- Electronic band structure
In

solid-state physics , the**electronic band structure**(or simply**band structure**) of asolid describes ranges ofenergy that anelectron is "forbidden" or "allowed" to have. It is due to the diffraction of the quantum mechanical electron waves in the periodiccrystal lattice . The band structure of a material determines several characteristics, in particular its electronic and optical properties.**Why bands occur in materials**The electrons of a single free-standing atom occupy

atomic orbital s, which form a discrete set ofenergy levels. If several atoms are brought together into a molecule, their atomic orbitals split, as in acoupled oscillation . This produces a number ofmolecular orbital s proportional to the number of atoms. When a large number of atoms (of order $10^\{20\}$ or more) are brought together to form a solid, the number of orbitals becomes exceedingly large, and the difference in energy between them becomes very small, so the levels may be considered to form continuous "bands" of energy rather than the discrete energy levels of the atoms in isolation. However, some intervals of energy contain no orbitals, no matter how many atoms are aggregated, forming "band gaps".Within an energy band, energy levels are so numerous as to be a near continuum. First, the separation between energy levels in a solid is comparable with the energy that electrons constantly exchange with

phonon s (atom ic vibrations). Second, it is comparable with the energy uncertainty due to theHeisenberg uncertainty principle , for reasonably long intervals of time. As a result, the separation between energy levels is of no consequence.Several approaches to finding band structure are discussed below

**Basic concepts**Any solid has a large number of bands. In theory, it can be said to have infinitely many bands (just as an atom has infinitely many energy levels). However, all but a few lie at energies so high that any electron that reaches those energies escapes from the solid. These bands are usually disregarded.

Bands have different widths, based upon the properties of the atomic orbitals from which they arise. Also, allowed bands may overlap, producing (for practical purposes) a single large band.

Figure 1 shows a simplified picture of the bands in a solid that allows the three major types of materials to be identified: metals, semiconductors and insulators.

"

Metal s" contain a band that is partly empty and partly filled regardless of temperature. Therefore they have very high conductivity.The lowermost, almost fully occupied band in an "insulator" or "

semiconductor " is called the "valence band " by analogy with thevalence electron s of individual atoms. The uppermost, almost unoccupied band is called the "conduction band " because only when electrons are excited to the conduction band can current flow in these materials. The difference between insulators and semiconductors is only that the forbiddenband gap between the valence band and conduction band is larger in an insulator, so that fewer electrons are found there and theelectrical conductivity is lower. Because one of the main mechanisms for electrons to be excited to the conduction band is due to thermal energy, the conductivity of semiconductors is strongly dependent on the temperature of the material.This band gap is one of the most useful aspects of the band structure, as it strongly influences the electrical and optical properties of the material. Electrons can transfer from one band to the other by means of

carrier generation and recombination processes. The band gap and defect states created in the band gap by doping can be used to createsemiconductor device s such assolar cell s,diode s,transistor s,laser diode s, and others.**ymmetry**A more complete view of the band structure takes into account the periodic nature of a crystal lattice using the symmetry operations that form a

space group . TheSchrödinger equation is solved for the crystal, which hasBloch wave s as solutions::$psi\_\{nmathbf\{k(mathbf\{r\})=e^\{imathbf\{k\}cdotmathbf\{ru\_\{nmathbf\{k(mathbf\{r\})$,

where

**k**is called the wavevector, and is related to the direction of motion of the electron in the crystal, and "n" is the band index, which simply numbers the energy bands. The wavevector**k**takes on values within theBrillouin zone (BZ) corresponding to the crystal lattice, and particular directions/points in the BZ are assigned conventional names like Γ, Δ, Λ, Σ, "etc." These directions are shown for the face-centered cubic lattice geometry in Figure 2.The available energies for the electron also depend upon

**k**, as shown in Figure 3 for silicon in the more complex energy band diagram at the right. In this diagram the topmost energy of the valence band is labeled $E\_v$ and the bottom energy in the conduction band is labeled $E\_c$. The top of the valence band is not directly below the bottom of the conduction band ($E\_v$ is for an electron traveling in direction Γ, $E\_c$ in direction X), so silicon is called an**indirect gap**material. For an electron to be excited from the valence band to the conduction band, it needs something to give it energy $E\_c\; -\; E\_v$ "and" a change in direction/momentum. In other semiconductors (for example GaAs) both are at Γ, and these materials are called**direct gap**materials (no momentum change required). Direct gap materials benefit the operation of semiconductorlaser diodes .Anderson's rule is used to align band diagrams between two different semiconductors in contact.**Band structures in different types of solids**Although electronic band structures are usually associated with

crystal line materials,quasi-crystal line andamorphous solid s may also exhibit band structures. However, the periodic nature and symmetrical properties of crystalline materials makes it much easier to examine the band structures of these materials theoretically. In addition, the well-defined symmetry axes of crystalline materials makes it possible to determine thedispersion relation ship between the momentum (a 3-dimension vector quantity) and energy of a material. 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**While the density of energy states in a band could be very large for some materials, it may not be uniform. It approaches zero at the band boundaries, and is generally highest near the middle of a band.The density of states for the

free electron model in three dimensions is given by,::$D(epsilon)=\; frac\{V\}\{2pi^2\}left(frac\; \{2m\}\{hbar^2\}\; ight)^\{3/2\}\; epsilon^\{1/2\}$**Filling of bands**Although the number of states in all of the bands is effectively infinite, in an uncharged material the number of electrons is equal only to the number of protons in the atoms of the material. Therefore not all of the states are occupied by electrons ("filled") at any time. The likelihood of any particular state being filled at any temperature is given by the

