Fano Resonance in a cavity-reflector hybrid system
Chengyu Yan, Sanjeev Kumar, Michael Pepper, Patrick See, Ian Farrer, David Ritchie, Jonathan Griffiths, Geraint Jones
FFano Resonance in a cavity-reflector hybrid system
Chengyu Yan,
1, 2, ∗ Sanjeev Kumar,
1, 2
Michael Pepper,
1, 2
Patrick See, Ian Farrer, David Ritchie, Jonathan Griffiths, and Geraint Jones London Centre for Nanotechnology, 17-19 Gordon Street, London WC1H 0AH, United Kingdom Department of Electronic and Electrical Engineering, University College London,Torrington Place, London WC1E 7JE, United Kingdom National Physical Laboratory, Hampton Road, Teddington, Middlesex TW11 0LW, United Kingdom Cavendish Laboratory, J.J. Thomson Avenue, Cambridge CB3 OHE, United Kingdom (Dated: November 12, 2018)We present the results of transport measurements in a hybrid system consisting of an arch-shaped quantum point contact (QPC) and a reflector; together, they form an electronic cavity inbetween them. On tuning the arch-QPC and the reflector, asymmetric resonance peak in resistanceis observed at the one-dimension to two-dimension transition. Moreover, a dip in resistance nearthe pinch-off of the QPC is found to be strongly dependent on the reflector voltage. These twostructures fit very well with the Fano line shape. The Fano resonance was found to get weakened onapplying a transverse magnetic field, and it smeared out at 100 mT. In addition, the Fano like shapeexhibited a strong temperature dependence and gradually smeared out when the temperature wasincreased from 1.5 to 20 K. The results might be useful in realising device for quantum informationprocessing.
There is a growing interest in realising the electricalanalog of photonic-cavity devices in condensed matteras they form the fundamental basis of tomorrow’s quan-tum technologies for realising building blocks for quan-tum information processing and quantum circuits. Muchsuccess with photonic cavity based system could beattributed to a better control over manipulating pho-tons and achieving entanglement over a larger distancefor information processing . Electronic devices havenot enjoyed a similar level of success due to limited ad-vancement in controlling the quantum states of electrons.Recently, an electronic cavity coupled to a quantum dotwas demonstrated with the observation of spin coher-ent state in the regime of dot-cavity coupling due toKondo effect . There have been experimental attemptsin the past in coupling an electronic cavity with one-dimensional (1D) electrons using a quantum point con-tact (QPC) , however, in such cases the oscillationsin the conductance were primarily due to quantuminterference rather than coupling between the 1D-cavitystates. Therefore, direct evidence of coupling betweenthe 1D states of a QPC and the cavity states would becrucial in creating a platform for realising cavity basedtunable electronic devices.In the present work, we demonstrate a cavity-reflectorhybrid quantum device consisting of a QPC as the sourceof 1D electrons and an electronic cavity of 2D electrons,and show that when these states couple, they give rise toa Fano resonance .The samples studied in the present work consist of apair of arch-shaped gates, with QPC forming in the cen-tre of the arch, and a reflector inclined at 75 ◦ against thecurrent flow direction such that centre of the QPC andthe reflector are aligned as shown in Fig. 1(a) . Withthis geometry the interference between incident and re-flected electrons which contribute to oscillations reportedpreviously is significantly reduced because the in- cident and reflected electrons are spatially separated,thus all the features observed here are due to the cou-pling effect. The hybrid devices were fabricated on a highmobility two-dimensional electron gas (2DEG) formedat the interface of GaAs/Al . Ga . As heterostructure.The 2DEG is situated 90 nm from the surface where thegates are deposited. The electron density (mobility) mea-sured at 1.5 K was 1.80 × cm − (2.1 × cm V − s − )therefore both the mean free path and phase coherencelength were over 10 µ m which is much larger than the dis-tance between the QPC and the reflector (1.5 µ m). Allthe measurements were performed with standard lock-intechnique in a cryofree pulsed-tube cryostat with a basetemperature of 1.5 K. For the four terminal resistancemeasurement, a 10 nA at 77 Hz ac current was appliedwhile an ac voltage of 10 µ V at 77 Hz was used for twoterminal conductance measurement. Figure 1(b) showsthe conductance plot of the QPC with well defined con-ductance plateaus; on the other hand, the conductance ofthe reflector drops, when measured separately, around -0.2 V, which indicates a sharp change in the transmissionprobability (Fig. 1(b), inset). The three regimes identi-fied from the characteristic of the reflector are