The electrothermal conductance and heat capacity of black phosphorus

Abstract

We study a thermal gradient induced current $\left(I_{th}\right)$ flow in potassium-doped two-dimensional anisotropic black phosphorus (BP) with semi-Dirac dispersion. The prototype device is a BP channel clamped between two contacts maintained at unequal temperatures. The choice of BP lies in the predicted efficient thermolectric behaviour. A temperature-induced difference in the Fermi levels of the two contacts drives the current (typified by the electro-thermal conductance) which we calculate using the Landauer transport equation. The current shows an initial rise when the device is operated at lower temperatures. The rise stalls at progressively higher temperatures and $I_{th}$ acquires a plateau-like flat profile indicating a competing effect between a larger number of transmission modes and a corresponding drop in the Fermi level difference between the contacts. The current is computed for both \textit{n}- and \textit{p}-type BP and the difference thereof is attributed to the particle-hole asymmetry. The utility of such calculations lie in conversion of the heat production and an attendant temperature gradient in miniaturized devices to useful electric power and a possible realization of solid-state Peltier cooling. Unlike the flow of $I_{th}$ that removes heat, the ability of a material to maintain a steady temperature is reflected in its heat capacity which is formulated in this work for BP via a Sommerfeld expansion. In the concluding part, we draw a microscopic connection between the two seemingly disparate processes of heat removal and absorption by pinning down their origin to the underlying density of states. Finally, a qualitative analysis of a Carnot-like efficiency of the considered thermoelectric engine is performed drawing upon the previous results on thermal current and heat capacity.