Distinguish electron-like and hole-like orbits in reciprocal space, with reference to the Brillouin Zones

So I have a solid state and thermodynamic exam next week and I've been going through some of the previous exams from years gone by to prepare. I came across this question "Distinguish electron-like and hole-like orbits in reciprocal space, with reference to the Brillouin Zones of squarium by sketching an example of each type." Apologies if this seems simple, solid state physics is not a strong point. Any help at all would be great! Also, what is squarium? I can't find anything in my notes on it and I'm wondering if it is simply referring to some sort of 2D square layout on which to sketch on. Thanks :)

• Squarium... yeah maybe it's just the square lattice? May 9, 2013 at 21:02
• Wow that was a quick reply! Yea i'm thinking that must be the case. Thanks very much :) any ideas about the distinguishing between electron-like and hole-like orbits in reciprocal space? I can't understand how a hole could have an orbit May 9, 2013 at 21:19

Think about a 2D Fermi surface (or a 2D section of a 3D Fermi surface). Now look in the extended Brillouin zone - thats the one where you take many copies of the first Brillouin zone and use it to tile the plane. Now we are in 2D so the Fermi surface is a bunch of curves in the extended BZ. There are three possibilities about the shape of the shape of this curve.

1. Your Fermi surface forms closed loops that enclose occupied electron states. This is an electronlike orbit.
2. Your Fermi surface froms closed loops that enclose unoccupied electron states. This is a holelike orbit.
3. Your Fermi does not form closed loops. This is an open orbit.

The reason for these names is that we are thinking of applying a perpendicular magnetic field $B_z$. In that case we have $$\frac{\partial k}{\partial t} = v_k \times B_z,$$ where the group velocity $v_k = \nabla_k E(k)$. Since $v_k$ is pointed in the opposite direction of $k$ in holelike orbits, the electrons orbits in the "wrong direction". That is it orbits in the direction that, naively, a positively charged particle would orbit in. Hence it is called holelike.

• Thank you very much! Very clearly explained. I tried to upvote you but unfortunately I just signed up for this so I still don't have the required reputation yet. Great answer, thanks again :) May 10, 2013 at 9:49
• I think you mean to say that we should look at the periodic zone scheme, where you take many copies of each brillouin zone and use it to tile the plane.
– DLV
Nov 27, 2017 at 17:02

For anyone who stumbles across this thread all these years later and is unclear on what exactly 'squarium' is...

Squarium: a two-dimensional material in which the atoms are arranged in a square crystal structure of lattice parameter 'a'. The lattice is therefore also square with the distance between lattice points being 'a'.

As for a reference, I am a final year Biology and Physics MSci at Durham University, UK, and this definition is credited to Professor Peter Hatton, Former Head of the Condensed Matter Research Group, Current Director of the Durham X-ray Group.