This week in cond-mat

This week the subject is boundary conditions. When we teach about statistical physics (as I am this semester), we often need to count allowed states of quantum particles or waves. The standard approach is to show how boundary conditions (for example, the idea that the tangential electric field has to go to zero at the walls of a conducting cavity) lead to restrictions on the wavelengths allowed. Boundary conditions = discrete list of allowed wavelengths. We then count up those allowed modes, converting the sum to an integral if we have to count many. The integrand is the density of states. One remarkable feature crops up when doing this for confined quantum particles: the resulting density of states is insensitive to the exact choice of boundary conditions. Hard wall boundary conditions (all particles bounce off the walls - no probability for finding the particle at or beyond the walls) and periodic boundary conditions (particles that leave one side of the system reappear on the other side, as in Asteroids) give the same density of states. The statistical physics in a big system is then usually relatively insensitive to the boundaries.

There are a couple of physical systems where we can really test the differences between the two types of boundary conditions.
arxiv:0811.1124 - Pfeffer and Zawadzki, "Electrons in superlattices: birth of the crystal momentum"
This paper considers semiconductor superlattices of various sizes. These structures are multilayers of nanoscale thickness semiconductor films that can be engineered with exquisite precision. The authors consider how the finite superlattice result (nonperiodic potential; effective hardwall boundaries) evolves toward the infinite superlattice result (immunity to details of boundary conditions). Very pedagogical.

arxiv:0811.0565, 0811.0676, 0811.0694 all concern themselves with graphene that has been etched laterally into finite strips. Now, we already have a laboratory example of graphene with periodic boundary conditions: the carbon nanotube, which is basically a graphene sheet rolled up into a cylinder. Depending on how the rolling is done, the nanotube can be metallic or semiconducting. In general, the larger the diameter of a semiconducting nanotube, the smaller the bandgap. This makes sense, since the infinite diameter limit would just be infinite 2d graphene again, which has no band gap. So, the question naturally arises, if we could cut graphene into narrow strips (hardwall boundary conditions transverse to the strip direction), would these strips have an electronic structure resembling that of nanotubes (periodic boundary conditions transverse to the tube direction), including a bandgap? The experimental answer is, yes, etched graphene strips to act like they have a bandgap, though it's clear that disorder from the etching process (and from having the strips supported by an underlying substrate) can dominate the electronic properties.