In condensed matter, what is a "valley", and why should you care?

One big challenge of talking about condensed matter physics to a general audience is that there are a lot of important physical concepts that don't have easy-to-point-to, visible consequences.  One example of this is the idea of "valleys" in the electronic structure of materials. 

To explain the basic concept, you first have to get across several ideas:

You've heard about wave-particle duality.  A free particle in in quantum mechanics can be described by a wavefunction that really looks like a wave, oscillating in space with some spatial frequency (\(k\ = 2 \pi\)/wavelength).  Momentum is proportional to that spatial frequency (\(p = \hbar k\)), and there is a relationship between kinetic energy and momentum (a "dispersion relation") that looks simple.  In the low-speed limit, K.E. \(= p^2/2m\), and in the relativistic limit, K.E. \( = pc \).

In a large crystal (let's ignore surfaces for the moment), atoms are arranged periodically in space.  This arrangement has lower symmetry than totally empty space, but can still have a lot of symmetries in there.  Depending on the direction one considers, the electron density can have all kinds of interesting spatial periodicities.  Because of the interactions between the electrons and that crystal lattice, the dispersion relation \(E(\mathbf{k})\) becomes direction-dependent (leading to spaghetti diagrams).  Some kinetic energies don't correspond to any allowed electronic states, meaning that there are "bands" in energy of allowed states, separated by gaps.  In a semiconductor, the highest filled (in the limit of zero temperature) band is called the valence band, and the lowest unoccupied band is called the conduction band.

Depending on the symmetry of the material, the lowest energy states in the conduction band might not be near where \(|\mathbf{k}| = 0\).  Instead, the lowest energy electronic states in the conduction band can be at nonzero \(\mathbf{k}\).  These are the conduction band valleys.  In the case of bulk silicon, for example, there are 6 valleys (!), as in the figure.

The six valleys in the Si conduction band, where the axes 

here show the different components of \(\mathbf{k}\), and 

the blue dot is at \(\mathbf{k}=0\).

One way to think about the states at the bottom of these valleys is that there are different wavefunctions that all have the same kinetic energy, the lowest they can and still be in the conduction band, but their actual spatial arrangements (how the electron probability density is arranged in the lattice) differ subtly. 

In the case of graphene, I'd written about this before.  There are two valleys in graphene, and the states at the bottom of those valleys differ subtly about how charge is arranged between the two "sublattices" of carbon atoms that make up the graphene sheet.  What is special about graphene, and why other some materials are getting a lot of attention, is that you can do calculations about the valleys using the same math that gets used when talking about spin, the internal angular momentum of particles.  Instead of being in one graphene valley or the other, you can write about having "pseudospin" up or down. 

Once you start thinking of valley-ness as a kind of internal degree of freedom of the electrons that is often conserved in many processes, like spin, then you can consider all sorts of interesting ideas.  You can talk about "valley ferromagnetism", where available electrons all hang out in one valley.  You can talk about the "valley Hall effect", where carriers of differing valleys tend toward opposite transverse edges of the material.   Because of spin-orbit coupling, these valley effects can link to actual spin physics, and therefore are of interest for possible information processing and optoelectronic ideas.