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*=

*p*²/2

*m*, 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*=

*ħ*²

*k*²/2

*m*.

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:

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:

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.

Hello, Bee and Stefan

ReplyDeletecongratulations to this great idea of a plot'l a day, and for the way you carry it out.

When I have time I find your thought-stimulating blog always worth reading.

Btw I noticed a little typo here in your text in the relation between momentum and wave-number.

Cheers,

Robert

Dear Robert,

ReplyDeletethanks for the cheering words :-)

btw I noticed a little typo here in your text in the relation between momentum and wave-number.Hum, how embarrassing... in fact, more than one typo. All these factors of 2π and ħ can at times really be confusing ;-). But thank you very much for the hint, I've fixed it now!

Best, Stefan

Hey, I noticed that typo as well, but I thought you did that because html is somewhat insufficient. I didn't even know there is an hbar! What is the html code for it?

ReplyDeleteYou are too kind :-)..

ReplyDeleteOriginally, I had missed 2π in the relation between wavevector and wavelength, multiplied momentum by

hinstead of dividing byħ, and puth² instead ofħ² in the expression for energy - that's a nice maximation of errors ;-)... OK, it was late when I typed that text.BTW, h-bar = ħ is ħ in HTML.

Best, Stefan

That's why I preferably set Planck's constant equal to one.

ReplyDeleteI learn something new every day:

ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ ħ

:-)

Bee and Stefan,

ReplyDeleteI'd like to echo robert's sentiments. Given all the silliness on physics blogs recently, your advent calender is a nice breath of fresh air! Solid results, well explained...

1) Thanks for this sequence, it's quite wonderful and enlightening.

ReplyDelete2) What, no citation of the infamous Kovar Report?

The Institute for Advanced Studies at Austin propose a mass modification experiment using GaAs by manipulating the effective electron masses mechanically. See: http://www.earthtech.org/experiments/index.html

ReplyDeleteHi anonymous,

ReplyDeleteWhat, no citation of the infamous Kovar Report?Thanks for the pointer! I have to admit, I was not aware of this seminal piece of work! However, one should keep in mind that it covers pure germanium - as we all know, adding arsenic makes life much more complicated ;-)

Best, Stefan

The best MathML entity markup reference I know of is here: http://www.w3.org/TR/1998/

ReplyDeleteREC-MathML-19980407/chapter6.html

(take out the line break I added, in the preview the URL was being truncated)

It has entities for everything from the Bernoulli function to quaternion integral operators.

A very interesting plot, thanks for this one.

Hadn't seen the Kovar Report in years and had forgotten about it. Very hilarious.

Ha, I didn't know that report, thanks for the pointer, I had a good laugh :-) It reminds me so much of the experiments we did as undergrads. We underwent great pains to randomize the 'results' around the textbook curves. The only problem was then to figure out how to pretend we actually knew what the equipment was good for.

ReplyDeleteAt least we didn't pay tuition fees.

nice explaining:-)

ReplyDeleteis Brillouin zones are the rasult of bloch theorm ?

and thabks