Electronic band structure: Difference between revisions
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:<math>\psi (\mathbf r ) =N \sum_{i,n} e^{i\mathbf{k\cdot R_i}}\varphi_n (\mathbf {r\ - \ R_i}) \ , </math> | :<math>\psi (\mathbf r ) =N \sum_{i,n} e^{i\mathbf{k\cdot R_i}}\varphi_n (\mathbf {r\ - \ R_i}) \ , </math> | ||
where ''N'' is a ''normalization factor'' adjusted to make the probability of finding an electron somewhere unity. The vector parameter '''k''' is the ''wavevector'', and the '''R<sub>i</sub>''' are the vector locations of the atomic centers, assumed to be on a periodic crystal lattice, which could be a simple cubic lattice, or a face-centered cubic lattice, or whatever. Assuming the electrons to respond only to some "effective" crystal potential made up of potentials somehow constructed from the atomic potentials and located at the atomic sites, the energy levels in the crystal can be found as a function of the wavevector '''k'''. For example, for a body-centered cubic crystal, an allowed band of energies is found to be:<ref name=Mott> | |||
where ''N'' is a ''normalization factor'' adjusted to make the probability of finding an electron unity. The vector parameter '''k''' is the ''wavevector'', and the '''R<sub>i</sub>''' are the vector locations of the atomic centers, assumed to be on a periodic crystal lattice, which could be a simple cubic lattice, or a face-centered cubic lattice, or whatever. Assuming the electrons to respond only to some "effective" crystal potential made up of potentials somehow constructed from the atomic potentials and located at the atomic sites, the energy levels in the crystal can be found as a function of the wavevector '''k'''. For example, for a body-centered cubic crystal, an allowed band of energies is found to be:<ref name=Mott> | |||
See the treatment in Mott & Jones, cited above, page 68. | See the treatment in Mott & Jones, cited above, page 68. | ||
Revision as of 09:16, 9 April 2011
Electronic band structure refers to very closely spaced energy levels available to electrons in solids by virtue of the small separation between atoms composing a solid. The small spacing of atoms means that the energy levels found in isolated atoms are disturbed by neighboring atoms, causing shifts in these energy levels. Thus, a single energy level for each of N atoms becomes a band of N closely spaced energy levels in a solid composed of these N atoms. These bands of allowed energies are separated in energy by energy gaps where no levels exist.
Material classification by band theory
According to the occupancy of these bands, a material will be an insulator if all the bands of lower energy are filled, and these fully occupied bands (the valence bands) are separated by an energy gap from higher energy bands (the conduction bands) that are completely unoccupied. The filled bands cannot conduct, because the electron configuration is fixed, having occupied all the available energy levels. Consequently these electrons cannot move in response to a field.
If a band is only partially filled, the material will conduct and be a metal. The partially filled band can conduct, by shifting occupancy of the levels within the band in response to an external field.
If the highest energy completely filled band is not widely separated from the next highest unoccupied band (that is, the intervening energy gap separating the bands is small), the electrons in the filled band can gain sufficient energy by heating to leave this lower-energy filled band (creating holes) and occupy the lower levels of the higher-energy band, making the material a semiconductor. The semiconductor can conduct because the unoccupied levels in the lower band can shift occupancy under a field (hole conduction) and the electrons in the partially occupied higher band can conduct by shifting energy levels within this band (electron conduction). Evidently, the number of holes and the number of higher-energy-band electrons is very sensitive to temperature, and more importantly, to any external field itself. This last property of semiconductors is the source of practical applications of semiconductors such as silicon in integrated circuits. The devices in these circuits control the conductivity of semiconductors using applied voltages to achieve switching and modulation of signals.
Calculations
The basic idea of the spreading of individual energy levels to form allowed and forbidden energy bands is illustrated in one-dimension by the Kronig-Penney model.[1] The model consists of a regularly spaced array of rectangular potential energy wells that are deep enough to trap one or more energy levels at wide separation. As the wells are made to approach one another, bands of allowed energy are formed, separated by energy gaps or forbidden bands of energy.
In three dimensions, the tight-binding model provides a more realistic approach that exhibits the role of different crystal symmetries.[2] In this model the assumption is made that the electron wavefunction in a crystal is made up of a linear combination of atomic orbitals with undetermined coefficients. Symmetry of the crystal demands the coefficients have exponential form, and the trial wavefunction ψ at location r is expressed in terms of the atomic orbitals φn (n labels the various atomic orbitals) as:[3]
where N is a normalization factor adjusted to make the probability of finding an electron somewhere unity. The vector parameter k is the wavevector, and the Ri are the vector locations of the atomic centers, assumed to be on a periodic crystal lattice, which could be a simple cubic lattice, or a face-centered cubic lattice, or whatever. Assuming the electrons to respond only to some "effective" crystal potential made up of potentials somehow constructed from the atomic potentials and located at the atomic sites, the energy levels in the crystal can be found as a function of the wavevector k. For example, for a body-centered cubic crystal, an allowed band of energies is found to be:[4]
where a is the lattice spacing, k = (kx, ky, kz) and α and γ are certain integrals describing the interaction with the potential. Today's calculations are based upon some of the same principles, but much more effort is placed upon tying the effective potential to first principles, approximating the electron-electron interactions very carefully.[5]
Notes
- ↑ This model is described by almost every book on quantum mechanics. For example, see Manijeh Razeghi (2009). Fundamentals of Solid State Engineering, 3rd ed. Springer, pp. 160 ff. ISBN 9780387921679.
- ↑ This model is described by almost every book on solid state physics. For example, see Gerald D. Mahan (2009). Quantum mechanics in a nutshell. Princeton University Press, pp. 163 ff. ISBN 0691137137. and the very lucid discussion of Sir Nevill Francis Mott, H. Jones (1958). “§4.4 Distance between the atoms large (approximation of tight binding)”, The theory of the properties of metals and alloys, Reprint of 1936 ed. Courier Dover Publications, pp. 65 ff. ISBN 048660456X.
- ↑ A rigorous version of this expression replaces the atomic orbitals with the so-called Wannier functions. See, for example, Robert Aleksandrovich Ėvarestov (2007). “§3.3 Symmetry of localized crystalline orbitals: Wannier functions”, Quantum chemistry of solids: the LCAO first principles treatment of crystals. Springer. ISBN 3540487468.
- ↑ See the treatment in Mott & Jones, cited above, page 68.
- ↑ See, for example, the book by Martin in the general references.