Electronic band structure

In solid state physics, the electronic band structure (or simply band structure) of a solid is the series of "forbidden" and "allowed" energy bands that it contains. The band structure determines a material's electronic properties, optical properties, and a variety of other properties.

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, producing a number of molecular orbitals 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. 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.

Basic concepts

Number, size, spacing of bands et al

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.

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.

Filling of bands

Not all of these 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}{kT}}}

where:

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

Regardless of the temperature, $f(E_F) = 1/2$ . At T=0, the distribution is a simple step function:

f(E) = \begin{cases} 1 & \mbox{if}\ 0 < E \le E_F \\

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.

Conductors, insulators, semiconductors, and special bands

Solids can be divided into three classes based upon their band structure:

Insulators contain a completely empty allowed band directly above a completely filled allowed band (at absolute zero, when the Fermi-Dirac distribution is not smoothed). These bands are called the conduction band and valence band, respectively. The Fermi level falls almost exactly in the middle of the forbidden band between them. For reasons that are explained at Electrical conduction, such a solid has very low conductivity.

Semiconductors are similar to insulators, but the conduction and valence bands are spaced closely enough together that, at room temperature, a nontrivial number of electrons is found in the conduction band. These materials have significant conductivity that is highly temperature-sensitive.

Metals contain a band that is partly empty and partly filled regardless of temperature. They have very high conductivity. In metals, the distinction between the valence band and conduction band is moot.

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.

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.

Applications of the band structure

One of the most useful aspects of the band structure is the feature known as the band gap. In semiconductor and insulator materials, this is the gap between the valence band and the conduction band. The band gap and defect states created in the band gap by doping can be used to create devices such as solar cells, laser diodes, transistors, and a range of other electronic devices.

Theory of band structures in crystals

Every crystal is a periodic structure 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 ( $\mathbf{b_1}$ , $\mathbf{b_2}$ , $\mathbf{b_3}$ ). Now, any periodic potential $V(\mathbf{r})$ which shares the same periodicity as the direct lattice can be expanded out as a Fourier Series whose only non-vanishing components are those associated with the reciprocal lattice vectors. So the expansion can be written as:

$V(\mathbf{r}) = \sum_{\mathbf{K}}{V_{\mathbf{K}}e^{i \mathbf{K}\cdot\mathbf{r}}}$

where $\mathbf{K} = m_1 \mathbf{b}_1 + m_2 \mathbf{b}_2 + m_3 \mathbf{b}_3$ for any set of integers $(m_1, m_2, m_3)$ .

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:

$\Psi(\mathbf{r}) = e^{i \mathbf{k}\cdot\mathbf{r}} u(\mathbf{r})$

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

(See for more detail Nearly-free electron model)

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.

Other

Calculating band structures is an important topic in theoretical solid state physics. A number of methods have been developed, including:

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.

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 0-47-141526-X
Electronic and Optoelectronic Properties of Semiconductor Structures - Chapter 2 and 3 by Jasprit Singh, ISBN 0-521-82379-X

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