labelledas regime 1-3, as shown in the inset.To highlight the effect of the cavity states, we studiedthe non-local four terminal resistance R , = V /I (hereafter denoted as R ) as shown in Fig. 2. This mea-surement configuration allows one to study the interfer-ence between reflected electrons which propagate towardsOhmic 3 directly without being affected by the cavitystates, and those go through cavity and get modulated.In regime 1, where the reflector voltage V r was sweptfrom - 0.30 (top trace) to - 0.27 V (bottom trace), a pro-nounced asymmetric resonance structure, which shows apeak at more negative arch-QPC voltage V a and valleyat less negative end, centred around V a = - 0.2 V was ob-served, which is also the centre of the 1D to 2D transition a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b FIG. 1. Schematic of the device and the experiment setup.(a) the yellow blocks are metallic gates and red squares areOhmics; excitation current is fed to Ohmic 1 while 2 isgrounded; Ohmics 3 and 4 are voltage probes. The open-ing angle of the arch is 45 ◦ and the radius (also the distancebetween arch-gate and reflector) is 1.5 µ m, both the lengthand width of the QPC embedded in the arch is 200 nm, thelength of inclined surface of the reflector is 3.0 µ m and thewidth is 300 nm. (b) Differential conductance measurementof the QPC (main plot) and the reflector (inset). To be notedthat in this plot when the QPC was measured, the reflec-tor was grounded, and visa-versa. The voltage applied tothe QPC (reflector) was V a (V r ). Three regimes are identi-fied from the reflector voltage characteristic and labelled asregime 1 (a-b), regime 2 (b-c) and regime 3 (c-d). regime of arch-QPC [as shown by a dotted rectangularbox in Fig. 2(a)].When the reflector was set to regime 2, V r increasesfrom -0.27 to -0.24 V (Fig. 2(b)), the asymmetric res-onance structure slowly evolves into a broad-shoulderstructure, and a dip gradually forms at the pinch-offregime of the QPC. It is interesting to notice that the dipstarts forming when the asymmetric resonance structurefully smears out. In addition, the centre of the shoul-der structure shifts towards a more negative V a with in-creasing V r (see the solid line), on the contrary, the dipstructure moves to a less negative V a (see the dashedline).In regime 3 (- 0.24 V ≤ V a ≤ FIG. 2. R as a function of arch-QPC voltage for variousV r . (a) Result in regime 1: a strong asymmetric resonancestructure highlighted by a dotted rectangle occurs around -0.2 V. (b) Result in regime 2: the resonance structure slowlyevolves into a broad shoulder structure (marked by the solidline), meanwhile a dip gradually forms at the pinch-off regimeof the arch-QPC (marked by the dashed line). (c) Result inregime 3: intensity of both the shoulder structure and the dipdecreases with less negative reflector voltage. Data have beenoffset vertically by 10 Ω in each plots for clarity. FIG. 3. Theoretical fitting of the resonance structures (a) anddip (b). It is remarkable that both the resonance structure(V r = - 0.30 V) and the dip (V r = - 0.25 V) follow well definedFano line shape. The horizontal black dotted line is a guideto the eye, reflecting the saturated value on the left end ofthe experimental data (the 1D regime of QPC) and that onthe right (the 2D regime) do not align. This may be due todramatic variation in DOS at the 1D-2D transition. Inset inplot (a) and (b) shows Fano factor q as a function of V r inthe range where the structures are observable. ). The asymmetric resonance structure observed at the1D-2D transition regime and the dip structure in thepinch-off regime seem to arise from dramatic change inthe density of states (DOS) in the QPC in these regimes.However, such changes in DOS is independent of V r whilethe resonance and the dip structure are highly sensitiveto V r . In the present device, the angle of the reflectoris properly designed to avoid back refection of the elec-trons, so that it is unlikely that reflected electrons wouldenter the QPC and perturb the DOS directly.A similar dip structure was reported in a double-QPC experiments when the intruder QPC was swept intopinch-off regime ; the effect was attributed to Fanoresonance which arises from the interference between the FIG. 4. R as a function of reflector voltage for various V a .(a) The arch-gate voltage was fixed at -0.23 V ( (cid:3) ), - 0.19 V( ♦ ) and 0 V ( (cid:52) ), respectively, while sweeping the reflectorvoltage; data have been offset vertically by 50 Ω for clarity. b , theoretical fitting of ∆R(V r ,V a ) = R(V r ,V a ) - R(V r ,0),where V a = -0.23 V, using Eq.