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Universal quantized thermal conductance in graphene

Universal quantized thermal conductance in graphene SCIENCE ADVANCES RESEARCH ARTICLE PHYSICS Copyright © 2019 The Authors, some rights reserved; exclusive licensee American Association 1 1 2 2 Saurabh Kumar Srivastav *, Manas Ranjan Sahu *, K. Watanabe , T. Taniguchi , for the Advancement 1 1† Sumilan Banerjee , Anindya Das of Science. No claim to original U.S. Government The universal quantization of thermal conductance provides information on a state's topological order. Recent mea- Works. Distributed surements revealed that the observed value of thermal conductance of the state is inconsistent with either Pfaffian under a Creative or anti-Pfaffian model, motivating several theoretical articles. Analysis has been made complicated by the presence Commons Attribution NonCommercial of counter-propagating edge channels arising from edge reconstruction, an inevitable consequence of separating License 4.0 (CC BY-NC). the dopant layer from the GaAs quantum well and the resulting soft confining potential. Here, we measured thermal conductance in graphene with atomically sharp confining potential by using sensitive noise thermometry on hex- agonal boron-nitride encapsulated graphene devices, gated by either SiO /Si or graphite back gate. We find the quantization of thermal conductance within 5% accuracy for n =1; ; 2 and 6 plateaus, emphasizing the universality of flow of information. These graphene quantum Hall thermal transport measurements will allow new insight into exotic systems like even-denominator quantum Hall fractions in graphene. INTRODUCTION [SU(4)], which is tunable by electric and magnetic field, and thus Measurement of the quantization of thermal conductance at its quan- exhibits a plethora of exciting phases, ranging from spontaneously 2 2 tum limit (k T; k ¼ p k =3h), and the demonstration of its univer- symmetry-broken states (28–35) to protected topological states such 0 0 sality irrespective of the statistics of the heat carriers, have been as quantum spin Hall state near the Dirac point (36). Compared with important quests in condensed matter physics. This is due to the fact GaAs, bilayer graphene has several additional even-denominator QH 1 3 5 7 that quantization can reveal the exotic topological nature of the car- fractions (37), such as ; ; ,and , which has topologically exotic 2 2 2 2 riers, which is not accessible via electrical conductance measurement ground states with possible non-Abelian excitations, and some of these (1, 2). Thermal conductance has been measured for phonons (3), exotic phases can be uniquely identified by thermal conductance mea- photons (4), and fermions (5, 6, 7). However, the definitive proof surement (1, 2, 28). of universality of the quantum limit of thermal conductance re- In this report, we carried out the thermal conductance measure- mained elusive for more than two decades (1, 8, 9, 10) until the recent ment in the integer and FQHE of graphene devices using sensitive measurements of thermal conductance in fractional quantum Hall noise thermometry setup. We first establish the quantum limit of effect (FQHE) of GaAs-based two-dimensional electron gas (11). The thermal conductance for integer plateaus of n =1,2,and 6in hBN- half of the quantum limit of thermal conductance (2.5k T)has also encapsulated monolayer graphene devices gated by an SiO /Si back 0 2 been reported (12) for the state, which has motivated many recent gate. We then further study the thermal conductance for fractional theoretical articles (13–16) based on earlier theoretical predictions plateau of n ¼ in a hBN-encapsulated graphene device gated by a (17–20). However, becauseof softconfining potential, theedge-statere- graphite back gate. We show that the values of thermal conductance construction leads to extra pairs of counterpropagating edges in the for n = and 2 are the same, although they have different electrical FQHE of GaAs (21–25) and makes it complicated to interpret the exact conductance. These results show the universality of thermal con- value of the thermal conductance. In this case, the measured value of ductance with its quantum limit as predicted by theory (1). Our work the thermal conductance can vary from the theoretically (1)predicted is an important step to measure half of a thermal conductance and to (N − N )k T to (N + N )k T depending on full thermal equilibration demonstrate the topological non-Abelian excitaton in graphene hy- d u 0 d u 0 to no thermal equilibration of the counterpropagating edges (11, 12), brids in the future. where N and N are the number of downstream and upstream edges, We used two SiO /Si back-gated devices and one graphite back-gated d u 2 respectively. Attaining the full thermal equilibration at very low tem- device for our measurements, where the hBN-encapsulated devices perature is quite challenging as the thermal relaxation length could be are fabricated using the standard dry transfer pickup technique (38) much bigger than the typical device dimensions (11, 12). Therefore, the followed by the edge contacting method (see Materials and Methods). precise measurement of universal thermal conductance requires a system The schematic is shown in Fig. 1A, where the floating metallic reservoir having no such edge reconstruction. Here, we demonstrate that graphene, in the middle connects both sides by edge contacts. The measurements a single carbon atomic layer, which offers unprecedented universal edge are performed in a cryofree dilution refrigerator having a base tempera- profile (26, 27) due to atomically sharp confining potential, is an ideal ture of ~12 mK. The thermal conductance was measured using noise platform to probe universal quantized thermal conductance and to un- thermometry based on LCR resonant circuit at resonance frequency ambiguously reveal the topological order of FQHE. The sharp edge of ~758 kHz, amplified by preamplifiers, and, lastly, measured by a potential profile in graphene is easily realized using few-nanometers- spectrum analyzer (fig. S2). The conductance measured at the source thick insulating spacer such as hexagonal boron nitride (hBN) between contact in Fig. 1A for device 1 has been plotted as a function of back-gate the graphene and the screening layer (26). Furthermore, the quantum voltage (V )at B = 9.8 T shown in Fig. 1B, where the clear plateaus at n = BG Hall (QH) state of graphene has higher symmetry in spin-valley space 1, 2, 4, 5, 6, and 10 are visible. The thermal noise (including amplifier noise) measured across the LCR circuit is plotted as a function of V in BG 1 2 Department of Physics, Indian Institute of Science, Bangalore 560012, India. Na- Fig. 1B, where the plateaus are also evident. tional Institute of Material Science, 1-1 Namiki, Tsukuba 305-0044, Japan. A DC current I, injected at the source contact (Fig. 1A), flows along *These authors contributed equally to this work. †Corresponding author. Email: [email protected] thechiraledgetowardthefloating reservoir. The outgoing current from Srivastav et al., Sci. Adv. 2019; 5 : eaaw5798 12 July 2019 1of5 | SCIENCE ADVANCES RESEARCH ARTICLE A B Spectrum analyzer B = 9.8 T N 9010 A T ~ 40 mK Room temperature M amplifier SD 3 (CG) 10 nF Cold amplifier L = 365 µH 0 5 10 