Electronic band structure

264: 2790:. 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. 520: 1358:, 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. 281: 3648: 951: 2531:. 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 4076: 4100: 536: 4112: 4088: 283: 288: 286: 282: 287: 2272: 285: 385:) 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. 2598:

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

267:

A hypothetical example of band formation when a large number of carbon atoms is brought together to form a diamond crystal. The right graph shows the energy levels as a function of the spacing between atoms. When the atoms are far apart (right side of graph) the eigenstates are the atomic orbitals of

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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

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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

744:) 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). 2653:

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

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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

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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

2579:(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. 2322:-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 2117: 687: 2063: 406:: 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 10 atoms, this is not a serious restriction; band theory even applies to microscopic-sized 268:

carbon. When the atoms come close enough (left side) that the orbitals begin to overlap, they hybridize into molecular orbitals with different energies. Since there are many atoms, the orbitals are very close in energy, and form continuous bands. The

1713: 2713:, 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. 1512: 2337: 510:) 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. 1316:

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:

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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

477:) 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 (see 284: 398:

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:

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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

2501:

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.

1188:). 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). 1824: 376:

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

1134: 424:: 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. 1587:

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.

594: 2623:, 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. 2599:

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

2535:) 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 1270: 1966: 2603:, 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, 1625: 1434: 2746: 2662:

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

304:, and are only slightly perturbed by the crystal lattice. This model explains the origin of the electronic dispersion relation, but the explanation for band gaps is subtle in this model. 1313:

s are also usually disregarded since they remain filled with electrons at all times, and are therefore inert. Likewise, materials have several band gaps throughout their band structure.

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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 (see

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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

2678:. 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 1961: 442:

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:

430:: 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 446:

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.,

1751: 2705:, 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 2098: 2304: 2267:{\displaystyle 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} };} 1919: 2682:

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.

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Stern, R.; Perry, J.; Boudreaux, D. (1969). "Low-Energy Electron-Diffraction Dispersion Surfaces and Band Structure in Three-Dimensional Mixed Laue and Bragg Reflections".

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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

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limits the number of electrons in a single orbital to two, and the bands are filled beginning with the lowest energy. At the actual diamond crystal cell size denoted by

3562: 1916: 2612: 2749:, a one-dimensional rectangular well model useful for illustration of band formation. While simple, it predicts many important phenomena, but is not quantitative. 243:

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

355:), the number of orbitals that hybridize with each other is very large. For this reason, the adjacent levels are very closely spaced in energy (of the order of 307:

The second model starts from the opposite limit, in which the electrons are tightly bound to individual atoms. The electrons of a single, isolated atom occupy

190: 2460:{\displaystyle \Psi _{n,\mathbf {k} }(\mathbf {r} )=\sum _{\mathbf {R} }e^{-i\mathbf {k} \cdot (\mathbf {R} -\mathbf {r} )}a_{n}(\mathbf {r} -\mathbf {R} ).} 1891:

The opposite extreme to the nearly free electron approximation assumes the electrons in the crystal behave much like an assembly of constituent atoms. This

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vectors. The consequences of periodicity are described mathematically by the Bloch's theorem, which states that the eigenstate wavefunctions have the form

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where it provides the number of mobile states, and in computing electron scattering rates where it provides the number of final states after scattering.

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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

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The NFE model works particularly well in materials like metals where distances between neighbouring atoms are small. In such materials the overlap of

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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

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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

2615:. 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 1195: 296:

The formation of electronic bands and band gaps can be illustrated with two complementary models for electrons in solids. The first one is the

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Paier, J.; Marsman, M.; Hummer, K.; Kresse, G.; Gerber, I. C.; Angyán, J. G. (2006). "Screened hybrid density functionals applied to solids".

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electron in each unit cell, and thus that the valence band is not fully occupied, making the material a conductor. However, materials such as

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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

1347:. The name "valence band" was coined by analogy to chemistry, since in semiconductors (and insulators) the valence band is built out of the 814:

in wavevector space, showing all of the states with energy equal to a particular value. The isosurface of states with energy equal to the

414:. With modifications, the concept of band structure can also be extended to systems which are only "large" along some dimensions, such as 90: 2941: 183: 2968: 4092: 2576: 583:

Band structure calculations take advantage of the periodic nature of a crystal lattice, exploiting its symmetry. The single-electron

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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,

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In the nearly free electron approximation, interactions between electrons are completely ignored. This approximation allows use of

914:, it provides both the number of excitable electrons and the number of final states for an electron. It appears in calculations of 112: 2539:

is approximated as a constant. Continuity of the potential between the atom-centered spheres and interstitial region is enforced.

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whose only non-vanishing components are those associated with the reciprocal lattice vectors. So the expansion can be written as:

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can be used to calculate the true band structure although in practice this is often difficult. A popular approach is the use of

2650: 2636: 124: 120: 3206: 300:, in which the electrons are assumed to move almost freely within the material. In this model, the electronic states resemble 3243: 3233: 3216: 3189: 3162: 3135: 2897: 2870: 2842: 2611:

holding for Kohn–Sham energies, as there is for Hartree–Fock energies, which can be truly considered as an approximation for

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The density of electrons in the material is simply the integral of the Fermi–Dirac distribution times the density of states:

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band gap (to distinguish it from the other band gaps in the band structure). The closest band above the band gap is called

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Assadi, M. Hussein. N.; Hanaor, Dorian A. H. (2013-06-21). "Theoretical study on copper's energetics and magnetism in TiO

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is an approximate theory that can include these interactions. It can be treated non-perturbatively within the so-called

555:

generated with tight binding model. Note that Si and Ge are indirect band gap materials, while GaAs and InAs are direct.

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levels, each with a different energy. Since the number of atoms in a macroscopic piece of solid is a very large number (

682:{\displaystyle \psi _{n\mathbf {k} }(\mathbf {r} )=e^{i\mathbf {k} \cdot \mathbf {r} }u_{n\mathbf {k} }(\mathbf {r} ),} 3152: 707:, the band index, which simply numbers the energy bands. Each of these energy levels evolves smoothly with changes in 4080: 3525: 3511: 3497: 3482: 3468: 3454: 3433: 3419: 2978: 2951: 2924: 2575:. DFT-calculated bands are in many cases found to be in agreement with experimentally measured bands, for example by 4143: 2058:{\displaystyle \Psi (\mathbf {r} )=\sum _{n,\mathbf {R} }b_{n,\mathbf {R} }\psi _{n}(\mathbf {r} -\mathbf {R} ),} 1376: 1019: 128: 2559:

In recent physics literature, a large majority of the electronic structures and band plots are calculated using

17: 4116: 4061: 3672: 3103: 1928: 3607: 1708:{\displaystyle \Psi _{n,\mathbf {k} }(\mathbf {r} )=e^{i\mathbf {k} \cdot \mathbf {r} }u_{n}(\mathbf {r} )} 1038: 897:

is defined as the number of electronic states per unit volume, per unit energy, for electron energies near

574: 503: 496:), there is no continuous band structure. The crossover between small and large dimensions is the realm of 2729: 2710: 2691: 1874: 1718: 1507:{\displaystyle V(\mathbf {r} )=\sum _{\mathbf {K} }{V_{\mathbf {K} }e^{i\mathbf {K} \cdot \mathbf {r} }}} 528: 158: 2068: 1275:

internal bulk of a material prefers to be charge neutral. The condition of charge neutrality means that

