# Fileset

[2025A00338G_Main_with_bib_V2.pdf](https://mdr.nims.go.jp/filesets/1fae2642-f3a5-4997-9716-08226ee040a4/download)

## Creator

Lingfei Zhao, Trevyn F. Q. Larson, Zubair Iftikhar, John Chiles, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), François Amet, Gleb Finkelstein

## Rights

 © 2025 American Physical Society[In Copyright](http://rightsstatements.org/vocab/InC/1.0/)

## Other metadata

[Thermal Properties of the Superconductor–Quantum Hall Interface](https://mdr.nims.go.jp/datasets/dc4c6df0-ce71-4594-8b6e-26316097c327)

## Fulltext

Thermal properties of the superconductor-quantum Hall interfacesLingfei Zhao,1, ∗ Trevyn F.Q. Larson,1, ∗ Zubair Iftikhar,1, ∗ John Chiles,1Kenji Watanabe,2 Takashi Taniguchi,2 François Amet,3 and Gleb Finkelstein1, †1Department of Physics, Duke University, Durham, NC 27708, USA2National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan3Department of Physics and Astronomy, Appalachian State University, Boone, NC 28607, USA(Dated: January 15, 2025)An important route of engineering topological states and excitations is to combine superconductors (SC) withthe quantum Hall (QH) effect, and over the past decade, significant progress has been made in this direction.While typical measurements of these states focus on electronic properties, little attention has been paid to theaccompanying thermal responses. Here, we examine the thermal properties of the interface between a type-IIsuperconducting electrodes and graphene in the QH regime. We use the thermal noise measurement to probethe local electron temperature of the superconductor next to the biased interface. Surprisingly, the measuredtemperature raise indicates that the superconductor provides a significant thermal conductivity, which is linearin temperature. This suggests electronic heat transport and may be unexpected, because the number of thequasiparticles in the superconductor should be exponentially suppressed. Instead, we attribute the measuredelectronic heat conductivity to the overlap of the normal states in the vortex cores.Over the past decade, significant progress has been made incombining quantum Hall states with superconductors [1–7].The experiments clearly demonstrate the coherent quasiparti-cle propagation in the proximitized chiral interfaces [3, 4, 6].While these works focussed on the time-averaged transport,theory predicts that thermal properties and fluctuations shouldcarry unique information, e.g. about the nature of the neutralexcitations [8]. However, with a few exceptions [9], little ex-perimental evidence exists about the thermal properties of theproximitized semiconductor interfaces.Thermal properties of the nanoscale quantum materialsemerged as an interesting novel area of research [10]. Whilesuch quantities as Nernst and Seebeck coefficients becamerelatively well known, thermal conductivity measurements re-main less explored [11]. Recent studies demonstrate that mea-surements of mK temperatures and fW powers are feasiblein appropriately designed QH samples via Johnson-Nyquistnoise [12–15]. We use a similar noise setup to examine thethermal response of a hybrid device made of graphene con-tacted by type-II thin film superconducting electrodes. In theQH regime, we bias the device forming hot spots – regionswhere the Joule heating is deposited – at the interface withthe contact, from where the heat can escape via several mech-anisms. We then measure the noise carried by the QH edgestates downstream of the superconductor to probe its localtemperature.We find that applying current on the scale of tens of nAcan heat the interfacial region of the contact to ∼ 1 K. Asthe magnetic field increases, the electron temperature gradu-ally decreases until becoming comparable to the temperatureof a similarly heated normal contact. We argue that this in-crease of the cooling efficiency is explained by the electronicheat conductivity of the superconductor, mediated by the nor-mal states in the vortex cores [16]. This mechanism becomesmore efficient as the distance between the vortices decreaseswith magnetic field. Further examination of the temperatureresponse