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We consider the heat equation for monolayer two-dimensional materials in thepresence of heat flow into a substrate and Joule heating due to electricalcurrent. We compare devices including a nanowire of constant width and a bowtie (or wedge) constriction of varying width, and we derive approximateone-dimensional heat equations for them; a bow tie constriction is described bythe modified Bessel equation of zero order. We compare steady state analyticsolutions of the approximate equations with numerical results obtained by afinite element method solution of the two-dimensional equation. Using thesesolutions, we describe the role of thermal conductivity, thermal boundaryresistance with the substrate and device geometry. The temperature in a deviceat fixed potential difference will remain finite as the width shrinks, but willdiverge for fixed current, logarithmically with width for the bow tie ascompared to an inverse square dependence in a nanowire.

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