Transmission of heat modes across a potential barrier
Amir Rosenblatt, Fabien Lafont, Ivan Levkivskyi, Ron Sabo, Itamar Gurman, Daniel Banitt, Moty Heiblum, Vladimir Umansky
TTransmission of heat modes across a potential barrier
Amir Rosenblatt , † , Fabien Lafont , † , Ivan Levkivskyi , Ron Sabo , Itamar Gurman , Daniel Banitt , Moty Heiblum and Vladimir Umansky Braun Center for Submicron Research, Dept. of Condensed Matter physics,Weizmann Institute of Science, Rehovot 76100, Israel and Institute of Ecology and Evolution, University of Bern, CH-3012 Bern, Switzerland ( † These authors contributed equally to this work) ∗ ABSTRACT
Controlling the transmission of electrical current us-ing a quantum point contact constriction paved a wayto a large variety of experiments in mesoscopic physics.The increasing interest in heat transfer in such sys-tems fosters questions about possible manipulationsof quantum heat modes that do not carry net charge(neutral modes). Here, we study the transmission ofupstream neutral modes through a quantum point con-tact in fractional hole-conjugate quantum Hall states.Employing two different measurement techniques, wewere able to render the relative spatial distribution ofthese chargeless modes with their charged counter-parts. In these states, which were found to harbormore than one downstream charge mode, the upstreamneutral modes are found to flow with the inner chargemode - as theoretically predicted. These results unveila universal upstream heat current structure and openthe path for more complex engineering of heat flowsand cooling mechanisms in quantum nano-electronicdevices.
INTRODUCTION
The intimate link between heat current, entropy flow andtherefore information transfer [1, 2], triggered recent in-terest in thermoelectric effects occurring at the nanoscale,such as measurements of the quantum limit of heat flowof a single quantum mode [3–6], heat Coulomb blockadeof a ballistic channel [7] or quantum limited efficiency ofheat engines and refrigerators [8, 9]. One main experimen-tal obstacle in measuring thermal effects is to decouplethe charge from heat currents. Such separation is madeeasier in the fractional quantum Hall effect (FQHE) [10],since, at least in hole-conjugate states (say, 1 / < ν < ∗ correspondence should be addressed to: [email protected] in describing various aspects of the FQHE at filling factors ν = 1 /m , with odd m , the structure of hole-conjugatefilling fractions, such as ν = 2 / ν = 3 /
5, is stillnot clear. Two edge-model structures had been proposedfor the most studied ν = 2 / ν = 1 / ν = 1 / ν = 1 electronic LL form a Laughlin conden-sate with ν = 1 / ν = 1 type condensate of fractional e ∗ = 1 / ν = 1 / ν = 2 / ν = 3 /
5) is in fact composed of two spatially separatedcharge modes [19], that would suggest the second pointof view to be more appropriate. Furthermore, topologicalarguments require to have a conserved total number of netmodes, therefore predicting two upstream neutral modesat ν = 2 /
3, and three at ν = 3 / RESULTSTransmission of neutral modes accross a QPC
The experimental setup, designed to map the transmis-sion of neutral modes through a QPC constriction, is pre-sented in Figs. 1a, 1c, 1e. The QPC (on the right) par-titions either the charge mode emitted from S2 (Fig. 1a),or the neutral modes excited by the hot spot at the backof contact H (Figs. 1c & 1e). In the latter case, twomethods were employed to convert the energy carried bythe neutral modes to a measurable charge current. Thefirst utilized a quantum dot (QD, on the left) to convert atemperature gradient to a net thermoelectric current [20– a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec -0.09 -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0-3036 I D ( p A ) µ m D1 µ m D1 chargedneutral µ m chargedneutral t = / D1 t = IV pl IV pl charged coldchargedhot a) c) e)b) d) f)g) V pl (V) S1 V pl H H
D2S2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 I H (nA) e x ce ss n o i s e ( − A / H z ) QD QD
Figure 1.