across a few µm-wide superconducting strip revealsthat indeed the heat spreads rather efficiently the across super-conductor. Eventually, at high magnetic field, the whole widthof the strip becomes uniformly heated. The estimated thermalconductivity varies by almost two orders of magnitude overthe studied range of magnetic field.The schematic and the image of our sample are shown inFigure 1a. The device is made of encapsulated graphene con-tacted by normal metal (Cr/Au) and superconductor (MoRe)electrodes. Detailes of the device can be found in Ref. [17].Throughout the measurement, we apply a field of several Teslato induce the QH effect in graphene. The MoRe has a criti-cal temperature Tc ∼ 10 K and stays superconducting at leastup to 12 T, the highest field in this measurement (Figure S1in [18]). The width of the superconducting strip (light grayin Figure 1a) is ∼ 2 µm, much longer than the MoRe co-herence length (ξ < 10 nm). Therefore, in the first part ofthis paper the two graphene-superconductor interfaces will beconsidered separately.The noise at the two interfaces is measured by two home-made cryogenic amplifiers (∼ 10 dB gain), A and B, whichare attached to the normal electrodes, EA and EB, at the topand bottom regions respectively (Figure 1a). The amplifiersare AC coupled to the sample electrodes via 10 nF blockingcapacitors and LC resonators centered around 1.5 MHz. Theoutputs of the cryogenic amplifiers are then fed to room tem-perature amplifiers (46 dB gain) and digitized.We start by exploring the thermalization of the edge statein contact with the superconductor. We work with the bottomregion of the sample (Figure 1a), and measure the spectraldensity SB of amplifier B. Depending on the field direction,the QH edge channels will arrive at the amplifier contacts ei-ther from the superconducting strip (positive field) or fromthe bottom normal contact (negative field). To enable a di-rect comparison, the length of the bottom normal metal andthe bottom superconducting interfaces are designed to be thesame (1 µm), and both contacts are cold grounded.Figure 1b plots the zero-bias SB measured vs. bath tem-2FIG. 1. Excess noise of superconducting and normal contacts. (a)Schematics of the device and measurement configuration. Normalcontacts (Cr/Au) are yellow and the superconducting contact (MoRe)is gray. The cross section (top) shows the hBN/graphene/hBN stacksitting on a graphite gate (outlined by black dashed lines in the op-tical image). The superconductor, the bottom and top-right normalcontacts are cold grounded. Current bias I is injected at the bottomleft normal contact. In the remainder of the figure, only the bot-tom region of graphene is measured. (b) Zero-bias SB of the normalmetal (red, B = –3 T) and superconductor (blue, B = 3 T) plottedvs. the bath temperature Tbath. The top and bottom curves are ob-tained on the ν = 2 and 6 plateaus. (c) Noise power SB measuredby amplifier B (after the amplification chain) plotted vs. B and VGat zero bias (top) and 50 nA (bottom). (d) Cuts of the maps in (c)measured along the dashed lines (middle of ν = 2 plateau). Insets:schematics of the edge state direction, and locations of the relevanthot spots and hot edges for negative and positive fields.perature Tbath for both ν = 2 and 6 at |B| = 3 T. The resultfor the superconductor and the normal contact are practicallyindistinguishable. The noise is linear vs. Tbath, allowing usto calibrate the gain of the amplification chain and convert SBto temperature [18]. Next, we measure SB at the base tem-perature Tbath = T0 as a function of VG and B (top panelof Figure 1c). The noise map is nearly symmetric with re-spect to the field direction, and the noise stays constant onthe plateaus, indicating that both the superconducting and thereference normal contact stay thermalized at the base temper-ature of the sample, T0 = 35 mK, throughout the full range ofmagnetic field. In the following, we disregard a narrow rangeof fields |B| < 1 T, where the conductance of the sample isno longer quantized (see Figure S2 in [18]).We next apply a current bias of I = 50 nA to the bottom-left