15 20 25 30 V (V) BG (CG) Fig. 1. Device configuration and QH response. (A) Schematic of the device with measurement setup. The device is set in integer QH regime at filling factor n = 1, where one chiral edge channel (line with arrow) propagates along the edge of the sample. The current I is injected (green line) through the contact SD S, which is absorbed in the floating reservoir (red contact). Chiral edge channel (red line) at potential V and temperature T leave the floating reservoir and M M terminate into two cold grounds (CGs). The cold edges (without any current) at temperature T are shown by the blue lines. The resulting increase in the electron temperature T of the floating reservoir is determined from the measured excess thermal noise at contact D.Aresonant (LC) circuit, situated at contact D,with resonance frequency f = 758 kHz, filters the signal, which is amplified by the cascade of amplification chain (preamplifier placed at 4K plate and a room temperature amplifier). Last, the amplified signal is measured by a spectrum analyzer. (B) Hall conductance measured at the contact S using lock-in amplifier at B = 9.8 T (black line). Thermal noise (including the cold amplifier noise) measured as a function of V at f = 758 kHz (red line). The plateaus for n =1,2,and 6 BG 0 are visible in both measurements. the floating reservoir splits into two equal parts, each propagating RESULTS AND DISCUSSION along the outgoing chiral edge from the floating reservoir to the cold In our experiment, for an integer filling factor n,the n chiral edge grounds. The floating reservoir reaches a new equilibrium potential modes impinge the current in the floating reservoir, and N =2n chiral V ¼ with the filling factor n of graphene determined by the V , edge modes leave the floating reservoir as shown in Fig. 1A. Figure 2 BG 2nG whereas the potential of the source contact isV ¼ .Thus, thepower (A to C) shows the measured excess thermal noise S for device 1 as a nG 1 I function of source current I for n = 1, 2, and 6 at B = 9.8 T. The SD input to the floating reservoir is P ¼ ðIV Þ¼ , where the prefac- in S 2 2nG increment in the temperature of the floating reservoir as a function tor of = results due to the fact that equal power dissipates at the source of I is exhibited in the increase of S . The x and y axes of Fig. 2 (A to SD I and the floating reservoirs (Fig. 1A). Similarly, the outgoing power C) are converted to J and T , respectively, and plotted in Fig. 2D for Q M 1 I I from the floating reservoir is P ¼ 2  V ¼ . Thus, the out M different n, where each solid circle is generated after averaging nine con- 2 2 4nG resultant power dissipation in the floating reservoir due to joule secutive data points (raw data in section S7). The T ~40mKwithout heating is J ¼ P  P ¼ , and as a result, the electrons in DC current was determined from the thermal noise measurement and Q in out 4nG the floating reservoir will get heated to a new equilibrium tempera- shown in section S3. As expected, the T is higher for lower filling factor ture (T ) such that the following heat balance equation is satisfied as less number of chiral edges are carrying the heat away from the floating reservoir. Thus, to maintain a constant T ,higher J is required M Q eph forhigherfilling factor.InFig.2E, we plotted l (= DJ /(0.5k ), where e Q 0 J ¼ J ðT ; T Þþ J ðT ; T Þ Q M 0 M 0 Q 2 DJ = J (n , T ) − J (n , T ), as a function of T for two different con- Q Q i M Q j M eph 2 2 ¼ 0:5Nk ðT  T Þþ J ðT ; TÞð1Þ 0 M 0 figurations (DN = 2 and 8) shown by solid circles. It can be seen that the M 0 Q l is proportional toT as expected from Eq. 1. The solid lines in Fig. 2E Here, J ðT ; T Þ is the heat current carried by the N chiral bal- represent the linear least square fits and give the values of 1.92 and 7.92 M 0 listic edge channels from the floating reservoir (T ) to the cold ground for DN =2and DN = 8, respectively. Similarly, we repeated the exper- eph (T ), and theJ ðT ; T Þis the heat loss rate from the hot electrons iment at B = 6 T for device 1 and device 2, and the linear fits give the 0 M 0 of the floating reservoir to the cold phonon bath. Note that the values of 7.76 and 8.64 (figs. S13 and S14) for DN = 8, respectively. From electronic contribution to the heat current in Eq. 1 is valid in the ab- these four linear fitting values, the average thermal conductance for a sence of heat Coulomb blockade, which is discussed in more detail in single edge mode is found to be g =(1±0.05)k T,where T =(T + Q 0 M eph section S10. In Eq. 1, T and J are the only unknowns to deter- T )/2and theerroristhe SD. M 0 mine the quantum limit of thermal conductance (k ). The T of the To measure the thermal conductance for the FQHE state, we used a 0 M floating reservoir in our experiment is obtained by measuring the graphite back-gated device (device 3), where the graphene channel is excess thermal noise, S = nk (T − T )G (7, 11, 12), along the out- isolated from the graphite gate by bottom hBN of thickness ~20 nm. I B M 0 0 going edge channels as shown in Fig. 1A. After measuring the T ac- For this device, the lower electron temperature T ~27 mK (section curately, one can determine k usingEq. 1bytuningthe number of S3) was achieved by introducing extra low-pass filters at the mixing outgoing channels (DN). chamber. The conductance plateaus and the thermal noise as a function Srivastav et al., Sci. Adv. 2019; 5 : eaaw5798 12 July 2019 2of5 T T 0 0 S/N 115016 C = 125 pF Cold ground (CG) G (e /h) –19 2 S (10 V /Hz) V | SCIENCE ADVANCES RESEARCH ARTICLE AB C 3.5 20 ν = 1 ν = 2 ν = 6 2.5 1.5 0.5 1 0 0 −2.5 −1.5 −0.5 0 0.5 1.5 2.5 −4 −3 −2 −1 01 2 3 4 −8 −6 −4 −2 0 2 4 6 8 I (nA) I (nA) I (nA) SD SD SD D E g κ Δ N = 8, = (0.99 ± 0.04) T ν = 1 0 ν = 2 g κ Δ N = 2, = (0.96 ± 0.02) T 40 Q ν = 6 T ~ 40 mK T ~ 40 mK 010 20 30 40 23 4 5 6 7 2 −3 2 J (fW) T (10 K ) Q M Fig. 2. Thermal conductance in integer QH. Excess thermal noise S is measured as a function of source current I at n =1(A), 2 (B), and 6 (C). (D) The increased I SD temperatures T of the floating reservoir are plotted (solid circles) as a function of dissipated power J for n =1 (N = 2), 2 (N = 4), and 6 (N = 12), respectively, where N = M Q 2n is the total outgoing channels from the floating reservoir. (E) The l = DJ /(0.5k ) is plotted as a function of T for DN = 2 (between n = 1 and 2) and DN = 8 (between Q 0 M n = 2 and 6), respectively, in red and black solid circles, where DJ = J (n , T ) − J (n , T ). The solid lines are the linear fittings to extract the thermal conductance values. Q Q i M Q j M Slope of these linear fits are 1.92 and 7.92 for DN = 2 and 8, respectively, which gives the g = 0.96k T and 0.99k T for the single edge mode, respectively. Q 0 0 AB 2.5 8 ν = 1 T ~ 27 mK ν = 2 T ~ 27 mK 2.0 5 6 1.5 g = (1.02 ± 0.01) κ 1.0 g = (2.02 ± 0.02) κ g = (2.08 ± 0.01) κ 0.5 0.2 0.4 0.6 0.8 1.0 01 2 3 4 5 22 −3 2 T − T (10 K ) V (V) BG M 0 Fig. 3. Thermal conductance in fractional QH. (A) Hall conductance (black line) and thermal noise (red line) measured in the graphite back-gated device plotted as a function of V at B = 7 T. The plateaus for n =1, , and 2 are visible in both the measurements. (B) Similar to the previous plots (Fig. 2), the excess thermal noise S is BG I measured as a function source current I , and the T is shown as a function of the dissipated power J in figs. S15 and S16, from which we have extracted the J (solid SD M Q Q 2 2 4 circles) as a function of T  T for n =1, , and 2 and shown up to T ~ 60 to 70 mK. The solid lines are the linear fits to extract the slopes, which give the thermal M 0 3 conductance values of 1.02, 2.08, and 2.02k T for n =1, , and 2, respectively. One can see that the thermal conductance values are quantized for n = 1 and 2, and the values are the same for both the n = and 2 plateaus. The inset shows the corresponding downstream charge modes for integer and fractional edges. The dashed curve represents the theoretically predicted (section S10) contribution of the heat Coulomb blockade (39, 40)for n = 1, showing its negligible contribution to the net thermal current. 