3852: 3761: 2511: 2277: 1597: 1005: 963: 945: 297: 142: 56: 48: 2735: 370:. The inner electron orbitals do not overlap to a significant degree, so their bands are very narrow. 340:, the atoms' atomic orbitals overlap with the nearby orbitals. Each discrete energy level splits into 154: 3801: 2560: 2548: 1042: 1028: 955: 269: 252: 105: 82: 3971: 3725: 3715: 3571: 2640: 2564: 1869:

of the electron can be approximated by a (modified) plane wave. The band structure of a metal like

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that electrons may have within it, as well as the ranges of energy that they may not have (called

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The density of states function is important for calculations of effects based on band theory. In

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Energy band gaps can be classified using the wavevectors of the states surrounding the band gap:

367: 70: 2563:(DFT), which is not a model but rather a theory, i.e., a microscopic first-principles theory of 1896: 584: 1618:

and energies which are periodic in wavevector up to a constant phase shift between neighboring

907: 470: 455: 244: 116: 2995:"NSM Archive - Aluminium Gallium Arsenide (AlGaAs) - Band structure and carrier concentration" 2887: 3868: 3847: 3781: 1176: 320: 3837: 3612: 3374: 3337: 3282: 3055: 3041: 2994: 2799: 2608: 2568: 2567:

that tries to cope with the electron-electron many-body problem via the introduction of an

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are orthogonal. The Wannier functions can be used to form the Schrödinger solution for the

1901: 810:, respectively. Another method for visualizing band structure is to plot a constant-energy 33: 2489:

for instance are well described by TB-Hamiltonians on the basis of atomic sp orbitals. In

315:. If two atoms come close enough so that their atomic orbitals overlap, the electrons can 8: 3791: 3771: 3756: 3705: 2811: 2722: 1892: 1601: 730: 570: 301: 204: 98: 41: 3378: 3341: 3286: 3059: 263: 3938: 3928: 3766: 3677: 3306: 3272: 2572: 1619: 1611: 1388: 1372: 1152: 911: 741: 560: 497: 411: 316: 248: 2832: 2732:: the "band structure" of a region of free space that has been divided into a lattice. 1299: 4099: 4046: 4011: 3918: 3842: 3530: 3521: 3507: 3493: 3478: 3464: 3450: 3429: 3415: 3390: 3298: 3239: 3212: 3185: 3158: 3131: 2974: 2947: 2920: 2893: 2866: 2838: 2646: 2620: 2490: 2111: 959: 881: 841: 431: 324: 3310: 4016: 3908: 3888: 3883: 3878: 3873: 3730: 3710: 3667: 3632: 3602: 3382: 3345: 3290: 3063: 3044:

physics, where the electrons can be injected into a material at high energies, see

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Animation of band formation and how electrons fill them in a metal and an insulator

162: 1391:, which encapsulates the periodicity in a set of three reciprocal lattice vectors 4041: 3996: 3662: 3579: 3179: 2775: 2725:. In addition to the models mentioned above, other models include the following: 2498: 2494: 1923: 1605: 1427:

which shares the same periodicity as the direct lattice can be expanded out as a

1384: 1380: 1348: 1336: 868: 829: 552: 337: 86: 2666:, so called from the mathematical form the self-energy takes as the product Σ = 1004:; however, in semiconductors the bands are near enough to the Fermi level to be 488:

Small systems: For systems which are small along every dimension (e.g., a small

4104: 4051: 3933: 3822: 2970:

Solid-State Physics: An Introduction to Principles of Materials Science, 4th Ed

2759: 2739: 2702: 2655: 2470: 2307: 1858: 1428: 737: 588: 564: 524: 507: 447: 378: 308: 3154:

The Many-Body Problem: Encyclopaedia of Exactly Solved Models in One Dimension

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derives these bands and band gaps by examining the allowed quantum mechanical

4132: 3976: 3958: 3943: 3923: 3827: 3796: 3627: 3350: 3325: 3302: 2752: 2706: 2604: 2595: 1886: 1866: 1322: 1009: 997: 864: 819: 240: 146: 806:

along straight lines connecting symmetry points, often labelled Δ, Λ, Σ, or

519: 3913: 3735: 3637: 3540: 3394: 3082: 2787: 2771: 2587: 2583: 2518: 2110:

refers to an atomic site. A more accurate approach using this idea employs

1615: 1343: 1296:), until it is at the correct equilibrium with respect to the Fermi level. 807: 482: 478: 459: 312: 220: 3102:

and also changes sign depending on the wave vector, as can be seen in the

1371:

is the special case of electron waves in a periodic crystal lattice using

713:, forming a smooth band of states. For each band we can define a function 4033: 3751: 3720: 3700: 3647: 3107: 2805: 2659: 2522: 2306:

is the periodic part of the Bloch's theorem and the integral is over the

1306: 1156: 983: 941: 844:: the closest states above and beneath the band gap do not have the same 815: 578: 493: 466: 382: 2473:

and potentials on neighbouring atoms. Band structures of materials like

1379:. Every crystal is a periodic structure which can be characterized by a 362:

This formation of bands is mostly a feature of the outermost electrons (

3948: 3786: 3622: 2100:

are selected to give the best approximate solution of this form. Index

950: 854: 811: 701:, there are multiple solutions to the Schrödinger equation labelled by 407: 3386: 3294: 4006: 3832: 3617: 3534: 2742:, and the parameters in the model are often determined by experiment. 1870: 1819:{\displaystyle u_{n}(\mathbf {r} )=u_{n}(\mathbf {r} -\mathbf {R} ).} 1355: 979: 544: 3518:

Electronic and Optoelectronic Properties of Semiconductor Structures

535: 1300:

Names of bands near the Fermi level (conduction band, valence band)

1001: 962:

for a certain energy in the material listed. The shade follows the

489: 373: 366:) in the atom, which are the ones involved in chemical bonding and 225: 3277: 2469:

The TB model works well in materials with limited overlap between

1129:{\displaystyle f(E)={\frac {1}{1+e^{{(E-\mu )}/{k_{\text{B}}T}}}}} 954:

Filling of the electronic states in various types of materials at

587:

is solved for an electron in a lattice-periodic potential, giving

393: 4056: 4023: 4001: 3981: 2721:

Calculating band structures is an important topic in theoretical

2486: 2474: 860: 540: 2758:

The band structure has been generalised to wavevectors that are

859:

Although electronic band structures are usually associated with

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To calculate the bands including electron-electron interaction

1368: 451: 435: 27:

Describes the range of energies of an electron within the solid

1329:, the Fermi level is surrounded by a band gap, referred to as 359:), and can be considered to form a continuum, an energy band. 3966: 3587: 1895:

assumes the solution to the time-independent single electron

975: 216: 336:

of identical atoms come together to form a solid, such as a

276:, two bands are formed, separated by a 5.5 eV band gap. 3504:

Elementary Solid State Physics: Principles and Applications

2478: 1862: 1041:, a thermodynamic distribution that takes into account the 415: 1265:{\displaystyle N/V=\int _{-\infty }^{\infty }g(E)f(E)\,dE} 872:

structure of solids has focused on crystalline materials.

832:: the lowest-energy state above the band gap has the same 3597: 1614:

which states that electrons in a periodic potential have

3461:

Electronic Structure: Basic Theory and Practical Methods

3364: 3231: 2913:

Halliday, David; Resnick, Robert; Walker, Jearl (2013).

