Lattice vacancies responsible for the linear dependence of the low-temperature heat capacity of insulating materials

Abstract

The linear dependence on temperature $(\ensuremath{\gamma}T)$ of the heat capacity at low temperatures $(Tl15\phantom{\rule{0.16em}{0ex}}\mathrm{K})$ is traditionally attributed to conduction electrons in metals; however, many insulators also exhibit a linear dependence that has been attributed to a variety of other physical properties. The property most commonly used to justify the presence of this linear dependence is lattice vacancies, but a correlation between these two properties has never been shown, to our knowledge. We have devised a theory that justifies a linear heat capacity as a result of lattice vacancies, and we provide measured values and data from the literature to support our arguments. We postulate that many small Schottky anomalies are produced by a puckering of the lattice around these vacancies, and variations in the lattice caused by position or proximity to some form of structure result in a distribution of Schottky anomalies with different energies. We present a mathematical model to describe these anomalies and their distribution based on literature data that ultimately results in a linear heat capacity. From these calculations, a quantitative relationship between the linear term and the concentration of lattice vacancies is identified, and we verify these calculations using values of $\ensuremath{\gamma}$ and vacancy concentrations for several materials. We have compiled many values of $\ensuremath{\gamma}$ and vacancy concentrations from the literature which show several significant trends that provide further evidence for our theory.