Spectroscopic study on hot-electron transport in a quantum Hall edge channel

Hot electron transport in a quantum Hall edge channel of an AlGaAs/GaAs heterostructure is studied by investigating the energy distribution function in the channel. Ballistic hot-electron transport, its optical-phonon replicas, weak electron-electron scattering, and electron-hole excitation in the Fermi sea are clearly identified in the energy spectra. The optical-phonon scattering is analyzed to evaluate the edge potential profile. We find that the electron-electron scattering is significantly suppressed with increasing the hot-electron's energy well above the Fermi energy. This can be understood with suppressed Coulomb potential with longer distance for higher energy. The results suggest that the relaxation can be suppressed further by softening the edge potential. This is essential for studying noninteracting chiral transport over a long distance.

Figure 1

(a) Schematic energy diagram of the lowest Landau levels (LLLs) for spin-up and -down electrons. Electron-electron scattering (i) and LO phonon scattering (ii) are illustrated. (b) Schematic energy diagram of hot electron spectroscopy. A hot electron injected from emitter E experiences relaxation associated with LO phonon emission and electron-electron scattering. The resulting energy distribution function is measured with tunable barrier ${\varepsilon }_{det}$. (c) The measurement setup with a scanning electron micrograph of an $L=5$ $\mu \mathrm{m}$ device. Other devices have slightly different gate patterns but are conceptually the same. Bias voltage $-V_{\mathrm{E}}$ (with positive $V_{\mathrm{E}}$) is applied on the emitter contact to inject hot electrons to the edge channel. A small voltage $V_b=0–10\mu \mathrm{V}$ was applied to the collector (see Appendix).

Figure 2

(a) $I_{det}$ and $I_{\mathrm{inj}}$, (b) $I_{det}/I_{\mathrm{inj}}$, and (c) $F=d(I_{det}/I_{\mathrm{inj}})/d{V}_{det}$ as a function of $V_{\det }$ for an $L=5\mu \mathrm{m}$ device on wafer W1. The equispaced peaks in (c) represent the ballistic transport (the leftmost peak) and its phonon replicas. The step height $P_0$ in (b) measures the probability of ballistic transport for $L=5\mu \mathrm{m}$. The peak with $I_{det}/I_{\mathrm{inj}}>1$ in (b) shows electron-hole excitation in the Fermi sea, and thus defines the condition for ${\varepsilon }_{det}=0$. The width $w$ (= 5 meV in energy) of the leftmost peak in (c) shows the energy resolution of this measurement. The reduced detector current ($I_{det}/I_{\mathrm{inj}}\simeq 0.5$) at $V_{det}>-0.5$ V is associated with large $Z_{\mathrm{m}}=1\mathrm{M}\mathrm{\Omega }$ at $V_{\mathrm{b}}=0$ (Appendix).

Figure 3

(a) Representative hot-electron spectrum $F$ as a function of ${\varepsilon }_{det}$ on the right axis, for various $V_{\text{E}}$. (b) Some traces $I_{det}/I_{\mathrm{inj}}$ at several $V_{\text{E}}$'s with horizontal offsets for clarity. The data were taken at $B$ = 9 T ($\nu =1.3$) and $V_{\mathrm{b}}=5$ $\mu \mathrm{V}$ with an $L=1.4$ $\mu \mathrm{m}$ device on wafer W2. The normalized peak height $\mathrm{\Delta }I/I_{\mathrm{inj}}$ at ${\varepsilon }_{det}=0$ in (b) is analyzed in Fig. 6.

Figure 4

(a) Phonon replicas of hot electrons in LLL and SLL seen in the color-scale plot of $F$, taken at $V_{\mathrm{E}}=180$ mV and $V_{\mathrm{SG}}=-1.2$ V with an $L=0.7$ $\mu \mathrm{m}$ device on wafer W2. The inset shows the direct LO phonon emission (dLO) within the LLL, inter-Landau-level (iLL) tunneling to the SLL, and inter-landau-level LO phonon (iLO) emission. (b) $B$ dependence of ${\ell }_{\mathrm{LO}}$, taken at $V_{\mathrm{E}}=100$ mV with an $L=0.7$ $\mu \mathrm{m}$ device on wafer W2 [different from the device in (a)]. The solid line is calculated for the dLO process, while the dashed line is calculated for the iLL transition.

Figure 5

(a) and (b) Energy $\varepsilon$ dependence of $\overline{E}$. The data with open squares were obtained from the $B$ -dependence, while other data with small dots were estimated from a single value of ${\ell }_{\mathrm{LO}}$. They were taken with a 0.7 $\mu \mathrm{m}$ device on wafer W2. The solid line in (a) is obtained from a self-consistent potential calculation shown in (c). The dashed lines represents the electric field for potential $\varphi \propto x^{\xi }$ ($x>0$) with $\xi =$ 2, 1.5, and 1.3. The lower-energy data follow $\xi =1.3$. (c) Self-consistent potential profile $\varphi$ for a realistic device geometry, shown in the upper inset, with $V_{\mathrm{SG}}=-0.85$ V and $V_{\mathrm{IPG}}=-0.2$ V. (d) $V_{\mathrm{SG}}$ dependence of ${\ell }_{\mathrm{LO}}$, taken with three devices ($L=5$, 10, and 15 $\mu \mathrm{m}$) on wafer W1.

Figure 6

(a) The color-scale plot of $F\left({\varepsilon }_{det},V_E\right)$. The peak deviates from the ballistic condition (the dashed line), and is explained well with the model calculation (the solid line at $\lambda =1$). (b) The obtained $\mathrm{\Delta }I/I_{\mathrm{inj}}$ (solid circles) as a function of $e{V}_E$. The data are consistent with the model calculation (the solid line at $\lambda =1$). (c) and (d) The model calculation of hot-electron energy ${〈\varepsilon 〉}_L$ in (c) and $\mathrm{\Delta }I/I_{\mathrm{inj}}$ in (d) for different $\lambda$.

Figure 7

$B$ dependence of $E_{\mathrm{th}}$ in (a) and $\mathrm{\Delta }{I}_{max}/I_{\mathrm{inj}}$ in (b) taken with several devices on wafer W2. The vertical dashed line shows the condition for $\nu$ = 2. The inset to (a) shows the interaction between a hot electron and electrons in the LLLs and the SLL at $\nu >$ 2. The right inset to (b) shows the interaction between the LLLs and the SLL. The left inset to (b) shows an equivalent circuit for relating $I_{det}$ and $I_{\mathrm{base}}$ with $I_{\mathrm{inj}}=I_{\mathrm{inj}}^{\left(\mathrm{rel}\right)}+I_{\mathrm{inj}}^{\left(\mathrm{hot}\right)}$.