4 4 of V at B = 7 T are shown in Fig. 3A, where the n =1, , and 2 are spectively. For n = , two downstream charge modes, one integer and one BG 3 3 1 e visible in both measurements. The T versus J plots for different fractional (inner n = with effective charge, e* ¼ ), are expected. The M Q 3 3 filling factors are shown in fig. S16. In Fig. 3B, we plotted the J (solid thermal conductance of n ¼ should be thesameas n =2having two 2 2 4 circles) as a function of T  T for n =1, ,and2overthetemperature integer downstream charge modes, which is observed in our experiment. M 0 3 window where the curve is linear, implying the dominance of the Thus, our result is consistent with the theory that the quantum limit of electronic contribution to the heat flow. The solid lines in Fig. 3B repre- thermal conductance is the same for both fractional and integer QH edges. sent the linear fits (in 0.5k ) and give the values of 2.04, 4.16, and 4.04, We would like to note that for device 3, the thermal conductance was which correspond to g = 1.02, 2.08, and 2.02k T for n =1, ,and 2, re- obtained without varying the number of outgoing channels (DN). This Q 0 Srivastav et al., Sci. Adv. 2019; 5 : eaaw5798 12 July 2019 3of5 G (e /h) −29 2 S (10 A /Hz) T (mK) −29 2 S (10 A /Hz) –19 2 S (10 V /Hz) J (fW) −3 2 λ (10 K ) −29 2 S (10 A /Hz) I | SCIENCE ADVANCES RESEARCH ARTICLE may lead to the inaccuracy in the extracted thermal conductance values (5/15/60 nm) was performed to make the contacts in an evaporator −7 −7 due to electron-phonon coupling and heat Coulomb blockade (39, 40). chamber having base pressure of ~1 × 10 to 2 × 10 mbar and However, measuring the right value of the thermal conductance within followed by lift-off procedure in acetone and IPA. The floating metallic 5% accuracy for device 3 corroborates the negligible contributions from reservoir in the middle was connected to both sides of the graphene part the electron-phonon coupling and heat Coulomb blockade. The latter is by the edge contacts. This procedure of making devices prevented con- discussed in more detail in section S10. The theoretical estimation (39, 40) tamination of exposed graphene edges with polymer residues, resulting of the heat Coulomb blockade for n = 1 is shown by a dash curve in in high-quality contacts. Fig. 3B. We discuss about the electron-phonon coupling, the accuracy of the measurements, and the effect of the heat Coulomb blockade in sections S8, S9, and S10, respectively. SUPPLEMENTARY MATERIALS Supplementary material for this article is available at http://advances.sciencemag.org/cgi/ In conclusion, we measured the thermal conductance for three content/full/5/7/eaaw5798/DC1 integer plateaus (1, 2, and 6) and one particle-like fractional plateau Section S1. Device characterization and measurement setup of graphene, and the values are consistent with the quantum limit 3 Section S2. Gain of the amplification chain 2 2 p k Section S3. Electron temperature (T ) determination T within 5% accuracy. These studies can be extended soon to 3h Section S4. Partition of current and contact resistance Section S5. Dissipated power in the floating reservoir measure the thermal conductance for the even-denominator QH Section S6. Determination of the temperature (T ) of floating reservoir plateaus in graphene (37) with atomically sharp confining potential Section S7. Extended excess thermal noise data to probe their non-Abelian nature. Section S8. Heat loss by electron-phonon cooling Section S9. Accuracy of the thermal conductance measurement Section S10. Discussion on heat Coulomb blockade Fig. S1. Optical image and device response at zero magnetic field. MATERIALS AND METHODS Fig. S2. Experimental setup for noise measurement. Device fabrication Fig. S3. Schematic used to derive the gain in section S2. Our encapsulated graphene devices were made using the following Fig. S4. Gain of amplification chain: Output voltage from a known input signal in QH state at resonance frequency. procedures similar to those used in previous reports (41, 42). First, an Fig. S5. Gain of amplification chain: From the temperature-dependent thermal noise. hBN/graphene/hBN stack was made using the “hot pickup” technique Fig. S6. Gain of amplification chain during measurement of device 3 (graphite back-gated (38). This involved the mechanical exfoliation of graphite and bulk hBN device). crystal on the SiO /Si wafer to obtain the single-layer graphene and thin 2 Fig. S7. RC filter assembly and thermal anchoring on the cold finger. hBN (~20 to 30 nm). Single-layer graphene and thin hBN (~20 to Fig. S8. Electron temperature (T ) determination. Fig. S9. Electron temperature (T ) determination: From shot noise measurement in a p-n 30 nm) were identified using an optical microscope. Fabrication of junction of graphene device. this hetrostructure assembly involved four steps. Step 1: We used a Fig. S10. Equipartition of current in left and right moving chiral states. poly-bisphenol-A-carbonate–coated polydimethylsiloxane block Fig. S11. Determination of contact resistance and source noise. mounted on a glass slide attached to tip of a micromanipulator to pick Fig. S12. Extended excess thermal noise raw data. Fig. S13. Extended data of device 1 at B =6 T. up theexfoliatedhBN flake. Theexfoliated hBN flake was picked up at Fig. S14. Extended data of device 2 at B =6 T. temperature of 90°C. Step 2: A previously picked-up hBN flake was Fig. S15. Extended data of device 3 (graphite back gate) at B =7 T. aligned over a graphene. Now, this graphene was picked up at tempera- Fig. S16. Extended data of device 3 (graphite back gate) at B =7 T. ture of 90°C. Step 3: The bottom hBN flake was picked up using the Fig. S17. Heat loss by electron-phonon coupling. previously picked-up hBN/graphene following step 2. Step 4: Last, this Table S1. Gain of amplification chain. Table S2. Electron temperature (T ). resulting hetrostructure (hBN/graphene/hBN) was dropped down on Table S3. Contact resistance and the source noise. topofanoxidizedsilicon wafer(p++doped siliconwithSiO thickness Table S4. Contact resistance and the source noise of device 3 (graphite back-gated device). of 285 nm) at temperature of 140°C, which served as a back gate (for the Table S5. Change in thermal conductance for different electron temperature T . graphite back-gated device after step 3, the graphite flake was picked up References (43–55) using the previously picked-up hBN/graphene/hBN following step 2; after this step, again, step 4 was followed). These final stacks were cleaned in chloroform (CHCl ) followed by acetone and isopropyl REFERENCES AND NOTES 1. C. Kane, M. P. Fisher, Quantized thermal transport in the fractional quantum Hall effect. alcohol (IPA). The next step involved electron-beam lithography Phys. Rev. B 55, 15832–15837 (1997). (EBL) to define the contact region. Poly-methyl-methacrylate was 2. T. Senthil, M. P. A. Fisher, Quasiparticle localization in superconductors with spin-orbit coated on the resulting hetrostructure. Contact region was defined scattering. Phys. Rev. B 61, 9690–9698 (2000). using EBL. Apart from conventional Hall probe geometry, we 3. K. Schwab, E. A. Henriksen, J. M. Worlock, M. L. Roukes, Measurement of the quantum of thermal conductance. Nature 404, 974–977 (2000). defined a region for floating reservoir of ~4- to 7-mm area. We used 4. M. Meschke, W. Guichard, J. P. Pekola, Single-mode heat conduction by photons. Nature two SiO /Si back-gated devices (device 1 and device 2) and one graphite 444, 187–190 (2006). back-gated device (device 3) for the thermal conductance measurement. 