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

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.

Thermoelectric Limitations of Graphene Nanodevices at Ultrahigh Current Densities

“…For such small bias currents, this temperature increase is mostly due to Joule heating and much less due to the Peltier effect. The temperature profile due to Joule heating is given by 49,52 i k j j j y { z z z…”

“…For such small bias currents, this temperature increase is mostly due to Joule heating and much less due to the Peltier effect. The temperature profile due to Joule heating is given by , T ( x ) = ( F 2 k ) ( s − x ) x + T b where T b is the temperature at the boundaries just after the jumps, s is the length of the device, k is the thermal conductivity of graphene, and F = PA –1 is the Joule heating generation term. Since this equation is extracted classically from Fourier’s law and is valid in the diffusive thermal transport regime, we choose to fit only the parabolic part of the temperature profile of the suspended regime (see the curve in the gray shaded part of Figure b) excluding the mini-peaks.…”

Enhanced Broadband Photodetection with Geometry and Interface Engineered Nanocrystalline Graphite

“…We have performed finite‐element simulations on the planar device geometry and found that 39% of the two‐probe resistance accounts for the central square‐shaped constriction, which is close to 47% derived by analytic expression. [ 29 ] The remaining resistance is due to the tapered NCG regions. Figure 2a shows the resistance of the device R versus the applied bias V for a substrate‐supported and suspended device.…”