Auger-spectroscopy in quantum Hall edge channels and the missing energy problem

Quantum Hall edge channels offer an efficient and controllable platform to study quantum transport in one dimension. Such channels are a prospective tool for the efficient transfer of quantum information at the nanoscale, and play a vital role in exposing intriguing physics. Electric current along the edge carries energy and heat leading to inelastic scattering, which may impede coherent transport. Several experiments attempting to probe the concomitant energy redistribution along the edge reported energy loss via unknown mechanisms of inelastic scattering. Here we employ quantum dots to inject and extract electrons at specific energies, to spectrally analyse inelastic scattering inside quantum Hall edge channels. We show that the missing energy puzzle could be untangled by incorporating non-local Auger-like processes, in which energy is redistributed between spatially separate parts of the sample. Our theoretical analysis, accounting for the experimental results, challenges common-wisdom analyses which ignore such non-local decay channels.

Here we discuss in detail the alignment of the Emitter and Detector QD electrochemical potentials at the specific points indicated by circled numbers in Transport through the Emitter QD shows an enhanced current when the Emitter QD electrochemical potential is aligned with the Fermi energy of either the Source or the Reservoir (as indicated by the small black arrows). This enhancement with respect to the current between the two arrows is due to a Fermi edge singularity, studied in detail elsewhere [1]. Except for a slightly larger emission current, the Fermi edge singularity does not qualitatively influence the transfer measurements. The schematic level alignment at the points indicated by the circled numbers in Supplementary  The current through the Sensor QD, shown in Supplementary Figure 3b is the same as that shown in Figure 4a of the main text. The simultaneously measured current through the Emitter QD is shown * corresponding author: tobiaskr@phys.ethz.ch in Supplementary Figure 3a. Due to the increased bias on the Source contact for this measurement, (V Source = −700 µV) as well as the separation between Sensor and Emitter QDs, which is smaller than that between Emitter and Detector QD, we see a slight tilt in the horizontal lines indicating the onset of current. The enhanced current for V EM ≈ −0.705 V stems from an excited state in the Emitter QD and does not show a qualitative influence on the effects described in the main text. The influence of excited states, especially in the Detector QD will be discussed in the next section. The schematic level alignment of those points indicated by the circled numbers in Supplementary Figure 3b  To keep the tunnel coupling of the QDs to the leads at the same order of magnitude the voltages applied to the surface gate electrodes have to be adapted accordingly. A quantitative comparison between the measurements is thus not feasible. However, one can detect both the direct transfer features, as well as the Auger-like recombination at varying filling factors.

Supplementary Note 3: Contribution of Excited States to the Detected Transfer
Here we show further experimental results which illustrate the influence of excited states in transfer experiments.
Supplementary Figure 5a  higher lying states. On top of the standard transfer features which we usually observe in the Detector QD current, we can clearly resolve a prominent positive (red) current in the region where we would expect to measure a negative signal. The gate voltages at which these positive features appear clearly correlate with the energies of the excited state in the Detector QD as seen when increasing the Source bias. To understand why an excited state can lead to a positive current signal, we have to look at the level alignment of the Detector QD, which is shown in the inset of Supplementary  Figure 5b. The depicted situation corresponds to a plunger gate voltage of roughly V DET ≈ −0.457 V (in all four panels). The electrochemical potentials of the ground state as well as the first excited state are below the Fermi energy of the Reservoir contact. Thus, in equilibrium the ground state should be occupied by an electron. As long as the ground state stays occupied, the excited state transitions are energetically forbidden due to Coulomb repulsion. We have seen that the electron of the ground state can tunnel into an empty state of the Reservoir, which was generated by electron-electron collisions. As soon as the ground state is emptied, all the excited state transitions suddenly are allowed, and conventional transfer of electrons (from the Reservoir to the Drain) through an excited state becomes possible as long as the ground state stays empty. If the ground state becomes populated all excited state transitions are again energetically forbidden. This phenomenon resembles the Pauli-Spin blockade where transport switches between a blocked and unblocked situation due to different spin states [2]. Here the switching happens between the two transport directions Drain to Reservoir (through the ground state) and vice versa (over an excited state). The measured sign of the net current depends on both the tunnel coupling to the different orbital states, as well as the amount of electrons which pass through the Emitter QD.

Supplementary Note 4: Electrical Isolation of the Sensor QD
To confirm that there is no charge transfer between the Source and Right contacts of the sample, we show the sum of the currents measured in terminals Left (corresponding to I Sens ) and Right in Supplementary  Figure 3a) and is due to a capacitive coupling of the output of the corresponding IV-converters used in the experiment.

Supplementary Note 5: Additional Information on the Parameters in the Theoretical Model
The main purpose of the theoretical model, treating Source, Reservoir, Drain, Right and Left regions as single and parallel channels, is to demonstrate that interactions between electrons in these different regions of the sample generate current in triangles 1 -2 -4 and 1 -4 -6 in Fig. 3b of the main text, and in all of the triangles in Fig. 4b of the main text. Thereby, our theoretical analysis supports the conclusion that these currents are a result of Auger-like recombination processes. Interactions between electrons on the same edge in the reservoir region qualitatively account for the current in 1 -3 -4 and 1 -4 -5 . The dominant relaxation mechanism for currents in the latter two triangles is likely due to interactions between electrons on different channels on the same edge, which goes beyond the scope of our treatment and has been considered in detail in Refs. [3][4][5][6][7][8]. Such interactions alone cannot account for inter-edge Auger recombination.
The model interaction matrix element between electrons at a distance |r| in the same channel is given by U (|r|) = ν exp(−|r|/λ)/2λ. The prefactor of this interaction is chosen such that in the limit λ → 0 a contact interaction with strength ν is obtained, as is regularly employed in theoretical studies of relaxation in quantum Hall edges [5][6][7][8]. Furthermore, the Fourier transform of the interaction U is compatible with the approximation for small photon momenta that is considered, e.g., in Refs. [3,6]. The specific form of the screened interaction does not have a qualitative impact on the calculated current. To take into account phenomenologically the additional separation between electrons in different channels, the interaction U is further suppressed by a factor exp(−d i /λ), where d i indicates the distance between the respective channels (estimated from the point of injection at the Emitter QD).
Parameter values for tunneling rates, bias, Fermi velocity and distances, that generate the plots in Figs. 3b and 4b of the main text, are oriented to values encountered in typical samples, and listed in Supplementary Table 1. All features in these Figures are stable with respect to variations of all listed parameters. The parameters for interaction strength ν and screening length λ, that enter the above-described screened electron-electron interaction, are not easily accessible in the experiment. The combination for ν and λ employed in Fig. 3b and Fig. 4b of the main text generates a comparable ratio of currents due to inelastic processes to Emitter current eΓ/2h as observed in the experiment, and assumes a screening length that is short in comparison to sample dimensions.