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

TL;DR

This paper models heat distribution in 2D materials with nanoconstrictions under Joule heating, deriving approximate equations and comparing steady state solutions to understand thermal behavior in different device geometries.

Contribution

It introduces approximate one-dimensional heat equations for nanoconstrictions and compares analytical solutions with numerical results, highlighting effects of geometry and thermal properties.

Findings

Temperature remains finite at fixed potential difference as width shrinks.

Temperature diverges logarithmically with width at fixed current.

Device geometry significantly influences thermal behavior and heat distribution.

Abstract

We consider the heat equation for monolayer 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; 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 the role of thermal conductivity, thermal boundary resistance with the substrate and device geometry. The temperature in a device at fixed potential difference will remain finite as the width shrinks, but will diverge for fixed current, logarithmically with width for the bow tie as compared to an inverse square dependence in a nanowire.