The heat equation for nanoconstrictions in 2D materials with Joule self-heating

Abstract

We consider the heat equation for two-dimensional materials in the presence of heat flow into a substrate and Joule heating due to electrical current. We compare devices including a nanowire of constant width and a bow tie (or wedge) constriction of varying width, and we derive approximate one-dimensional heat equations for them. For example, a bow tie constriction is described by the modified Bessel equation of zero order. We compare steady state analytic solutions of the approximate equations with numerical results obtained by a finite element method solution of the two-dimensional equation. Using these solutions, we describe how the thermal conductivity, thermal resistance to the substrate and device geometry combine to produce different peak temperature values and spatial profiles of the temperature.