contact and plot the resulting SB(VG, B) in the bottompanel of Figure 1c. At negative B, when we locally heat thebottom normal contact, the noise on the ν = 2 plateau in-creases compared to zero bias. This indicates that the normalelectrode is locally heated by the applied power, and its ther-mal noise is then detected by the amplifier B located down-stream. Note that in this case, the excess noise stays con-stant in B. At positive B, when the superconductor is locallyheated, the excess noise on the ν = 2 plateau stays constantin VG, but strongly dependent on magnetic field.In Figure 1d we plot the noise measured along the dashedlines corresponding to the ν = 2 plateaus in Figure 1c. Thisnoise (left axis) is converted to the temperature (right axis) viathe calibration developed in Figure 1b. At I = 0, the noise onthe ν = 2 plateau stays constant in B, except for the smallvicinity of zero field. At finite bias, the elevated noise staysconstant at negative fields, indicating that the normal contactis heated to about 100 mK independent of the field. At posi-tive field corresponding to biasing the superconductor, we ob-serve a strong enhancement of the noise followed by a gradualdecrease.In principle, the increased noise at the biasedsuperconductor-QH interface could have originated fromthe shot noise [19] (see [18] for a discussion of other noisesources). Namely, in the case of an ideal superconductor, anincoming electron would have to undergo either a normal orAndreev reflection. Strong current fluctuations could thenbe expected, corresponding to the electrons or holes beingemitted downstream (toward the amplifier). However, theprobabilities of the normal and Andreev processes stronglydepend on the gate voltage [4], while experimentally, thenoise at a fixed field stays flat across the plateaus (Figure 1c).(Very small variations of the noise, on the scale of 1%, havebeen observed as a function of gate voltage.) Furthermore,our previous studies of the superconductor-QH interfaces,including this very sample [17], indicate that most of theelectrons arriving at the superconducting contact are ab-sorbed, likely by the Caroli-de Gennes-Matricon (CdGM)states of the superconducting vortices [20]. We argue thatas a result, the interfacial region of the superconductor islocally heated, and its thermal noise is carried downstreamto the amplifier, similar to the case of the normal contact. In[18], we independently verified the temperature rise at thesuperconductor-QH interface by measuring the suppressionof the non-local resistances.In the following, we convert the excess noise to the localelectron temperature using the calibration of Figure 1b. (No-tice that additional factor of 2 has to be applied in this con-version to account for the fact that the amplifier contact stayscold [12, 18].) In Figure 2a, we plot the temperature of thesuperconductor next to the bottom interface T (I), as the fieldis stepped from B = 1 to 12 T. The gate voltage is adjustedto stay in the middle of the ν = 2 plateau. At B = 1 T,the temperature reaches ∼ 1 K at 100 nA. Even at a 10 nAbias (corresponding to a voltage drop of ∼ 130µV), the elec-tron temperature increases to about 300 mK. As B increases,T decreases precipitously, as show by the T (B) graphs mea-sured at fixed I in Figure 2b. At the highest field of 12 T,the temperature of the superconductor next to the interfacebecomes comparable to that of the reference normal contact,whose T (I) is plotted in Figure 2a as a dashed line. Interest-ingly, at that point MoRe is still superconducting (Figure S1in [18]).3FIG. 2. Heating of the bottom interface. (a) Temperature increase∆TB plotted vs. bias current I for ν = 2. The different curvescorresponds to magnetic fields B is stepped by 0.1 T (1–4 T range)and 0.5 T (4 – 12 T range). (b) ∆TB vs. B for ν = 2 is plotted atconstant bias currents of 10, 20, 40 and 80 nA, showing the gradualdecay of temperature with field. Variations of temperature observedat small fields for all curves are likely caused by changes of the cool-ing pathways due to vortex rearrangements, see Figure S3 in [18].