(1) where q = 1.47, it is clearthat the data follow a Fano line shape. discrete states and the continuum , where the role of theintruder was to provide the continuum. In our system,the electronic cavity which is defined using the arch-gateand the reflector, having its states filled up to chemi-cal potential, hosts the continuum while the electrons in-jected from the arch-QPC are energetically discrete; thusthe coupling between the QPC and the cavity states re-sults in Fano resonance at the two regimes, (1) near thepinch-off and (2) at the 1D-2D transition. In addition,weak oscillations are observed in the resistance in the 1Dregime of the arch-QPC (Fig. 2(a)) which could be a con-sequence of interference of 1D electrons with the cavitystates. A detailed study in this regime will be publishedelsewhere .Figure 3 shows the fitting of the asymmetric resonanceand the dip structures. It is remarkable that both the FIG. 5. Effect of perpendicular magnetic field. In plot (a)and (b) the transverse magnetic field is incremented from 0(bottom trace) to ±
150 mT (top trace), respectively, it isseen that both the peak (marked by the solid line) and thedip (highlighted by the dashed line) rapidly weaken againstthe field, they almost smears out at ±
100 mT. Data havebeen offset vertically by 20 Ω for clarity. structures follow well defined Fano line shape , R = R ( q + γ ( V − V )) γ ( V − V ) + R inc (1)where R is the measured resistance, R is a constant rep-resenting the amplitude of the resonance, q is the Fanofactor which decides the asymmetry of the line shape, γ = 20 V − (estimated from the Fermi energy and thepinch-off voltage), V is the arch-gate voltage at the cen-ter of the resonance (dip), and R inc denotes the intrinsiccontribution from the background . However, it is notedthat the fitting matches with the experimental result inthe centre while it diverges at both ends for Fig. 3(a)and (b), which is likely due to the fact that R changesas a function of V a even in the absence of the quan-tum correction, because electrons are collimated whenthe QPC is operating in the 1D regime, whereas they FIG. 6. Temperature dependence of R . (a) R in regime 1 (V r = - 0.3 V) as a function of temperature from 1.5 K (top trace)to 20 K (bottom trace); it may be seen that the intensity ofboth the peak and the dip structures decreases with increasingtemperature. Data have been offset vertically by 20 Ω forclarity. (b) The temperature dependence of R for V a = - 0.50( (cid:3) ), - 0.23 ( ♦ ), - 0.19 ( (cid:53) ) and 0 V ( (cid:13) ); solid lines are a guideto the eye. (c) Mott fitting for temperature dependence of thepeak structure, ∆ R ( V a , T ) = R ( V a , T ) − R (0 , T ), where V a =- 0.23 V, R = -0.2656 kΩ, R = 0.4238 kΩ, T = 1.2 K. spread out when the QPC is in the 2D regime . More-over, the divergence in the pinch-off regime is due tothe fact that once the 1D channel is pinched-off the de-tected signal will not change any further. The inset inFig. 3(a) and (b) shows Fano factor q as a function ofV r . It was seen that q , which represents the coupling be-tween the QPC and the cavity, changes rapidly in boththe regimes, which is most likely due to a sharp changein DOS of the QPC in the 1D-2D transition andpinch-off regime, respectively. Concerning the fitting inFig. 3(b), the structure occurs in the pinch-off regimewhere making V a even more negative will not change thesignal, therefore, this leaves an open question whetherthe data at the pinch-off are the dip of Fano resonanceor do they a Breit-Wigner line shape. It is to be notedthat an asymmetric Fano line shape is observed when-ever resonant and nonresonant scattering paths interfere.However, when there is no interference between them, asymmetric Breit-Wigner resonance is expected . If welook at Fig. 2(b) and focus on the dip in resistance atV a = - 0.8 V as a function of V r , it seems that the dipstructure takes a symmetric shape before finally settlingat V r =- 0.24 V to reflect an asymmetric line shape. Itmay be possible that the lower dip at the pinch-off ismoving through the Breit-Wigner line shape to the Fanoline shape, depending on the coupling factor q. However,such observations need a detailed study and it is difficultto comment on them at the moment.The Fano line shape arising from coupling between theQPC and the cavity states can be modulated by eithertuning the QPC states while fixing the cavity states asalready shown in Fig. 2, or in a complementary way byadjusting the cavity states and holding the QPC statesas shown in Fig. 4(a). In this measurement, the reflectorvoltage was swept which allows one to control the energyspacing of the cavity states while V a was set to - 0.23 (thepeak of asymmetric resonance structure), - 0.19 (the dipof asymmetric resonance structure) and 0 V, respectively,as shown in figure 4(a). When V a was set to - 0.19 and 0V, respectively, where the cavity was not defined, it wasseen that the resistance R was initially almost constantwhen the reflector voltage V r > − .
