Local signatures of electron-electron scattering in an electronic cavity
Carolin Gold, Beat A. Bräm, Richard Steinacher, Tobias Krähenmann, Andrea Hofmann, Christian Reichl, Werner Wegscheider, Mansour Shayegan, Klaus Ensslin, Thomas Ihn
LLocal signatures of electron-electron scattering in an electronic cavity
Carolin Gold, ∗ Beat A. Br¨am, Richard Steinacher, Tobias Kr¨ahenmann, Andrea Hofmann, Christian Reichl, Werner Wegscheider, Mansour Shayegan, Klaus Ensslin, and Thomas Ihn Solid State Laboratory, ETH Zurich, 8093 Zurich, Switzerland Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA (Dated: January 15, 2021)We image equilibrium and non-equilibrium transport through a two-dimensional electronic cavityusing scanning gate microscopy (SGM). Injecting electrons into the cavity through a quantum pointcontact close to equilibrium, we raster-scan a weakly invasive tip above the cavity regions and mea-sure the modulated conductance through the cavity. Varying the electron injection energy between ± ± . et al. [Jura et al.,Phys. Rev. B , 155328 (2010)] who used a strongly invasive tip potential to study electron in-jection into an open two-dimensional electron gas. This resemblance suggests a similar microscopicorigin based on electron-electron interactions. I. INTRODUCTION
Electron–electron interactions and their role in elec-tron transport are a topic of continuing interest inmesoscopic physics. Due to momentum conservation inelectron–electron scattering processes, the latter do notinfluence the electron mobility unless paired with anotherscattering mechanism [1]. Interactions have been foundto impact e.g. the conductivity of disordered systemsvia Friedel oscillations around screened impurities [2, 3],and are the key low-temperature decoherence mecha-nism in quantum transport experiments [4] such as theAharonov–Bohm effect [5–7] or weak localization [8]. Re-cently, renewed interest in viscous effects observed in elec-tron liquids at elevated temperatures has arisen [9, 10].The rich variety of existing experiments includes at-tempts to probe electron-electron scattering by injectingnon-equilibrium electrons into an equilibrium Fermi sea[11, 12]. Among them is a publication by Jura et al. [13],which inspired the experiments to be presented in thispaper. In this publication, the flow of electrons injectedthrough a quantum point contact is imaged at energiesabove the thermal smearing of the Fermi–Dirac distribu-tion. Raster-scanning a locally depleting scanning gatetip above the open electron gas downstream of the in-jection point, the authors observed a contrast inversionof the branched electron flow signal at elevated source–drain bias voltages. They interpreted this contrast inver-sion as a manifestation of electron–electron scattering inthe electron gas.Our experiments aim at finding this effect for so-calledweakly-invasive tip potentials induced by the scanninggate. In general, most scanning gate experiments (includ-ing branched electron flow measurements behind a pointcontact [14]) require a tip induced potential which de-pletes the electron gas locally (strongly-invasive regime). ∗ [email protected] However, we recently found a method to significantly en-hance the sensitivity at non-depleting voltages (weakly-invasive regime) [15], thus reducing the influence of thetip onto the unperturbed system. This method utilizes agate-defined open cavity structure [16–18], which concen-trates the scattering density of states behind the quan-tum point contact and thereby enables scanning gate ex-periments at strongly reduced voltages applied to thescanning gate. In this paper, we operate such a struc-ture in the nonlinear bias regime and find the interactioneffects previously observed for electron injection into anopen two-dimensional electron gas [13] in this modifiedsetting. Our finding may help to unravel the microscopicdetails of this effect by theoretical means beyond the ex-planation given in Ref. 13.
