Effective medium theory for the low-temperature heat capacity of a metasolid plate

Nanopatterning can be used to strongly control the thermal properties of solids, but theoretical understanding relies often on complex numerical simulations. Here, an analytical theory is derived for the low temperature heat capacity of a nanopatterned phononic crystal plate, focusing on the geometry of a square lattice of cylindrical holes in an isotropic matrix material. Its quasistatic elastic properties were studied using an anisotropic effective medium theory, that is, considering it as a homogenized metasolid. The effective elastic parameters can then be used as an input for an anisotropic plate theory, yielding analytical expressions for the dispersion relations of the three lowest phonon modes that are dominant in the low temperature limit below 1K. Those results were then used to derive a simple analytical formula for the heat capacity, which was compared numerically with the exact results for an example material. The effects of material and geometric design parameters in the formula are also discussed, giving simple guidelines how to tune the heat capacity up to an order of magnitude or more.