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Ultrafast energy relaxation of quantum dot-generated 2D hot electrons

Through a series of transverse magnetic focusing experiments, we show that hot electrons in a two-dimensional electron gas system undergo an ultrafast relaxation when generated by a quantum dot (QD) instead of a quantum point contact (QPC). We find here that QPC hot electrons were well described by the non-interacting Fermi gas model for excitations up to 1.5 meV above the Fermi level of 7.44 meV, whereas QD hot electrons exhibited an energy loss quadratic to the excitation. The energy relaxation was a sizeable fraction of the tested excitations, up to about 55%. With the proposal that the hot electrons are relaxed by the QD immediately after emission, we present a toy model in which a capacitive coupling between the QD and its leads results in a finite, ultrafast energy relaxation.

Quasiparticles form the basis upon which a vast array of many-body systems, ranging from band metals to topological insulators, are understood in condensed matter physics [1][2][3]. When extending beyond the ground state, the properties of quasiparticle excitations become crucial in describing system behaviors. In particular, the single electron excited above a Fermi sea has become the archetype of fermionic excitations. These excitations are commonly referred to as hot electrons and have been continuously studied over the past few decades due to their role in understanding coherent quantum devices [4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Specifically, their relaxation mechanisms are a central yet controversial topic, still remaining to be fully explained [18][19][20][21][22][23].
In mesoscopic physics, single hot electrons have been realized using the discrete energy levels of a quantum dot (QD) [24]. Naturally, QDs have found widespread use as energy filters, as both single hot electron sources and energy spectrometers [6,[13][14][15][16][17]25,26]. In the context of energy relaxation, these applications presume that QD-generated hot electrons have sufficiently long lifetimes. However, this presumption has recently been challenged by reports of ultrafast relaxation of QD hot electrons in integer quantum Hall systems [25,26]. Such relaxations had previously been veiled, possibly by the conventional use of quantum point contacts (QPC) in generating the hot electrons [4][5][6][7][10][11][12]16,17,27]. This posits the question of how QPC and QD hot electrons differ, and whether the latter have non-vanishing lifetimes.
Here, we report an ultrafast relaxation of QD hot electrons in the low magnetic field regime.  Fig. 1(a) was fabricated on a GaAs/AlGaAs heterojunction containing a two-dimensional electron gas (2DEG) 75 nm below the surface with density = 2.08 × 10 11 cm −2 and mobility = 3.8 × 10 6 cm 2 Vs ⁄ . Metallic Schottky gates of 75 nm width were deposited on the surface using standard electron beam lithography.
A QD was formed by depleting the 2DEG underneath four neighboring gates, and the plunger gate voltage ( ) was used to modulate the QD energy level. The QPCs were defined using three gates rather than the traditional two. A trench gate screens the electric field of the split gates and sharpens the confinement potential [28,29], which widens the inter-subband energy separation and allows the QPC to retain its conductance quantization for a wider range of bias voltage. An AC excitation of ≤ 10 μV rms at 987.6 Hz and a DC voltage bias ( ) were summed through a bias-tee and supplied to the relevant source reservoirs ( Fig. 1(a), red). This force deflects the free particle into a circular orbit that has a radius dependent on the particle's momentum and the magnetic field, i.e. the cyclotron radius. Consider a free electron emitted with momentum in the direction orthogonal to a collector located at distance . The electron is collected when a perpendicular magnetic field of strength 0 is present ( Fig. 1(a), yellow curve): where is the absolute charge of the electron. If there is a reflecting barrier between the emitter and the collector, the electron skips along the barrier and is collected at multiples of 0 . When a collimated beam of electrons is emitted, the collected current plotted against the magnetic field is called the focusing spectrum of the beam. Figure 1(b) is a typical example, measured from our sample using the QPC emitter ( Fig. 1(b), inset). The roughly periodic appearance of peaks corresponds to the multiples of some 0 , which we call the focusing field.
At the same field, however, electrons with greater kinetic energy have a larger cyclotron radius and will not be collected ( Fig. 1(a), red curve). In the massive Fermi gas model, an electron only has kinetic energy, i.e. ∝ 2 . Therefore, hot electrons with energy − above the Fermi level will have a modified focusing field B 0 ( ) related to that of equilibrium electrons by where we have assumed the effective mass to be constant [10]. By the above principle, TMF can be used as an energy spectrometer, akin to the optical monochromator [10,27]. Energy loss in hot electrons can then be observed as a deviation from Eq. (2).
QD Transverse Magnetic Focusing. We replaced the conventional QPC emitter with a QD to test if QD hot electrons retain their energies during emission and propagation in the 2D reservoir. However, performing QD TMF is nontrivial due to the effect of the magnetic field on the QD [24]. A magnetic field shifts the QD energy levels and rearranges the order of the orbitals. Consequently, the valence electron's energy becomes unpredictable, as seen in the non-monotonic shifts in QD conductance by the magnetic field in Fig. 2(a). Naïve use of a QD emitter leads to undesirable changes in the hot electron energy on the order of the QD excitation energy. Here, the magnetic energy shifts were cancelled by continually adjusting to align the transmitting QD level with the Fermi level of the leads. The Fermi level of a 2DEG does not change in low magnetic fields and thus serves as an appropriate energy reference. At the appropriate , the QD level aligns with the Fermi levels, and QD conductance is maximized.
We retrieved the focusing spectrum for a fixed energy level by examining , which is approximately the product of the focusing spectrum and the QD conductance ( Fig. 2(b)), at values where + exhibited Coulomb blockade peaks (Fig. 2(c), inset). Figure 2(c) is the resulting spectrum, normalized for changes in QD coupling strength by plotting against the magnetic field. This normalized focusing spectrum, which we call the focusing ratio, was used for the remainder of our analysis.
The QD focusing spectrum was similar to that from the QPC TMF. The first peak of the focusing spectrum was well predicted by Eq. (1) with the expected lineshape [31], but the skipping orbit peaks exhibited several deviations. First, the focusing field slightly deviated from the predicted value; the dominant source of error in this case can be attributed to geometric interference between multiple paths [31], boundary specularity and roughness issues in skipping orbits [32], and the increasing relevance of quantum Hall edge states [33]. Therefore, we restricted our interest to the first peak in order to avoid such irrelevant effects.

