Three-omega thermal-conductivity measurements with curved heater geometries

Abstract

The three-omega method, a powerful technique to measure the thermal conductivity of nanometer-thick films and the interfaces between them, has historically employed straight conductive wires to act as both heaters and thermometers. When investigating stochastically prepared samples such as two-dimensional materials and nanomembranes, residue and excess material can make it difficult to fit the required millimeter-long straight wire on the sample surface. There are currently no available criteria for how diverting three-omega heater wires around obstacles affects the validity of the thermal measurement. In this Letter, we quantify the effect of the wire curvature by performing three-omega experiments with a wide range of frequencies using both curved and straight heater geometries on SiO2/Si samples. When the heating wire is curved, we find that the measured Si substrate thermal conductivity changes by only 0.2%. Similarly, we find that wire curvature has no significant effect on the determination of the thermal resistance of an ∼65 nm SiO2 layer, even for the sharpest corners considered here, for which the largest measured ratio of the thermal penetration depth of the applied thermal wave to radius of curvature of the heating wire is 4.3. This result provides useful design criteria for three-omega experiments by setting a lower bound for the maximum ratio of the thermal penetration depth to wire radius of curvature.

Thickness-Dependent Cross-Plane Thermal Conductivity Measurements of Exfoliated Hexagonal Boron Nitride

“…The thermal conductivity of hBN, κ (units of W m –1 K –1 ), as a function of thickness, d , can be expressed as κ ( d ) = d R normalt normalh normale normalr normalm ( d ) × A where R therm ( d ) (units of m 2 K GW –1 ) is the total cross-plane thermal resistance and A is the surface area of a hBN flake of thickness d . We measure R therm ( d ) using a differential three-omega measurement where the temperature of a reference heating wire on the bare sample substrate is subtracted from the temperature rise of a heating wire on an hBN flake, as seen in Figure b. − This temperature difference can be converted to a measure of thermal resistance, R therm , from which the thermal conductivity can be calculated using eq . For this technique, a metal four-probe wire is fabricated across the film of interest and is then used as both a heater and a thermometer.…”

A Simple Numerical Method for Evaluating Heat Dissipation from Curved Wires with Periodic Applied Heating

“…A recent experimental work has demonstrated that, to some degree, the straight wire rule can be bent: thermal conductivity measurements of SiO 2 thin films on Si substrates performed with both straight and curved heating wires were found to agree to within the experimental uncertainty. 17 This a) Electronic mail: gjaffe@wisc.edu study provided a lower bound for how curved a heating wire can be without noticeably affecting the accuracy of a measurement. What is currently not known is at what point bends or curves in a wire cause significant deviations in the wire temperature from the straight-wire prediction, and how these deviations scale with parameters such as wire bend angle, wire curvature, and the thermal penetration depth of the applied thermal wave.…”

Long Phonon Mean Free Paths Observed in Cross-plane Thermal-Conductivity Measurements of Exfoliated Hexagonal Boron Nitride

“…These measurements are performed in a differential configuration, with the signal from a heating wire placed directly on the substrate subtracted from the signal measured with a heating wire on a pillar of hBN. [23][24][25][26][27] Fig. 1(a) shows an optical image of a heating wire on an hBN flake.…”