II. SAMPLE AND EXPERIMENTAL SETUP
Our measurements are performed on the open res-onator structure depicted in Fig. 1(a) at tempera-ture T = 270 mK. The sample is based on aGa(Al)As-heterostructure [dark grey in Fig. 1(a)] inwhich a two-dimensional electron gas (2DEG) with elec-tron density n = 1 . × cm − and mobility µ =4 . × cm / Vs is formed 90 nm below the surface.Negative gate voltages, applied to the 300 nm-wide quan-tum point contact (QPC) and arc-shaped cavity gate[light grey in Fig. 1(a)], form a 2 µ m-long resonator withan opening angle of 90 ◦ centered around the QPC.Applying a bias voltage V sd = V sd , ac + V sd , dc be-tween the source (S) and grounded drain (D) contact, weperform both equilibrium and non-equilibrium measure-ments of the differential conductance G = I sd , ac /V sd , ac through the sample. Here, I SD is the measured source-drain current, V sd , ac = 50 µ V rms for all measure-ments, and the dc-voltage is varied between V sd , dc =[ − , a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n V qpc (a) V cav SD 1 μ m V qpc –2 0 2–700–600–500 V sd,dc (mV) V q p c ( m V ) d G /d V qpc (a.u.) (b) y ( μ m ) x ( μ m) G ( e / h ) (c) FIG. 1. (a) Scanning electron micrograph of the open resonator structure with Schottky gates (light gray) on a GaAs-surface(dark gray). The cavity area is depicted by the blue standing wave. The black squares denote the source (S) and drain (D)ohmic contacts. (b) Numerical derivative dG/dV qpc of the differential conductance G as a function of V sd , dc and V qpc . Redarrows denote the cavity modes. The dashed orange line corresponds to the QPC-voltage used in all following measurements.The data for | V sd , dc | > G ( x, y ) in the cavity as a function of the tipposition for V tip = − microscopy measurements. To this end, we raster-scana voltage-biased metallic tip approximately 65 nm abovethe open resonator structure while measuring the differ-ential conductance G ( x, y ) as a function of the tip po-sition ( x, y ). Unless stated otherwise, the tip is biasedat a voltage V tip = − E F [15]. Electrons interacting with this weakly-invasivetip-induced potential are not backscattered by a hard-wall potential [14, 19, 20] but rather experience gentleelectron lensing. III. CHARACTERIZATION OF THE CAVITYA. Characterization of the cavity in absence of theSGM tip
We first characterize the open resonator in absenceof the tip by measuring the differential conductance G ( V sd , dc , V qpc ). The numerical derivative dG/dV qpc ofthe latter is depicted in Fig. 1(b) and exhibits the char-acteristic diamond-shaped pattern associated with non-equilibrium measurements of QPCs. The dark rhombiwith dG/dV qpc ≈ sb = 1 . V qpc = −
570 mV [c.f. orange dashed line inFig. 1(b)]. Additionally to the diamond-shaped pattern,we observe parallel and equally spaced lines in the dif-ferential conductance in the region of the QPC-plateaus[cf red arrows in Fig. 1(b)]. These lines are observed for nonzero cavity gate voltages only and are a manifes-tation of cavity modes with an average energy spacing∆ E cav = (236 ± µ eV. [21, 22] B. Characterization of the cavity in presence of theSGM tip
To study the conductance through the cavity on a localscale, we perform SGM-measurements above the wholecavity area limited by the QPC-gates on one side andthe cavity gate on the other side. The differential con-ductance G ( x, y ) measured at different tip positions ( x, y )within this area is depicted in Fig 1(c). In agreement withprevious work [15] it exhibits a distinct spatial structureof fine conductance modulations which emanate from theQPC radially. These conductance modulations arise dueto the influence of the tip-induced potential onto the localdensity of scattering states, which emanate from the QPCinto the cavity and are concentrated in the latter [23].The average conductance G ( x, y ) in Fig. 1(c) is re-duced with respect to the conductance of G = 3 × e /h on the third QPC plateau due to the capacitive action ofthe cavity gate and the tip on the QPC-channel. IV. FINITE BIAS MEASUREMENTSA. Finite bias measurements in presence of theSGM tip