Description of the experimental device and methods: a)
Scanning electron microscope (SEM) image of thedevice. In this configuration the current is sourced from S1, reaching to the quantum dot (QD). Sweeping the plunger gatevoltage results in a succession of Coulomb peaks as shown by the blue curve on g). b) Schematic of the equilibrium distributionon each side of the QD. When the plunger gate is tuned, electron can tunnel into the dot from the high occupation numberregion to the lower one, creating a positive current measured at D1. c) Neutral mode heat detection configuration: The currentsourced in H is directed to the ground and plays no part in the experiment. A hot spot present at the upstream side of contactH excites the neutral heat modes flowing upstream towards the QD, creating a thermal gradient across the QD. The producedthermoelectric current then flows to D1. d) Sketch of the equilibrium distribution on both side of the dot in the case depictedin c). In this case the tunneling direction will depend if an energy level of the QD is placed bellow or above the center of thedistribution. This induces an alternating current when the plunger gate is tuned. e) Noise measurement configuration: Herethe input QPC of the QD is set to half transmission while the second one is fully open. This turns the QD into an effectivesingle QPC device. The heat carried by the neutral modes increase the electron temperature at the input QPC of the dot,which increases the Johnson-Nyquist noise, measured at D1. f )
Excess noise measured at D1 as function of current injected inH, as described in e). g) Measurement corresponding to the configuration a) in blue and c) in orange.
24] as sketched in Fig. 1c. With the thermal distributionat the input of the QD being hotter than at the output,scanning the energy level of the QD leads to a net ther-moelectric current through the QD with an alternating po-larity (see Fig. 1d and Fig. 1g, and experimental details inMethods Section). The second approach used excess noisemeasurements, resulting from the upstream heat currentimpinging on a QPC (Fig. 1e) [11, 25, 26]. Here, theinput-QPC of the QD was tuned to transmission half whilethe output-QPC was fully open; therefore turning the QD to a single QPC. The impinging neutral mode increasesthe electron temperature at the QPC and accordingly theJohnson-Nyquist noise measured at D1 (Fig. 1f).The thermopower of a QD, subjected to such a thermalgradient, was studied both theoretically and experimentally[21–23, 27]. On the theoretical side, one needs to ob-tain the thermoelectric current while considering a specificmodel of the edge modes structure (see below). On theexperimental side, we can directly relate the thermoelec-tric current to the heat current carried by the transmit-ted neutral modes (Fig. 2a). Since half of the injectedpower into contact H, P H = 1 / × IV , is dissipated onthe hot spot at the back of the ohmic contact H , it isproportional to the heat current (more details in sup. ma-terial), P ∝ ( T − T ) × π k / h , carried away by theneutral modes which allows us to map the thermoelectriccurrent to the heat flow[3, 4]. Utilizing this correspon-dence, the spatial distribution of the neutral modes canbe ascertained by measuring their transmission through aQPC constriction. Applying constant power ( ∼
20 fW) tocontact H and measuring the thermoelectric (TE) currentacross the QD as we gradually pinch the right-QPC (seedetails in Measurement technique), we find the evolution ofheat current carried by the neutral modes (“neutral trans-mission”), that we present on Fig. 2c (blue curve) to-gether with the conductance of the QPC (black curve).The neutral-transmission was extracted by taking the cor-responding power for a given measured TE current fromthe mapping in Fig.2a (blue curve); normalized to a fullyopen right-QPC. A second measurement, using the noisethermometry technique described above was used to vali-date the neutral transmission. The results are presented onFig. 2c-red curve, where we plot the evolution of T − T (which is proportional to the heat current carried by theneutral modes, see sup. material), where T is measuredand T = 30 mK is the base fridge temperature. It is clearfrom Fig.2c that the two methods led to consistent re-sults. Comparing the neutral transmission with the one ofthe charge modes, one notices that ∼