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It is commonly believed that DFT is a theory to predict

2493:

a mixed TB-NFE model is used to describe the broad NFE

1842:

is related to the direction of motion of the electron,

514: 251:, and forms the foundation of the understanding of all 3177: 1340:, and the closest band beneath the band gap is called 3204: 3150: 2912: 2802:– the process of altering a material's band structure 2340: 2280: 2120: 2071: 1969: 1931: 1904: 1759: 1721: 1628: 1591: 1437: 1198: 1051: 597: 3005: 2943:

Optical Metamaterials: Fundamentals and Applications

855:

Asymmetry: Band structures in non-crystalline solids

3045: 2946:. Springer Science and Business Media. p. 12. 2766:, which is of interest at surfaces and interfaces. 2973:. Springer Science and Business Media. p. 2. 2607:electronic structure of a system, and there is no 2459: 2298: 2266: 2092: 2057: 1955: 1910: 1818: 1745: 1707: 1506: 1264: 1128: 681: 3323: 3181:Electronic Structure and the Properties of Solids 838:as the highest-energy state beneath the band gap. 4130: 258: 2525:and Rostocker, and is often referred to as the 1753:is periodic over the crystal lattice, that is, 780:. In scientific literature it is common to see 465:Along the same lines, most electronic effects ( 394:Assumptions and limits of band structure theory 3123: 3081:Low-energy bands are however important in the 2685: 1037:being filled with an electron is given by the 3556: 736:The wavevector takes on any value inside the 184: 3570: 3490:Computational Methods in Solid State Physics 3258: 2542: 958:. Here, height is energy while width is the 695:is called the wavevector. For each value of 531:showing labels for special symmetry points. 91:Multi-configurational self-consistent field 3646: 3563: 3549: 2939: 2916:Fundamentals of Physics, Extended, 10th Ed 2860: 319:between the atoms. This tunneling splits ( 191: 177: 3488:Nemoshkalenko, V. V., and N. V. Antonov, 3349: 3276: 3119: 3117: 3115: 2966: 2865:. Cambridge: Cambridge University Press. 2814:– pioneer in the theory of band structure 2808:– pioneer in the theory of band structure 2674:and the dynamically screened interaction 2577:angle-resolved photoemission spectroscopy 1255: 3440:Pseudopotentials in the theory of metals 3171: 2940:Cai, Wenshan; Shalaev, Vladimir (2009). 2856: 2854: 949: 534: 518: 279: 262: 113:Time-dependent density functional theory 75:Semi-empirical quantum chemistry methods 2906: 2885: 1956:{\displaystyle \psi _{n}(\mathbf {r} )} 740:, which is a polyhedron in wavevector ( 14: 4131: 3225: 3198: 3144: 3112: 3108:https://www.phys.ufl.edu/fermisurface/ 2879: 2586:properties of a system only (e.g. the 1880: 1865:is relatively large. In that case the 1179:, this quantity is more often denoted 1031:, the likelihood of a state of energy 125:Linearized augmented-plane-wave method 121:Orbital-free density functional theory 3544: 3358: 3130:(Seventh ed.). New York: Wiley. 2919:. John Wiley and Sons. p. 1254. 2861:Girvin, Steven M.; Yang, Kun (2019). 2851: 2830: 2104:refers to an atomic energy level and 1362: 4087: 3438:Harrison, Walter A.; W. A. Benjamin 3410:Ashcroft, Neil and N. David Mermin, 3040:High-energy bands are important for 2960: 1848:is the position in the crystal, and 875: 515:Crystalline symmetry and wavevectors 4111: 3324:Hohenberg, P; Kohn, W. (Nov 1964). 3208:Impurity Scattering in Metal Alloys 3127:Introduction to Solid State Physics 2837:. Oxford: Oxford University Press. 2637:Green's function (many-body theory) 1854:is the location of an atomic site. 1746:{\displaystyle u_{n}(\mathbf {r} )} 992:lies inside at least one band. In 935: 95:Quantum chemistry composite methods 24: 3404: 2967:Ibach, Harald; Lüth, Hans (2009). 2933: 2342: 2093:{\displaystyle b_{n,\mathbf {R} }} 1970: 1905: 1630: 1592:Nearly free electron approximation 1226: 1221: 255:(transistors, solar cells, etc.). 79:Møller–Plesset perturbation theory 25: 4155: 3535:Tutorial on Bandstructure Methods 3232:Kuon Inoue, Kazuo Ohtaka (2004). 3157:. World Scientific. p. 340. 2889:Understanding Solid State Physics 2627:Green's function methods and the 2299:{\displaystyle u_{n\mathbf {k} }} 388: 4110: 4098: 4086: 4075: 4074: 2781: 2762:, resulting in what is called a 2447: 2439: 2416: 2408: 2397: 2379: 2363: 2353: 2290: 2255: 2237: 2229: 2218: 2202: 2143: 2135: 2084: 2045: 2037: 2017: 1999: 1977: 1946: 1806: 1798: 1774: 1736: 1698: 1678: 1670: 1651: 1641: 1497: 1489: 1474: 1461: 1445: 921:For energies inside a band gap, 910:, a calculation for the rate of 669: 659: 644: 636: 617: 607: 416:two-dimensional electron systems 3473:Millman, Jacob; Arvin Gabriel, 3426:Elementary Electronic Structure 3317: 3252: 3178:Walter Ashley Harrison (1989). 3088: 3075: 2892:. CRC Press. pp. 177–178. 2863:Modern Condensed Matter Physics 1861:and potentials on neighbouring 1377:dynamical theory of diffraction 886:The density of states function 539:Fig 2. Band structure plot for 129:Projector augmented wave method 3205:Joginder Singh Galsin (2001). 3151:Daniel Charles Mattis (1994). 