  A theory and mechanistic understanding of the thermal properties of solids
under nanoscale confinement are currently missing. We develop a theoretical
quantum confinement description of thin films which predicts a new physical law
for the heat capacity. In particular, due to the suppression of vibrational
modes caused by the thin film confinement, the vibrational density of states
(VDOS) deviates from the Debye quadratic law in frequency and is, instead,
cubic in frequency. This leads to a temperature dependence of the heat capacity
which is $\sim T^4$ instead of Debye's $\sim T^3$ law. Furthermore, the new
theory predicts a linear increase of the heat capacity upon increasing the
nanometric film thickness. Both dependencies are found in excellent agreement
with recent experimental data on NbTiN thin films.