80% of the neutralmode power is reflected when the inner charge mode, withconductance of e / h , is reflected (at the beginning ofthe conductance plateau, around V QPC = − . ν = 3 / Theoretical Model
We now compare our experimental results to a theoret-ical model of the ν = 2 / e /h flowing downstream and e / h flowingupstream. Due to the absence of any experimental evi-dence of upstream current [28], Kane, Fisher and Polchin-ski [16, 29] introduced scattering between the above men-tioned channels leading to a single downstream chargemode with conductance 2 e / h and an upstream neutralmode. Later, Meir et al. [17, 18] refined this model fora soft edge potential (which is the case for gate-definededge, like presently) and proposed an edge modes picturepresented on Fig. 2b. A smooth confining potential in- duces a non-trivial density variation near the edge: Startingfrom the bulk, the local filling factor goes from 2/3 to 1creating an upstream charge mode of conductance e / h .The subsequent filling factor drop goes from unity to 0and therefore creates a charge mode going downstreamwith an associated conductance equal to e /h . Finally,an extra density hump reaching ν = 1 / G = e / h counter propagating channels. Taking intoaccount interactions and scattering between the channelsleads to the presence of a decoupled channel with conduc-tance e / h of width ξ close to the edge, flowing nextto an inner, wider channel of width ξ , also with conduc-tance e / h accompanied by two neutral modes flowingupstream. Note, that there are several alternative pic-tures of QH states at filling factors 2/3 and 3/5 both withand without neutral upstream edge modes (see, e.g., [30]).Moreover, edge reconstruction phenomena [31, 32] (e.g.additional humps in the density) can affect both the mi-croscopic and effective behavior of edge states. Therefore,we focus our discussion on the simplest effective model ofedge states that is less sensitive to the details of exper-imental situation. The theoretical model presented hereutilizes the “Meir et al. ” edge structure where couplingsbetween channels leads to different excitation. The outeredge mode is completely decoupled from all other chan-nels, while the inner channel gives rise to excitations, eachcharacterized by a charge e ∗ and a scaling dimension ∆(see details in Supplementary note 4), which allow us tocalculate the TE current through a single energy level (cid:15) of the QD for this particular edge state picture and fordominant excitation with e ∗ = 1 / I TE ∝ [ f out ( T out , (cid:15) ) f in ( T in , − (cid:15) ) − f in ( T in , (cid:15) ) f out ( T out , − (cid:15) )] (1) where f out , f in , T out , T in are the occupation numbers andtemperatures corresponding to the Input and Output sideof the quantum dot. Considering that the two upstreammodes originate from a hot reservoir at temperature T and the downstream ones at temperature T , it is possibleto express the QD’s input/output occupation number as, f in ( (cid:15) ) = Z dt e i(cid:15) t (cid:20) T sinh( πT ( t − iη )) (cid:21) δ (cid:20) T sinh( πT ( t − iη )) (cid:21) δ (2) with δ = 1 / , δ = 2 / δ + δ = 1 and f out ( (cid:15) ) = Z dt e i(cid:15) t T sinh( πT ( t − iη )) = 1 e (cid:15) /T + 1 (3) Inserting Eqs. (2) & (3) in Eq. (1) one can calculate theevolution of the TE current in a QD with a single energylevel for this particular edge modes picture (see Supplemen-tary note 4). The results either using effective temperatureapproximation or exact numeric integration are plotted inFig.2a. The good agreement between the theoretical andexperimental results strengthen the validity of an edge pic-ture that consists of two charge modes accompanied bytwo upstream neutral modes. Using the same theoretical f ) ν = 1 ν = 2 / ν = 1 / ρ ( y ) φ φ φ φ c ξ φ ξ yφ (cid:48) n φ (cid:48)(cid:48) n G QPC ( e /h ) n e u tr a l tr a n s m i ss i o n ν B = 2 / ν B = 3 / t h e r m o - e l ec tr i cc u rr e n t( p A ) V QPC (a.u) tt n ν = 2 / a ) b ) c ) d ) e ) ν = 2 / ν = 3 / Injected power (10 − W) Numerical integration ν QPC = 1 / tr a n s m i ss i o n -1 -0.5 0 0.500.20.40.6 00.511.522.500.20.40.60.81 c o ndu c t a n ce ( e / h ) V QPC (V) × − n e u tr a l tr a n s m i ss i o n T − T ( K ) -1 -0.5 0 0.500.20.40.6 00.511.522.533.544.500.20.40.60.81 × − n e u tr a l tr a n s m i ss i o n T − T ( K ) ν = 3 / c o ndu c t a n ce ( e / h ) V QPC (V) ν B = 2 / ν B = 3 / Temperatureapproximation
Figure 2.