3034: 2987: 2824: 2649:, one can resort to so-called 2571:term in the functional of the 2451: 2435: 2420: 2404: 2367: 2359: 2241: 2225: 2176: 2166: 2147: 2131: 2049: 2033: 1981: 1973: 1950: 1942: 1810: 1794: 1778: 1770: 1740: 1732: 1702: 1694: 1655: 1647: 1449: 1441: 1416:. Now, any periodic potential 1252: 1246: 1240: 1234: 1097: 1085: 1061: 1055: 673: 665: 621: 613: 13: 1: 3673:Spontaneous symmetry breaking 2834:The Oxford Solid State Basics 2818: 2528:Korringa–Kohn–Rostoker method 504:Strongly correlated materials 330:Similarly, if a large number 259:Why bands and band gaps occur 167:Korringa–Kohn–Rostoker method 3326:"Inhomogeneous Electron Gas" 3094:In copper, for example, the 2886:Holgate, Sharon Ann (2009). 2505: 1375:as treated generally in the 1017: 1000:the Fermi level is inside a 733:for electrons in that band. 575:Crystallographic point group 7: 3485:, Tata McGraw-Hill Edition. 3013:"Electronic Band Structure" 2793: 2730:Empty lattice approximation 2711:dynamical mean-field theory 2692:Dynamical mean-field theory 2686:Dynamical mean-field theory 1875:empty lattice approximation 1836:th energy band, wavevector 960:density of available states 529:face-centered cubic lattice 323:) the atomic orbitals into 159:Empty lattice approximation 10: 4160: 4139:Electronic band structures 3853:Spin gapless semiconductor 3762:Nearly free electron model 3265:Journal of Applied Physics 2689: 2634: 2552: 2546: 2512:Multiple scattering theory 2509: 1918:is well approximated by a 1884: 1598:Nearly free electron model 1595: 939: 879: 568: 558: 298:nearly free electron model 143:Nearly free electron model 57:Modern valence bond theory 4070: 4032: 3957: 3901: 3861: 3810: 3802:Density functional theory 3777:electronic band structure 3744: 3693: 3686: 3655: 3644: 3578: 3104:De Haas–Van Alphen effect 3068:10.1103/RevModPhys.41.275 3048:Reviews of Modern Physics 2831:Simon, Steven H. (2013). 2716: 2561:density-functional theory 2549:Density functional theory 2543:Density-functional theory 1715:where the Bloch function 1043:Pauli exclusion principle 1029:thermodynamic equilibrium 784:which show the values of 327:with different energies. 302:free electron plane waves 270:Pauli exclusion principle 209:electronic band structure 136:Electronic band structure 106:Density functional theory 83:Configuration interaction 3972:Bogoliubov quasiparticle 3716:Quantum spin Hall effect 3608:Bose–Einstein condensate 3572:Condensed matter physics 3447:Condensed Matter Physics 3351:10.1103/PhysRev.136.B864 3238:. Springer. p. 66. 3211:. Springer. Appendix C. 2670:of the Green's function 2565:condensed matter physics 1559:for any set of integers 1169:total chemical potential 1039:Fermi–Dirac distribution 1014:intrinsic semiconductors 964:Fermi–Dirac distribution 475:electric-field screening 151:Muffin-tin approximation 64:Molecular orbital theory 53:Generalized valence bond 4144:Solid state engineering 3271:(23): 233913–233913–5. 3124:Charles Kittel (1996). 2736:k·p perturbation theory 2065:where the coefficients 1873:even gets close to the 1012:. "intrin." indicates 974:: no state filled). In 916:electrical conductivity 368:electrical conductivity 219:describes the range of 155:k·p perturbation theory 3184:. Dover Publications. 3022:. Springer. p. 24 2764:complex band structure 2613:quasiparticle energies 2461: 2300: 2268: 2094: 2059: 1957: 1912: 1820: 1747: 1709: 1508: 1266: 1151:is the product of the 1130: 1024: 970:: all states filled, 946:Fermi–Dirac statistics 683: 556: 532: 471:electrical conductance 293: 277: 245:electrical resistivity 49:Coulson–Fischer theory 3848:Topological insulator 3782:Anderson localization 3424:Harrison, Walter A., 2462: 2301: 2269: 2095: 2060: 1958: 1913: 1911:{\displaystyle \Psi } 1821: 1748: 1710: 1509: 1387:we can determine the 1267: 1177:semiconductor physics 1131: 953: 684: 538: 522: 291: 266: 3726:Aharonov–Bohm effect 3613:Fermionic condensate 3445:Marder, Michael P., 3042:electron diffraction 2800:Band-gap engineering 2641:Green–Kubo relations 2569:exchange-correlation 2338: 2334:-th energy band as: 2278: 2118: 2069: 1967: 1929: 1902: 1897:Schrödinger equation 1757: 1719: 1626: 1435: 1309:). These low-energy 1196: 1049: 782:band structure plots 595: 585:Schrödinger equation 404:Infinite-size system 34:Electronic structure 4117:Physics WikiProject 3792:tight binding model 3772:Fermi liquid theory 3757:Free electron model 3706:Quantum Hall effect 3687:Electrons in solids 3412:Solid State Physics 3379:2006JChPh.124o4709P 3342:1964PhRv..136..864H 3287:2013JAP...113w3913A 3060:1969RvMP...41..275S 2812:Alan Herries Wilson 2747:Kronig–Penney model 2723:solid state physics 2555:Kohn–Sham equations 1893:tight binding model 1881:Tight binding model 1602:Free electron model 1230: 1006:thermally populated 908:Fermi's Golden Rule 731:dispersion relation 571:Symmetry in physics 434:, other electrons, 412:integrated circuits 253:solid-state devices 205:solid-state physics 99:Quantum Monte Carlo 71:Hartree–Fock method 42:Valence bond theory 3678:Critical phenomena 3531:Vasileska, Dragica 3520:Chapters 2 and 3, 2621:hybrid functionals 2617:time-dependent DFT 2601:Kohn–Sham energies 2594:, etc.), and that 2573:electronic density 2537:screened potential 2457: 2384: 2296: 2264: 2090: 2055: 2004: 1953: 1920:linear combination 1908: 1816: 1743: 1705: 1620:reciprocal lattice 1504: 1466: 1389:reciprocal lattice 1363:Theory in crystals 1262: 1213: 1153:Boltzmann constant 1126: 1025: 1008:with electrons or 912:optical absorption 742:reciprocal lattice 679: 557: 533: 498:mesoscopic physics 432:lattice vibrations 422:Homogeneous system 325:molecular orbitals 294: 278: 249:optical absorption 117:Thomas–Fermi model 4126: 4125: 4012:Exciton-polariton 3897: 3896: 3869:Thermoelectricity 3459:Martin, Richard, 3442:, (New York) 1966 3387:10.1063/1.2187006 3336:(3B): B864–B871. 