LO-Phonon Emission Rate of Hot Electrons from an On-Demand Single-Electron Source in a GaAs/AlGaAs Heterostructure

Using a recent time-of-flight measurement technique with 1 ps time resolution and electron-energy spectroscopy, we develop a method to measure the longitudinal-optical-phonon emission rate of hot electrons traveling along a depleted edge of a quantum Hall bar. Comparison to a single-particle model implies the scattering mechanism involves a two-step process via an intra-Landau-level transition. We show that this can be suppressed by control of the edge potential profile, and a scattering length $>1\mathrm{mm}$ can be achieved, allowing the use of this system for scalable single-electron device applications.

Figure 1

(a) SEM image of the device and electrical connections. The 2DEG region is shaded in light gray. Metallic gates are colored—electron pump $G_1$ and $G_2$, red; detector gate $G_3$, blue; depletion gate $G_5$, yellow; deflection gate $G_4$, green. Note that the area encircled by the yellow gate, $G_5$, is etched away. Long and short paths are marked with dashed and solid lines, respectively. (b) Measurement of the phonon emission probability as a fraction of the pumped current, for the case of the current traveling short path $S$ (black) and long path $L$ (blue). The survival fraction $1-P_0^{L(S)}$ is defined as the fraction of the detector current ($I_d$) at the phonon plateau against $ef$. Above, the derivative $d{I}_d/d{V}_{G3}^{\mathrm{dc}}$ of the long path trace shows the energy spectrum of electrons with original emission energy (left peak) and the ones that have emitted one LO phonon (right peak), which are separated by the LO-phonon energy of 36 meV. (c)–(e) Establishing the LO-phonon emission rate at $B=11\mathrm{T}$. (c) Survival fraction $1-P_{\mathrm{LO}}^l$ as a function of electron emission energy $E$, for different values of $V_{G5}$. (d) Electron drift velocity $v_d$ as a function of $E$, for different values of $V_{G5}$. (e) LO-phonon emission rate ${\mathrm{\Gamma }}_{\mathrm{LO}}=-(v_d/l)\ln (1-P_0^l)$ as a function of $E$, for different values of $V_{G5}$. (f) LO-phonon emission rate as a function of $E$, measured at different magnetic fields $B$ and at $V_{G5}=-0.25$ V. (g) Scattering length $L_{\mathrm{LO}}=v_d/{\mathrm{\Gamma }}_{\mathrm{LO}}$, for an energy $E=-25.2\mathrm{meV}$. A representative fit is shown in red. If we consider the strongest depletion $V_{G5}=-0.35\mathrm{V}$ at this energy, we find $L_{\mathrm{LO}}=1.4\mathrm{mm}$ (marked with a cross).

Figure 2

(a) The arrival time distributions $A_{\mathrm{LO}}(t)$ for electrons emitting 0 (black) and 1 (red) LO phonon. The good match to the expected distribution (blue) implies that the phonon emission rate is uniform. (b) The edge potential profile $\varphi$, measured for different $V_{G5}$. We take $(y,\varphi )=(0,0)$ as the point of highest electron emission energy attainable by the pump. Solid lines show a parabolic fit, which is used to calculate the phonon emission rate.

Figure 3

(a) A comparison of measured LO-phonon emission rates [squares; colors match Fig. 1] with the calculated rates for $m=0\to 0$ LO-phonon mode (solid lines) shows poor agreement. (b) Two lowest Landau levels at the sample edge ($m=0$ and 1), showing the phonon scattering paths. Red arrow, $m=0\to 0$ LO-phonon emission; green arrow, $m=0\to 1$ LADP phonon emission; blue arrow, $m=1\to 0$ LO-phonon emission. (c) The discrepancy fraction $\beta =\mathrm{expected}\mathrm{rate}/\mathrm{measured}\mathrm{rate}$ of phonon emission for the direct ($m=0\to 0$) mode. (d) As in (a), with the combined rate of emission from the $m=0\to 0$ LO mode and the $m=0\to 1$ LADP emission, the $m=1\to 0$ LO-emission mode (solid line) shows better agreement with the data. (e) The discrepancy fraction $\beta$ for the combined mode of direct and the two-step process. In this combined case, we see that we are much closer to unity.