(c) ∆TB plotted vs. the Joule power P at Tbath = 35 mK. Differentcolors correspond to magnetic fields of 2, 3 & 4 T, while ν = 2 and6 are represented by the solid and dashed lines. The corresponding∆TB(P ) curves of the normal contact are nearly independent on Band ν (black lines). (d) The difference in power between the ν = 2and 6 curves of panel (c) plotted against T 5B − T 5bath. The lineardependence indicates that the difference can be attributed to the con-tribution of phonons.The T (I) curves in Figure 2a have a V-shape typical for hotelectrons in a metal that are cooled via diffusion (Wiedemann-Franz mechanism) and the emission of phonons [12]. Namely,at low temperatures, the phonon emission is negligible, andthe applied power P ∝ I2 is balanced by electronic cool-ing ∝ (T 2 − T 20 ), resulting in the roughly linear slope of theT (I) curves visible in Figure 2a. At higher temperatures, theemission of phonons by hot electrons results in the sublinearT (I) [21].Next, we plot T (P ) – the temperature of the superconduc-tor next to the interface as a function of applied power forthree magnetic field values (2, 3 & 4 T) in Figure 2c. HereP = I2R/2, where the factor of 1/2 appears because theJoule heating is evenly distributed between two hot spots. Forcomparison, we also plot the T (P ) of the normal metal at 2 T,which are nearly independent of B.The focus of this figure is to compare the behavior for thefilling factors ν = 2 (solid lines) and ν = 6 (dashed lines).At a given field, the pairs of the T (P ) curves for ν = 2 and6 overlap at low power, but diverge at high power. Surpris-ingly, the temperature at ν = 6 is hotter, ruling out the moreefficient cooling via QH edges. Moreover, the difference isnoticeable at higher power, where the phonon emission is thedominant cooling mechanism. To prove this point, we plotthe difference in power, ∆P , needed to reach the same T vs.T δ − T δ0 and find a linear relation for δ = 5 (Figure 2d).∆P ∝ (T 5−T 50 ) is the typical cooling power of the hot elec-trons via phonon emission in diffusive metals [21].While we cannot pinpoint the exact origin of the differencebetween the two filling factors, we attribute it to the differencein the size of the QH hot spot, which is determined by thelength over which the QH edge channel is equilibrated withthe contact. Notice that a similar difference is also observedfor the normal contact (black lines) – ν = 2 is slightly coolerthan ν = 6 – so this effect is not specific to the superconduc-tor.Finally, we use the data in Figure 2 to bring forward two ar-guments further supporting the attribution of the excess noiseto the electron temperature. First, T (P ) curves for ν = 2and 6 nearly coincide at low power. However, a significantlyenhanced shot noise could have been expected for ν = 6 vs.ν = 2, because a given power P would correspond to a√3times higher bias current. Second, in principle we could stillattempt to fit the S(I) curves with the shot noise expression,S ∝ I × coth(eV/2kBTe), with V = hI/νe2. The curvatureat the minimum of the S(I) curves would then correspond tothe electrons’ base temperature Te treated as a fitting param-eter. Te would then depend on both ν and B, increasing withmagnetic field to∼ 200 mK at 12 T. This result would both bephysically unreasonable and contradict Figure 1d. We there-fore rule out the shot noise as an explanation of the measuredS(I) curves.To further analyze the purely electronic cooling, in Fig-ure 3a we plot T 2 vs. applied power in the range of P < 500pW and T < 100 mK – at which temperatures the phononemission should be negligible [21]. The data points show aclear linear dependence, indicating that the superconductor isindeed cooled via the diffusion of hot electrons and the emis-sion of phonons can be neglected, with a possible exceptionof the lowest fields (B <∼ 1.5 T). We further extract the slopeof the P = α(T 2B − T 20 ) dependence (Figure 3b). The coeffi-cient α is presented in the quantized units of αQ = π2kB/6h,corresponding to the heat flow in a single QH channel [12].Even at the lowest field, the measured α greatly exceeds thatof the quantum Hall. We therefore