# What is an electron/hole pocket and what is the significance?

What is an electron/hole pocket and what is the significance? I'm trying to get my head around this. I've read what Ashcroft and Mermin have to say on the subject, but it's a little convoluted. They talk about the 1st, 2nd, 3rd, etc Brillouin Zones (BZ), and then show that, as you increase the radius of the free electron Fermi sphere (FS), parts of its surface will lie within certain BZ's.

From there, you do this kind of strange process of taking the parts of the FS that lie within the nth BZ, and translate it with reciprocal vectors so they're within the 1st BZ. It seems like (in the example they use of a FCC metal) the "electron pockets" come from the 4th BZ being translated into the first.

I understand what they're doing, but I don't really see why. I also don't really see the significance of the pockets, but I keep seeing them mentioned in papers. I have a suspicion -- do they (reduced) BZ's (i.e., after you translate the appropriate part of the nth BZ into the 1st BZ) represent different bands (n = 1,2,etc), and the electron/hole pockets represent the highest band?

Any other elucidation would be greatly welcomed as well.

• Yes, the reduced BZ's can be labeled by band index $n$ if there is only a single site in the unit cell. But we do not care this index very much. I don't think it will make any difference if the electron pocket is coming from the 14th BZ instead of the 4th. All the bands that are fully filled below the Fermi energy are irrelevant to low-energy physics. The only band we care about is the highest band, where the Fermi surface rest in. The Fermi pockets are just the Fermi surface in this highest band. – Everett You Apr 6 '14 at 5:11

Fermi pockets (or Fermi surfaces) are contours of Fermi energy in the Brillouin zone. Depending on the effective mass $m^*$ of quasi-particles, the Fermi pockets can be divided into electron pockets (if $m^*>0$) and hole pockets (if $m^*<0$).
(1) Nesting Instability. If an electron pocket can be translated in the Brillouin zone by a wave vector $\vec{Q}$ to coincide with another hole pocket, then the system is subject to strong nesting instability and can develop SDW/CDW order with the ordering momentum $\vec{Q}$. So you can determine the order pattern by just looking at the position of the electron/hole pockets.
(2) Pairing Instability. Pairing instability is a special kind of nesting instability between the pocket and its own particle-hole conjugate (which is always perfectly nested), which happens right on the Fermi surface. So the pairing pattern is usually very sensitive the shape of the Fermi pocket. To gain the most energy from pairing gap opening, the nodal line of the pairing order must usually avoid the Fermi pockets, or avoid the van-Hove singularities on the Fermi pockets. This helps to intuitively understand the $s^{\pm}$-wave pairing in the iron-base superconductors and the $d$-wave pairing in copper-based superconductors.