Based on previous measurements on an open 2DEG be-hind a QPC [13], we perform SGM measurements at fi- (a) (b) V s d , d c ( m V ) x ( μ m) G (2e /h) x ( μ m) Δ G V sd,dc (mV) . e / h FIG. 2. Differential conductance G ( x, V sd , dc ) along thedashed red line in Fig. 1(c). (a) Raw data. (b) Linewise con-ductance difference ∆ G ( x ) as a function of the tip position x along the red line in (a) for small steps in V sd , dc . The redcurve denotes the conductance at V sd , dc = 0 mV and the linesare offset with respect to each other. Black dashed guides tothe eye denote three exemplary minimum-to-maximum tran-sitions. nite source–drain voltages in order to probe e-e-scatteringin the open resonator on a local scale.The thus measured differential conductance G ( x, V sd , dc ) for tip positions x along the dashedred line in Fig 1(c) and bias voltages V sd , dc is depictedin Fig. 2(a).At zero source-drain-bias we recover the conductancemodulations G ∈ [2 . , . · e /h already observed alongthe dashed red line in the spatial cavity map [cf. Fig.1(c)]. With increasing source-drain bias V sd , dc , the over-all conductance G through the sample increases whilemaintaining its distinct spatial modulation. Minima (ormaxima) of the conductance occur at exactly the sametip positions x for source-drain biases of up to approxi-mately V trans . sd , dc ≈ . | V sd , dc | > . G ( x ) = G ( x ) − (cid:104) G ( x ) (cid:105) at equally spaced V sd , dc between V sd , dc = 0 mV and 2 mV in Fig. 2(b). Here, (cid:104) G ( x ) (cid:105) isthe conductance averaged along x for fixed dc source-drain voltage V sd , dc . At specific fixed tip-positions x , thedifferential conductance shows a transition from minimaat | V sd , dc | < . | V sd , dc | > . V tip = − –750 –500–2–1012 (a) (b) V s d , d c ( m V ) G (2e /h) Δ G V sd,dc (mV) . e / h V cav (mV) –800 –600 –400 V cav (mV) FIG. 3. Differential conductance G ( V cav , V sd , dc ) in absenceof the tip. (a) Raw data. (b) ∆ G ( V cav ) = G ( V cav ) −(cid:104) G ( V cav ) (cid:105) for small steps in V sd , dc , where (cid:104) G ( V cav ) (cid:105) is the average con-ductance in V cav at fixed V sd , dc . The conductance at zerodc-bias ( V sd , dc = 0 mV) is denoted in red and the lines areoffset with respect to each other. equilibrium carriers are injected through a QPC into anopen 2DEG region. In the latter, the observed contrastinversion in regions of branched electron flow at source-drain voltages of up to V sd , dc = 2 . ee depends on the square ofthe excess energy ∆ above the Fermi-energy at whichan electron is injected into the system (Γ ee ∝ ∆ , seediscussion below). Increasing the source-drain-bias V sd , dc from zero to 2 mV, e–e scattering in the cavity will thusbecome particularly relevant for electrons injected at thehighest energies.As reported in Ref. 13, other inelastic scattering mech-anism for hot electrons in 2DEGs (among which the mostimportant ones are plasmon emission [24] and the exci-tation of acoustic phonons [25]) are irrelevant at the in-jection energies under investigation (see Appendix A inRef. 13 for details). B. Finite bias measurements in absence of theSGM-tip
Signatures of e–e-scattering have been shown to bepresent in many transport experiments in absence of ascanning tip, ranging e.g from hydrodynamic flow ex-periments [26–28] to Young’s double slit experiments[29]. Even though the tip-induced potential in ourSGM-measurements is smaller than the Fermi energy, itdoes influence the scattering states in the cavity [cf Fig.1(c)]. This raises the question of whether the observedminimum-maximum-transition is observable only in thepresence of the SGM-tip or if it is an intrinsic signatureof e–e-scattering in the cavity also present in the absenceof the tip.In an attempt to resolve this question, we measure thedifferential conductance G ( V cav , V sd , dc ) in the absence ofthe tip for a fully formed cavity [ V cav below the pinch-offvoltage]. The changing cavity gate voltage V cav therebyreplaces the varying tip-induced potential. Figure 3(a)shows the resulting differential conductance. Again, weobserve a clear transition between regions with maximaland minimal conductance but at a slightly different dcvoltage V trans , , dc ≈ µ V. The shift of the position ofthe minima/maxima in V cav arises due to the gating ofthe cavity modes by the bias voltage. This gating ef-fect is even more obvious in Fig. 3(b) which is obtainedby the same analysis that was done to obtain Fig. 2(b)[here, the average was taken along V cav ]. Due to this gat-ing effect, it is impossible to identify whether minima inthe conductance turn into maxima at higher bias. Fig-ure 3(b) also shows that the shift of the minima/maximais not linear and therefore cannot be accounted for easily.Therefore, a minimum-to-maximum-transition could notbe conclusively observed in the absence of the tip. V. DISCUSSION
