Analytical solutions of the heat diffusion equation for 3ω method geometry

Abstract

"3ω" experiments aim at measuring thermal conductivities and diffusivities. Data analysis relies on integral expressions of the temperature. In this paper, we derive new explicit analytical formulations of the solution of the heat diffusion equation, using Bessel, Struve and Meijer-G functions, in the 3ω geometry for bulk solids. These functions are available in major computational tools. Therefore numerical integrations can be avoided in data analysis. Moreover, these expressions enable rigorous derivations of the asymptotic behaviors. We also underline that the diffusivity can be extracted from the phase data without any calibration while the conductivity measurement requires a careful one.

3-omega method for thermal properties of thin film multilayers

“…With proper substitutions and calculations, the last expression is related to the Meijer G-function by (9) :…”

“…Assuming that the heater circuit is stable, i.e., that all the transient perturbations decay over time, the steady-state harmonic temperature oscillations in the metal filament produce harmonic variations in the resistance given by (9) where R h,0 is the nominal -room temperature-resistance of the heater, ∆T DC is the steady-state temperature increase due to the rms power dissipated by the filament, ∆T AC is the magnitude of the steady-state temperature oscillations due to the sinusoidal component of the power and φ is the phase angle between the temperature oscillations and the excitation current. The resulting voltage across the sensor is obtained by multiplying the input current be the heater resistance yielding…”

Thermal Transport in Extremely Confined Metallic Nanostructures: TET Characterization

“…This is the famous Wiedemann–Franz (WF) law, but the WF law is not applicable to nanoscale metal film materials [ 11 , 12 , 13 , 14 , 15 , 16 ]. Based on the theoretical works related to the optimization of WF law and electrical conductivity [ 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 ], a series of methods have been developed to experimentally measure the thermal transport properties of metallic nanofilms and metallic nanowires. These methods include the 3ω method [ 25 , 26 , 27 , 28 ], the photothermal reflection technique [ 29 , 30 ], the femtosecond laser pumping detection method [ 31 ] and the non-stationary electrical heating method [ 32 , 33 , 34 , 35 ].…”