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During the standard ac lock-in measurement of the resistance of atwo-dimensional electron gas (2DEG) applying an ac current $I = \sqrt{2} I_0\sin(\omega t)$, the electron temperature $T_e$ oscillates with the angularfrequency $2 \omega$ due to the Joule heating $\propto I^2$. We have shown thatthe highest ($T_\mathrm{H}$) and the lowest ($T_\mathrm{L}$) temperaturesduring a cycle of the oscillations can be deduced, at cryogenic temperatures,exploiting the third-harmonic (3$\omega$) component of the voltage dropgenerated by the ac current $I$ and employing the amplitude of the Shubnikov-deHaas oscillations as the measure of $T_e$. The temperatures $T_\mathrm{H}$ and$T_\mathrm{L}$ thus obtained allow us to roughly evaluate the thermalconductivity $\kappa_{xx}$ of the 2DEG via the modified 3$\omega$ method, inwhich the method originally devised for bulk materials is modified to beapplicable to a 2DEG embedded in a semiconductor wafer. The $\kappa_{xx}$ thusdeduced is found to be consistent with the Wiedemann-Franz law. The methodprovides a convenient way to access $\kappa_{xx}$ using only a standardHall-bar device and the simple experimental setup for the resistancemeasurement.

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