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

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.