Measurement of Electron-Hole Friction in an $n$-Doped $\mathrm{GaAs}/\mathrm{AlGaAs}$ Quantum Well Using Optical Transient Grating Spectroscopy

We use phase-resolved transient grating spectroscopy to measure the drift and diffusion of electron-hole density waves in a semiconductor quantum well. The unique aspects of this optical probe allow us to determine the frictional force between a two-dimensional Fermi liquid of electrons and a dilute gas of holes. Knowledge of electron-hole friction enables prediction of ambipolar dynamics in high-mobility electron systems.

Figure 1

(a) Relative amplitude of $e\mathrm{\text{−}}h$ density wave as function of time for several values of the wave vector $q$ measured at 50 K. Inset: The decay rate $\gamma$ of the amplitude, plotted as a function of $q^2$; the slope of the solid line through the data points is the ambipolar diffusion coefficient $D_a$, and the intercept is the inverse of the $e\mathrm{\text{−}}h$ recombination time ${\tau }_{\mathrm{rec}}$. (b) Linear advance of the phase of the $e\mathrm{\text{−}}h$ density wave with time for several values of $q$, at 50 K. The applied electric field is $E\approx 2\mathrm{V}/\mathrm{cm}$. Inset: The rate of phase change $\dot{\varphi }$ as a function of $q$.

Figure 2

(a) Ambipolar mobility ${\mu }_a$ at three different pump intensities compared with electron mobility ${\mu }_e$, as a function of $T$. Inset: ${\mu }_a$ as a function of laser intensity at 5 K, the solid line is a fit (see text). (b) Ambipolar mobility ${\mu }_a$ and sample resistance $R$ as a function of gate voltage $V_{BT}$ at 50 K. Inset: ${\mu }_a$ as a function of laser intensity at fixed gate bias $V_{BT}=1.5\mathrm{V}$ at 50 K; solid line is a linear fit showing that ${\mu }_a\propto 1/I$.

Figure 3

(a) Comparison of electron mobility ${\mu }_e$; ambipolar mobility ${\mu }_{a0}$ and packet mobility ${\mu }_p$ as a function of $T$. (b) The $e\mathrm{\text{−}}h$ drag transresistivity ${\rho }_{eh}$ as a function of $T$; solid line is a theoretical prediction of ${\rho }_{eh}$ based on the RPA. Inset: $D_a/{\mu }_p$ compared with the Einstein relation prediction in the nondegenerate regime $k_{B}T/e$.