5. L. W. Molenkamp, Th. Gravier, H. van Houten, O. J. A. Buijk, M. A. A. Mabesoone, The edge contacts were achieved by reactive ion etching (a mixture of C. T. Foxon, Peltier coefficient and thermal conductance of a quantum point contact. CHF and O gas was used with a flow rate of 40 and 4 sccm, respec- Phys. Rev. Lett. 68, 3765–3768 (1992). 3 2 6. O. Chiatti, J. T. Nicholls, Y. Y. Proskuryakov, N. Lumpkin, I. Farrer, D. A. Ritchie, Quantum tively, at 25°C with radio frequency power of 60 W), where the etching thermal conductance of electrons in a one-dimensional wire. Phys. Rev. Lett. 97, 056601 time has been varied from 100 to 50 s for the SiO /Si and graphite back- (2006). gated devices, respectively, such that for the SiO /Si device, the bottom 7. S. Jezouin, F. D. Parmentier, A. Anthore, U. Gennser, A. Cavanna, Y. Jin, F. Pierre, Quantum hBN is being etched completely, whereas for the graphite back-gated limit of heat flow across a single electronic channel. Science 342, 601–604 (2013). 8. J. B. Pendry, Quantum limits to the flow of information and entropy. J. Phys. A Math. Gen. device, the bottom hBN is partially etched to isolate the contacts from 16, 2161–2171 (1983). the bottom graphite back gate. Last, the thermal deposition of Cr/Pd/Au Srivastav et al., Sci. Adv. 2019; 5 : eaaw5798 12 July 2019 4of5 | SCIENCE ADVANCES RESEARCH ARTICLE 9. C. L. Kane, M. P. A. Fisher, Thermal transport in a Luttinger liquid. Phys. Rev. Lett. 76, 39. E. Sivre, A. Anthore, F. D. Parmentier, A. Cavanna, U. Gennser, A. Ouerghi, Y. Jin, F. Pierre, 3192–3195 (1996). Heat Coulomb blockade of one ballistic channel. Nat. Phys. 14, 145–148 (2018). 10. L. G. C. Rego, G. Kirczenow, Fractional exclusion statistics and the universal quantum of 40. A. O. Slobodeniuk, I. P. Levkivskyi, E. V. Sukhorukov, Equilibration of quantum Hall edge thermal conductance: A unifying approach. Phys. Rev. B 59, 13080–13086 (1999). states by an Ohmic contact. Phys. Rev. B 88, 165307 (2013). 11. M. Banerjee, M. Heiblum, A. Rosenblatt, Y. Oreg, D. E. Feldman, A. Stern, V. Umansky, 41. L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, Observed quantization of anyonic heat flow. Nature 545,75–79 (2017). L. M. Campos, D. A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, C. R. Dean, 12. M. Banerjee, M. Heiblum, V. Umansky, D. E. Feldman, Y. Oreg, A. Stern, Observation of One-dimensional electrical contact to a two-dimensional material. Science 342, 614–617 half-integer thermal Hall conductance. Nature 559, 205–210 (2018). (2013). 13. D. E. Feldman, Comment on “Interpretation of thermal conductance of the n = 5/2 edge”. 42. A. V. Kretinin, Y. Cao, J. S. Tu, G. L. Yu, R. Jalil, K. S. Novoselov, S. J. Haigh, A. Gholinia, Phys. Rev. B 98, 167401 (2018). A. Mishchenko, M. Lozada, T. Georgiou, C. R. Woods, F. Withers, P. Blake, G. Eda, A. Wirsig, 14. S. H. Simon, Interpretation of thermal conductance of the n = 5/2 edge. Phys. Rev. B 97, C. Hucho, K. Watanabe, T. Taniguchi, A. K. Geim, R. V. Gorbachev, Electronic properties of 121406 (2018). graphene encapsulated with different two-dimensional atomic crystals. Nano Lett. 14, 15. C. Wang, A. Vishwanath, B. I. Halperin, Topological order from disorder and the quantized 3270–3276 (2014). Hall thermal metal: Possible applications to the n = 5/2 state. Phys. Rev. B 98, 045112 43. A. Venugopal, J. Chan, X. Li, C. W. Magnuson, W. P. Kirk, L. Colombo, R. S. Ruoff, E. M. Vogel, (2018). Effective mobility of single-layer graphene transistors as a function of channel dimensions. 16. D. F. Mross, Y. Oreg, A. Stern, G. Margalit, M. Heiblum, Theory of disorder-induced J. Appl. Phys. 109, 104511 (2011). half-integer thermal hall conductance. Phys. Rev. Lett. 121, 026801 (2018). 44. B.-R. Choi, A. E. Hansen, T. Kontos, C. Hoffmann, S. Oberholzer, W. Belzig, 17. D. T. Son, Is the composite fermion a dirac particle? Phys. Rev. X 5, 031027 (2015). C. Schönenberger, T. Akazaki, H. Takayanagi, Shot-noise and conductance measurements 18. P. T. Zucker, D. E. Feldman, Stabilization of the particle-hole Pfaffian order by of transparent superconductor/two-dimensional electron gas junctions. Phys. Rev. B 72, Landau-level mixing and impurities that break particle-hole symmetry. Phys. Rev. Lett. 024501 (2005). 117, 096802 (2016). 45. M. Büttiker, Absence of backscattering in the quantum Hall effect in multiprobe 19. M. Levin, B. I. Halperin, B. Rosenow, Particle-hole symmetry and the Pfaffian state. conductors. Phys. Rev. B 38, 9375–9389 (1988). Phys. Rev. Lett. 99, 236806 (2007). 46. N. Kumada, F. D. Parmentier, H. Hibino, D. C. Glattli, P. Roulleau, Shot noise generated by 20. S.-S. Lee, S. Ryu, C. Nayak, M. P. Fisher, Particle-hole symmetry and the n ¼ quantum Hall graphene p–n junctions in the quantum Hall effect regime. Nat. Commun. 6, 8068 (2015). state. Phys. Rev. Lett. 99, 236807 (2007). 47. S. Matsuo, S. Takeshita, T. Tanaka, S. Nakaharai, K. Tsukagoshi, T. Moriyama, T. Ono, 21. D. B. Chklovskii, B. I. Shklovskii, L. I. Glazman, Electrostatics of edge channels. Phys. Rev. B K. Kobayashi, Edge mixing dynamics in graphene p–n junctions in the quantum Hall 46, 4026–4034 (1992). regime. Nat. Commun. 6, 8066 (2015). 22. C. d. C. Chamon, X. Wen, Sharp and smooth boundaries of quantum Hall liquids. 48. U. Sivan, Y. Imry, Multichannel Landauer formula for thermoelectric transport with Phys. Rev. B 49,8227–8241 (1994). application to thermopower near the mobility edge. Phys. Rev. B 33, 551–558 (1986). 23. H. Inoue, A. Grivnin, Y. Ronen, M. Heiblum, V. Umansky, D. Mahalu, Proliferation of neutral 49. J.-H. Jiang, Y. Imry, Linear and nonlinear mesoscopic thermoelectric transport with modes in fractional quantum Hall states. Nat. Commun. 5, 4067 (2014). coupling with heat baths. C. R. Phys. 17, 1047–1059 (2016). 24. Y. Zhang, Y.-H. Wu, J. A. Hutasoit, J. K. Jain, Theoretical investigation of edge 50. C. W. J. Beenakker, M. Büttiker, Suppression of shot noise in metallic diffusive conductors. 5 7 reconstruction in the v ¼ and fractional quantum Hall states. Phys. Rev. B 90, 165104 Phys. Rev. B 46, 1889–1892 (1992). 2 3 (2014). 