Joule heating and the thermal conductivity of a two-dimensional electron gas at cryogenic temperatures studied by modified 3$\omega$ method

TL;DR

This paper introduces a modified 3ω method to measure the thermal conductivity of a 2D electron gas at cryogenic temperatures by analyzing Joule heating effects and voltage harmonics during resistance measurements.

Contribution

It develops a novel approach to estimate the thermal conductivity of a 2DEG using standard resistance measurements and harmonic analysis, adapting the 3ω method for two-dimensional systems.

Findings

Thermal conductivity $c$ of 2DEG consistent with Wiedemann-Franz law.

Method allows simple measurement of $c$ with standard Hall-bar devices.

Temperatures during Joule heating oscillations can be deduced from third-harmonic voltage.

Abstract

During the standard ac lock-in measurement of the resistance of a two-dimensional electron gas (2DEG) applying an ac current $I = \sqrt{2} I_0 \sin(\omega t)$, the electron temperature $T_e$ oscillates with the angular frequency $2 \omega$ due to the Joule heating $\propto I^2$. We have shown that the highest ($T_\mathrm{H}$) and the lowest ($T_\mathrm{L}$) temperatures during a cycle of the oscillations can be deduced, at cryogenic temperatures, exploiting the third-harmonic (3$\omega$) component of the voltage drop generated by the ac current $I$ and employing the amplitude of the Shubnikov-de Haas oscillations as the measure of $T_e$. The temperatures $T_\mathrm{H}$ and $T_\mathrm{L}$ thus obtained allow us to roughly evaluate the thermal conductivity $\kappa_{xx}$ of the 2DEG via the modified 3$\omega$ method, in which the method originally devised for bulk materials is modified to be applicable to a 2DEG embedded in a semiconductor wafer. The $\kappa_{xx}$ thus deduced is found to be consistent with the Wiedemann-Franz law. The method provides a convenient way to access $\kappa_{xx}$ using only a standard Hall-bar device and the simple experimental setup for the resistance measurement.