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

Oliver M G Ward,Edward Mccann

doi:10.1088/1361-6463/ac21fe

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.

Highlights

There is huge interest in the thermal properties of twodimensional (2D) materials, motivated by applications to thermal management [1,2,3,4], interconnects in integrated circuits [5,6,7], thermoelectric devices [8,9,10,11] and nanoscale fabrication [12,13,14,15]
We compare devices including a nanowire of constant width and a bow tie 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
Joule heating is the local source of temperature increase that is generally dependent on position r as determined by conservation of current and the sample shape

Summary

Introduction

There is huge interest in the thermal properties of twodimensional (2D) materials, motivated by applications to thermal management [1,2,3,4], interconnects in integrated circuits [5,6,7], thermoelectric devices [8,9,10,11] and nanoscale fabrication [12,13,14,15]. We include two opposing sources of energy generation p(r) = pJ(r) + pB(r) where pJ(r) = j · E describes Joule heating with electrical current density j and electric field E, and pB(r) describes heat loss from the 2D monolayer sample in the vertical direction through sample-substrate or sample-air interfaces (depending on the particular setup) This may be parameterised with a thermal boundary resistance RB such that pB(r) = −[T(r) − T0]/RB for ambient temperature of the environment T0 [39,40,41, 55, 59, 60]. Geometries, figure 1, forms of Joule heating TJ(r), and values of LH

Rectangular sample

Nanowire

Bow tie constriction

Temperature dependent thermal conductivity

Conclusions

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