Notes on the Brillouin zone

by Brendan James Arnold
Reciprocal space – where condensed matter physicists live
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The Brillouin zone is a concept that I found tricky to get my head around. Listed here are a few questions that bothered me, hopefully you will find them helpful.

If \(k\) points in the Brillouin zone are given by \(2\pi/L\), does that mean atoms defined at the origin in real-space are now at infinity?

The reciprocal space maps out the wavelengths (and directions) of waves not real-space distributions (or at least not directly). To map out a real-space distribution in reciprocal space, we first take the Fourier transform of the distribution - the distribution of the Fourier transform would give you the wavevectors of the waves needed to recreate the real-space distribution.

An atom defined at the origin, also exists at the corner of the cell and the next cell and so on. As such, this atom would be represented in reciprocal space by a blob at the corner of the Brillouin zone (the wavelength of the wave that defines the atoms 'probability amplitude'), and because of the translation symmetry, the blob would also be at the origin of the Brillouin zone too.

Why do we use reciprocal space?

The chief reason for using reciprocal space is to map out waves and interactions with waves. Since waves are dealt with a lot in crystals, for example in neutron and x-ray scattering and calculating the wavefunctions of electrons etc. it makes sense to use a reciprocal lattice. Moreover since crystals are periodic themselves, the Fourier transforms are generally not too messy.

Fermi surface topology is another case where reciprocal space shines. Determining nesting conditions, for example, is easier to do with an electron wavefunction map in the Brillouin zone.

I don't know if this fully captures the reasons why we use reciprocal space and would be interested in knowing if the problems they tackle could be dealt with in real space.

If the distribution of \(k\) points is determined by the size of the entire crystal, doesn't splitting a crystal in half affects the distribution?

The size and shape of crystal affects density of k-states very little - see Lebowitz and Lieb PRL 22, 631 (1969) for a rigorous proof

Why do \(k\) vectors not necessarily point in the same direction as the electron velocity?

Take the example of a Fermi surface for electrons that only move in the \(ab\) plane (a cylinder reaching from the bottom to the top of the Brillouin zone). We know the electrons only move in the \(ab\) plane (with no vertical movement in the \(c\) direction), yet some of the \(k\) vectors at the Fermi surface have a \(k_z\) component - how can the electrons move in a different direction to their \(k\) vector?

The answer is due to the fact that Bloch electrons move in whichever way the probability amplitude moves and not the plane-wave. In the case of the Bloch wavefunction, \(\psi(x, t)=e^{(ik.x - \omega t)}u(k)\), where the wavefunction is actually given by a plane wave multiplied by function of the same periodicity of the lattice potential, the modulation of the plane wave by the lattice potential can cause a group velocity in another direction. In the 2D example, the lattice potential is 'layered' (holding the electrons in the plane). The direction of the group velocity of a plane wave with some component along the layered potential is shifted into the direction of the layers.

The animation below demonstrates this – push 'Play' to start it. It begins with what is clearly a plane wave in dark grey moving with a \(k\) vector which is at about 45 degrees up from the \(ab\) plane (n.b. at this stage the plane wave is artificially truncated in its extent so that you can easily distinguish it). Under the plane wave in light grey is a schematic layered lattice potential for a 2D crystal (we are looking along the layers). Holding down the mouse will extend the plane wave to infinity (no longer truncated, its 'true' form) and it becomes clear that the motion of the wave amplitude is now along the layers.

(Click the animation to extend the plane wave)
Light grey represents the lattice potential part of the Bloch function, dark grey represents the lattice plane wave part of the Bloch function

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