electrons and the lattice vibrations. As of 2016 this prediction is yet to be experimentally verified.

Magnetic properties

All known high-T_c superconductors are Type-II superconductors. In contrast to Type-I superconductors, which expel all magnetic fields due to the Meissner effect, Type-II superconductors allow magnetic fields to penetrate their interior in quantized units of flux, creating "holes" or "tubes" of normal metallic regions in the superconducting bulk called vortices. Consequently, high-T_c superconductors can sustain much higher magnetic fields.

Ongoing research

Superconductor timeline. BCS superconductors are displayed as green circles, cuprates as blue diamonds, and iron-based superconductors as yellow squares. (YBaCuO should be at 93K according to table below.)

 

The question of how superconductivity arises in high-temperature superconductors is one of the major unsolved problems of theoretical condensed matter physics. The mechanism that causes the electrons in these crystals to form pairs is not known. Despite intensive research and many promising leads, an explanation has so far eluded scientists. One reason for this is that the materials in question are generally very complex, multi-layered crystals (for example, BSCCO), making theoretical modelling difficult. 

Improving the quality and variety of samples also gives rise to considerable research, both with the aim of improved characterisation of the physical properties of existing compounds, and synthesizing new materials, often with the hope of increasing T_c. Technological research focuses on making HTS materials in sufficient quantities to make their use economically viable and optimizing their properties in relation to applications.

Possible mechanism

There have been two representative theories for high-temperature or unconventional superconductivity. Firstly, weak coupling theory suggests superconductivity emerges from antiferromagnetic spin fluctuations in a doped system. According to this theory, the pairing wave function of the cuprate HTS should have a d_x²-y² symmetry. Thus, determining whether the pairing wave function has d-wave symmetry is essential to test the spin fluctuation mechanism. That is, if the HTS order parameter (pairing wave function) does not have d-wave symmetry, then a pairing mechanism related to spin fluctuations can be ruled out. (Similar arguments can be made for iron-based superconductors but the different material properties allow a different pairing symmetry.) Secondly, there was the interlayer coupling model, according to which a layered structure consisting of BCS-type (s-wave symmetry) superconductors can enhance the superconductivity by itself. By introducing an additional tunnelling interaction between each layer, this model successfully explained the anisotropic symmetry of the order parameter as well as the emergence of the HTS. Thus, in order to solve this unsettled problem, there have been numerous experiments such as photoemission spectroscopy, NMR, specific heat measurements, etc. Up to date the results were ambiguous, some reports supported the d symmetry for the HTS whereas others supported the s symmetry. This muddy situation possibly originated from the indirect nature of the experimental evidence, as well as experimental issues such as sample quality, impurity scattering, twinning, etc. 

This summary makes an implicit assumption: superconductive properties can be treated by mean field theory. It also fails to mention that in addition to the superconductive gap, there is a second gap, the pseudogap. The cuprate layers are insulating, and the superconductors are doped with interlayer impurities to make them metallic. The superconductive transition temperature can be maximized by varying the dopant concentration. The simplest example is La(2)CuO(4), which consist of alternating CuO(2) and LaO layers which are insulating when pure. When 8% of the La is replaced by Sr, the latter act as dopants, contributing holes to the CuO(2) layers, and making the sample metallic. The Sr impurities also act as electronic bridges, enabling interlayer coupling. With this realistic picture of the electronic and atomic structure of high temperature superconductors, one can show from thousands of experiments that the basic pairing interaction is still interaction with phonons, just as in the old metallic superconductors with Cooper pairs. While the undoped materials are antiferromagnetic, even a few % impurity dopants introduce a smaller pseudogap in the CuO2 planes which is also caused by phonons (technically charge density waves). This gap decreases with increasing charge carriers, and as it nears the superconductive gap, the latter reaches its maximum. This picture has been extended to answer the most obvious question of high temperature superconductivity, why are the transition temperatures so high? The carriers follow zig-zag percolative paths, largely in metallic domains in the CuO(2) planes, until blocked by charge density wave domain walls, where they use dopant bridges to cross over to a metallic domain of an adjacent CuO(2) plane. This model of self-organized networks of percolative paths describes all known maximum transition temperatures with no adjustable parameters. The transition temperature maxima are reached when the host lattice has weak bond-bending forces, which produce strong electron-phonon interactions at the interlayer dopants.^[69]

