Scalable Sondheimer oscillations driven by commensurability between two quantizations

The electrical conductivity of metallic crystals exhibits size effects when the electron mean free path exceeds the sample thickness. One such phenomenon, known as Sondheimer oscillations, was discovered decades ago. These oscillations, periodic in magnetic field, have been hitherto treated with no reference to Landau quantization. Here, we present a study of longitudinal and transverse conductivity in cadmium single crystals with thicknesses ranging from 12.6 to 475 $\mu$m, and demonstrate that the amplitude of the first ten oscillations is determined by the quantum of conductance and a length scale that depends on the sample thickness, the magnetic length and the Fermi surface geometry. We argue that this scaling is unexpected in semiclassical scenarios and it arises from the degeneracy of the momentum derivative of the cross-sectional area $A$ along the orientation of the magnetic field $\frac{\partial A}{\partial k_z}$ in cadmium, which couples Landau quantization to the discretization of $k_z$ imposed by the finite sample thickness. We show that the oscillating component of the conductivity is uniquely governed by fundamental constants and the ratio of two degeneracies, which acts as an inverted filling factor. Our conjecture is supported by the absence of such scaling in thin copper crystals.