attribute it to the electronicheat conductivity of the superconductor, which appears to belinear in temperature, like in a normal metal.At zero field, the electronic heat conductivity of a super-conductor is strongly suppressed due to the gap in the single-particle spectrum. (Though violation of this dependence hasbeen reported see e.g. [22].) However, at finite field, theCdGM states should behave like a normal metal, and the tun-neling of the quasipartciels between the vortex cores can ef-ficiently conduct heat. The resulting thermal conductivity islinear in temperature and rapidly grows with magnetic fielddue to the increasing overlap between the vortex cores [16].4FIG. 3. Thermal conductivity of the superconducting film. (a)T 2B plotted vs. Joule heating power P from B =1 to 11.5 T (shownin steps of 0.75 T). Linear dependence is clearly visible indicatingelectronic thermal conductivity. (b) The slope α extracted from theT 2B(P ) curves in panel (a) as a function of B. α is represented inunits of αQ = π2kB/6h, corresponding to a single QH channel. (c)∆TA plotted vs. ∆TB on the ν = 2 plateau fromB = 1 to 12 T. TheTB data is the same as in Figure 2a. The black line is ∆TA = ∆TB.(d) ∆TA/∆TB ratio extracted from the lower range of temperaturesin panel (c), plotted vs. B for ν = 2 (blue circles) and 6 (red crosses).The quantity converge to unity at high magnetic field. (e) Schematicsof the heat flow across the superconductor showing the heat balanceat the top interface. (f) β, plotted in units of αQ, represents the elec-tronic thermal conductivity across the top interface (please see maintext). β(B) dependencies nearly overlap for ν = 2 (blue circles) andν = 6 (red crosses).This regime was tentatively reported in the seventies [23], andmore recently explored in Ref. [24]. Closer to Hc2, the or-der parameter becomes nearly uniformly suppressed, whichincreases the density of quasiparticles [25]. In this regime,α is linearly approaching the value corresponding to the nor-mal state of the metal [26]. Both regimes – the initial rapidrise followed by flattening close to Hc2 – are clearly visible inFigure 3b, which may be the first such demonstration in thinfilm samples.We are now interested to find out what happens once theheat spreads from the QH hot spot through the superconduc-tor. To that end, we use amplifier A (Figure 1a) to study thetemperature of the edge state which originates at the top su-perconducting interface, TA. (We also reinstate subscript Bto indicate the temperature measured by the bottom amplifierinterface, TB.) In Figure S4 of [18], we plot TA vs. P whilethe power is still applied by biasing the bottom left contact,the same way as in Figure 2. Evidently, the heat spreads effi-ciently to the opposite side of the superconducting strip.In Figure 3c we plot ∆TA = TA−T0 vs. ∆TB. Experimen-tally, we find that ∆TA scales almost linearly with ∆TB. Therobust nature of this relation, which extends over the range oftemperature changes much larger than the base temperature,is presently not clear. We extract the ∆TA/∆TB slope at lowpower and plot it in Figure 3d as a function of B. Similar datafor several applied powers is presented in Figure S5 of [18].Independent of P , the slope increases monotonically with Band saturates at 1 for B >∼ 4 T, directly confirming that thewhole width of the superconducting strip gets uniformly hot.To understand the behavior in the lower field range, B <∼4T, we note the difference between the plots of ∆TA/∆TBmeasured at ν = 2 and ν = 6 (lower curve in Figure 3d). Weknow that the superconductor is highly thermally conductivefor the whole range of fields B > 1 T (Figure 3b), and the2 vs. 6 additional quantum channels of graphene should notchange the temperature of the metal itself. We therefore arguethat TA is the temperature of the QH edge running along therelatively short top interface, which has not fully equilibratedwith the hot electrons in the superconductor.We analyze the heat flow at the top superconducting in-terface as sketched in Figure 3e. Given the high heat con-ductivity of the superconductor (Figure 3b), we approximatethe electron