Junction experiment supporting the d symmetry

The Meissner effect or a magnet levitating above a superconductor (cooled by liquid nitrogen)

 

An experiment based on flux quantization of a three-grain ring of YBa₂Cu₃O₇ (YBCO) was proposed to test the symmetry of the order parameter in the HTS. The symmetry of the order parameter could best be probed at the junction interface as the Cooper pairs tunnel across a Josephson junction or weak link. It was expected that a half-integer flux, that is, a spontaneous magnetization could only occur for a junction of d symmetry superconductors. But, even if the junction experiment is the strongest method to determine the symmetry of the HTS order parameter, the results have been ambiguous. J. R. Kirtley and C. C. Tsuei thought that the ambiguous results came from the defects inside the HTS, so that they designed an experiment where both clean limit (no defects) and dirty limit (maximal defects) were considered simultaneously. In the experiment, the spontaneous magnetization was clearly observed in YBCO, which supported the d symmetry of the order parameter in YBCO. But, since YBCO is orthorhombic, it might inherently have an admixture of s symmetry. So, by tuning their technique further, they found that there was an admixture of s symmetry in YBCO within about 3%. Also, they found that there was a pure d_x²-y² order parameter symmetry in the tetragonal Tl₂Ba₂CuO₆.

Qualitative explanation of the spin-fluctuation mechanism

Despite all these years, the mechanism of high-T_c superconductivity is still highly controversial, mostly due to the lack of exact theoretical computations on such strongly interacting electron systems. However, most rigorous theoretical calculations, including phenomenological and diagrammatic approaches, converge on magnetic fluctuations as the pairing mechanism for these systems. The qualitative explanation is as follows: 

In a superconductor, the flow of electrons cannot be resolved into individual electrons, but instead consists of many pairs of bound electrons, called Cooper pairs. In conventional superconductors, these pairs are formed when an electron moving through the material distorts the surrounding crystal lattice, which in turn attracts another electron and forms a bound pair. This is sometimes called the "water bed" effect. Each Cooper pair requires a certain minimum energy to be displaced, and if the thermal fluctuations in the crystal lattice are smaller than this energy the pair can flow without dissipating energy. This ability of the electrons to flow without resistance leads to superconductivity.

In a high-T_c superconductor, the mechanism is extremely similar to a conventional superconductor, except, in this case, phonons virtually play no role and their role is replaced by spin-density waves. Just as all known conventional superconductors are strong phonon systems, all known high-T_c superconductors are strong spin-density wave systems, within close vicinity of a magnetic transition to, for example, an antiferromagnet. When an electron moves in a high-T_c superconductor, its spin creates a spin-density wave around it. This spin-density wave in turn causes a nearby electron to fall into the spin depression created by the first electron (water-bed effect again). Hence, again, a Cooper pair is formed. When the system temperature is lowered, more spin density waves and Cooper pairs are created, eventually leading to superconductivity. Note that in high-T_c systems, as these systems are magnetic systems due to the Coulomb interaction, there is a strong Coulomb repulsion between electrons. This Coulomb repulsion prevents pairing of the Cooper pairs on the same lattice site. The pairing of the electrons occur at near-neighbor lattice sites as a result. This is the so-called d-wave pairing, where the pairing state has a node (zero) at the origin.