Transmission of neutral modes: a)
Evolution of the thermoelectric current as function of the Joule powerapplied on H at ν = 2 / (blue curve) and at ν = 3 / (green curve). Theoretical prediction of the thermoelectric currentthrough the QD, orange dashed curve shows effective temperature approximation, orange solid curve shows result of exactnumeric integration. b) Theoretical model of the edge structure of the ν = 2 / state. Before equilibration four charge modes φ , , , are considered with respective velocity set by ∂ρ ( y ) /∂y . After renormalization the system consists of two downstreamcharge modes of different width ξ and ξ and two upstream neutral modes attached to the inner mode. c) black curve-leftaxis: Conductance at the QPC constriction as function of the QPC split gate voltage at ν = 2 / , blue curve-right blue axis:Evolution of the neutral transmission as function of the QPC split gate voltage. Red curve-red axis: evolution of the excessnoise measured at D1 as function of the QPC split gate voltage. d) Theoretical charge and neutral transmission. e) Similar toc) at filling factor ν = 3 / . f ) Neutral transmission as function of the conductance in the QPC constriction for the bulk filingfactors ν B = 2 / and ν B = 3 / . The plain and open symbols designate two different samples. transmission ∆ S ( − A / H z ) -1 -0.5 0 0.500.20.40.6 G Q P C ( e / h ) V QPC (V) t = 1 / µ m D1 H S2S2 D2D2 I T E ( p A ) ∆ S ( − A / H z ) a ) b ) c ) d ) e ) f ) g ) S2 → D2 H → D2 S2 → D1 α × (H → D2) ∆ S ( − A / H z ) -3 -2 -1 0 1 2 3051015 t = 1 / S2 → D2H → D2 α × (H → D2) I SD (nA) Figure 3.
Noise measurements: a)
Sketch of noise measurement at D2 when sourcing charge current from S2 b) Sketchof thermoelectric measurement at D1 when sourcing from S2, the impinging current generates heat carried upstream by theneutral modes and converted to thermoelectric current at the QD. c) Sketch of noise measurement at D2 when sourcing fromH, the upstream neutral modes excited at the hot spot increase the Johnson–Nyquist noise measured at D2 d) Conductanceof the QPC as function of the split gate voltage. e) Green: Current fluctuations measured at D2 when sourcing from S2 asfunction of the transmission of the QPC. Blue: Current fluctuations measured at D2 when sourcing from H. black: Same asblue multiplied by a scaling factor α = 2 . Red: Thermoelectric current measured at D1 when sourcing from S2. f ) Excess noisemeasured at D2 as function of the current at transmission half when sourcing from S2 (green) or H (blue). The black curverepresents the noise from H to D2 (blue curve) multiplied by the scaling factor α = 2 . g) Grey dots: Shot noise contribution N SN = N tot − αN th = N S2 → D2 − αN H → D2 . Orange curve: expected shot noise from 2 separated ballistic charge channels. picture, we have modeled the neutral and charge transmis-sions trough the QPC constriction using a quasi-classicalapproximation [34]: t i ( V ) = 1 / (cid:16) e ( V − V i ) /δV i (cid:17) (4)where i =inner, outer. In our model there are two modesthat are spatially separated, therefore they will be centered around different gate voltage V i of the split-gate QPC,and with different width corresponding to δV i . The totalcharge transmission t charge ( V ) = ( t inner ( V ) + t outer ( V )) / t neutral ( V ) = t inner ( V ), where both the neutral modes arelocated near the inner charge mode. As visible on Fig.2d,this simple model is able to qualitatively explain the trans-mission of the neutral modes; and in particular the observedvanishing of the neutral transmission when the conduc-tance reaches G QPC = 1 /
3. To compare in more detailsthe measured transmission of the neutral modes in the twobulk fractions, ν B = 2 / ν B = 3 /
5, we have plottedeach one as function of the conductance of the QPC con-striction G QPC (Fig.2f). Surprisingly, they present a nearlyidentical behavior, which strongly suggests that, at leastfor these two hole-conjugate states, the transmission ofthe neutral modes is dictated by the conductance of theQPC and is poorly dependent on the bulk filling factor.We have plotted on the same figure (open symbols) theresults of another similar sample (presented in Supplemen-tary note 1). It is clear that the curves present an overlapon a large G QPC region which indicates that the structureof the neutral modes is universal and do not depend on thebulk state as well as the particular mobility or disorder inthe QPC constriction.