Fermi-Dirac statistics . The probability is given by the following::$f(E)\; =\; frac\{1\}\{1\; +\; e^\{frac\{E-E\_F\}\{k\_B\; T\}$

where:

* $k\_B$ isBoltzmann's constant ,

* "T" is thetemperature ,

* $E\_F$ is theFermi energy (or 'Fermi level').The Fermi level naturally is the level at which the electrons and protons are balanced.

At "T=0", the distribution is a simple

step function ::$f(E)\; =\; egin\{cases\}\; 1\; mbox\{if\}\; 0\; E\; le\; E\_F\; \backslash 0\; mbox\{if\}\; E\_F\; E\; end\{cases\}$

At nonzero temperatures, the step "smooths out", so that an appreciable number of states below the Fermi level are empty, and some states above the Fermi level are filled.

**Band structure of crystals****Brillouin zone**Because electron momentum is the reciprocal of space, the dispersion relation between the energy and momentum of electrons can best be described in reciprocal space. It turns out that for crystalline structures, the dispersion relation of the electrons is periodic, and that the

Brillouin zone is the smallest repeating space within this periodic structure. For an infinitely large crystal, if the dispersion relation for an electron is defined throughout the Brillouin zone, then it is defined throughout the entire reciprocal space.**Theory of band structures in crystals**The

ansatz is the special case of electron waves in a periodic crystal lattice usingBloch waves as treated generally in thedynamical theory of diffraction . Every crystal is a periodic structure which can be characterized by aBravais lattice , and for eachBravais lattice we can determine thereciprocal lattice , which encapsulates the periodicity in a set of three reciprocal lattice vectors ($mathbf\{b\_1\}$, $mathbf\{b\_2\}$, $mathbf\{b\_3\}$). Now, any periodic potential $V(mathbf\{r\})$ which shares the same periodicity as the direct lattice can be expanded out as aFourier series whose only non-vanishing components are those associated with the reciprocal lattice vectors. So the expansion can be written as:$V(mathbf\{r\})\; =\; sum\_\{mathbf\{K\{V\_\{mathbf\{Ke^\{i\; mathbf\{K\}cdotmathbf\{r\}$

where $mathbf\{K\}\; =\; m\_1\; mathbf\{b\}\_1\; +\; m\_2\; mathbf\{b\}\_2\; +\; m\_3\; mathbf\{b\}\_3$ for any set of integers $(m\_1,\; m\_2,\; m\_3)$.

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 in solid state physics interactions between electrons are completely ignored. This approximation allows use of

Bloch's Theorem which states that electrons in a periodic potential havewavefunction s and energies which are periodic in wavevector up to a constant phase shift between neighboringreciprocal lattice vectors. The consequences of periodicity are described mathematically by the Bloch wavefunction::$\{Psi\}\_\{n,mathbf\{k\; (mathbf\{r\})\; =\; e^\{i\; mathbf\{k\}cdotmathbf\{r\; u\_n(mathbf\{r\})$

where the function $u\_n(mathbf\{r\})$ is periodic over the crystal lattice, that is,

:$u\_n(mathbf\{r\})\; =\; u\_n(mathbf\{r-R\})$.

Here index "n" refers to the "n-th" energy band, wavevector

**k**is related to the direction of motion of the electron,**r**is position in the crystal, and**R**is location of an atomic site.cite book

author=Charles Kittel

title=Introduction to Solid State Physics

year= 1996

edition=Seventh Edition

publisher=Wiley

page=pp. 179

location=New York

isbn=0-471-11181-3

url=http://worldcat.org/isbn/0471111813] .(For more detail see

nearly-free electron model andpseudopotential method).**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 electronSchrödinger equation $Psi$ is well approximated by a linear combination ofatomic orbitals $psi\_n(mathbf\{r\})$.cite book

author=Charles Kittel

title=Introduction to Solid State Physics

year= 1996

edition=Seventh Edition

publisher=Wiley

page=pp. 245-248

location=New York

isbn=0-471-11181-3

url=http://worldcat.org/isbn/0471111813] .:$Psi(mathbf\{r\})\; =\; sum\_\{n,mathbf\{R\; b\_\{n,mathbf\{R\; psi\_n(mathbf\{r-R\})$,

where the coefficients $b\_\{n,mathbf\{R$ 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 cite book