This summary assumes superconductive properties can be treated by mean field theory. It also fails to mention that in addition to the superconductive gap, there is a second gap, the pseudogap. The cuprate layers are insulating, and the superconductors are doped with interlayer impurities to make them metallic. The superconductive transition temperature can be maximized by varying the dopant concentration. The simplest example is La(2)CuO(4), which consist of alternating CuO(2) and LaO layers which are insulating when pure. When 8% of the La is replaced by Sr, the latter act as dopants, contributing holes to the CuO(2) layers, and making the sample metallic. The Sr impurities also act as electronic bridges, enabling interlayer coupling. With this realistic picture of the electronic and atomic structure of high temperature superconductors, one can show from thousands of experiments that the basic pairing interaction is still interaction with phonons, just as in the old metallic superconductors with Cooper pairs. While the undoped materials are antiferromagnetic, even a few % impurity dopants introduce a smaller pseudogap in the CuO2 planes which is also caused by phonons (technically charge density waves). This gap decreases with increasing charge carriers, and as it nears the superconductive gap, the latter reaches its maximum. This picture has been extended to answer the most obvious question of high temperature superconductivity, why are the transition temperatures so high? The carriers follow zig-zag percolative paths, largely in metallic domains in the CuO(2) planes, until blocked by charge density wave domain walls, where they use dopant bridges to cross over to a metallic domain of an adjacent CuO(2) plane. This model describes all known maximum transition temperatures with no adjustable parameters. The maxima are reached when the host lattice has weak bond-bending forces.

Examples

Examples of high-T_c cuprate superconductors include La_1.85Ba_0.15CuO₄, and YBCO (yttrium-barium-copper oxide), which is famous as the first material discovered to achieve superconductivity above the boiling point of liquid nitrogen. 

Transition temperatures of well-known superconductors (Boiling point of liquid nitrogen for comparison)

Transition temperature (in kelvins)	Transition temperature (in degrees Celsius)	Material	Class
203	−70	H2S (at 150 GPa pressure)	Hydrogen-based superconductor
195	−78	Sublimation point of dry ice
184	−89.2	Lowest temperature recorded on Earth
145	−128	Boiling point of tetrafluoromethane
133	−140	HgBa2Ca2Cu3Ox(HBCCO)	Copper-oxide superconductors
110	−163	Bi2Sr2Ca2Cu3O10(BSCCO)
93	−180	YBa2Cu3O7 (YBCO)
90	−183	Boiling point of liquid oxygen
77	−196	Boiling point of liquid nitrogen
55	−218	SmFeAs(O,F)	Iron-based superconductors
41	−232	CeFeAs(O,F)
26	−247	LaFeAs(O,F)
20	−253	Boiling point of liquid hydrogen
18	−255	Nb3Sn	Metallic low-temperature superconductors
10	−263	NbTi
9.2	−263.8	Nb
4.2	−268.8	Boiling point of liquid helium
4.2	−268.8	Hg (mercury)	Metallic low-temperature superconductors

Electronic band structure

From Wikipedia, the free encyclopedia

In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energies that an electron within the solid may have (called energy bands, allowed bands, or simply bands) and ranges of energy that it 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.).

Why bands and band gaps occur

Showing how electronic band structure comes about by the hypothetical example of a large number of carbon atoms being brought together to form a diamond crystal. The graph (right) shows the energy levels as a function of the spacing between atoms. When the atoms are far apart (right side of graph) each atom has valence atomic orbitals p and s which have the same energy. However when the atoms come closer together their orbitals begin to overlap. Due to the Pauli Exclusion Principle each atomic orbital splits into N molecular orbitals each with a different energy, where N is the number of atoms in the crystal. Since N is such a large number, adjacent orbitals are extremely close together in energy so the orbitals can be considered a continuous energy band. a is the atomic spacing in an actual crystal of diamond. At that spacing the orbitals form two bands, called the valence and conduction bands, with a 5.5 eV band gap between them.

 

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 into a molecule, their atomic orbitals overlap. The Pauli exclusion principle dictates that no two electrons can have the same quantum numbers in a molecule. So if two identical atoms combine to form a diatomic molecule, each atomic orbital splits into two molecular orbitals of different energy, allowing the electrons in the former atomic orbitals to occupy the new orbital structure without any having the same energy. 