The electron-electron scattering length l ee in the sys-tem is given as l ee = v F τ ee , where v F is the electron ve-locity at the Fermi-energy and τ ee is the electron-electronscattering time. The latter can be estimated to be [24, 27]1 τ ee = E F h (cid:18) ∆ E F (cid:19) (cid:20) ln (cid:18) E F ∆ (cid:19) + ln (cid:18) q TF k F (cid:19) + 1 (cid:21) . Here, ∆ is the excess energy with respect to the Fermi-energy E F , k F is the Fermi wave number and q TF is thetwo-dimensional Thomas-Fermi screening wave vector.Taking the excess energy ∆ = −| e | V sd , dc = ± . l ee = 3 . µ m. This lengthis smaller than the path-length of the round-trip betweenQPC and cavity gate. The life-time broadening of thecavity modes in the open resonator thus becomes signif-icant with respect to the cavity mode spacing at theseinjection energies. This may be the reason for the de-creasing amplitude of the conductance modulations inFig. 2(a) with increasing V sd , dc .Our measurements differ from those in Ref. 13 in twoways. First, our sample consists of an open resonatorformed between a quantum point contact and an arc-shaped cavity gate instead of an open 2DEG behind aquantum point contact. Second, our measurements areobtained with tip-induced potentials lower than E F in-stead of the the strongly-invasive tip potential used inthe experiment by Jura et al. [13]. The tip-induced po-tential in our experiments thus does not backscatter elec-trons but rather gently lenses the propagating electrons.Surprisingly, despite these differences, our data yield aminimum-to-maximum transition at finite bias voltage,similar to the measurements in Ref. 13. Due to the complex scattering dynamics in the cav-ity, the exact microscopic origin of the minimum-to-maximum transition remains elusive. However, the qual-itatively similar phenomenology of our data with the re-sults in Ref. 13, as well as the estimates of l ee givenabove, suggest the relevance of e–e interactions in thecavity involving the injected non-equilibrium electrons.Due to the design of our structure, an electron remainswithin the cavity for several roundtrips between the QPCand cavity-gates. This is consistent with the model of acollective ac motion of electrons in the cavity, which orig-inates in the the ac electron flow injected through theQPC at energies ∆ = −| e | V sd , dc , as proposed in Ref. 13 VI. CONCLUSIONS
In conclusion, we measure non-equilibrium transportthrough an electronic cavity with scanning gate mi-croscopy. We observe a minimum-to-maximum transi-tion as a function of the source-drain bias V sd , dc in thedifferential conductance modulation caused by the tip-induced potential. Our measurements show that gentleelectron lensing due to a tip-induced potential below theFermi-energy [15] is sufficient to observe this transition.However, data taken in the absence of the tip show therelevance of the tip-induced potential for the observationof the transition. Despite significant experimental differ-ences, our observations are phenomenologically similarto strongly-invasive scanning gate measurements on elec-trons injected through a point contact into an open two-dimensional electron gas [13]. This suggests a similar mi-croscopic origin of the minimum-to-maximum transitionin both experiments, which is based on electron-electronscattering. The detailed microscopic mechanisms of theelaborate scattering processes of electrons in the elec-tronic cavity remains an interesting open question thatwill require further theoretical and experimental work. ACKNOWLEDGMENTS
We thank Leonid Levitov and Vadim Khrapai for fruit-ful discussions and Peter M¨arki, Thomas B¨ahler as wellas the staff of the ETH cleanroom facility FIRST fortheir technical support. We also acknowledge financialsupport by the ETH Zurich grant ETH-38 17-2 and theSwiss National Science Foundation via NCCR QuantumScience and Technology.