3295:10.1063/1.4811539 3245:978-3-540-20559-3 3235:Photonic Crystals 3218:978-0-306-46574-1 3191:978-0-486-66021-9 3164:978-981-02-1476-0 3137:978-0-471-11181-8 2899:978-1-4200-1232-3 2872:978-1-107-13739-4 2844:978-0-19-150210-1 2647:many-body effects 2609:Koopmans' theorem 2491:transition metals 2373: 2195: 2186: 2112:Wannier functions 1987: 1455: 1171:of electrons, or 1124: 1114: 1022: 882:Density of states 876:Density of states 865:quasi-crystalline 842:Indirect band gap 428:Non-interactivity 364:valence electrons 289: 201: 200: 16:(Redirected from 4151: 4114: 4113: 4102: 4090: 4089: 4078: 4077: 4017:Phonon polariton 3909:Amorphous magnet 3889:Electrostriction 3884:Flexoelectricity 3879:Ferroelectricity 3874:Piezoelectricity 3731:Josephson effect 3711:Spin Hall effect 3691: 3690: 3668:Phase transition 3650: 3633:Luttinger liquid 3580:States of matter 3565: 3558: 3551: 3542: 3541: 3516:Singh, Jasprit, 3475:Microelectronics 3399: 3398: 3362: 3356: 3355: 3353: 3321: 3315: 3314: 3280: 3256: 3250: 3249: 3229: 3223: 3222: 3202: 3196: 3195: 3175: 3169: 3168: 3148: 3142: 3141: 3121: 3110: 3092: 3086: 3079: 3073: 3071: 3038: 3032: 3031: 3029: 3027: 3020:www.springer.com 3017: 3009: 3003: 3002: 2991: 2985: 2984: 2964: 2958: 2957: 2937: 2931: 2930: 2910: 2904: 2903: 2883: 2877: 2876: 2858: 2849: 2848: 2828: 2664:GW approximation 2651:Green's function 2631:GW approximation 2592:atomic structure 2466: 2464: 2463: 2458: 2450: 2442: 2434: 2433: 2424: 2423: 2419: 2411: 2400: 2383: 2382: 2366: 2358: 2357: 2356: 2333: 2327: 2321: 2315: 2305: 2303: 2302: 2297: 2295: 2294: 2293: 2273: 2271: 2270: 2265: 2260: 2259: 2258: 2245: 2244: 2240: 2232: 2221: 2205: 2197: 2196: 2193: 2187: 2185: 2184: 2183: 2164: 2163: 2154: 2146: 2138: 2130: 2129: 2109: 2103: 2099: 2097: 2096: 2091: 2089: 2088: 2087: 2064: 2062: 2061: 2056: 2048: 2040: 2032: 2031: 2022: 2021: 2020: 2003: 2002: 1980: 1962: 1960: 1959: 1954: 1949: 1941: 1940: 1917: 1915: 1914: 1909: 1853: 1847: 1841: 1835: 1831: 1825: 1823: 1822: 1817: 1809: 1801: 1793: 1792: 1777: 1769: 1768: 1752: 1750: 1749: 1744: 1739: 1731: 1730: 1714: 1712: 1711: 1706: 1701: 1693: 1692: 1683: 1682: 1681: 1673: 1654: 1646: 1645: 1644: 1583: 1558: 1513: 1511: 1510: 1505: 1503: 1502: 1501: 1500: 1492: 1479: 1478: 1477: 1465: 1464: 1448: 1426: 1415: 1349:valence orbitals 1295: 1284: 1271: 1269: 1268: 1263: 1229: 1224: 1206: 1187: 1166: 1150: 1135: 1133: 1132: 1127: 1125: 1123: 1122: 1121: 1120: 1116: 1115: 1112: 1105: 1100: 1068: 1036: 1018: 936:Filling of bands 931: 902: 896: 869:amorphous solids 849: 837: 818:is known as the 805: 799: 779: 770: 761: 752: 728: 712: 706: 700: 694: 688: 686: 685: 680: 672: 664: 663: 662: 649: 648: 647: 639: 620: 612: 611: 610: 358: 354: 352: 345: 335: 290: 193: 186: 179: 163:GW approximation 30: 29: 21: 4159: 4158: 4154: 4153: 4152: 4150: 4149: 4148: 4129: 4128: 4127: 4122: 4066: 4047:Granular matter 4042:Amorphous solid 4028: 3953: 3939:Antiferromagnet 3929:Superparamagnet 3902:Magnetic phases 3893: 3857: 3806: 3767:Bloch's theorem 3740: 3682: 3663:Order parameter 3656:Phase phenomena 3651: 3642: 3574: 3569: 3407: 3405:Further reading 3402: 3363: 3359: 3322: 3318: 3262: 3257: 3253: 3246: 3230: 3226: 3219: 3203: 3199: 3192: 3176: 3172: 3165: 3149: 3145: 3138: 3122: 3113: 3093: 3089: 3080: 3076: 3039: 3035: 3025: 3023: 3015: 3011: 3010: 3006: 2993: 2992: 2988: 2981: 2965: 2961: 2954: 2938: 2934: 2927: 2911: 2907: 2900: 2884: 2880: 2873: 2859: 2852: 2845: 2829: 2825: 2821: 2796: 2784: 2760:complex numbers 2719: 2694: 2688: 2643: 2635:Main articles: 2633: 2557: 2551: 2545: 2514: 2508: 2499:pseudopotential 2495:conduction band 2484: 2471:atomic orbitals 2446: 2438: 2429: 2425: 2415: 2407: 2396: 2389: 2385: 2378: 2377: 2362: 2352: 2345: 2341: 2339: 2336: 2335: 2329: 2323: 2317: 2311: 2289: 2285: 2281: 2279: 2276: 2275: 2254: 2250: 2246: 2236: 2228: 2217: 2210: 2206: 2201: 2192: 2188: 2179: 2175: 2165: 2159: 2155: 2153: 2142: 2134: 2125: 2121: 2119: 2116: 2115: 2105: 2101: 2083: 2076: 2072: 2070: 2067: 2066: 2044: 2036: 2027: 2023: 2016: 2009: 2005: 1998: 1991: 1976: 1968: 1965: 1964: 1945: 1936: 1932: 1930: 1927: 1926: 1924:atomic orbitals 1903: 1900: 1899: 1889: 1883: 1859:atomic orbitals 1849: 1843: 1837: 1833: 1829: 1805: 1797: 1788: 1784: 1773: 1764: 1760: 1758: 1755: 1754: 1735: 1726: 1722: 1720: 1717: 1716: 1697: 1688: 1684: 1677: 1669: 1665: 1661: 1650: 1640: 1633: 1629: 1627: 1624: 1623: 1612:Bloch's Theorem 1608: 1606:pseudopotential 1596:Main articles: 1594: 1581: 1574: 1567: 1560: 1557: 1551: 1544: 1538: 1531: 1525: 1515: 1496: 1488: 1484: 1480: 1473: 1472: 1468: 1467: 1460: 1459: 1444: 1436: 1433: 1432: 1417: 1413: 1406: 1399: 1392: 1385:Bravais lattice 1383:, and for each 1381:Bravais lattice 1373:Bloch's theorem 1365: 1337:conduction band 1302: 1286: 1276: 1225: 1217: 1202: 1197: 1194: 1193: 1186: 1180: 1162: 1146: 1140: 1111: 1107: 1106: 1101: 1084: 1083: 1079: 1072: 1067: 1050: 1047: 1046: 1032: 1023: 991: 948: 940:Main articles: 938: 922: 898: 887: 884: 878: 857: 845: 833: 830:Direct band gap 801: 793: 785: 777: 772: 768: 763: 759: 754: 748: 729:, which is