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. Since the Pauli exclusion principle dictates that no two electrons in the solid have the same quantum numbers, each atomic orbital splits into N discrete molecular orbitals, 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 eV). The energy of 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 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:

• 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 10²² atoms, this is not a serious restriction; band theory even applies to microscopic-sized transistors 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 vibrations, other electrons, photons, 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 or dopant 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.
• Along the same lines, most electronic effects (capacitance, electrical conductance, electric-field screening) 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.
• Small systems: For systems which are small along every dimension (e.g., a small molecule or a quantum dot), there is no continuous band structure. The crossover between small and large dimensions is the realm of mesoscopic physics.
• Strongly correlated materials (for example, Mott insulators) 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

Fig 1. Brillouin zone of a face-centered cubic lattice showing labels for special symmetry points.

 

Fig 2. Band structure plot for Si, Ge, GaAs and InAs generated with tight binding model. Note that Si and Ge are indirect band gap materials, while GaAs and InAs are direct.

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 waves as solutions:

${\displaystyle \psi _{n\mathbf {k} }(\mathbf {r} )=e^{i\mathbf {k} \cdot \mathbf {r} }u_{n\mathbf {k} }(\mathbf {r} )}$,

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 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 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:

• Direct band gap: 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 crystalline materials, quasi-crystalline and amorphous solids may also exhibit band structures. 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

Filling of the electronic states in various types of materials at equilibrium. Here, height is energy while width is the density of available states for a certain energy in the material listed. The shade follows the Fermi–Dirac distribution (black = all states filled, white = no state filled). In metals and semimetals the Fermi level E_F lies inside at least one band. In insulators and semiconductors the Fermi level is inside a band gap; however, in semiconductors the bands are near enough to the Fermi level to be thermally populated with electrons or holes. 

 

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:

${\displaystyle f(E)={\frac {1}{1+e^{{(E-\mu )}/{k_{\rm {B}}T}}}}}$

where:

• k_BT is the product of Boltzmann's constant and temperature, and
• µ is the total chemical potential of electrons, or Fermi level (in semiconductor physics, 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:

${\displaystyle N/V=\int _{-\infty }^{\infty }g(E)f(E)\,dE}$

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 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:

• In a semiconductor 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. The name "valence band" was coined by analogy to chemistry, since in semiconductors (and insulators) the valence band is built out of the valence orbitals.
• In a metal or semimetal, 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 is the special case of electron waves in a periodic crystal lattice using Bloch waves 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:

${\displaystyle V(\mathbf {r} )=\sum _{\mathbf {K} }{V_{\mathbf {K} }e^{i\mathbf {K} \cdot \mathbf {r} }}}$

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.

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 wavefunction:

${\displaystyle {\Psi }_{n,\mathbf {k} }(\mathbf {r} )=e^{i\mathbf {k} \cdot \mathbf {r} }u_{n}(\mathbf {r} )}$

where the function ${\displaystyle u_{n}(\mathbf {r} )}$ is periodic over the crystal lattice, that is,

${\displaystyle u_{n}(\mathbf {r} )=u_{n}(\mathbf {r-R} )}$.

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.

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 ${\displaystyle \Psi }$ is well approximated by a linear combination of atomic orbitals ${\displaystyle \psi _{n}(\mathbf {r} )}$.

${\displaystyle \Psi (\mathbf {r} )=\sum _{n,\mathbf {R} }b_{n,\mathbf {R} }\psi _{n}(\mathbf {r-R} )}$,

where the coefficients ${\displaystyle b_{n,\mathbf {R} }}$ 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:

${\displaystyle a_{n}(\mathbf {r-R} )={\frac {V_{C}}{(2\pi )^{3}}}\int \limits _{\text{BZ}}d\mathbf {k} e^{-i\mathbf {k} \cdot (\mathbf {R-r} )}u_{n\mathbf {k} }}$;

in which ${\displaystyle u_{n\mathbf {k} }}$ is the periodic part of the Bloch wave 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:

${\displaystyle \Psi _{n,\mathbf {k} }(\mathbf {r} )=\sum _{\mathbf {R} }e^{-i\mathbf {k} \cdot (\mathbf {R-r} )}a_{n}(\mathbf {r-R} )}$.

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 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.

A variational implementation was suggested by Korringa and by Kohn and Rostocker, and is often referred to as the KKR model.

Density-functional theory

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 Koopman's 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.

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 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.

Mott insulators

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.

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 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 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).

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 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.