Quantum Size Effect on the Lattice Specific Heat of Nanostructures

For structures consisting of a few atomic layers (in one, two, or three dimensions), discretized modes of lattice vibration must be considered and their contributions need to be evaluated by summation (in the confined directions) rather than integration over the wavevector components as done for bulk materials in the Debye theory. Existing studies on the specific heat of nanostructures have left a question as to whether the lattice specific heat of nanostructures will be suppressed or enhanced over the corresponding bulk value at cryogenic temperatures. This study presents explicit analytical formulations of the lattice specific heat of nanostructures under the assumption of linear dispersion. The specific heat at sufficiently low temperatures is dominated by the lowest-energy vibration modes. Subsequently, the planar modes (for which one of the wavevector components becomes zero) or axial modes (for which two of the wavevector components become zero) are responsible to the T2 or T dependence of specific heat for thin films or nanowires, respectively, resulting in a specific-heat enhancement over the bulk materials (whose specific heat depends on T3 at low temperatures). As the temperature is reduced, however, the specific heat decays faster for cubic nanoparticles than for bulk materials, because the contribution of the lowest vibration modes in nanoparticles is proportional to (1/T2) exp(−1/T) at extremely low temperatures. The effects of the aspect ratio and shape on the specific heat of nanoparticles as well as boundary conditions are also discussed. This work clarifies the trends in the lattice specific heat of nanostructures at low temperatures.