the 722: 714: 708: 702: 696: 690: 668: 658: 654: 650: 643: 635: 631: 627: 616: 606: 602: 598: 596: 593: 592: 589:Bloch electrons 581: 567: 561:Bloch's theorem 559:Main articles: 517: 508:Mott insulators 396: 391: 379:atomic orbitals 356: 348: 347: 341: 338:crystal lattice 331: 309:atomic orbitals 280: 261: 231:forbidden bands 197: 165: 161: 157: 153: 149: 145: 127: 123: 119: 115: 97: 93: 89: 87:Coupled cluster 85: 81: 77: 73: 55: 51: 28: 23: 22: 15: 12: 11: 5: 4157: 4147: 4146: 4141: 4124: 4123: 4121: 4120: 4108: 4105:Physics Portal 4096: 4084: 4071: 4068: 4067: 4065: 4064: 4059: 4054: 4052:Liquid crystal 4049: 4044: 4038: 4036: 4030: 4029: 4027: 4026: 4021: 4020: 4019: 4014: 4004: 3999: 3994: 3989: 3984: 3979: 3974: 3969: 3963: 3961: 3959:Quasiparticles 3955: 3954: 3952: 3951: 3946: 3941: 3936: 3931: 3926: 3921: 3919:Superdiamagnet 3916: 3911: 3905: 3903: 3899: 3898: 3895: 3894: 3892: 3891: 3886: 3881: 3876: 3871: 3865: 3863: 3859: 3858: 3856: 3855: 3850: 3845: 3843:Superconductor 3840: 3835: 3830: 3825: 3823:Mott insulator 3820: 3814: 3812: 3808: 3807: 3805: 3804: 3799: 3794: 3789: 3784: 3779: 3774: 3769: 3764: 3759: 3754: 3748: 3746: 3742: 3741: 3739: 3738: 3733: 3728: 3723: 3718: 3713: 3708: 3703: 3697: 3695: 3688: 3684: 3683: 3681: 3680: 3675: 3670: 3665: 3659: 3657: 3653: 3652: 3645: 3643: 3641: 3640: 3635: 3630: 3625: 3620: 3615: 3610: 3605: 3600: 3595: 3590: 3584: 3582: 3576: 3575: 3568: 3567: 3560: 3553: 3545: 3539: 3538: 3528: 3514: 3502:Omar, M. Ali, 3500: 3486: 3471: 3457: 3443: 3436: 3422: 3406: 3403: 3401: 3400: 3373:(15): 154709. 3357: 3316: 3260: 3251: 3244: 3224: 3217: 3197: 3190: 3170: 3163: 3143: 3136: 3111: 3096:effective mass 3087: 3074: 3033: 3004: 2986: 2979: 2959: 2952: 2932: 2925: 2905: 2898: 2878: 2871: 2850: 2843: 2822: 2820: 2817: 2816: 2815: 2809: 2803: 2795: 2792: 2783: 2780: 2756: 2755: 2750: 2743: 2740:semiconductors 2733: 2718: 2715: 2703:Mott insulator 2690:Main article: 2687: 2684: 2656:Dyson equation 2632: 2625: 2547:Main article: 2544: 2541: 2510:Main article: 2507: 2504: 2482: 2456: 2453: 2449: 2445: 2441: 2437: 2432: 2428: 2422: 2418: 2414: 2410: 2406: 2403: 2399: 2395: 2392: 2388: 2381: 2376: 2372: 2369: 2365: 2361: 2355: 2351: 2348: 2344: 2316:refers to the 2308:Brillouin zone 2292: 2288: 2284: 2263: 2257: 2253: 2249: 2243: 2239: 2235: 2231: 2227: 2224: 2220: 2216: 2213: 2209: 2204: 2200: 2191: 2182: 2178: 2174: 2171: 2168: 2162: 2158: 2152: 2149: 2145: 2141: 2137: 2133: 2128: 2124: 2114:, defined by: 2086: 2082: 2079: 2075: 2054: 2051: 2047: 2043: 2039: 2035: 2030: 2026: 2019: 2015: 2012: 2008: 2001: 1997: 1994: 1990: 1986: 1983: 1979: 1975: 1972: 1952: 1948: 1944: 1939: 1935: 1907: 1885:Main article: 1882: 1879: 1832:refers to the 1815: 1812: 1808: 1804: 1800: 1796: 1791: 1787: 1783: 1780: 1776: 1772: 1767: 1763: 1742: 1738: 1734: 1729: 1725: 1704: 1700: 1696: 1691: 1687: 1680: 1676: 1672: 1668: 1664: 1660: 1657: 1653: 1649: 1643: 1639: 1636: 1632: 1593: 1590: 1579: 1572: 1565: 1555: 1549: 1542: 1536: 1529: 1523: 1499: 1495: 1491: 1487: 1483: 1476: 1471: 1463: 1458: 1454: 1451: 1447: 1443: 1440: 1429:Fourier series 1411: 1404: 1397: 1364: 1361: 1360: 1359: 1354:In a metal or 1352: 1327:band insulator 1301: 1298: 1261: 1258: 1254: 1251: 1248: 1245: 1242: 1239: 1236: 1233: 1228: 1223: 1220: 1216: 1212: 1209: 1205: 1201: 1190: 1189: 1184: 1160: 1144: 1119: 1110: 1104: 1099: 1096: 1093: 1090: 1087: 1082: 1078: 1075: 1071: 1066: 1063: 1060: 1057: 1054: 998:semiconductors 989: 937: 934: 880:Main article: 877: 874: 856: 853: 852: 851: 839: 800:for values of 789: 775: 766: 757: 738:Brillouin zone 718: 678: 675: 671: 667: 661: 657: 653: 646: 642: 638: 634: 630: 626: 623: 619: 615: 609: 605: 601: 565:Brillouin zone 525:Brillouin zone 516: 513: 512: 511: 506:(for example, 501: 486: 463: 448:surface states 440: 439: 425: 419: 395: 392: 390: 389:Basic concepts 387: 311:with discrete 260: 257: 241:wave functions 213:band structure 199: 198: 196: 195: 188: 181: 173: 170: 169: 139: 138: 132: 131: 109: 108: 102: 101: 67: 66: 60: 59: 45: 44: 38: 37: 26: 18:Band structure 9: 6: 4: 3: 2: 4156: 4145: 4142: 4140: 4137: 4136: 4134: 4119: 4118: 4109: 4107: 4106: 4101: 4097: 4095: 4094: 4085: 4083: 4082: 4073: 4072: 4069: 4063: 4060: 4058: 4055: 4053: 4050: 4048: 4045: 4043: 4040: 4039: 4037: 4035: 4031: 4025: 4022: 4018: 4015: 4013: 4010: 4009: 4008: 4005: 4003: 4000: 3998: 3995: 3993: 3990: 3988: 3985: 3983: 3980: 3978: 3975: 3973: 3970: 3968: 3965: 3964: 3962: 3960: 3956: 3950: 3947: 3945: 3942: 3940: 3937: 3935: 3932: 3930: 3927: 3925: 3922: 3920: 3917: 3915: 3912: 3910: 3907: 3906: 3904: 3900: 3890: 3887: 3885: 3882: 3880: 3877: 3875: 3872: 3870: 3867: 3866: 3864: 3860: 3854: 3851: 3849: 3846: 3844: 3841: 3839: 3836: 3834: 3831: 3829: 3828:Semiconductor 3826: 3824: 3821: 3819: 3816: 3815: 3813: 3809: 3803: 3800: 3798: 3797:Hubbard model 3795: 3793: 3790: 3788: 3785: 3783: 3780: 3778: 3775: 3773: 3770: 3768: 3765: 3763: 3760: 3758: 3755: 3753: 3750: 3749: 3747: 3743: 3737: 3734: 3732: 3729: 3727: 3724: 3722: 3719: 3717: 3714: 3712: 3709: 3707: 3704: 3702: 3699: 3698: 3696: 3692: 3689: 3685: 3679: 3676: 3674: 3671: 3669: 3666: 3664: 3661: 3660: 3658: 3654: 3649: 3639: 3636: 3634: 3631: 3629: 3626: 3624: 3621: 3619: 3616: 3614: 3611: 3609: 3606: 3604: 3601: 3599: 3596: 3594: 3591: 3589: 3586: 3585: 3583: 3581: 3577: 3573: 3566: 3561: 3559: 3554: 3552: 3547: 3546: 3543: 3536: 3532: 3529: 3527: 3526:0-521-82379-X 3523: 3519: 3515: 3513: 3512:0-201-60733-6 3509: 3505: 3501: 3499: 3498:90-5699-094-2 3495: 3491: 3487: 3484: 3483:0-07-463736-3 3480: 3476: 3472: 3470: 3469:0-521-78285-6 3466: 3462: 3458: 3456: 3455:0-471-17779-2 3452: 3448: 3444: 3441: 3437: 3435: 3434:981-238-708-0 3431: 3427: 3423: 3421: 3420:0-03-083993-9 3417: 3413: 3409: 3408: 3396: 3392: 3388: 3384: 3380: 3376: 3372: 3368: 3361: 3352: 3347: 3343: 3339: 3335: 3331: 3327: 3320: 3312: 3308: 3304: 3300: 3296: 3292: 3288: 3284: 3279: 3274: 3270: 3266: 3263:polymorphs". 