Electronic band structure
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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. The band structure determines a material's electronic properties, optical properties, and a variety of other properties.
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[edit] 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 due to the Pauli exclusion principle. This produces a number of molecular orbitals proportional to the number of atoms. When a large number of atoms (of order <math>10^{20}</math> 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.
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.
[edit] Basic concepts
Image:Electronic band diagram.svgAny 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.
Metals contain a band that is partly empty and partly filled regardless of temperature. Therefore they have very high conductivity.
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 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.
[edit] 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.
[edit] 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,
- <math>
D(\epsilon)= \frac{V}{2\pi^2}(\frac {2m}{\hbar^2})^{3/2} \epsilon^{1/2} </math>
[edit] 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:
- <math>f(E) = \frac{1}{1 + e^{\frac{E-E_F}{kT}}}</math>
where:
- k is Boltzmann's constant,
- T is the temperature,
- <math>E_F</math> is the Fermi energy (or 'Fermi level').
The Fermi level naturally is the level at which the electrons and protons are balanced.
Regardless of the temperature, <math>f(E_F) = 1/2</math>. At T=0, the distribution is a simple step function:
- <math>f(E) = \begin{cases} 1 & \mbox{if}\ 0 < E \le E_F \\
0 & \mbox{if}\ E_F < E \end{cases}</math>
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.
[edit] Band structure of crystals
[edit] 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.
[edit] Theory of band structures in crystals
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 (<math>\mathbf{b_1}</math>, <math>\mathbf{b_2}</math>, <math>\mathbf{b_3}</math>). Now, any periodic potential <math>V(\mathbf{r})</math> 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:
<math>V(\mathbf{r}) = \sum_{\mathbf{K}}{V_{\mathbf{K}}e^{i \mathbf{K}\cdot\mathbf{r}}}</math>
where <math>\mathbf{K} = m_1 \mathbf{b}_1 + m_2 \mathbf{b}_2 + m_3 \mathbf{b}_3 </math> for any set of integers <math>(m_1, m_2, m_3)</math>.
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.
[edit] 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:
<math>\Psi(\mathbf{r}) = e^{i \mathbf{k}\cdot\mathbf{r}} u(\mathbf{r}) </math>
where the function <math>u(\mathbf{r})</math> is periodic over the crystal lattice.
(See for more detail Nearly-free electron model)
[edit] 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.
[edit] 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 of 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.
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).
[edit] References
- Kotai no denshiron (The theory of electrons in solids), 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,
- 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
[edit] See also
cs:Pásová struktura de:Bandstruktur ja:バンド構造 pl:Pasmowa teoria przewodnictwa
