In solid state physics, the electronic band structure (or simply band structure) of a solid describes ranges of energy that an electron is "forbidden" or "allowed" to have. It is due to the diffraction of the quantum mechanical electron waves in the periodic crystal lattice. The band structure of a material determines several characteristics, in particular its electronic and optical properties. Solid-state physics, the largest branch of condensed matter physics, is the study of rigid matter, or solids. ... For other uses, see Solid (disambiguation). ... e- redirects here. ... The dynamical theory of diffraction describes the interaction of wave fields with a regular lattice. ... In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. ...

Why bands occur

The electrons of a single free-standing atom occupy atomic orbitals, which form a discrete set of energy levels. If several atoms are brought together into a molecule, their atomic orbitals split like in a coupled oscillation. This produces a number of molecular orbitals proportional to the number of atoms. When a large number of atoms (of order 10²⁰ 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. However, some intervals of energy contain no orbitals, no matter how many atoms are aggregated. In chemistry, an atomic orbital is the region in which an electron may be found around a single atom. ... Oscillation is the periodic variation, typically in time, of some measure as seen, for example, in a swinging pendulum. ... In chemistry, a molecular orbital is a region in which an electron may be found in a molecule. ...

These energy levels are so numerous as to be indistinct. First, the separation between energy levels in a solid is comparable with the energy that electrons constantly exchange with phonons (atomic vibrations). Second, it is comparable with the energy uncertainty due to the Heisenberg uncertainty principle, for reasonably long intervals of time. Normals modes of vibration progression through a crystal. ... Properties In chemistry and physics, an atom (Greek ἄτομος or átomos meaning indivisible) is the smallest particle still characterizing a chemical element. ... Oscillation is the variation, typically in time, of some measure as seen, for example, in a swinging pendulum. ... In quantum physics, the Heisenberg uncertainty principle, sometimes called the Heisenberg indeterminacy principle, expresses a limitation on accuracy of (nearly) simultaneous measurement of observables such as the position and the momentum of a particle. ...

A view popular in physics is to start with uncharged electrons and cores, which are therefore both free and plane waves and can have any energy, and then fade in the charge. This leads to Bragg reflection and therefore bands. In physics, Braggs law is the result of experiments into the diffraction of X-rays or neutrons off crystal surfaces at certain angles, derived by physicists Sir W.H. Bragg and his son Sir W.L. Bragg in 1912, and first presented on 1912-11-11 to the Cambridge...

Basic concepts

Simplified diagram of the electronic band structure of an insulator or semiconductor.

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. Image File history File links Electronic_band_diagram. ... Image File history File links Electronic_band_diagram. ...

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.

Metals contain a band that is partly empty and partly filled regardless of temperature. Therefore they have very high conductivity. Hot metal work from a blacksmith In chemistry, a metal (Greek: Metallon) is an element that readily loses electrons to form positive ions (cations) and has metallic bonds between metal atoms. ...

The uppermost occupied band in an insulator or semiconductor is called the valence band by analogy to the valence electrons of individual atoms. The lowermost 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 forbidden band gap between the valence band and conduction band is larger in an insulator, so that fewer electrons are found there and the electrical conductivity is less. 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 article or section is in need of attention from an expert on the subject. ... A semiconductor is a solid whose electrical conductivity can be controlled over a wide range, either permanently or dynamically. ... In solids, the valence band is the highest range of electron energies where electrons are normally present at zero temperature. ... In chemistry, valence electrons are the electrons contained in the outermost, or valence, electron shell of an atom. ... In semiconductors and insulators, the conduction band is the range of electron energy, higher than that of the valence band, sufficient to make the electrons free to accelerate under the influence of an applied electric field and thus constitute an electric current. ... This article or section does not adequately cite its references or sources. ... Electrical conductivity or specific conductivity is a measure of a materials ability to conduct an electric current. ...

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 create semiconductor devices such as solar cells, diodes, transistors, laser diodes, and others. In the solid state physics of semiconductors, carrier generation and recombination are processes by which mobile electrons and electron holes are created and eliminated. ... In semiconductor production, doping refers to the process of intentionally introducing impurities into an intrinsic semiconductor in order to change its electrical properties. ... Semiconductor devices are electronic components that exploit the electronic properties of semiconductor materials, principally silicon, germanium, and gallium arsenide. ... A solar cell, made from a monocrystalline silicon wafer A solar cell or photovoltaic cell is a device that converts light energy into electrical energy. ... Types of diodes closeup, showing germanium crystal In electronics, a diode is a component that restricts the direction of movement of charge carriers. ... Assorted discrete transistors A transistor is a semiconductor device, commonly used as an amplifier. ... A packaged laser diode with penny for scale. ...

Anderson's rule is used to create band diagrams between two semi-conductors. Andersons rule is used for the construction of energy band diagrams of the heterojunction between two semiconductor materials. ...

Band structures in different types of solids

Although electronic band structures are usually associated with crystalline materials, quasi-crystalline and amorphous solids 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 the dispersion relationship 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. Quartz crystal Synthetic bismuth crystal Insulin crystals Gallium, a metal that easily forms large single crystals A huge monocrystal of potassium dihydrogen phosphate grown from solution by Saint-Gobain for the megajoule laser of CEA. In chemistry and mineralogy, a crystal is a solid in which the constituent atoms, molecules... Quasicrystals are a peculiar form of solid in which the atoms of the solid are arranged in a seemingly regular, yet non-repeating structure. ... An amorphous solid is a solid in which there is no long-range order of the positions of the atoms. ... The relation between the energy of a system and its corresponding momentum is known as its dispersion relation. ...