Excess noise subtraction
Noise measurements reveal even more about energyexchange mechanisms between the charge and neutralmodes. We first start from the configuration presented onFig.3a, where we measure excess-noise in D2 when sourc-ing DC current from S2. As shown on Figs. 3e & 3f (greencurves) and reported previously [25, 26, 35, 36], the noiseremain finite when the QPC is tuned to the conductanceplateau at t QPC = 1 / G QPC = e / h (Fig. 3d).This finite noise is even more puzzling since in recent ex-periment [19] it was shown that two spatially separatedchannels are present, excluding the presence of conven-tional shot noise at such transmission. Taking advantageof this device we were able to show that a thermal contri-bution is added to the shot noise. In order to distinguishbetween the thermal noise and the partition noise of thequasiparticles (shot noise), the TE current in the QD wasmeasured as function of the right-QPC, but this time thecurrent was sourced in S2 (Fig. 3b). The finite measuredthermoelectric current presented on Fig. 3e (red curve)is a clear signature of heat dissipation occurring at theright-QPC constriction where the charge modes exchangeenergy with the neutral mode. Similarly, exciting neutralmodes using contact H as shown in Fig. 3c, increases lo-cally the electronic temperature at the QPC – as discussedbefore (Fig. 3f, blue curve). Indeed, a similar dependenceon the right-QPC transmission, in both measurements, isobserved in Fig. 3e. Subsequently, it is interesting to ex-clude the thermal contribution from the total excess noisemeasured at D2 when sourcing from S2 and be left withthe shot-noise contribution solely. In order to do that, wesubtracted the thermal noise N th multiplied by a constant α , that characterizes the neutral mode decay, from the to-tal noise N tot . The constant α was chosen such that the shot noise contribution at half-transmission would be zero- as expected from a full transmission of one channel and afull reflection of the second. The result of this subtraction N sn = N tot − αN th is plotted on Fig. 3g together withthe expected excess noise (orange curve) for two chargedchannels giving a “double hump” shape (due to the t (1 − t )dependence of the noise of each channel). The shot noise N SN follows the expected behavior in the range 0 − . DISCUSSION
Via studying the transmission of neutral modes througha QPC constriction at hole conjugate states, ν = 2 / ν = 3 / METHODS
Sample fabrication
The samples were fabricated in GaAs–AlGaAs het-erostructures, embedding a 2DEG, with an areal den-sity of (1 . . × cm − and a 4.2 K “dark”mobility(3 . . × cm V − s − , 70–116 nm below thesurface. The different gates were defined with electronbeam lithography, followed by deposition of Ti/Au. Ohmiccontacts were made from annealed Au/Ge/Ni. The samplewas cooled to 30 mK in a dilution refrigerator. Measurement technique
Conductance measurements were done by applying ana.c. signal with ∼ µ V r.m.s. excitation around 1.3 MHzin the relevant source. The drain voltage was filtered us-ing an LC resonant circuit and amplified by homemadevoltage preamplifier (cooled to 1 K) followed by a room-temperature amplifier (NF SA-220F5).For the thermoelectric current measurement, an alter-nating voltage V H at frequency f ≈
650 kHz was applied tocontact H, giving rise to an upstream neutral modes withtemperature proportional to | V H | , producing a series of har-monics starting at 2 f . The heat reaching the upper sideof the QD generates an alternating thermoelectric currentflowing to drain D1. The signal is then filtered using an LCcircuit with a resonant frequency at 2 f = 1 . V H amplitude by averaging the peak-to-peakvoltage of more then 20 consecutive oscillations. The QDis in the metallic regime ∆ E (cid:28) k B T (cid:28) e /C where thetemperature is above the level spacing ∆ E and below thecharging energy e /k B C = 290 mK (see Coulomb diamondmeasurement in Supplementary note 3). Data availability
The data that support the plots within this paper andother findings of this study are available from the corre-sponding author upon request.
Acknowledgments
A.R and F.L acknowledge Robert Whitney, Ady Stern,Yuval Gefen, Jinhong Park and Kyrylo Snizhko for fruit-ful discussions. M.H. acknowledges the partial support ofthe Minerva Foundation, grant no. 711752, the GermanIsraeli Foundation (GIF), grant no. I-1241- 303.10/2014,the Israeli Science Foundation, grant no. ISF- 459/16 ,and the European Research Council under the EuropeanCommunity’s Seventh Framework Program (FP7/2007-2013)/ERC Grant agreement No. 339070.