51. Y. M. Blanter, E. V. Sukhorukov, Semiclassical theory of conductance and noise in open 25. R. Sabo, I. Gurman, A. Rosenblatt, F. Lafont, D. Banitt, J. Park, M. Heiblum, Y. Gefen, chaotic cavities. Phys. Rev. Lett. 84, 1280–1283 (2000). V. Umansky, D. Mahalu, Edge reconstruction in fractional quantum Hall states. Nat. Phys. 52. A. Sergeev, V. Mitin, Electron-phonon interaction in disordered conductors: Static and 13, 491–496 (2017). vibrating scattering potentials. Phys. Rev. B 61, 6041–6047 (2000). 26. Z.-X. Hu, R. N. Bhatt, X. Wan, K. Yang, Realizing universal edge properties in graphene 53. S. S. Kubakaddi, Interaction of massless Dirac electrons with acoustic phonons in fractional quantum Hall liquids. Phys. Rev. Lett. 107, 236806 (2011). graphene at low temperatures. Phys. Rev. B 79, 075417 (2009). 27. G. Li, A. Luican-Mayer, D. Abanin, L. Levitov, E. Y. Andrei, Evolution of Landau levels into 54. F. Pierre, A. B. Gougam, A. Anthore, H. Pothier, D. Esteve, N. O. Birge, Dephasing of edge states in graphene. Nat. Commun. 4, 1744 (2013). electrons in mesoscopic metal wires. Phys. Rev. B 68, 085413 (2003). 28. F. Pientka, J. Waissman, P. Kim, B. I. Halperin, Thermal transport signatures of 55. P. W. Brouwer, M. Büttiker, Charge-relaxation and dwell time in the fluctuating broken-symmetry phases in graphene. Phys. Rev. Lett. 119, 027601 (2017). admittance of a chaotic cavity. Europhys. Lett. 37, 441–446 (1997). 29. K. Yang, S. D. Sarma, A. H. MacDonald, Collective modes and skyrmion excitations in graphene SU(4) quantum Hall ferromagnets. Phys. Rev. B 74, 075423 (2006). Acknowledgments: We would like to thank J. Jain for the critical inputs and help in writing 30. I. Sodemann, A. H. MacDonald, Broken SU(4) symmetry and the fractional quantum Hall the manuscript. We would also like to thank M. Heiblum for the fruitful discussion. We also effect in graphene. Phys. Rev. Lett. 112, 126804 (2014). thank R. Pandit, H. Choi, Y. Ronen, and V. Singh for the useful discussions. Fundings: The 31. M. Kharitonov, Phase diagram for the n = 0 quantum Hall state in monolayer graphene. authors acknowledge device fabrication and characterization facilities in CeNSE, IISc, Phys. Rev. B 85, 155439 (2012). Bangalore. A.D. thanks the Department of Science and Technology (DST), government of India, 32. B. E. Feldman, J. Martin, A. Yacoby, Broken-symmetry states and divergent resistance in under grant nos. DSTO1470 and DSTO1597. K.W. and T.T. acknowledge support from the suspended bilayer graphene. Nat. Phys. 5, 889–893 (2009). Elemental Strategy Initiative conducted by the MEXT, Japan and the CREST (JPMJCR15F3), JST. 33. R. T. Weitz, M. T. Allen, B. E. Feldman, J. Martin, A. Yacoby, Broken-symmetry states in Author contributions: S.K.S. contributed to the device fabrication, data acquisition, and doubly gated suspended bilayer graphene. Science 330, 812–816 (2010). analysis. M.R.S. contributed in the noise setup, data acquisition, and analysis. A.D. contributed 34. A. F. Young, C. R. Dean, L. Wang, H. Ren, P. Cadden-Zimansky, K. Watanabe, T. Taniguchi, in conceiving the idea and designing the experiment, data interpretation, and analysis. J. Hone,K. L. Shepard,P.Kim,Spin and valleyquantum Hall ferromagnetismin graphene. S.B. contributed in the data interpretation and theoretical understanding of the manuscript. Nat. Phys. 8,550–556 (2012). K.W. and T.T. synthesized the hBN single crystals. All authors contributed in writing the 35. P. Maher, C. R. Dean, A. F. Young, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, P. Kim, manuscript. Competing interests: The authors declare that they have no competing interests. Evidence for a spin phase transition at charge neutrality in bilayer graphene. Nat. Phys. 9, Data and materials availability: All data needed to evaluate the conclusions in the paper 154–158 (2013). are present in the paper and/or the Supplementary Materials. Additional data related to 36. A. F. Young, J. D. Sanchez-Yamagishi, B. Hunt, S. H. Choi, K. Watanabe, T. Taniguchi, this paper may be requested from the authors. R. C. Ashoori, P. Jarillo-Herrero, Tunable symmetry breaking and helical edge transport in a graphene quantum spin Hall state. Nature 505, 528–532 (2014). Submitted 7 January 2019 37. J. I. A. Li, C. Tan, S. Chen, Y. Zeng, T. Taniguchi, K. Watanabe, J. Hone, C. R. Dean, Accepted 31 May 2019 Even-denominator fractional quantum Hall states in bilayer graphene. Science 358, Published 12 July 2019 648–652 (2017). 10.1126/sciadv.aaw5798 38. F. Pizzocchero, L. Gammelgaard, B. S. Jessen, J. M. Caridad, L. Wang, J. Hone, P. Bøggild, T. J. Booth, The hot pick-up technique for batch assembly of van der Waals Citation: S. K. Srivastav, M. R. Sahu, K. Watanabe, T. Taniguchi, S. Banerjee, A. Das, Universal heterostructures. Nat. Commun. 7, 11894 (2016). quantized thermal conductance in graphene. Sci. Adv. 5, eaaw5798 (2019). Srivastav et al., Sci. Adv. 2019; 5 : eaaw5798 12 July 2019 5of5 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Science Advances Pubmed Central

Universal quantized thermal conductance in graphene

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SCIENCE ADVANCES RESEARCH ARTICLE PHYSICS Copyright © 2019 The Authors, some rights reserved; exclusive licensee American Association 1 1 2 2 Saurabh Kumar Srivastav *, Manas Ranjan Sahu *, K. Watanabe , T. Taniguchi , for the Advancement 1 1† Sumilan Banerjee , Anindya Das of Science. No claim to original U.S. Government The universal quantization of thermal conductance provides information on a state's topological order. Recent mea- Works. Distributed surements revealed that the observed value of thermal conductance of the state is inconsistent with either Pfaffian under a Creative or anti-Pfaffian model, motivating several theoretical articles. Analysis has been made complicated by the presence Commons Attribution NonCommercial of counter-propagating edge channels arising from edge reconstruction, an inevitable consequence of separating License 4.0 (CC BY-NC). the dopant layer from the GaAs quantum well and the resulting soft confining potential. Here, we measured thermal conductance in graphene with atomically sharp confining potential by using sensitive noise thermometry on hex- agonal boron-nitride encapsulated graphene devices, gated by either SiO /Si or graphite back gate. We find the quantization of thermal conductance within 5% accuracy for n =1; ; 2 and 6 plateaus, emphasizing the universality of flow of information. These graphene quantum Hall thermal transport measurements will allow new insight into exotic systems like even-denominator quantum Hall fractions in graphene. INTRODUCTION [SU(4)], which is tunable by electric and magnetic field, and thus Measurement of the quantization of thermal conductance at its quan- exhibits a plethora of exciting phases, ranging from spontaneously 2 2 tum limit (k T; k ¼ p k =3h), and the demonstration of its univer- symmetry-broken states (28–35) to protected topological states such 0 0 sality irrespective of the statistics of the heat carriers, have been as quantum spin Hall state near the Dirac point (36). Compared with important quests in condensed matter physics. This is due to the fact GaAs, bilayer graphene has several additional even-denominator QH 1 3 5 7 that quantization can reveal the exotic topological nature of the car- fractions (37), such as ; ; ,and , which has topologically exotic 2 2 2 2 riers, which is not accessible via electrical conductance measurement ground states with