3255: 3247: 3241: 3237: 3236: 3228: 3220: 3214: 3210: 3209: 3201: 3193: 3187: 3183: 3182: 3174: 3166: 3160: 3156: 3155: 3147: 3139: 3133: 3129: 3128: 3120: 3118: 3116: 3109: 3105: 3101: 3097: 3091: 3084: 3078: 3069: 3065: 3061: 3057: 3053: 3049: 3043: 3037: 3021: 3014: 3008: 3000: 2996: 2990: 2982: 2980:9783540938040 2976: 2972: 2971: 2963: 2955: 2953:9781441911513 2949: 2945: 2944: 2936: 2928: 2926:9781118230619 2922: 2918: 2917: 2909: 2901: 2895: 2891: 2890: 2882: 2874: 2868: 2864: 2857: 2855: 2846: 2840: 2836: 2835: 2827: 2823: 2813: 2810: 2807: 2804: 2801: 2798: 2797: 2791: 2789: 2782:Band diagrams 2779: 2777: 2773: 2767: 2765: 2761: 2754: 2753:Hubbard model 2751: 2748: 2744: 2741: 2737: 2734: 2731: 2728: 2727: 2726: 2724: 2714: 2712: 2708: 2707:Hubbard model 2704: 2700: 2693: 2683: 2681: 2677: 2673: 2669: 2665: 2661: 2657: 2652: 2648: 2642: 2638: 2630: 2624: 2622: 2618: 2614: 2610: 2606: 2605:quasiparticle 2602: 2597: 2596:excited state 2593: 2589: 2585: 2580: 2578: 2574: 2570: 2566: 2562: 2556: 2550: 2540: 2538: 2534: 2530: 2529: 2524: 2520: 2513: 2503: 2500: 2496: 2492: 2488: 2480: 2476: 2472: 2467: 2454: 2443: 2430: 2426: 2412: 2401: 2393: 2390: 2386: 2374: 2370: 2349: 2346: 2332: 2326: 2320: 2314: 2310:. Here index 2309: 2286: 2282: 2261: 2251: 2247: 2233: 2222: 2214: 2211: 2207: 2198: 2189: 2180: 2172: 2169: 2160: 2156: 2150: 2139: 2126: 2122: 2113: 2108: 2080: 2077: 2073: 2052: 2041: 2028: 2024: 2013: 2010: 2006: 1995: 1992: 1988: 1984: 1937: 1933: 1925: 1921: 1898: 1894: 1888: 1887:Tight binding 1878: 1876: 1872: 1868: 1867:wave function 1864: 1860: 1855: 1852: 1846: 1840: 1826: 1813: 1802: 1789: 1785: 1781: 1765: 1761: 1727: 1723: 1689: 1685: 1674: 1666: 1662: 1658: 1637: 1634: 1621: 1617: 1616:wavefunctions 1613: 1607: 1603: 1599: 1589: 1585: 1578: 1571: 1564: 1554: 1548: 1541: 1535: 1528: 1522: 1518: 1493: 1485: 1481: 1469: 1456: 1452: 1438: 1430: 1424: 1420: 1410: 1403: 1396: 1390: 1386: 1382: 1378: 1374: 1370: 1357: 1353: 1350: 1346: 1345: 1339: 1338: 1332: 1328: 1324: 1323:semiconductor 1320: 1319: 1318: 1314: 1312: 1308: 1297: 1293: 1289: 1283: 1279: 1272: 1259: 1256: 1249: 1243: 1237: 1231: 1218: 1214: 1210: 1207: 1203: 1199: 1183: 1178: 1174: 1170: 1165: 1161: 1158: 1154: 1149: 1143: 1139: 1138: 1137: 1117: 1108: 1102: 1094: 1091: 1088: 1080: 1076: 1073: 1069: 1064: 1058: 1052: 1044: 1040: 1035: 1030: 1021: 1015: 1011: 1007: 1003: 999: 995: 988: 985: 981: 977: 973: 969: 965: 961: 957: 952: 947: 943: 933: 929: 925: 919: 917: 913: 909: 904: 901: 894: 890: 883: 873: 870: 866: 862: 848: 843: 840: 836: 831: 828: 827: 826: 823: 821: 820:Fermi surface 817: 813: 809: 804: 797: 792: 788: 783: 778: 769: 760: 751: 745: 743: 739: 734: 732: 726: 721: 717: 711: 705: 699: 693: 676: 655: 651: 640: 632: 628: 624: 603: 599: 591:as solutions 590: 586: 580: 576: 572: 566: 562: 554: 550: 546: 542: 537: 530: 526: 521: 509: 505: 502: 499: 495: 491: 487: 484: 480: 476: 472: 468: 464: 461: 457: 453: 449: 445: 444: 443: 437: 433: 429: 426: 423: 420: 417: 413: 409: 405: 402: 401: 400: 386: 384: 380: 375: 371: 369: 365: 360: 351: 344: 339: 334: 328: 326: 322: 318: 314: 313:energy levels 310: 305: 303: 299: 275: 271: 265: 256: 254: 250: 246: 242: 238: 234: 232: 228: 227: 222: 221:energy levels 218: 214: 210: 206: 194: 189: 187: 182: 180: 175: 174: 172: 171: 168: 164: 160: 156: 152: 148: 147:Tight binding 144: 141: 140: 137: 134: 133: 130: 126: 122: 118: 114: 111: 110: 107: 104: 103: 100: 96: 92: 88: 84: 80: 76: 72: 69: 68: 65: 62: 61: 58: 54: 50: 47: 46: 43: 40: 39: 35: 32: 31: 19: 4115: 4103: 4091: 4079: 3997:Pines' demon 3776: 3736:Kondo effect 3638:Time crystal 3517: 3503: 3489: 3474: 3460: 3446: 3439: 3425: 3411: 3370: 3366: 3360: 3333: 3329: 3319: 3268: 3264: 3254: 3234: 3227: 3207: 3200: 3180: 3173: 3153: 3146: 3125: 3090: 3083:Auger effect 3077: 3051: 3047: 3036: 3024:. Retrieved 3019: 3007: 2999:www.ioffe.ru 2998: 2989: 2969: 2962: 2942: 2935: 2915: 2908: 2888: 2881: 2862: 2833: 2826: 2788:band diagram 2785: 2774:salts (e.g. 2772:metal halide 2768: 2763: 2757: 2720: 2695: 2679: 2675: 2671: 2667: 2644: 2628: 2588:total energy 2584:ground state 2581: 2558: 2532: 2526: 2515: 2468: 2330: 2324: 2318: 2312: 2106: 1890: 1856: 1850: 1844: 1838: 1827: 1609: 1586: 1576: 1569: 1562: 1552: 1546: 1539: 1533: 1526: 1520: 1516: 1422: 1418: 1408: 1401: 1394: 1366: 1344:valence band 1341: 1334: 1330: 1315: 1310: 1307:1s electrons 1303: 1291: 1287: 1281: 1277: 1273: 1191: 1181: 1172: 1163: 1147: 1141: 1033: 1026: 986: 971: 967: 927: 923: 920: 905: 899: 892: 888: 885: 858: 846: 834: 824: 802: 795: 790: 786: 781: 773: 764: 755: 749: 746: 735: 724: 719: 715: 709: 703: 697: 691: 582: 483:band bending 479:space charge 460:band bending 441: 427: 421: 403: 397: 383:1s electrons 372: 361: 349: 342: 332: 329: 306: 295: 273: 236: 235: 230: 224: 212: 208: 202: 135: 4034:Soft matter 3934:Ferromagnet 3752:Drude model 3721:Berry phase 3701:Hall effect 3367:J Chem Phys 3026:10 November 2806:Felix Bloch 2660:self-energy 2533:muffin tins 1828:Here index 1173:Fermi level 1157:temperature 984:Fermi level 956:equilibrium 942:Fermi level 863:materials, 861:crystalline 816:Fermi level 579:Space group 494:quantum dot 467:capacitance 408:transistors 237:Band theory 211:(or simply 4133:Categories 3949:Spin glass 3944:Metamagnet 3924:Paramagnet 3811:Conduction 3787:BCS theory 3628:Superfluid 3623:Supersolid 3054:(2): 275. 