Interaction-induced charge transfer in a mesoscopic electron spectrometer

In this paper, we theoretically investigate a mesoscopic electron spectrometer that allows for the probing of relaxation processes in quantum Hall edge channels, which has recently been experimentally realized. The device is composed of an emitter quantum dot that injects energy-resolved electrons into the channel closest to the sample edge, to be subsequently probed downstream by a detector quantum dot of the same type. In addition to inelastic processes in the sample that stem from interactions inside the region between the quantum dot energy filters (inner region), anomalous signals are measured when the detector energy exceeds the emitter energy. Considering finite-range Coulomb interactions in the sample, we find that energy exchange between electrons in the current-inducing source channel and the inner region, similar to Auger recombination processes, is responsible for such anomalous currents. In addition, our perturbative treatment of interactions shows that electrons emitted from the source, which dissipate energy to the inner region before entering the detector, contribute to the current most strongly when emitter-detector energies are comparable. Charge transfer in which the emitted electron is exchanged for a charge carrier from the Fermi sea, on the other hand, preferentially occurs close to the Fermi level.

Figure 1

(From Refs. [15, 16]) (a) Mesoscopic sample for the spectral selection of charge carriers. Electrons from a source electrode are injected at a well-defined energy into a reservoir region via an emitter quantum dot. Subsequent energy selection at a detector quantum dot allows for the probing of intermediate relaxation of the electrons in the reservoir region. In the presence of a strong magnetic field, the electrons propagate along chiral quantum Hall edge channels (blue line). (b) Drain lead current of the sample displayed in (a) in a strong magnetic field, as a function of the voltages applied to emitter and detector quantum dot (note that emitter energies increase toward the bottom and detector energies increase toward the left). Depending on the detector quantum dot voltage, the energy introduced by electrons from the source electrode gives rise to either an electron current (red) or a hole current (blue) through the detector into the drain. Currents are also measured when the detector energy exceeds the emitter energy (relative to the reservoir chemical potential, triangles III and IV).

Figure 2

Sketch of the experimental sample: source ($L$), drain ($R$), and reservoir region ($I$) channels are held at the chemical potentials ${\mu }_L, {\mu }_R$, and ${\mu }_I$, respectively. The source and drain channels are coupled to the reservoir region via the emitter quantum dot at energy ${\omega }_L$ and the detector quantum dot at energy ${\omega }_R$, with tunneling coupling strength $\mathrm{\Gamma }$.

Figure 3

Current (21) through drain in absence of interactions, for large bias between source and drain channels, and for (a) slightly positive, (b) zero, and (c) slightly negative bias between reservoir and drain channels. Note that dot energies increase from top to bottom and from right to left and that the aspect ratio has been adjusted, so as to match the conventions used in the experimental data shown in Fig. 1. [Parameters are ${\mu }_L={\mu }_R+400\mathrm{\Gamma }, {\mu }_I=$ (a): ${\mu }_R+30\mathrm{\Gamma }$, (b): ${\mu }_R$, (c): ${\mu }_R-30\mathrm{\Gamma }$.]

Figure 4

Tunneling-dressed (a) polarization and (b) exchange self-energy diagrams. Diagrams that determine the current through the drain are terminated by Green's functions of this channel (red lines). Tunneling (black crosses) from the intermediate region (blue lines) to the source (green lines) and drain channels through emitter and detector quantum dot, respectively, generates 27 diagrams for both self-energies. These 27 diagrams correspond to 27 distinct physical processes. Limiting tunneling as described in Sec. 6, only three of those processes remain.

Figure 5

Corrections due to interactions between electrons cause the elastic current (33) to decrease from the maximum value $I_R^{\text{max}}$ as the distance $\omega ={\omega }_R={\omega }_L$ of emitter and detector quantum dot energies is increased from the Fermi level ${\mu }_R$. Elastic current close to zero indicates parameter values for which the perturbative approach to second order in interactions becomes unphysical [cf. (33) and (34)]. [Plot includes (31), (32), and interchannel correction (51), parameters are ${\mu }_L={\mu }_R+400\mathrm{\Gamma }, v=260\lambda \mathrm{\Gamma }, {\nu }_0=640\lambda \mathrm{\Gamma }, \mathrm{\Delta }x=8\lambda , d=2.8\lambda$.]

Figure 6

Introducing interactions within the reservoir channel of the sample, three second-order processes contribute to the inelastic current: (a) In the first process, an incoming electron generates an electron-hole pair, and subsequently enters the detector quantum dot at a lower energy. (b) In the second process, the electron of the pair enters the detector, while the initial electron equilibrates in the reservoir region. Due to indistinguishability of the electrons, processes (a) and (b) interfere. (c) In the third process, the hole of the pair can escape into the detector when the latter's energy is tuned below the Fermi level.

Figure 7

Diagram corresponding to the amplitude of the inelastic process depicted in Fig. 6. This process gives rise to contribution (42) to the current through the drain channel.