Density of states

While the density of energy states in a band is very great, it is not uniform. It approaches zero at the band boundaries, and is generally greatest near the middle of a band. The density of states for the free electron model is given by, Density of states (DOS) is a property in statistical and condensed matter physics that quantifies how closely packed energy levels are in some physical system. ... In solid-state physics, the free electron model is a simple model for the behaviour of valence electrons in a crystal structure of a metallic solid. ...

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: Fermi-Dirac distribution as a function of ε/μ plotted for 4 different temperatures. ...

where:

• k is Boltzmann's constant,
• T is the temperature,
• E_F is the Fermi energy (or 'Fermi level').

The Fermi level naturally is the level at which the electrons and protons are balanced. The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... Fig. ... The Fermi energy is a concept in quantum mechanics referring to the energy of the highest occupied quantum state in a system of fermions at zero temperature. ...

Regardless of the temperature, f(E_F) = 1 / 2. At T=0, the distribution is a simple step function: In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of half-open intervals. ...

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. In mathematics and solid state physics, the first Brillouin zone is the primitive cell in the reciprocal lattice in momentum space. ...

Theory of band structures 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 (, , ). Now, any periodic potential 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: Ansatz (Ger. ... A Bloch wave or Bloch state is the wavefunction of a particle (usually, an electron) placed in a periodic potential. ... The dynamical theory of diffraction describes the interaction of wave fields with a regular lattice. ... In geometry and crystallography, a Bravais lattice, named after Auguste Bravais, is an infinite set of points generated by a set of discrete translation operations. ... In geometry and crystallography, a Bravais lattice, named after Auguste Bravais, is an infinite set of points generated by a set of discrete translation operations. ... In crystallography, the reciprocal lattice of a Bravais lattice is the set of all vectors K such that for all lattice point position vectors R. The reciprocal lattice is itself a Bravais lattice, and the reciprocal of the reciprocal lattice is the original lattice. ... The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ...

where 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

The nearly-free electron approximation in solid state physics is similar in some respects to the Hydrogen-like atom of quantum mechanics in that interactions between electrons are completely ignored. This allows us to use 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. This can be described mathematically by: A Bloch wave or Bloch state is the wavefunction of a particle (usually, an electron) placed in a periodic potential. ... This article discusses the concept of a wavefunction as it relates to quantum mechanics. ... In crystallography, the reciprocal lattice of a Bravais lattice is the set of all vectors K such that for all lattice point position vectors R. The reciprocal lattice is itself a Bravais lattice, and the reciprocal of the reciprocal lattice is the original lattice. ...

where the function is periodic over the crystal lattice.

(See for more detail Nearly-free electron model) In solid-state physics, the nearly free electron model is a model of electron behavior in solids that enables understanding the electronic band structure of crystalline materials. ...

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 new theories, such as the Hubbard model, to explain the discrepancy. A Mott Insulator is a metal that naturally does not conduct electricity, however under certain conditions the metal can be made to conduct electricity. ... This article is in need of attention from an expert on the subject. ...

Other

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 tight binding model, which assumes that each electron is usually associated with only one atom at a time, and treats the other atoms in the solid as perturbations.
• The Kronig-Penney model, which depicts the atoms as barriers to electron motion, while the electrons are otherwise free and independent. While simple, it predicts many important phenomena, but is not quantitatively accurate.
• Bands may also be viewed as the large-scale limit of molecular orbital theory. A solid creates a large number of closely spaced molecular orbitals, which appear as a band.
• Methods involving Green's function
• Hubbard model
• Density functional theory

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. In the tight binding model, electrons are treated as highly localised, and expanded as single electron wavefunctions in terms of atomic orbitals. ... In quantum mechanics, the particle in a one-dimensional lattice is an idealised system that can be solved completely with some simplifications. ... In quantum chemistry, molecular orbitals are the statistical states electrons can have within molecules. ... In mathematics, a Greens function is a type of function used to solve inhomogeneous differential equations subject to boundary conditions. ... This article is in need of attention from an expert on the subject. ... Density functional theory (DFT) is a quantum mechanical method used in physics and chemistry to investigate the electronic structure of many-body systems, in particular molecules and the condensed phases. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...

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). Metal halide lamps are similar to mercury vapor lamps, but instead of just mercury, they also contain all metals in the halide group of the periodic table (Hence the name). ... Sodium chloride, also known as common salt, table salt, or halite, is a chemical compound with formula NaCl. ...

References

1. Kotai no denshiron (The theory of electrons in solids), by Hiroyuki Shiba, ISBN 4-621-04135-5
2. Microelectronics, by Jacob Millman and Arvin Gabriel, ISBN 0-07-463736-3, Tata McGraw-Hill Edition.
3. Solid State Physics, by Neil Ashcroft and N. David Mermin, ISBN 0-03-083993-9,
4. Introduction to Solid State Physics by Charles Kittel, ISBN 0-471-41526-X
5. Electronic and Optoelectronic Properties of Semiconductor Structures - Chapter 2 and 3 by Jasprit Singh, ISBN 0-521-82379-X

See also

• Fermi surface
• Effective mass
• Dynamical theory of diffraction
• Solid state physics
• Ralph Kronig
• Anderson's rule