Energy relaxation in hot electron quantum optics via acoustic and optical phonon emission

We study theoretically the relaxation of hot quantum-Hall edge-channel electrons under the emission of both acoustic and optical phonons. Aiming to model recent experiments with single-electron sources, we describe simulations that provide the distribution of electron energies and arrival times at a detector a fixed distance from the source. From these simulations we extract an effective rate of emission of optical phonons that contains contributions from both a direct emission process as well as one involving inter-edge-channel transitions that are driven by the sequential emission of first an acoustic and then an optical phonon. Furthermore, we consider the mean energy loss due to acoustic phonon emission and resultant broadening of the electron energy distribution and derive an effective drift-diffusion model for this process.

Figure 1

The rate densities ${\tilde{\mathrm{\Gamma }}}_{n^{\prime }n}^{\mathrm{LA}}(E-\epsilon ,E)$ as a function of energy loss $\epsilon$ for transitions induced by acoustic phonon emission in the outermost two edge channels $n,n^{\prime }=0,1$. The initial energy $E$ was 100 meV above the $n=0$ band bottom, and results are shown for two values of the magnetic field: $B=6\mathrm{T}$ (black solid), $B=12\mathrm{T}$ (blue dashed). Other parameters are as set out in Appendix pp1. At these fields, the subband spacings are respectively $\hslash \mathrm{\Omega }=10.3\mathrm{meV}$ and $\hslash \mathrm{\Omega }=20.9\mathrm{meV}$. The rate densities are defined such that the area under each curve gives the total rate out of a state with energy $E$ and subband $n$ into subband $n^{\prime }$. The curves for off-diagonal rates end abruptly at low $\epsilon$ due to the cut-off Eq. (13).

Figure 2

Energy-resolved arrival-time distribution (ATD) $A(E,\tau ,x_D)$ plotted as a function of energy loss $\delta E=E-E_0$, with $E_0$ the initial energy, and the arrival time $\tau$. White indicates absence of electrons, darker colors greater electron probability density. Clustered around $\delta E=0$ (horizontal line) is the ATD for electrons that arrive at the detector without having emitted any LO phonons. Around $\delta E=-\hslash {\omega }_{\mathrm{LO}}=-36\mathrm{meV}$ we see the ATD of the first LO-phonon replica. Parameters: $B=6\mathrm{T}$, $E_0=70\mathrm{meV}$, $x_D=28\mu \mathrm{m}$, ${\sigma }_E=1\mathrm{meV}$, and ${\sigma }_{\tau }=5\mathrm{ps}$. The number of electron trajectories followed was ${10}^5$.

Figure 3

As in Fig. 2, but here with the higher injection energy of $E_0=120\mathrm{meV}$. Distributions corresponding to three LO-phonon replicas (the third is faint), plus that of the directly transmitted electrons, are observed.

Figure 4

(a) The survival probabilities $P_0$ and $P_1$ as a function of injection energy $E_0$ for two magnetic field strengths, $B=6,12\mathrm{T}$. (b) Mean time of flight for electrons in the $A^{\left(0\right)}$ distribution, i.e., those that reach the detector without having emitted a LO phonon. ${\langle \tau \rangle }_0$ counts only those electrons detected in the outermost edge channel, whereas ${\langle \tau \rangle }_{01}$ counts those in both $n=0,1$ states. The difference between these two quantities is seen to be minor.

Figure 5

The extracted LO emission rates ${\gamma }_0$ (solid circles) and ${\gamma }_{01}$ (empty circles) as a function of injection energy for (a) $B=6\mathrm{T}$ and (b) $B=12\mathrm{T}$. Also plotted are the calculated LO emission rates ${\mathrm{\Gamma }}_{00}^{\mathrm{LO}\mathrm{tot}}$, ${\mathrm{\Gamma }}_{01}^{\mathrm{LO}\mathrm{tot}}$ and total LA emission rate ${\mathrm{\Gamma }}_{10}^{\mathrm{LA}\mathrm{tot}}$.

Figure 6

Drift diffusion of the $A^{\left(0\right)}$ electron distribution. (a) The scaled energy-drift parameter ${\tilde{v}}_E\equiv v_E/v_0$, which gives the mean energy loss of the electron per distance traveled (in $\mathrm{meV}/\mu \mathrm{m}$). (b) The similarly scaled diffusion parameter ${\tilde{D}}_E\equiv 2D_E/v_0$, which gives how much the variance increases per distance traveled (in ${\mathrm{meV}}^2/\mu \mathrm{m}$). Solid lines are from the full integrals of Eq. (26), whereas the dashed lines show the approximate forms from Eqs. (27) and (28). (c) The variance ${\langle E^2\rangle }_c$ of the $A^{\left(0\right)}$ distribution as a function of mean time-of-flight-squared ${\langle \tau \rangle }_0^2$ for $B=6,12$ T. The symbols represent results extracted from simulations; the straight lines are the analytic result from Eq. (29) with coefficients calculated from the full integrals.

Figure 7

Phonon emission rate densities ${\tilde{\mathrm{\Gamma }}}_{n^{\prime }n}(E-\epsilon ,E)$ as a function of energy loss $\epsilon$ for the three acoustic phonon interactions: LADP (solid black lines), LAPZ (blue dashed), and TAPZ (violet dashed-dot). Left: transitions within the outermost LL $n=0\to n^{\prime }=0$. Right: first inward transition $n=0\to n^{\prime }=1$. The magnetic field was $B=12\mathrm{T}$, and the starting energy was $E=100\mathrm{meV}$.

Figure 8

Survival probabilities (a) $P_0$ and (b) $P_1$ as a function of injection energy $E_0$ for two magnetic field strengths, $B=6,12$ T. Solid symbols show results with LO + LADP interactions only. Open symbols show results with LO + all three acoustic phonon interactions.