temperature of the the superconducting metalclose to the top interface as TB. The edge channels arriveat the interface with temperature T0 and acquire heat fromthe superconductor, eventually reaching a temperature of TA.This temperature should satisfy the heat balance equation,β(T 2B − T 2A) = ναQ (T 2A − T 20 ), where β is proposed torepresent the effective electronic heat conductivity across thetop interface. Using this expression, we extract β by fittingT 2B− T 2A vs. T 2A− T 20 and find that the values nearly coincidefor the two filling factors ν = 2 and 6 (Figure 3f). We notethat β is different from α in Figure 3b, which characterizesthe electronic heat conductivity of the superconductor itself.In contrast, β characterizes the heat conductivity between themetal and the edge state at the top interface.Summarizing our results, we find that the dissipation at thehot spot results in a substantial overheating of the supercon-ducting metal close to the SC-QH interfaces, which is partic-ularly noticeable at low magnetic fields. Carefully engineeredheat sinks would be necessary for quantum applications ofsuch systems e.g. for quantum information processing. In-creasing the magnetic field rapidly thermalizes the supercon-ductor, resulting in a change of the thermal conductivity bymore than one order of magnitude (Figure 3b). By ∼ Hc2/2,the heat dissipation by the superconductor becomes compa-rable to that of the normal electrode. Finally, our observa-tions are unfavorable for measuring the heat conductivity ofthe proximitized QH edge states along the SC interface – un-fortunately, they would be thermally shunted by the thermalconductivity of the superconductor.We thank C. Beenakker for explaining the reason for thesuppression of the shot noise in our measurement. G.F. ap-preciates the technical discussions of the amplification setupwith M. Heiblum, F. Lafont, M. Reznikov, and Y. Ronen. Thework at Duke University was supported by the Division ofMaterials Sciences and Engineering, Office of Basic EnergySciences, U.S. Department of Energy, under Award No. DE-SC0002765. The deposition of MoRe was performed by F.A.at the Appalachian State University. K.W. and T.T. acknowl-edge support from the Elemental Strategy Initiative conducted5by the MEXT, Japan, (grant no. JPMXP0112101001), JSPSKAKENHI (grant no. JP20H00354) and CREST (no. JP-MJCR15F3, JST). The sample fabrication was performed atthe Duke University Shared Materials Instrumentation Facil-ity (SMIF), a member of the North Carolina Research TriangleNanotechnology Network.∗ These three authors contributed equally† gleb@duke.edu[1] F. Amet, C. T. Ke, I. V. Borzenets, J. Wang, K. Watan-abe, T. Taniguchi, R. S. Deacon, M. Yamamoto, Y. Bomze,S. Tarucha, and G. Finkelstein, Science 352, 966 (2016).[2] G.-H. Lee, K.-F. Huang, D. K. Efetov, D. S. Wei, S. Hart,T. Taniguchi, K. Watanabe, A. Yacoby, and P. Kim, Nat. Phys.13, 693 (2017).[3] A. Seredinski, A. W. Draelos, E. G. Arnault, M.-T. Wei, H. Li,T. Fleming, K. Watanabe, T. Taniguchi, F. Amet, and G. Finkel-stein, Science Advances 5, eaaw8693 (2019).[4] L. Zhao, E. G. Arnault, A. Bondarev, A. Seredinski, T. F. Q.Larson, A. W. Draelos, H. Li, K. Watanabe, T. Taniguchi,F. Amet, H. U. Baranger, and G. Finkelstein, Nature Physics16, 862 (2020).[5] O. Gül, Y. Ronen, S. Y. Lee, H. Shapourian, J. Zauberman, Y. H.Lee, K. Watanabe, T. Taniguchi, A. Vishwanath, A. Yacoby,and P. Kim, Phys. Rev. X 12, 021057 (2022).[6] H. Vignaud, D. Perconte, W. Yang, B. Kousar, E. Wagner,F. Gay, K. Watanabe, T. Taniguchi, H. Courtois, Z. Han, H. Sel-lier, and B. Sacp, Nature 624, 545 (2023).[7] A. Uday, G. Lippertz, K. Moors, H. F. Legg, R. Joris,A. Bliesener, L. M. C. Pereira, A. A. Taskin, and Y. Ando,Nature Physics (2024), 10.1038/s41567-024-02574-1.[8] C. W. J. Beenakker, Rev. Mod. Phys. 87, 1037 (2015).[9] A. Denisov, A. Bubis, S. Piatrusha, N. Titova, A. Nasibulin,J. Becker, J. Treu, D. Ruhstorfer, G. Koblmüller, E. Tikhonov,and V. Khrapai, Nanomaterials 12, 1461 (2022).[10] J. P. Pekola and B. Karimi, Rev. Mod. Phys. 93, 041001 (2021).