Author contributions
A.R., F.L., R.S., I.G., and M.H. designed the exper-iment, A.R., F.L., R.S., I.G., D.B. preformed the mea-surements, A.R., F.L., R.S., I.G., D.B. and M.H. did theanalysis, I.L. developed the theoretical model. A.R., F.L.,I.L. and M.H. wrote the paper. V.U. grew the 2DEG.
Competing Financial Interest
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Here we present a second sample studied in order to probe if any contribution to the heat measured by the quantum dotwas propagating through the bulk and not only through the edge modes. The device is very similar to the one from themain text. A deflector gate is placed before the quantum dot that allows to redirect the neutral modes to the grounddirectly. The results of the thermoelectric current measured versus the plunger gate voltage for the two configurations,deflector open and close, are presented on Supplementary Figure 1. When the QPC is closed a net thermoelectriccurrent is measured similarly to the main paper sample. Nevertheless, one can notice that the thermoelectric currentis only positive, which is due to the fact that we used a spectrum analyzer, only sensitive to the absolute value of thesignal. Conversely, when the deflector is open, no thermoelectric current is measured by the quantum dot, which showsthat no measurable contribution arise from the bulk. H ν = 2 / ν = 3 / chargedneutral QPC deflector D V pl QD t pl (V) -0.42 -0.415 -0.41 -0.40500.050.10.150.2 H ν = 2 / ν = 3 / chargedneutral QPC deflector D V pl QD t pl (V) -0.42 -0.415 -0.41 -0.40500.050.10.150.2 S ν = 2 / ν = 3 / charged QPC deflector D V pl QD t pl (V) -0.42 -0.415 -0.41 -0.40500.050.10.150.2 a ) b ) c ) d ) e ) f ) Supplementary Figure 1. Measurement of the second sample: a)
Configuration corresponding to the measurementof the Coulomb blockade peaks. both sides of the quantum dot are at base temperature. Sourcing current from Sresults in transmission peaks measured in D1 when changing the plunger gate voltage, as visible on b) . c) Configurationcorresponding to the measurement of the neutral transmission using the thermoelectric current created through thequantum dot when the deflector gate is closed. d) Evolution of the thermoelectric current across the quantum dot asfunction of the plunger gate voltage. e) Configuration corresponding to the measurement of the neutral transmissionusing the thermoelectric current created through the quantum dot when the deflector gate is open f) No thermoelectriccurrent is measurable through the QD when the deflector is open showing that no heat transport is happening in thebulk of the sample. 1 a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec upplementary Note 2: Upstream neutral modes at other filling factors In Supplementary Figures 2 we present thermoelectric measurements at other filling factors, ν = 2 , , / , / , / , / V H ∼ µ V RMS and scanned the plunger over a range of CB peaks where the QPC if fullyopen and the deflector gate is energized. At fillings ν = 2 , / / ν = 1 , / / ν = 2 / / ν = 1 upstream heat mode was measured before [1, 2] and it was attribute to a state of ν = 2 / ν = 1. -0.7 -0.69 -0.68 -0.67 -0.66 -0.65 -0.64 -0.63 -0.62 -0.61 -0.600.5 0500.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.0900.2 050-0.78 -0.77 -0.76 -0.75 -0.74 -0.7300.2 050-0.24 -0.22 -0.2 -0.18 -0.16 -0.14 -0.12 -0.100.2 050-0.85 -0.84 -0.83 -0.82 -0.81 -0.8 -0.79 -0.78 -0.77 -0.76 -0.7500.2 050-0.64 -0.63 -0.62 -0.61 -0.6 -0.59 -0.58 -0.5700.2 050 C o ndu c t a n ce ( e / h ) T E S i gn a l ( n V ) V pl (V) ν = 2 ν = 1 ν = 2 / ν = 3 / ν = 2 / ν = 1 / Supplementary Figure 2. Evolution of the Coulomb peaks and thermoelectric currents for several filling factors:
Thermoelectric voltage at other filling factors, ν = 2 , , / , / , / , /
