Low-dimensional nanostructures such as nanotubes, nanowires, and quantum dots are promising building blocks for electronic, optical, sensing, and energy conversion applications. For effective device design it is important to understand how the basic thermal properties of nanostructures differ from those of bulk materials. For example, the measured thermal conductivity of silicon nanowires  can be understood with a 3-dimensional dispersion relation  for diameters down to about 40 nm, although at 22 nm diameter the experiment and modeling diverge sharply.
Low-Dimensional Phonon Heat Capacity of Titanium Dioxide Nanotubes
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Dames, C, Chen, G, Poudel, B, Wang, W, Huang, J, Ren, Z, Sun, Y, Oh, JI, Opeil, C, & Naughton, MJ. "Low-Dimensional Phonon Heat Capacity of Titanium Dioxide Nanotubes." Proceedings of the ASME 2004 3rd Integrated Nanosystems Conference. Design, Synthesis, and Applications. Pasadena, California, USA. September 22–24, 2004. pp. 71-72. ASME. https://doi.org/10.1115/NANO2004-46080
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