[11] J. Waissman, L. E. Anderson, A. V. Talanov, Z. Yan, Y. J.Shin, D. H. Najafabadi, M. Rezaee, X. Feng, D. G. Nocera,T. Taniguchi, K. Watanabe, B. Skinner, K. A. Matveev, andP. Kim, Nature Nanotechnology 17, 166 (2021).[12] S. Jezouin, F. D. Parmentier, A. Anthore, U. Gennser, A. Ca-vanna, Y. Jin, and F. Pierre, Science 342, 601 (2013).[13] E. Sivre, A. Anthore, F. D. Parmentier, A. Cavanna, U. Gennser,A. Ouerghi, Y. Jin, and F. Pierre, Nature Physics 14, 145148(2017).[14] M. Banerjee, M. Heiblum, V. Umansky, D. E. Feldman,Y. Oreg, and A. Stern, Nature 559, 205210 (2018).[15] R. A. Melcer, S. Konyzheva, M. Heiblum, and V. Umansky,Nature Physics 19, 327 (2023).[16] A. A. Golubov and A. E. Koshelev, Phys. Rev. B 83, 094521(2011).[17] L. Zhao, Z. Iftikhar, T. F. Q. Larson, E. G. Arnault, K. Watan-abe, T. Taniguchi, F. Amet, and G. Finkelstein, Phys. Rev. Lett.131, 176604 (2023).[18] See Supplemental Material (which includes Ref. [27–29]) at[URL] for the additional figures and notes on 1) Thermal NoiseCalibration, 2) Noise Sources, 3) Superconducting transitionof the film, 4) DC conductance, 5) Thermal conductivity es-timations, 6) Variations of cooling efficiency due to vortex re-arrangement, 7) Temperature of the top interface vs. power, 8)∆TA/∆TB at different applied bias current, 9) Additional datafor heating the top interface, and 10) CAES thermometry.[19] M. R. Sahu, A. K. Paul, J. Sutradhar, K. Watanabe,T. Taniguchi, V. Singh, S. Mukerjee, S. Banerjee, and A. Das,Phys. Rev. B 104, L081404 (2021).[20] C. Caroli, P. De Gennes, and J. Matricon, Physics Letters 9,307 (1964).[21] B. Huard, H. Pothier, D. Esteve, and K. E. Nagaev, Phys. Rev.B 76, 165426 (2007).[22] A. V. Feshchenko, O.-P. Saira, J. T. Peltonen, and J. P. Pekola,Scientific Reports 7, 41728 (2017).[23] W. Vinen, E. Forgan, C. Gough, and M. Hood, Physica 55,94113 (1971).[24] E. Boaknin, M. A. Tanatar, J. Paglione, D. Hawthorn, F. Ron-ning, R. W. Hill, M. Sutherland, L. Taillefer, J. Sonier, S. M.Hayden, and J. W. Brill, Phys. Rev. Lett. 90, 117003 (2003).[25] C. Caroli and M. Cyrot, Physik der Kondensierten Materie 4,285 (1965).[26] J. O. Willis and D. M. Ginsberg, Phys. Rev. B 14, 1916 (1976).[27] Z. Iftikhar, A. Anthore, S. Jezouin, F. D. Parmentier, Y. Jin,A. Cavanna, A. Ouerghi, U. Gennser, and F. Pierre, NatureCommunications 7, 12908 (2016).[28] M. Büttiker, Phys. Rev. B 46, 12485 (1992).[29] L. Zhao, Andreev conversion in the quantum Hall regime,Ph.D. thesis, Duke University (2022).mailto:gleb@duke.eduhttp://dx.doi.org/10.1126/science.aad6203http://dx.doi.org/10.1038/nphys4084http://dx.doi.org/10.1038/nphys4084http://dx.doi.org/10.1126/sciadv.aaw8693http://dx.doi.org/ 10.1038/s41567-020-0898-5http://dx.doi.org/ 10.1038/s41567-020-0898-5http://dx.doi.org/10.1103/PhysRevX.12.021057http://dx.doi.org/ 10.1038/s41586-023-06764-4http://dx.doi.org/10.1038/s41567-024-02574-1http://dx.doi.org/10.1103/RevModPhys.87.1037http://dx.doi.org/10.3390/nano12091461http://dx.doi.org/10.1103/RevModPhys.93.041001http://dx.doi.org/10.1038/s41565-021-01015-xhttp://dx.doi.org/ 10.1126/science.1241912http://dx.doi.org/10.1038/nphys4280http://dx.doi.org/10.1038/nphys4280http://dx.doi.org/ 10.1038/s41586-018-0184-1http://dx.doi.org/10.1038/s41567-022-01885-5http://dx.doi.org/10.1103/PhysRevB.83.094521http://dx.doi.org/10.1103/PhysRevB.83.094521http://dx.doi.org/ 10.1103/PhysRevLett.131.176604http://dx.doi.org/ 10.1103/PhysRevLett.131.176604http://dx.doi.org/10.1103/PhysRevB.104.L081404http://dx.doi.org/https://doi.org/10.1016/0031-9163(64)90375-0http://dx.doi.org/https://doi.org/10.1016/0031-9163(64)90375-0http://dx.doi.org/ 10.1103/PhysRevB.76.165426http://dx.doi.org/ 10.1103/PhysRevB.76.165426http://dx.doi.org/10.1038/srep41728http://dx.doi.org/ 10.1016/0031-8914(71)90245-xhttp://dx.doi.org/ 10.1016/0031-8914(71)90245-xhttp://dx.doi.org/ 10.1103/PhysRevLett.90.117003http://dx.doi.org/10.1007/bf02422842http://dx.doi.org/10.1007/bf02422842http://dx.doi.org/10.1103/PhysRevB.14.1916http://dx.doi.org/10.1038/ncomms12908http://dx.doi.org/10.1038/ncomms12908http://dx.doi.org/10.1103/PhysRevB.46.12485 Thermal properties of the superconductor-quantum Hall interfaces Abstract Acknowledgments References