Figure 8

Temperature dependence of inelastic contribution (40), stemming from the process depicted in Fig. 6, at fixed emitter energy. At temperatures comparable to the experimental value (red dashed line), the current is in good agreement with its value at $T=0$ (green dashed-dotted line), except within a range of $T$ to triangle outline (42). The black dashed-dotted line indicates the maximum of the current as $T$ is varied between 0 and $\infty$. (Parameters are ${\mu }_L={\mu }_R+400\mathrm{\Gamma }, {\omega }_L={\mu }_R+350\mathrm{\Gamma }, v=260\lambda \mathrm{\Gamma }, {\nu }_0=720\lambda \mathrm{\Gamma }, x=8\lambda$.)

Figure 9

Diagram corresponding to the process depicted in Fig. 6, in which the source electron is swapped for an electron from the Fermi sea, subsequently entering the detector quantum dot.

Figure 10

Diagram corresponding to the process in Fig. 6, in which the hole of the electron-hole pair that is generated by the source electron escapes into the detector quantum dot.

Figure 11

The exchange diagrams for the electron current are obtained upon combining the process diagrams in Figs. 7 and 9. The diagrams describe interference of these processes.

Figure 12

Exchange diagram corresponding to the hole current process depicted in Fig. 6. The diagram reduces the current as a consequence of the indistinguishability of electrons.

Figure 13

Simple model for interchannel interaction between reservoir and source. To close the channels into a single edge with given chirality, a point $x_C$ downstream of the emitter at $x_L^{\prime }$ in the source channel is identified with a point upstream of the emitter at $x_L$ in the reservoir channel. The interaction is subsequently defined to be maximal when the distance of electrons in the reservoir channel measured from $x_L$ coincides with the distance of electrons in the source channel measured from $x_L^{\prime }$.

Figure 14

Polarization contribution generated by interactions between charge carriers in the source lead and the reservoir region, showing relevant tunneling processes only.

Figure 15

Diagram describing the process in which the source electron, after tunneling into the reservoir region, creates an electron-hole pair in the source lead, before entering the detector quantum dot. The diagram is structurally equivalent to the diagram in Fig. 7, and reflects that the source lead provides a further decay channel when interactions between source and reservoir are taken into account.

Figure 16

(See also Ref. [16]) Diagram corresponding to the process in Fig. 17, in which the source electron interacts with the reservoir region where an electron-hole pair is created. The electron of the pair subsequently passes through the detector. The diagram creates current in triangle III in the experimental data shown in Fig. 1.

Figure 17

Introducing interactions between reservoir and source channels of the sample, three further inelastic second-order processes generate inelastic current: the first process (not shown) is equivalent to the process displayed in Fig. 6, with the exception that the electron-hole pair is generated in the source lead. (a) In the second process, (1.) an electron of the source lead enters the reservoir region. This electron is (2.) subsequently replaced by a further source electron dissipating energy. This energy (3.) creates an electron-hole pair in the reservoir region. The electron of the pair then (4.) enters the drain lead, where the absolute distance of the detector quantum dot energy from the Fermi level can be larger than the corresponding distance for the emitter. This process generates the upper left triangle in Fig. 1. (b) In the third process, the hole of the pair enters the drain when the detector energy is located below the Fermi level.

Figure 18

Diagram corresponding to the process of Fig. 17. In this process, the hole of an electron-hole pair, created in the reservoir by a virtual photon from the source, enters the detector. The diagram generates current in triangle IV in Fig. 1.

Figure 19

(See also Ref. [16]) Drain current ${\text{I}}_R$ on logarithmic scale. The processes depicted in Figs. 6, 6, and 6, caused by interactions in the reservoir region, generate current in the lower red and blue triangles [compare triangles I and II in Fig. 1]. The processes of Figs. 17 and 17, due to interactions between the reservoir region and the source lead, generate current in the inverted red and blue triangles [compare triangles III and IV in Fig. 1]. The present approach does not account for strong enough interactions to completely suppress the elastic current, as seen in the experimental data. (Parameters are ${\mu }_L={\mu }_R+400\mathrm{\Gamma }, v=260\lambda \mathrm{\Gamma }, {\nu }_0=720\lambda \mathrm{\Gamma }, \mathrm{\Delta }x=8\lambda , d=2.8\lambda$.)

Figure 20

Detector current ${\text{I}}_R$ for increasing spatial separation $\mathrm{\Delta }x$ between emitter and detector quantum dots. As the elastic peak at ${\omega }_L={\omega }_R$ diminishes for increasing separation, inelastic contributions intensify. The curve for the value $x=9\lambda$ (full line) is close to the boundary of validity of the approach [cf. (34)], after which the elastic current takes on negative values and the approximation becomes unphysical. While the peak next to the elastic contribution is associated with processes in which the initial electron is directly transferred [see Fig. 6], the peaks close to the Fermi energy ${\mu }_R$ are associated with processes in which a charge carrier from the Fermi sea enters the detector, both for the regular triangles in Figs. 6 and 6, as well as for the inverted triangles, Figs. 17 and 17. (Parameters are ${\mu }_L={\mu }_R+400\mathrm{\Gamma }, {\omega }_L={\mu }_R+350\mathrm{\Gamma }, v=260\lambda \mathrm{\Gamma }, {\nu }_0=720\lambda \mathrm{\Gamma }, d=2.8\lambda$.)