  Knowledge of the mean free path distribution of heat-carrying phonons is key
to understanding phonon-mediated thermal transport. We demonstrate that thermal
conductivity measurements of thin membranes spanning a wide thickness range can
be used to characterize how bulk thermal conductivity is distributed over
phonon mean free paths. A non-contact transient thermal grating technique was
used to measure the thermal conductivity of suspended Si membranes ranging from
15 to 1500 nm in thickness. A decrease in the thermal conductivity from 74% to
13% of the bulk value is observed over this thickness range, which is
attributed to diffuse phonon boundary scattering. Due to the well-defined
relation between the membrane thickness and phonon mean free path suppression,
combined with the range and accuracy of the measurements, we can reconstruct
the bulk thermal conductivity accumulation vs. phonon mean free path, and
compare with theoretical models.
