Heat flow at the mesoscale

When we teach about thermal physics at the macroscopic scale, we talk in terms of the thermal conductivity, k.  For the 1d problem of a homogeneous rod of cross sectional area A and length L, the rate that energy flows from one end of the rod to the other is given by (kA/L)(T_h-T_c), where T_h and T_c are the temperatures of the hot and cold ends of the rod, respectively.  Built into this approach is the tacit assumption that the phonons, the quantized vibrational modes of the lattice that carry what we consider to be the thermal energy of the atoms in the solid, move in a diffusive way.  That is, if a phonon is launched, it bounces many times in a random walk sort of motion before it traverses across our region of interest.  Phonons can scatter off disorder in the lattice, or mobile charge carriers (or even each other, if the vibrations aren't perfectly harmonic).  

However, phonon motion doesn't have to be diffusive!   If phonons don't scatter while propagating a certain length scale, their motion is said to be "ballistic".  In this paper, the authors have done a very clever experiment to look at whether there is a significant contribution of ballistic phonons to heat transport in silicon at room temperature on scales considerably longer than the "textbook" mean free path for phonon scattering under those conditions, about 40 nm.  The authors use the interference pattern between two "pump" lasers to produce a (sin^2) intensity pattern (and thus, because of absorption and the electron-lattice coupling, a (sin^2) pattern of elevated temperature) in a suspended Si membrane.  The change in local temperature leads to a small change in local index of refraction.  A low intensity "probe" laser can diffract off the grating pattern set up by the temperature variation.  Depending on how long one waits between pump and probe, the temperature pattern can wash itself out due to phonon transport.  So, by varying the delay between pump and probe and looking at the strength of the diffracted probe signal, they can monitor the time evolution of the temperature profile.  By changing the pitch of the initial interferogram, they can look at thermal transport over different length scales.   They find that there are significant deviations from the expectations of diffusive phonon transport (originally worked out by Klaus Fuchs, among others) up to micron scales, which is pretty darn cool, and important for understanding heat flow in, e.g., computer chips.   Very elegantly done.