3. Blue curves - left axes - conductancethrough the QD tuned to Coulomb blockade regime. Orange curves - right axes - thermoelectric voltage at the sameregime. Measured with the deflector gate energized and V H ∼ µ V RMS upplementary Note 3: Quantum Dot Coulomb diamonds Here is presented the evolution of the conductance in the quantum dot as function of the voltage applied in S1 and theplunger gate. The extracted charging energy is ∼ µe V which corresponds to an equivalent temperature of ∼ -1.305 -1.3 -1.295 -1.29 -1.285 -1.28 -1.275 -1.27 -1.265 -1.26V pl (V)-4-2024 V S ( V ) -5 /h) Supplementary Figure 3. Coulomb Diamonds:
Evolution of the conductance of the quantum dot as function of V S1 and V pl . Coulomb diamonds are visible with a typical charging energy of 25 µe V with corresponds to an equivalenttemperature of ∼
290 mK
Supplementary Note 4:Theoretical model
Fractional quantum Hall states at simple Landau level filling fractions like ν = 1 /m with odd m could be described bythe Laughlin wave function [3]: h z , ..., z n | Ψ L i ∝ Y i 5, etc., it is still not completely clear what is the structure thewave function. Here we focus on the ν = 2 / ν = 2 / ν = 1 / ν = 1 form a Laughlin condensate with ν = 1 / 3. Another oneconsiders ν = 2 / ν = 1 type condensate of fractional e ∗ = 1 / ν = 1 / | Ψ i ∼ Z J ( ζ, ¯ ζ ) d M ζd M ¯ ζ Y i 1. This is consistent with the fact that the correlation length inthe inner condensate of (3) is larger. Statistical phases for electronic excitation are indeed fermionic θ αβ = πK αβ , α, β = 0 , ..., 3. ˆ K = (6)so that electron vectors form an integral lattice that has a hexagonal projection on the neutral sector. Quasi-particlesmust have single-valued wave functions as, e.g., Laughlin quasi-particles (2) discussed above. In the effective modellanguage this translates into integer statistical phases with respect to all electronic excitation. In other words, quasi-particles form a dual lattice: χ n = n √ φ + n − n + n √ φ c + n − n √ φ n + n − n + n √ φ n (7)The charges of the corresponding excitation are e ∗ n = ( n + n − n + n ) / n = 13 n + n − n + n + 43 n n − n ( n − n + n ) . (8)It is interesting to investigate the most relevant particles content in the inner channel, we index them as χ n n n .There is one excitation with e ∗ = 1 / / χ = φ c √ e ∗ = 0 and ∆ = 2 / χ , χ − , χ (10)and conjugate. Interestingly there are six quasi-particles with e ∗ = 1 / χ , χ , χ , χ , χ , χ (11)These could be a signature of ν = 1 like condensate (plasmon waves on the boundary of inner condensate of e ∗ =1 / Quantitative results: Thermo-electric current Upstream thermo-electric current through a single level with energy (cid:15) can be estimated as I th − el ∝ [ f D ( (cid:15) ) f U ( − (cid:15) ) − f U ( (cid:15) ) f D ( − (cid:15) )] (12)4 ) ⌫ = 1 ⌫ = 2 / ⌫ = 1 / ⇢ ( y ) c ⇠ ⇠ y n n G QPC ( e /h ) n e u tr a l tr a n s m i ss i o n ⌫ B = 2 / ⌫ B = 3 / t h e r m o - e l ec tr i cc u rr e n t( p A ) V QPC (a.u) tt n ⌫ = 2 / a ) b ) c ) d ) e ) ⌫ = 2 / ⌫ = 3 / Injected power (10 W) Numerical integration ⌫ QPC = 1 / tr a n s m i ss i o n -1 -0.5 0 0.500.20.40.6 00.511.522.500.20.40.60.81 c o ndu c t a n ce ( e / h ) V QPC (V) ⇥ n e u tr a l tr a n s m i ss i o n T T ( K ) -1 -0.5 0 0.500.20.40.6 00.511.522.533.544.500.20.40.60.81 ⇥ n e u tr a l tr a n s m i ss i o n T T ( K ) ⌫ = 3 / c o ndu c t a n ce ( e / h ) V QPC (V) ⌫ B = 2 / ⌫ B = 3 / Temperatureapproximation Supplementary Figure 4. Theoretical edge state structure: Schematic of the effective edge model. The blue curveshows possible transversal profile of charge density near the boundary of two-dimensional electron gas. In the effectivetheory, the inner modes are described in a universal way by three boson fields φ c , φ n , φ n with action (4). Note thatthese modes have larger width than the outermost charged mode.Here effective occupation numbers f i ( (cid:15) ) are simply Fourier transforms of the corresponding correlation functions: f ( (cid:15) ) = Z dte i(cid:15) t K ( t ) (13a) K ( t ) = h exp[ − iχ n ( t )] exp[ iχ n (0)] i (13b)We take into account that downstream modes on the upper edge of quantum dot originate from a cold reservoirwith base temperature T , while upstream modes are from ”hot” Ohmic contact with temperature T . Assuming thatthe dominant cooling mechanism is by four edge states [9, 10], and taking into account that every chiral bosonic modeat temperature T carries a heat flux πT / 12 we could write down the heat balance equation between the heat producedin the contact and heat dissipated (carried away by edge modes): I ∆ µ = π (cid:0) T − T (cid:1) . (14)Therefore we find that T = p T + (∆ µ/π ) for ν = 2 / f U ( (cid:15) ) = Z dte i(cid:15) t (cid:20) T sinh( πT ( t − iη )) (cid:21) δ (cid:20) T sinh( πT ( t − iη )) (cid:21) δ . (15)In our model δ = 1 / , δ = 2 / δ + δ = 1. For the lower edge effective occupation coincides with freefermions, since ∆ = 1: f D ( (cid:15) ) = Z dte i(cid:15) t T sinh( πT ( t − iη )) = 1 e (cid:15) /T + 1 (16)We could also estimate the upper effective occupation (13) taking into account that at small energies the integralcomes from the large times, where the correlation function decays exponentially: K ( t ) ’ exp[ − πT eff | t | ] , T eff = T + 2 T . (17)5hus one could make an estimation: f U ( (cid:15) ) ∼ e (cid:15) /T eff + 1 (18)Results of this approximation as compared to exact numeric integration are shown in Fig 2d in the main text . Theyalso agree well with the experimental data.The asymptotic behavior of the thermo-electric current (12) in effective temperature approximation is I th − el ∼ ∆ µ /T , ∆ µ (cid:28) T (19a) I th − el ∼ const , ∆ µ (cid:29) T (19b)The saturation constant itself does not depend on base temperature T when it is sufficiently low, but is suppressed as δ(cid:15) /T when temperature becomes comparable with the level spacing δ(cid:15) of the quantum dot. Qualitative discussion: QPC charge and neutral transmissions The key ingredient for the results of previous section is the electron-like scaling behavior of fractional quasi-particles(11). Here we also use the argument that the fractional quasi-particles coupled to neutral modes behave like freeelectrons. It is important to note that the effective action (4) is an intermediate fixed point, it has relevant neutralons∆ = 2 / 3. In contrast, low-energy fixed point of Polchinski-Kane-Fisher model [11] has only irrelevant neutralons with∆ = 2. However, quantum point contacts and quantum dots can give additional energy scale that pins the intermediatefixed point.There are three wide inner states in our model: one charged and two upstream neutral and one narrow chargedouter. The transmission of a single channel could be modeled in the quasi-classical approximation [12, 13] as t i ( V ) = 11 + e ( V i − V ) /δV i (20)where i = in , out. In our model there are two channels that are spatially separated, therefore they will be pinched offat different QPC voltages: t charge ( V ) = 12 (cid:20) 11 + e ( V in − V ) /δV in + 11 + e ( V out − V ) /δV out (cid:21) (21)Analogously, both neutral modes are located at inner channel, so that: t neutral ( V ) = 11 + e ( V in − V ) /δV in (22)It is natural to assume that δV in /δV out ∝ ξ/ξ (cid:29) • Appearance of quasi-particles with charge e ∗ = 1 / • Couplings of the above quasi-particles to the changed and neutral modes are universal δ = 1 / δ = 2 / • Inner and outer edge channels have significantly different widths ξ/ξ (cid:29) • Upstream neutral excitation only appear in the inner channel.Most of these features will appear in all other filling fractions that allow hierarchic condensates. And indeed, theexperimental data for ν = 3 / Supplementary References [1] Vivek Venkatachalam, Sean Hart, Loren Pfeiffer, Ken West, and Amir Yacoby. 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