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Many interesting information about the properties of electrons in a crystal is encoded in the so-called band structure, which, for the semiconducting material gallium arsenide (GaAs), looks like this:

[Image]
Source: Michael Rohlfing, Peter Krüger, and Johannes Pollmann: Quasiparticle band-structure calculations for C, Si, Ge, GaAs, and SiC using Gaussian-orbital basis sets, Phys. Rev. B48 (1993) 17791-17805 (doi: 10.1103/PhysRevB.48.17791), Fig. 3.

The band structure plot shows the energy of electrons in the crystal as a function of momentum. Energy, along the vertical axis, is measured in units of electron Volt (eV), while momentum, on the horizontal axis, is given along specific directions in the crystal, which are denoted by the symbols Γ, Δ, X, Λ, and L.

For free electrons under conditions as typically encountered in solid-state physics, the relation between Energy E and momentum p is simply given by the classical, Newtonian formula E = p²/2m, where m is the mass of the electron. In a plot, energy as a function of momentum would be represented by a simple parabola. If we take into account quantum mechanics, we know for example that the energy states of electrons in atoms come in discrete steps. The energy levels of free electrons, however, are not quantised: According to quantum mechanics, free electrons are described just by plane waves. The inverse of the wavelength λ is, up to a factor of 2π, the so-called wavevector k = 2π/λ, which is proportional to momentum, k = p/ħ. Here, ħ, called "h-bar", is Planck's constant h divided by 2π - there are a lot of factors 2π floating around in this business...). Hence, for the free electron the relation between energy and wavevector is also given by a parabola, E = ħ²k²/2m.

We know, however, that electrons of atoms in a crystal are not free, but subject to the periodic potential of the positively charged atomic cores. As a consequence, the simple quadratic relation between wavevector respectively momentum and energy will be modified. Here, for example, is the crystal structure of gallium arsenide:

[Image]
Source: Wikipedia on Gallium Arsenide.

The brown balls mark the positions of the gallium atoms, the arsenic atoms are shown in violet. The crystal structure is called fcc (face-centred cubic), because the gallium atoms are located the corners and at the centres of the faces of a cube. Electrons in gallium arsenide have to reflect the symmetry of this crystal structure. Their wavefunctions are given by so-called Bloch waves.

Moreover, the wavelength of the electrons should not be shorter than the lattice constants of the crystal, i.e. the smallest periodicity length. This is because the same electron state could be described as well by a function with a longer wavelength. As a consequence of the minimal wavelength, the wavevectors have a maximal length. The possible wavevectors, then, all lay within a geometrical shape which is called the Brillouin zone. The Brillouin zone of the gallium arsenide crystal is shown here:

[Image]
Source: Wikipedia on the Brillouin Zone.

The centre of the Brillouin zone, corresponding to the wavevector k = 0 with no momentum, is denoted by Γ. Moreover, we see that the points L and X denote the centres of the hexagonal and quadratic faces of the Brillouin zone, thus corresponding to the maximal wavevectors in directions of the GaAs crystal normal to the faces of the cube and along the diagonal. The points Λ and Δ just denote half of these maximal wavevectors.

Now, we can understand the band structure plot. The figure shows the energy for electrons with wavevectors between the centre and the border of the Brillouin zone along the directions Γ-L and Γ-X. Lines are calculated, and the dots represent data points measured by photoemission spectroscopy.

We see that the energy of the electrons can be well described by a parabola close to the Γ point, but not for the whole Brillouin zone. Moreover, remarkably, there is not just one curve, but many. And there is a certain energy region which is not covered by any wavevector in the Brillouin zone - there are just no states available at these energies. This is the so-called band gap. States below the the gap are called states in the valence band, states above the gap are in the conduction band. Depending on whether there are electrons in the conduction band or not, the material can conduct electricity or not - it is a metal, or an insulator. In gallium arsenide, the valence band is completely full, and a few electrons are usually thermally excited to the conduction band. That's typical for a semiconductor.

Something that makes gallium arsenide special is the feature marked in orange in the band structure plot: The maximum of the valence band is at the same wavevector as the minimum of the conduction band. This is called a direct gap, and it means that electrons can be promoted from the conduction band to the valence band by the absorption of light. Similarly, the transition of an electron across the direct gap is accompanied by the emission of light. At a direct bandgap, a crystal can absorb and emit light, much like an isolated atom. The band gap of gallium arsenide at room temperature is 1.43 eV, corresponding to light of the wavelength 870 nm in the near infrared.

Thus, gallium arsenide with its direct band gap was one of the first materials used to build light emitting diodes (LEDs) and solid-state lasers.


Here is a nice, a bit more elaborate introduction to the concepts behind band structure plots (beware, pop stars pop up on the page).


This post is part of our 2007 advent calendar A Plottl A Day.