possible non-Abelian excitations, and some of these (1, 2). Thermal conductance has been measured for phonons (3), exotic phases can be uniquely identified by thermal conductance mea- photons (4), and fermions (5, 6, 7). However, the definitive proof surement (1, 2, 28). of universality of the quantum limit of thermal conductance re- In this report, we carried out the thermal conductance measure- mained elusive for more than two decades (1, 8, 9, 10) until the recent ment in the integer and FQHE of graphene devices using sensitive measurements of thermal conductance in fractional quantum Hall noise thermometry setup. We first establish the quantum limit of effect (FQHE) of GaAs-based two-dimensional electron gas (11). The thermal conductance for integer plateaus of n =1,2,and 6in hBN- half of the quantum limit of thermal conductance (2.5k T)has also encapsulated monolayer graphene devices gated by an SiO /Si back 0 2 been reported (12) for the state, which has motivated many recent gate. We then further study the thermal conductance for fractional theoretical articles (13–16) based on earlier theoretical predictions plateau of n ¼ in a hBN-encapsulated graphene device gated by a (17–20). However, becauseof softconfining potential, theedge-statere- graphite back gate. We show that the values of thermal conductance construction leads to extra pairs of counterpropagating edges in the for n = and 2 are the same, although they have different electrical FQHE of GaAs (21–25) and makes it complicated to interpret the exact conductance. These results show the universality of thermal con- value of the thermal conductance. In this case, the measured value of ductance with its quantum limit as predicted by theory (1). Our work the thermal conductance can vary from the theoretically (1)predicted is an important step to measure half of a thermal conductance and to (N − N )k T to (N + N )k T depending on full thermal equilibration demonstrate the topological non-Abelian excitaton in graphene hy- d u 0 d u 0 to no thermal equilibration of the counterpropagating edges (11, 12), brids in the future. where N and N are the number of downstream and upstream edges, We used two SiO /Si back-gated devices and one graphite back-gated d u 2 respectively. Attaining the full thermal equilibration at very low tem- device for our measurements, where the hBN-encapsulated devices perature is quite challenging as the thermal relaxation length could be are fabricated using the standard dry transfer pickup technique (38) much bigger than the typical device dimensions (11, 12). Therefore, the followed by the edge contacting method (see Materials and Methods). precise measurement of universal thermal conductance requires a system The schematic is shown in Fig. 1A, where the floating metallic reservoir having no such edge reconstruction. Here, we demonstrate that graphene, in the middle connects both sides by edge contacts. The measurements a single carbon atomic layer, which offers unprecedented universal edge are performed in a cryofree dilution refrigerator having a base tempera- profile (26, 27) due to atomically sharp confining potential, is an ideal ture of ~12 mK. The thermal conductance was measured using noise platform to probe universal quantized thermal conductance and to un- thermometry based on LCR resonant circuit at resonance frequency ambiguously reveal the topological order of FQHE. The sharp edge of ~758 kHz, amplified by preamplifiers, and, lastly, measured by a potential profile in graphene is easily realized using few-nanometers- spectrum analyzer (fig. S2). The conductance measured at the source thick insulating spacer such as hexagonal boron nitride (hBN) between contact in Fig. 1A for device 1 has been plotted as a function of back-gate the graphene and the screening layer (26). Furthermore, the quantum voltage (V )at B = 9.8 T shown in Fig. 1B, where the clear plateaus at n = BG Hall (QH) state of graphene has higher symmetry in spin-valley space 1, 2, 4, 5, 6, and 10 are visible. The thermal noise (including amplifier noise) measured across the LCR circuit is plotted as a function of V in BG 1 2 Department of Physics, Indian Institute of Science, Bangalore 560012, India. Na- Fig. 1B, where the plateaus are also evident. tional Institute of Material Science, 1-1 Namiki, Tsukuba 305-0044, Japan. A DC current I, injected at the source contact (Fig. 1A), flows along *These authors contributed equally to this work. †Corresponding author. Email: [email protected] thechiraledgetowardthefloating reservoir. The outgoing current from Srivastav et al., Sci. Adv. 2019; 5 : eaaw5798 12 July 2019 1of5 | SCIENCE ADVANCES RESEARCH ARTICLE A B Spectrum analyzer B = 9.8 T N 9010 A T ~ 40 mK Room temperature M amplifier SD 3 (CG) 10 nF Cold amplifier L = 365 µH 0 5 10 15 20 25 30 V (V) BG (CG) Fig. 1. Device configuration and QH response. (A) Schematic of the device with measurement setup. The device is set in integer QH regime at filling factor n = 1, where one chiral edge channel (line with arrow) propagates along the edge of the sample. The current I is injected (green line) through the contact SD S, which is absorbed in the floating reservoir (red contact). Chiral edge channel (red line) at potential V and temperature T leave the floating reservoir and M M terminate into two cold grounds (CGs). The cold edges (without any current) at temperature T are shown by the blue lines. The resulting increase in the electron temperature T of the floating reservoir is determined from the measured excess thermal noise at contact D.Aresonant (LC) circuit, situated at contact D,with resonance frequency f = 758 kHz, filters the signal, which is amplified by the cascade of amplification chain (preamplifier placed at 4K plate and a room temperature amplifier). Last, the amplified signal is measured by a spectrum analyzer. (B) Hall conductance measured at the contact S using lock-in amplifier at B = 9.8 T (black line). Thermal noise (including the cold amplifier noise) measured as a function of V at f = 758 kHz (red line). The plateaus for n =1,2,and 6 BG 0 are visible in both measurements. the floating reservoir splits into two equal parts, each propagating RESULTS AND DISCUSSION along the outgoing chiral edge from the floating reservoir to the cold In our experiment, for an integer filling factor n,the n chiral edge grounds. The floating reservoir reaches a new equilibrium potential modes impinge the current in the floating reservoir, and N =2n chiral V ¼ with the filling factor n of graphene determined by the V , edge modes leave the floating reservoir as shown in Fig. 1A. Figure 2 BG 2nG whereas the potential of the source contact isV ¼ .Thus, thepower (A to C) shows the measured excess thermal noise S for device 1 as a nG 1 I function of source current I for n = 1, 2, and 6 at B = 9.8 T. The SD input to the floating reservoir is P ¼ ðIV Þ¼ , where the prefac- in S 2 2nG increment in the temperature of the floating reservoir as a function tor of = results due to the fact that equal power dissipates at the source of I is exhibited in the increase of S . The x and y axes of Fig. 2 (A to SD I and the floating reservoirs (Fig. 1A). Similarly, the outgoing power C) are converted to J and T , respectively, and plotted in Fig. 2D for Q M 1 I I from the floating reservoir is P ¼ 2  V ¼ . Thus, the out M different n, where each solid circle is generated after averaging nine con- 2 2 4nG resultant power dissipation in the floating reservoir due to joule secutive data points (raw data in section S7). The T ~40mKwithout heating is J ¼ P  P ¼ , and as a result, the electrons in DC current was determined from the thermal noise measurement and Q in out 4nG the floating reservoir will get heated to a new equilibrium tempera- shown in section S3. As expected, the T is higher for lower filling factor ture (T ) such that the following heat balance equation is satisfied as less number of chiral edges are carrying the heat away from the floating reservoir. Thus, to maintain a constant T ,higher J is required M Q eph forhigherfilling factor.InFig.2E, we plotted l (= DJ /(0.5k ), where e Q 0 J ¼ J ðT ; T Þþ J ðT ; T Þ Q M 0 M 0 Q 2 DJ = J (n , T ) − J (n , T ), as a function of T for two different con- Q Q i M Q j M eph 2 2 ¼ 0:5Nk ðT  T Þþ J ðT ; TÞð1Þ 0 M 0 figurations (DN = 2 and 8) shown by solid circles. It can be seen that the M 0 Q l is proportional toT as expected from Eq. 1. The solid lines in Fig. 2E Here, J ðT ; T Þ is the heat current carried by the N chiral bal- represent the linear least square fits and give the values of 1.92 and 7.92 M 0 listic edge channels from the floating reservoir (T ) to the cold ground for DN =2and DN = 8, respectively. Similarly, we repeated the exper- eph (T ), and theJ ðT ; T Þis the heat loss rate from the hot electrons iment at B = 6 T for device 1 and device 2, and the linear fits give the 0 M 0 of the floating reservoir to the cold phonon bath. Note that the values of 7.76 and 8.64 (figs. S13 and S14) for DN = 8, respectively. From electronic contribution to the heat current in Eq. 1 is valid in the ab- these four linear fitting values, the average thermal conductance for a sence of heat Coulomb blockade, which is discussed in more detail in single edge mode is found to be g =(1±0.05)k T,where T =(T + Q 0 M eph section S10. In Eq. 1, T and J are the only unknowns to deter- T )/2and theerroristhe SD. M 0 mine the quantum limit of thermal conductance (k ). The T of the To measure the thermal conductance for the FQHE state, we used a 0 M floating reservoir in our experiment is obtained by measuring the graphite back-gated device (device 3), where the graphene channel is excess thermal noise, S = nk (T − T )G (7, 11, 12), along the out- isolated from the graphite gate by bottom hBN of thickness ~20 nm. I B M 0 0 going edge channels as shown in Fig. 1A. After measuring the T ac- For this device, the lower electron temperature T ~27 mK (section curately, one can determine k usingEq. 1bytuningthe number of S3) was achieved by introducing extra low-pass filters at the mixing outgoing channels (DN). chamber. The conductance plateaus and the thermal noise as a function Srivastav et al., Sci. Adv. 2019; 5 : eaaw5798 12 July 2019 2of5 T T 0 0 S/N 115016 C = 125 pF Cold ground (CG) G (e /h) –19 2 S (10 V /Hz) V | SCIENCE ADVANCES RESEARCH ARTICLE AB C 3.5 20 ν = 1 ν = 2 ν = 6 2.5 1.5 0.5 1 0 0 −2.5 −1.5 −0.5 0 0.5 1.5 2.5 −4 −3 −2 −1 01 2 3 4 −8 −6 −4 −2 0 2 4 6 8 I (nA) I (nA) I (nA) SD SD SD D E g κ Δ N = 8, = (0.99 ± 0.04) T ν = 1 0 ν = 2 g κ Δ N = 2, = (0.96 ± 0.02) T 40 Q ν = 6 T ~ 40 mK T ~ 40 mK 010 20 30 40 23 4 5 6 7 2 −3 2 J (fW) T (10 K ) Q M Fig. 2. Thermal conductance in integer QH. Excess thermal noise S is measured as a function of source current I at n =1(A), 2 (B), and 6 (C). (D) The increased I SD temperatures T of the floating reservoir are plotted (solid circles) as a function of dissipated power J for n =1 (N = 2), 2 (N = 4), and 6 (N = 12), respectively, where N = M Q 2n is the total outgoing channels from the floating reservoir. (E) The l = DJ /(0.5k ) is plotted as a function of T for DN = 2 (between n = 1 and 2) and DN = 8 (between Q 0 M n = 2 and 6), respectively, in red and black solid circles, where DJ = J (n , T ) − J (n , T ). The solid lines are the linear fittings to extract the thermal conductance values. Q Q i M Q j M Slope of these linear fits are 1.92 and 7.92 for DN = 2 and 8, respectively, which gives the g = 0.96k T and 0.99k T for the single edge mode, respectively. Q 0 0 AB 2.5 8 ν = 1 T ~ 27 mK ν = 2 T ~ 27 mK 2.0 5 6 1.5 g = (1.02 ± 0.01) κ 1.0 g = (2.02 ± 0.02) κ g = (2.08 ± 0.01) κ 0.5 0.2 0.4 0.6 0.8 1.0 01 2 3 4 5 22 −3 2 T − T (10 K ) V (V) BG M 0 Fig. 3. Thermal conductance in fractional QH. (A) Hall conductance (black line) and thermal noise (red line) measured in the graphite back-gated device plotted as a function of V at B = 7 T. The plateaus for n =1, , and 2 are visible in both the measurements. (B) Similar to the previous plots (Fig. 2), the excess thermal noise S is BG I measured as a function source current I , and the T is shown as a function of the dissipated power J in figs. S15 and S16, from which we have extracted the J (solid SD M Q Q 2 2 4 circles) as a function of T  T for n =1, , and 2 and shown up to T ~ 60 to 70 mK. The solid lines are the linear fits to extract the slopes, which give the thermal M 0 3 conductance values of 1.02, 2.08, and 2.02k T for n =1, , and 2, respectively. One can see that the thermal conductance values are quantized for n = 1 and 2, and the values are the same for both the n = and 2 plateaus. The inset shows the corresponding downstream charge modes for integer and fractional edges. The dashed curve represents the theoretically predicted (section S10) contribution of the heat Coulomb blockade (39, 40)for n = 1, showing its negligible contribution to the net thermal current. 4 4 of V at B = 7 T are shown in Fig. 3A, where the n =1, , and 2 are spectively. For n = , two downstream charge modes, one integer and one BG 3 3 1 e visible in both measurements. The T versus J plots for different fractional (inner n = with effective charge, e* ¼ ), are expected. The M Q 3 3 filling factors are shown in fig. S16. In Fig. 3B, we plotted the J (solid thermal conductance of n ¼ should be thesameas n =2having two 2 2 4 circles) as a function of T  T for n =1, ,and2overthetemperature integer downstream charge modes, which is observed in our experiment. M 0 3 window where the curve is linear, implying the dominance of the Thus, our result is consistent with the theory that the quantum limit of electronic contribution to the heat flow. The solid lines in Fig. 3B repre- thermal conductance is the same for both fractional and integer QH edges. sent the linear fits (in 0.5k ) and give the values of 2.04, 4.16, and 4.04, We would like to note that for device 3, the thermal conductance was which correspond to g = 1.02, 2.08, and 2.02k T for n =1, ,and 2, re- obtained without varying the number of outgoing channels (DN). This Q 0 Srivastav et al., Sci. Adv. 2019; 5 : eaaw5798 12 July 2019 3of5 G (e /h) −29 2 S (10 A /Hz) T (mK) −29 2 S (10 A /Hz) –19 2 S (10 V /Hz) J (fW) −3 2 λ (10 K ) −29 2 S (10 A /Hz) I | SCIENCE ADVANCES RESEARCH ARTICLE may lead to the inaccuracy in the extracted thermal conductance values (5/15/60 nm) was performed to make the contacts in an evaporator −7 −7 due to electron-phonon coupling and heat Coulomb blockade (39, 40). chamber having base pressure of ~1 × 10 to 2 × 10 mbar and However, measuring the right value of the thermal conductance within followed by lift-off procedure in acetone and