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

The linear dependence on temperature $\left(\gamma T\right)$ of the heat capacity at low temperatures $(T<15\mathrm{K})$ 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 $\gamma$ and vacancy concentrations for several materials. We have compiled many values of $\gamma$ and vacancy concentrations from the literature which show several significant trends that provide further evidence for our theory.

Figure 1

Selected linear terms $\gamma$ from fits to the low-temperature $(T<15\mathrm{K})$ heat capacity data of insulating materials. Hollow symbols represent the nanophase of the material; solid represents bulk.

Figure 2

Various possible distributions of energy gaps $n_{\mathrm{Sch}}\left(\theta \right)$ for the Schottky heat capacity arising from lattice vacancies. (a) Gaussian distribution. (b) Skewed Gaussian distribution. (c) Two Gaussian distributions summed. (d) Step distribution. See text for more details.

Figure 3

Heat capacities generated by summing Schottky distributions that have $n_{\mathrm{Sch}}$ and $\theta$ values corresponding to the distributions: (a) Gaussian, (b) skewed Gaussian, (c) sum of two Gaussians, and (d) step (as seen in Fig. 2). Plots are of $C/T$; therefore, a linear heat capacity will be a constant in these plots.

Figure 4

$\gamma$ versus $n_{\mathrm{vac}}$ of several samples. From left to right: $\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3$ (n), $\mathrm{C}{\mathrm{o}}_3{\mathrm{O}}_4,\mathrm{CuO}\left(\mathrm{n}\right),\mathrm{C}{\mathrm{o}}_3{\mathrm{O}}_4\left(\mathrm{n}\right),\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4\left(\mathrm{n}\right)$, and $\mathrm{CoO}\left(\mathrm{n}\right)$. Also shown are the lines derived from the four distributions of Schottky anomalies with the slopes (units of millijoules per mole per Kelvin) shown in parentheses in the legend.