2819:References 2553:See also: 994:insulators 980:semimetals 812:isosurface 569:See also: 357:10 eV 321:hybridizes 4007:Polariton 3914:Diamagnet 3862:Couplings 3838:Conductor 3833:Semimetal 3818:Insulator 3694:Phenomena 3618:Fermi gas 3330:Phys. Rev 3303:0021-8979 3278:1304.1854 2680:ab initio 2658:once the 2629:ab initio 2506:KKR model 2444:− 2413:− 2402:⋅ 2391:− 2375:∑ 2343:Ψ 2274:in which 2234:− 2223:⋅ 2212:− 2190:∫ 2173:π 2140:− 2042:− 2025:ψ 1989:∑ 1971:Ψ 1934:ψ 1906:Ψ 1871:aluminium 1803:− 1675:⋅ 1631:Ψ 1494:⋅ 1457:∑ 1356:semimetal 1311:core band 1227:∞ 1222:∞ 1219:− 1215:∫ 1095:μ 1092:− 641:⋅ 600:ψ 374:Band gaps 226:band gaps 4081:Category 4062:Colloids 3395:16674253 3311:94599250 2794:See also 2519:Korringa 1002:band gap 808:, , and 490:molecule 4093:Commons 4057:Polymer 4024:Polaron 4002:Plasmon 3982:Exciton 3375:Bibcode 3338:Bibcode 3283:Bibcode 3056:Bibcode 2487:diamond 1167:is the 1136:where: 523:Fig 1. 436:photons 215:) of a 36:methods 3992:Phonon 3987:Magnon 3745:Theory 3603:Plasma 3593:Liquid 3537:(2008) 3524: 3510: 3496: 3481: 3467: 3453: 3432: 3418: 3393: 3309: 3301: 3242: 3215: 3188: 3161: 3134: 3106:; see 3100:tensor 2977: 2950: 2923: 2896: 2869: 2841: 2717:Others 2590:, the 1604:, and 1514:where 1369:ansatz 976:metals 850:value. 689:where 577:, and 456:doping 452:dopant 438:, etc. 317:tunnel 207:, the 3967:Anyon 3588:Solid 3307:S2CID 3273:arXiv 3098:is a 3016:(PDF) 2481:, SiO 1863:atoms 1321:In a 1159:, and 1010:holes 972:white 968:black 930:) = 0 527:of a 492:or a 217:solid 3977:Hole 3522:ISBN 3508:ISBN 3494:ISBN 3479:ISBN 3465:ISBN 3451:ISBN 3430:ISBN 3416:ISBN 3391:PMID 3299:ISSN 3240:ISBN 3213:ISBN 3186:ISBN 3159:ISBN 3132:ISBN 3028:2016 2975:ISBN 2948:ISBN 2921:ISBN 2894:ISBN 2867:ISBN 2839:ISBN 2776:NaCl 2745:The 2639:and 2523:Kohn 2485:and 2479:GaAs 1367:The 1342:the 1335:the 1175:(in 1155:and 1020:edit 996:and 982:the 978:and 944:and 867:and 753:vs. 563:and 553:InAs 551:and 549:GaAs 353:≈ 10 247:and 3598:Gas 3383:doi 3371:124 3346:doi 3334:136 3291:doi 3269:113 3064:doi 2778:). 2699:CoO 1922:of 1331:the 1325:or 1027:At 450:or 410:in 233:). 229:or 203:In 4135:: 3533:, 3506:, 3492:, 3477:, 3463:, 3449:, 3428:, 3414:, 3389:. 3381:. 3369:. 3344:. 3332:. 3328:. 3305:. 3297:. 3289:. 3281:. 3267:. 3114:^ 3062:. 3052:41 3050:. 3018:. 2997:. 2853:^ 2668:GW 2521:, 2477:, 2475:Si 2194:BZ 1963:. 1877:. 1600:, 1584:. 1575:, 1568:, 1545:+ 1532:+ 1519:= 1407:, 1400:, 1045:: 1016:. 932:. 903:. 822:. 771:, 762:, 573:, 547:, 545:Ge 543:, 541:Si 485:). 481:, 473:, 469:, 462:). 458:, 3564:e 3557:t 3550:v 3397:. 3385:: 3377:: 3354:. 3348:: 3340:: 3313:. 3293:: 3285:: 3275:: 3261:2 3248:. 3221:. 3194:. 3167:. 3140:. 3085:. 3072:. 3070:. 3066:: 3058:: 3030:. 3001:. 2983:. 2956:. 2929:. 2902:. 2875:. 2847:. 2676:W 2672:G 2483:2 2455:. 2452:) 2448:R 2440:r 2436:( 2431:n 2427:a 2421:) 2417:r 2409:R 2405:( 2398:k 2394:i 2387:e 2380:R 2371:= 2368:) 2364:r 2360:( 2354:k 2350:, 2347:n 2331:n 2325:R 2319:n 2313:n 2291:k 2287:n 2283:u 2262:; 2256:k 2252:n 2248:u 2242:) 2238:r 2230:R 2226:( 2219:k 2215:i 2208:e 2203:k 2199:d 2181:3 2177:) 2170:2 2167:( 2161:C 2157:V 2151:= 2148:) 2144:R 2136:r 2132:( 2127:n 2123:a 2107:R 2102:n 2085:R 2081:, 2078:n 2074:b 2053:, 2050:) 2046:R 2038:r 2034:( 2029:n 2018:R 2014:, 2011:n 2007:b 2000:R 1996:, 1993:n 1985:= 1982:) 1978:r 1974:( 1951:) 1947:r 1943:( 1938:n 1851:R 1845:r 1839:k 1834:n 1830:n 1814:. 1811:) 1807:R 1799:r 1795:( 1790:n 1786:u 1782:= 1779:) 1775:r 1771:( 1766:n 1762:u 1741:) 1737:r 1733:( 1728:n 1724:u 1703:) 1699:r 1695:( 1690:n 1686:u 1679:r 1671:k 1667:i 1663:e 1659:= 1656:) 1652:r 1648:( 1642:k 1638:, 1635:n 1582:) 1580:3 1577:m 1573:2 1570:m 1566:1 1563:m 1561:( 1556:3 1553:b 1550:3 1547:m 1543:2 1540:b 1537:2 1534:m 1530:1 1527:b 1524:1 1521:m 1517:K 1498:r 1490:K 1486:i 1482:e 1475:K 1470:V 1462:K 1453:= 1450:) 1446:r 1442:( 1439:V 1425:) 1423:r 1421:( 1419:V 1414:) 1412:3 1409:b 1405:2 1402:b 1398:1 1395:b 1393:( 1351:. 1294:) 1292:E 1290:( 1288:g 1282:V 1280:/ 1278:N 1260:E 1257:d 1253:) 1250:E 1247:( 1244:f 1241:) 1238:E 1235:( 1232:g 1211:= 1208:V 1204:/ 1200:N 1185:F 1182:E 1164:µ 1148:T 1145:B 1142:k 1118:T 1113:B 1109:k 1103:/ 1098:) 1089:E 1086:( 1081:e 1077:+ 1074:1 1070:1 1065:= 1062:) 1059:E 1056:( 1053:f 1034:E 990:F 987:E 966:( 928:E 926:( 924:g 900:E 895:) 893:E 891:( 889:g 847:k 835:k 803:k 798:) 796:k 794:( 791:n 787:E 776:z 774:k 767:y 765:k 758:x 756:k 750:E 727:) 725:k 723:( 720:n 716:E 710:k 704:n 698:k 692:k 677:, 674:) 670:r 666:( 660:k 656:n 652:u 645:r 637:k 633:i 629:e 625:= 622:) 618:r 614:( 608:k 604:n 500:. 418:. 350:N 343:N 333:N 274:a 192:e 185:t 178:v 20:)

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