Appendix A: Finite bias measurements withstrongly invasive tip potentials
We evaluate the influence of a strongly invasive tip po-tential on the observation of the minimum-to-maximumtransitions by repeating the measurement depicted inFig. 2 for a strongly invasive tip potential ( V tip = − (a) (b) V s d , d c ( m V ) x ( μ m) G (2e /h) x ( μ m) Δ G V sd,dc (mV) . e / h FIG. 4. Differential conductance G ( x, V sd , dc ) for a stronglyinvasive tip potential ( V tip = − G ( x ) for tip positions x along the red line in (a)and small steps in V sd , dc . The red curve corresponds to zerosource-drain bias and the lines are offset with respect to eachother. Black dashed guides to the eye denote three exemplaryminimum-to-maximum transitions. The thus obtained data is depicted in Fig. 4. In ac-cordance with previous experiments [15], the additionalscattering of electrons off the strongly invasive tip poten-tial results into sharper and denser conductance mod-ulations in the cavity area. Nonetheless, Fig. 4 de-picts the same minima-maxima transition observed inFig. 2. Therefore, the observation of the minima-to-maxima transitions is independent of the strength of thetip-induced potential. [1] K. Seeger,
Semiconductor Physics An Introduction , 5thed., Springer Series in Solid-State Sciences, Vol. 40(Springer Berlin Heidelberg, Berlin, Heidelberg, 1991).[2] A. L. ˙Efros and M. Pollak, eds.,
Electron-Electron Inter-actions in Disordered Systems , Modern Problems in Con-densed Matter Sciences, Vol. 10 (North-Holland, Amster-dam, 1985).[3] G. Zala, B. N. Narozhny, and I. L. Aleiner, Interac-tion corrections at intermediate temperatures: Longitu-dinal conductivity and kinetic equation, Phys. Rev. B ,214204 (2001).[4] Y. Imry, Introduction to Mesoscopic Physics , 2nd ed.,Mesoscopic Physics and Nanotechnology 2, Ed. 2009(University Press, Oxford, 2009).[5] Y. Aharonov and D. Bohm, Significance of Electromag-netic Potentials in the Quantum Theory, Phys. Rev. ,485 (1959).[6] R. A. Webb, S. Washburn, C. P. Umbach, and R. B. Lai-bowitz, Observation of he Aharonov-Bohm Oscillations inNormal-Metal Rings, Phys. Rev. Lett. , 2696 (1985).[7] A. E. Hansen, A. Kristensen, S. Pedersen, C. B. Sørensen,and P. E. Lindelof, Mesoscopic decoherence in Aharonov-Bohm rings, Phys. Rev. B , 045327 (2001).[8] G. Bergmann, Weak localization in thin films: A time-of-flight experiment with conduction electrons, PhysicsReports , 1 (1984).[9] D. A. Bandurin, I. Torre, R. K. Kumar, M. B. Shalom,A. Tomadin, A. Principi, G. H. Auton, E. Khestanova,K. S. Novoselov, I. V. Grigorieva, L. A. Ponomarenko,A. K. Geim, and M. Polini, Negative local resistancecaused by viscous electron backflow in graphene, Science , 1055 (2016).[10] B. A. Braem, F. M. D. Pellegrino, A. Principi, M. R¨o¨osli,C. Gold, S. Hennel, J. V. Koski, M. Berl, W. Dietsche, W. Wegscheider, M. Polini, T. Ihn, and K. Ensslin, Scan-ning gate microscopy in a viscous electron fluid, Phys.Rev. B , 241304 (2018).[11] D. M. Zumb¨uhl, C. M. Marcus, M. P. Hanson, and A. C.Gossard, Asymmetry of Nonlinear Transport and Elec-tron Interactions in Quantum Dots, Phys. Rev. Lett. ,206802 (2006).[12] D. Taubert, C. Tomaras, G. J. Schinner, H. P. Tranitz,W. Wegscheider, S. Kehrein, and S. Ludwig, Relaxationof hot electrons in a degenerate two-dimensional electronsystem: Transition to one-dimensional scattering, Phys.Rev. B , 235404 (2011).