Hot Electron Transverse Magnetic Focusing. A clear difference between QD TMF and QPC
TMF was observed in hot electron experiments. The energies of QD hot electrons were controlled by aligning the QD level to the biased electrochemical potential − ( Fig.   3(a), Supplementary Fig. 1). The excitation levels were tuned to be greater than the estimated value of the thermal energy, so that only the main QD energy level would tunnel electrons. The resulting focusing spectra for from 0 mV to −1.5 mV are shown in Fig. 3(c), with redder lines corresponding to hotter electrons. At small , the focusing peak shifted linearly with , while at larger , the peak shift exhibited a visible nonlinearity (Fig. 3(c)). As a reference to traditional results, QPC TMF was also performed on the same device with care taken to maintain only one channel between the biased and grounded electrochemical potentials ( Fig. 3(b), Supplementary Fig. 2). This condition was satisfied by setting during transit from the QD to the collector, calculated from the relative focusing peak shift and Eq. (3). The 2 μm QD TMF data was measured using the previous emitter QPC as the collector; the 2 μm and 1 μm dots are horizontally offset by 0.01 mV for clarity. Error bars were assigned by summing the Coulomb peak width and the focusing peak position fitting error range by full width at half maximum; error bars for the QPC data are smaller than the marker. Unlike the QPC TMF, both QD TMF cases exhibit a similar, increasing energy loss sizeable to − . A quadratic fit to the 1 μm data is provided: Very little energy deficit was present in the QPC TMF spectra (Fig. 4(a), blue circles), which agrees with past reports [10,13,34]. However, a sizeable portion of the hot electron energy was lost in the QD TMF case (Fig. 4(a), red crosses). Moreover, the energy deficit in QD TMF was insensitive to doubling the focusing distance from 1 μm to 2 μm. Similar results were obtained for skipping orbit peaks as well.
From this discrepancy, we can speculate two possible scenarios: either QPC electrons relax less, or QD electrons relax more. Specifically, the first scenario implies that QPC-generated hot electrons are suppressed from relaxation in the 2D reservoir. This is unlikely to be the discriminant, since a relaxation mechanism in the 2DEG would lead to greater relaxations for longer focusing distances. In contrast, our results show that the QD hot electrons still retain a sizeable portion of their initial energy, as if the relaxation had abruptly stopped before depleting the electrons of their excess energy. Therefore, the second scenario is more likely, in which QD-generated hot electrons experience stronger relaxations. In particular, we expect the hot electrons to lose energy near the QD at a length scale much shorter than 1 μm.
Toy Model. The above phenomena can be qualitatively captured by the following toy model.
We extend the usual QD model to incorporate a capacitive interaction between the leads and the QD. An electron tunnels through the QD with period 0 = ⁄ ≳ 260 ps ( Fig. 4(b), blue).
The electron dwells inside the QD for a duration on the order of the QD level lifetime 1 = ℏ⁄ ≈ 65 ps ( Fig. 4(b), red), where is the QD-leads tunneling strength (Fig. 4(c)). Right after tunneling, the hot electron may capacitively interact with the QD within some interaction length scale . This length scale can be larger than the bulk 2D screening length ≈ 10 nm but must be much shorter than the focusing length ≈ 1.5 μm-the local screening length may be larger than the bulk 2D value due to gating effects, such as 2DEG depletion and lowered electron density. Such interaction occurs within an ultrafast timescale 2 = ⁄ ≪ 10 ps ( Fig. 4(b), orange), where is the Fermi velocity. During this time, the hot electron can transfer part of its energy to the QD (Fig. 4(d)) before propagating away. The excited QD is then left to relax with the leads (Fig. 4(e)) for 0 ≫ 1 , 2 before returning to its ground state to repeat the process ( Fig. 4(b), green). The expected hot electron energy loss is given by the expected QD excitation during 2 . Predictions from a semi-classical rate equation for a reasonable set of parameter values qualitatively resemble the observed energy loss (Supplementary Note 1 and Fig. 6).
After a careful comparison between QD and QPC emitters in magnetic focusing experiments, we conclude that an ultrafast relaxation exists for QD-generated 2D hot electrons on the order of the excitation. Such a relaxation was absent in QPC TMF, as reported in previous studies.
Although a more detailed model is required for better quantitative explanation, we found that a simple toy model can present a similar relaxation through a QD-mediated capacitive interaction. In particular, our relaxation model does not invoke the presence of a magnetic field and is symmetric under time reversal, suggesting that a QD may not only relax the hot electrons leaving it, but also those entering it. This relaxation may have gone unnoticed, since the energy loss becomes increasingly pronounced with higher excitations.
Our results suggest that QDs may not be reliable energy filters in 2D systems for energies larger than the QD level spacing. If a QD has limitations in creating a monoenergetic beam of electrons, then it may be possible that a QD also has similar limitations in accurately measuring the energies of lead electrons. In future research, a better non-QD spectrometer will be necessary in order to eliminate the inherent limitations of TMF; the peak broadening from beam collimation heavily burdens the task of tracking the precise hot electron energy. Nevertheless, we believe that our experiment confirms the presence of a large and ultrafast energy relaxation in QD hot electrons, an observation that will be important to further studies on quasiparticle relaxations, especially near local potential traps or impurities.