author=Charles Kittel

title=Introduction to Solid State Physics

year= 1996

edition=Seventh Edition

publisher=Wiley

page=Eq. 42 p. 267

location=New York

isbn=0-471-11181-3

url=http://worldcat.org/isbn/0471111813] $^,$cite book

author=Daniel Charles Mattis

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

year= 1994

publisher=World Scientific

page=p. 340

isbn=9810214766

url=http://books.google.com/books?id=BGdHpCAMiLgC&pg=PA332&dq=wannier+functions&sig=6ESVld_3xysxYR1o1DdUPJGyCCY#PPA340,M1] .:$a\_n(mathbf\{r-R\})\; =\; frac\{V\_\{C\{(2pi)^\{3\; int\_\{BZ\}\; dmathbf\{k\}\; e^\{-imathbf\{k\}cdot(mathbf\{R-r\})\}u\_\{nmathbf\{k$;in which $u\_\{nmathbf\{k$ 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::$Psi\_\{n,mathbf\{k\; (mathbf\{r\})\; =\; sum\_\{mathbf\{R\; e^\{-imathbf\{k\}cdot(mathbf\{R-r\})\}a\_n(mathbf\{r-R\})$.

**Density-functional theory**In present days physics literature, the large majority of the electronic structures and band plots is calculated using the

density-functional theory (DFT) which is not a model but rather a theory, i.e. a microscopic first-principle theory ofcondensed matter physics that tries to cope with the electron-electron many-body problem via the introduction of anexchange-correlation term in the functional of theelectronic density . DFT calculated bands are found in many cases in agreement with experimental measured bands, for example by angle-resolved photoemission spectroscopy (ARPES). In particular, the band shape seems well reproduced by DFT. But also there are systematic errors of DFT bands with respect to the experiment. In particular, DFT seems to underestimate systematically by a 30-40% the band gap in insulators and semiconductors.It must be said that DFT is in principle an exact theory to reproduce and predict

ground state properties (e.g. thetotal energy , theatomic structure , etc.). However DFT is not a theory to addressexcited state properties, such as the band plot of a solid that represents the excitation energies of electrons injected or removed from the system. What in literature is quoted as a DFT band plot is a representation of the DFTKohn-Sham energies , that is the energies of a fictive non-interacting system, theKohn-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 noKoopman's theorem holding for Kohn-Sham energies, like on the other hand for Hartree-Fock energies that can be truly considered as an approximation forquasiparticle energies . Hence in principle DFT is not a band theory, not a theory suitable to calculate bands and band-plots.**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, the knowledge of the Green's function of a system provides both ground (the total energy) and also excited state observables of the system. Thepoles of the Green's function are the quasiparticle energies, the bands of a solid. The Green's function can be calculated by solving theDyson equation once theself-energy of the system is known. For real systems like solids, the self-energy is a very complex quantity and usually approximations are needed to solve the problem. One of such approximations is theGW approximation , so called from the mathematical form the self-energy takes as product $Sigma=GW$ of the Green's function $G$ and thedynamically screened interaction $W$. This approach is more pertinent to address the calculation of band plots (and also quantities beyond, such as thespectral function ) and can be also formulated in a completely "ab initio" way. The GW approximation seems to provide band gaps of insulators and semiconductors in agreement with the 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. TheHubbard model is an approximate theory that can include these interactions.**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:

* 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.

* Bands may also be viewed as the large-scale limit ofmolecular orbital theory . A solid creates a large number of closely spaced molecular orbitals, which appear as a band.

*Hubbard model The band structure has been generalised to wavevectors that are

complex number s, resulting in what is called acomplex band structure , which is of interest at surfaces and interfaces.Each model describes some types of solids very well, and others poorly. The nearly-free electron model works well for metals, but poorly for non-metals. The tight binding model is extremely accurate for ionic insulators, such as

metal halide salts (e.g. NaCl).**References****Further reading**# Kotai no denshiron (The theory of electrons in solids), by Hiroyuki Shiba, ISBN 4-621-04135-5

# "Microelectronics", by Jacob Millman and Arvin Gabriel, ISBN 0-07-463736-3, Tata McGraw-Hill Edition.

# "Solid State Physics", by Neil Ashcroft and N. David Mermin, ISBN 0-03-083993-9

# "Elementary Solid State Physics: Principles and Applications", by M. Ali Omar, ISBN 0-20-160733-6

# "Introduction to Solid State Physics" by Charles Kittel, ISBN 0-471-41526-X

# "Electronic and Optoelectronic Properties of Semiconductor Structures - Chapter 2 and 3" by Jasprit Singh, ISBN 0-521-82379-X**See also***

Bloch waves

*Nearly-free electron model

*Fermi surface

*Band Gap

*Effective mass

*Kronig-Penney model

*Anderson's rule

*k·p perturbation theory

* Tight binding model

*Local-density approximation

*Dynamical theory of diffraction

*Solid state physics

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