20 V. On decreasingV r < − .
20 V a rise in resistance occurs simultaneouslywhen the reflector conductance drops where the reflec-tion probability r increases rapidly, and then R saturateswhen r becomes unity (see inset, Fig. 1(b)). When V a was set to - 0.23 V , R follows a similar trend as previouscase when V r > − .
20 V where the cavity was not acti-vated. After the cavity was switched on, an anomalouspeak in R occurs when - 0.28 V ≤ V r ≤ - 0.25 V. The rel-ative peak intensity, ∆R(V r ,V a ) = R(V r ,V a ) - R(V r ,0),fits very well with Fano line shape, as shown in Fig. 4(b)using Eq.(1), here R(V r ,0) is taken as background signalbecause it accounts for the change in R due to the changein reflection probability only, and it is always present asthe background signal regardless of V a , and the resonancestructure superposes on such a background; therefore, tohighlight the resonance, we subtracted the background. Coupling and decoupling between discrete and con-tinuum states lead to Fano resonance, thus parame-ters which affect the coupling, apart from electrostaticconfinement, are expected to influence the line shapedramatically . Here, we present the effect of both neg-ative and positive transverse magnetic field as shown inFig. 5(a) and (b). The intensity of the asymmetric reso-nance structure is highly sensitive to the magnetic fieldin both field orientations, and almost smears out at afield of ±
100 mT. The reason for the the reduction ofthe intensity against magnetic field is three-fold: first,the chance of electron entering the cavity is magneticfield dependent; second, the cavity states, which repre-sent the quantization of standing waves in the cavity,are highly dependent on the trajectory of electrons, withlarge magnetic field the electrons in the cavity becomemore localized and therefore coupling between 1D andcavity state is weakened; third, the perpendicular mag-netic field also leads to a reduction of phase coherencelength and thereby weakens the interference effect .A slight difference in the line shape for negative and pos-itive field is likely to arise from the fact that both theinclined reflector and the negative field favor Ohmic 3,while a positive field prefers guiding electrons to Ohmic4. We studied temperature dependence of R as shown inFig. 6. The reflector voltage was set to - 0.3 V, whereboth peak and dip of the asymmetric resonance structureare pronounced. It is found that the intensity of the peakand the dip decreases against increasing temperature, thedip structure smears out at around 16 K while the peaksurvives up to 20 K (although the height drops signifi-cantly). The evolution of line shape against temperatureis clearer in Fig. 6(b), it is seen that R increases slowlyand almost linearly against temperature when V a is setto - 0.50 or 0 V, while the peak structure (V a = - 0.23V) and the dip structure (V a = - 0.19 V ) change rapidlywith increasing temperature. The fitting of temperaturedependence of relative intensity of the peak structure,∆ R ( V a , T ) = R ( V a , T ) − R (0 , T ) where V a = -0.23 V, re-sembles a Mott line shape as shown in Fig. 6(c), usingthe relation, R = R e ( T T ) + R e ( T T ) (2)where R , R are fitting parameters while T is definedas T ∝ k B N ( E F ) ξ (3)where k B is Boltzmann constant, N ( E F ) is the density ofstate in the absence of electron-electron interaction and ξ is the localization length , T = 1.2 K is extracted fromthe fitting. The and terms arise from the contributionof 1D and 2D, respectively. This agrees with the factthe Fano resonance is a direct manifestation of couplingbetween the QPC and the cavity states.Mott’s law was initially proposed for random hoppingin disordered system , whereas the clean systems gener-ally do not follow Mott’s law explicitly (e.g. see the tracefor V a = 0 V in Fig. 6(b)). Random hopping accounts forthe compromise between the spatial and energetic sepa-ration. It is possible that with the cavity switched onthe multiple reflection process is a mimic of random hop-ping process, because the energy changing process in thecavity is accompanied with the spacial change (statesin the cavity are sensitive to the confinement length, theconfinement length, in turn, is coordinate sensitive).In conclusions, we have shown the operation of a hy-brid quantum device consisting of an arch-shaped QPC coupled to an electronic cavity, whose states can be tunedusing a reflector gate. We have shown that, upon reduc-ing the width of the QPC and increasing the reflectionprobability, Fano resonance can be produced at the 1D-2D transition and near the pinch off regime. The Fanoresonant structure is very sensitive to the transverse mag-netic field, and exhibits Mott line-shape temperature de-pendence. 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