[13] M. P. Jura, M. Grobis, M. A. Topinka, L. N. Pfeiffer,K. W. West, and D. Goldhaber-Gordon, Spatially probedelectron-electron scattering in a two-dimensional electrongas, Phys. Rev. B , 155328 (2010).[14] M. A. Topinka, B. J. LeRoy, R. M. Westervelt, S. E. J.Shaw, R. Fleischmann, E. J. Heller, K. D. Maranowski,and A. C. Gossard, Coherent branched flow in a two-dimensional electron gas, Nature , 183 (2001).[15] R. Steinacher, C. P¨oltl, T. Kr¨ahenmann, A. Hofmann,C. Reichl, W. Zwerger, W. Wegscheider, R. A. Jalabert,K. Ensslin, D. Weinmann, and T. Ihn, Scanning gateexperiments: From strongly to weakly invasive probes,Phys. Rev. B , 075426 (2018).[16] C. Yan, S. Kumar, M. Pepper, P. See, I. Farrer,D. Ritchie, J. Griffiths, and G. Jones, Interference Ef-fects in a Tunable Quantum Point Contact Integratedwith an Electronic Cavity, Phys. Rev. Applied , 024009(2017).[17] C. Yan, S. Kumar, M. Pepper, P. See, I. Farrer,D. Ritchie, J. Griffiths, and G. Jones, Incipient singlet-triplet states in a hybrid mesoscopic system, Phys. Rev.B , 241302 (2018). [18] C. Yan, S. Kumar, P. See, I. Farrer, D. Ritchie, J. P.Griffiths, G. a. C. Jones, and M. Pepper, Magnetoresis-tance in an electronic cavity coupled to one-dimensionalsystems, Appl. Phys. Lett. , 112101 (2018).[19] M. A. Eriksson, R. G. Beck, M. A. Topinka, J. A. Ka-tine, R. M. Westervelt, K. L. Campman, and A. C. Gos-sard, Effect of a charged scanned probe microscope tip ona subsurface electron gas, Superlattices and Microstruc-tures , 435 (1996).[20] M. A. Topinka, B. J. LeRoy, S. E. J. Shaw, E. J. Heller,R. M. Westervelt, K. D. Maranowski, and A. C. Gossard,Imaging Coherent Electron Flow from a Quantum PointContact, Science , 2323 (2000).[21] J. A. Katine, M. A. Eriksson, A. S. Adourian, R. M.Westervelt, J. D. Edwards, A. Lupu-Sax, E. J. Heller,K. L. Campman, and A. C. Gossard, Point Contact Con-ductance of an Open Resonator, Phys. Rev. Lett. ,4806 (1997).[22] C. R¨ossler, D. Oehri, O. Zilberberg, G. Blatter, M. Kar-alic, J. Pijnenburg, A. Hofmann, T. Ihn, K. Ensslin,C. Reichl, and W. Wegscheider, Transport Spectroscopyof a Spin-Coherent Dot-Cavity System, Phys. Rev. Lett. , 166603 (2015). [23] C. Gold, B. A. Br¨am, M. S. Ferguson, T. Kr¨ahenmann,A. Hofmann, K. Fratus, C. Reichl, W. Wegscheider,D. Weinmann, K. Ensslin, and T. Ihn, Imaging signa-tures of the local density of states in an electronic cavity,arXiv:2011.13869 [cond-mat] (2020), arXiv:2011.13869[cond-mat].[24] G. F. Giuliani and J. J. Quinn, Lifetime of a quasiparticlein a two-dimensional electron gas, Phys. Rev. B , 4421(1982).[25] T. Ihn, Semiconductor Nanostructures: Quantum Statesand Electronic Transport (Oxford University Press, Ox-ford, New York, 2009).[26] L. W. Molenkamp and M. J. M. de Jong, Observationof Knudsen and Gurzhi transport regimes in a two-dimensional wire, Solid-State Electronics , 551 (1994).[27] M. J. M. de Jong and L. W. Molenkamp, Hydrodynamicelectron flow in high-mobility wires, Phys. Rev. B ,13389 (1995).[28] G. M. Gusev, A. D. Levin, E. V. Levinson, andA. K. Bakarov, Viscous electron flow in mesoscopictwo-dimensional electron gas, AIP Advances , 025318(2018).[29] A. Yacoby, U. Sivan, C. P. Umbach, and J. M. Hong, In-terference and dephasing by electron-electron interactionon length scales shorter than the elastic mean free path,Phys. Rev. Lett.66