A theoretical study of the specific heat and Debye temperature of low-dimensional materials

A theoretical model is proposed in this work for an evaluation of the specific heat and Debye temperature of low-dimensional materials. In the model, the allowed discrete vibration modes in the confined direction(s) are first obtained by solving the elastic vibration equation. The acoustic specific heat is then calculated by summing over these discrete, excited, phonon modes and integrated over the continuous wave numbers in the unconfined directions. An effective Debye temperature is then defined as the one appearing in the conventional Debye model that gives a same value for the specific heat. It is found that the so-defined Debye temperature of the mixed polarization associated with nanowires is about half the longitudinal Debye temperature of bulk materials at room temperature. This agrees with the experimental observations. Those of both the dilatational and flexural polarizations associated with the thin films on the other hand are about 28% smaller than the bulk longitudinal Debye temperature. When the temperature is so low that there are only a few phonon modes excited, these low-dimensional materials show two-dimensional behavior, excluding the flexural polarization of the thin films, which shows one-dimensional behavior instead due to its parabolic dispersion relation at small dimensionless wave numbers.