IPA. The floating metallic 5% accuracy for device 3 corroborates the negligible contributions from reservoir in the middle was connected to both sides of the graphene part the electron-phonon coupling and heat Coulomb blockade. The latter is by the edge contacts. This procedure of making devices prevented con- discussed in more detail in section S10. The theoretical estimation (39, 40) tamination of exposed graphene edges with polymer residues, resulting of the heat Coulomb blockade for n = 1 is shown by a dash curve in in high-quality contacts. Fig. 3B. We discuss about the electron-phonon coupling, the accuracy of the measurements, and the effect of the heat Coulomb blockade in sections S8, S9, and S10, respectively. SUPPLEMENTARY MATERIALS Supplementary material for this article is available at http://advances.sciencemag.org/cgi/ In conclusion, we measured the thermal conductance for three content/full/5/7/eaaw5798/DC1 integer plateaus (1, 2, and 6) and one particle-like fractional plateau Section S1. Device characterization and measurement setup of graphene, and the values are consistent with the quantum limit 3 Section S2. Gain of the amplification chain 2 2 p k Section S3. Electron temperature (T ) determination T within 5% accuracy. These studies can be extended soon to 3h Section S4. Partition of current and contact resistance Section S5. Dissipated power in the floating reservoir measure the thermal conductance for the even-denominator QH Section S6. Determination of the temperature (T ) of floating reservoir plateaus in graphene (37) with atomically sharp confining potential Section S7. Extended excess thermal noise data to probe their non-Abelian nature. Section S8. Heat loss by electron-phonon cooling Section S9. Accuracy of the thermal conductance measurement Section S10. Discussion on heat Coulomb blockade Fig. S1. Optical image and device response at zero magnetic field. MATERIALS AND METHODS Fig. S2. Experimental setup for noise measurement. Device fabrication Fig. S3. Schematic used to derive the gain in section S2. Our encapsulated graphene devices were made using the following Fig. S4. Gain of amplification chain: Output voltage from a known input signal in QH state at resonance frequency. procedures similar to those used in previous reports (41, 42). First, an Fig. S5. Gain of amplification chain: From the temperature-dependent thermal noise. hBN/graphene/hBN stack was made using the “hot pickup” technique Fig. S6. Gain of amplification chain during measurement of device 3 (graphite back-gated (38). This involved the mechanical exfoliation of graphite and bulk hBN device). crystal on the SiO /Si wafer to obtain the single-layer graphene and thin 2 Fig. S7. RC filter assembly and thermal anchoring on the cold finger. hBN (~20 to 30 nm). Single-layer graphene and thin hBN (~20 to Fig. S8. Electron temperature (T ) determination. Fig. S9. Electron temperature (T ) determination: From shot noise measurement in a p-n 30 nm) were identified using an optical microscope. Fabrication of junction of graphene device. this hetrostructure assembly involved four steps. Step 1: We used a Fig. S10. Equipartition of current in left and right moving chiral states. poly-bisphenol-A-carbonate–coated polydimethylsiloxane block Fig. S11. Determination of contact resistance and source noise. mounted on a glass slide attached to tip of a micromanipulator to pick Fig. S12. Extended excess thermal noise raw data. Fig. S13. Extended data of device 1 at B =6 T. up theexfoliatedhBN flake. Theexfoliated hBN flake was picked up at Fig. S14. Extended data of device 2 at B =6 T. temperature of 90°C. Step 2: A previously picked-up hBN flake was Fig. S15. Extended data of device 3 (graphite back gate) at B =7 T. aligned over a graphene. Now, this graphene was picked up at tempera- Fig. S16. Extended data of device 3 (graphite back gate) at B =7 T. ture of 90°C. Step 3: The bottom hBN flake was picked up using the Fig. S17. Heat loss by electron-phonon coupling. previously picked-up hBN/graphene following step 2. Step 4: Last, this Table S1. Gain of amplification chain. Table S2. Electron temperature (T ). resulting hetrostructure (hBN/graphene/hBN) was dropped down on Table S3. Contact resistance and the source noise. topofanoxidizedsilicon wafer(p++doped siliconwithSiO thickness Table S4. Contact resistance and the source noise of device 3 (graphite back-gated device). of 285 nm) at temperature of 140°C, which served as a back gate (for the Table S5. Change in thermal conductance for different electron temperature T . graphite back-gated device after step 3, the graphite flake was picked up References (43–55) using the previously picked-up hBN/graphene/hBN following step 2; after this step, again, step 4 was followed). These final stacks were cleaned in chloroform (CHCl ) followed by acetone and isopropyl REFERENCES AND NOTES 1. C. Kane, M. P. Fisher, Quantized thermal transport in the fractional quantum Hall effect. alcohol (IPA). The next step involved electron-beam lithography Phys. Rev. B 55, 15832–15837 (1997). (EBL) to define the contact region. Poly-methyl-methacrylate was 2. T. Senthil, M. P. A. Fisher, Quasiparticle localization in superconductors with spin-orbit coated on the resulting hetrostructure. Contact region was defined scattering. Phys. Rev. B 61, 9690–9698 (2000). using EBL. Apart from conventional Hall probe geometry, we 3. K. Schwab, E. A. Henriksen, J. M. Worlock, M. L. Roukes, Measurement of the quantum of thermal conductance. Nature 404, 974–977 (2000). defined a region for floating reservoir of ~4- to 7-mm area. We used 4. M. Meschke, W. Guichard, J. P. Pekola, Single-mode heat conduction by photons. Nature two SiO /Si back-gated devices (device 1 and device 2) and one graphite 444, 187–190 (2006). back-gated device (device 3) for the thermal conductance measurement. 5. L. W. Molenkamp, Th. Gravier, H. van Houten, O. J. A. Buijk, M. A. A. Mabesoone, The edge contacts were achieved by reactive ion etching (a mixture of C. T. Foxon, Peltier coefficient and thermal conductance of a quantum point contact. CHF and O gas was used with a flow rate of 40 and 4 sccm, respec- Phys. Rev. Lett. 68, 3765–3768 (1992). 3 2 6. O. Chiatti, J. T. Nicholls, Y. Y. Proskuryakov, N. Lumpkin, I. Farrer, D. A. Ritchie, Quantum tively, at 25°C with radio frequency power of 60 W), where the etching thermal conductance of electrons in a one-dimensional wire. Phys. Rev. Lett. 97, 056601 time has been varied from 100 to 50 s for the SiO /Si and graphite back- (2006). gated devices, respectively, such that for the SiO /Si device, the bottom 7. S. Jezouin, F. D. Parmentier, A. Anthore, U. Gennser, A. Cavanna, Y. Jin, F. Pierre, Quantum hBN is being etched completely, whereas for the graphite back-gated limit of heat flow across a single electronic channel. Science 342, 601–604 (2013). 8. J. B. Pendry, Quantum limits to the flow of information and entropy. J. Phys. A Math. Gen. device, the bottom hBN is partially etched to isolate the contacts from 16, 2161–2171 (1983). the bottom graphite back gate. Last, the thermal deposition of Cr/Pd/Au Srivastav et al., Sci. Adv. 2019; 5 : eaaw5798 12 July 2019 4of5 | SCIENCE ADVANCES RESEARCH ARTICLE 9. C. L. Kane, M. P. A. Fisher, Thermal transport in a Luttinger liquid. Phys. Rev. Lett. 76, 39. E. Sivre, A. Anthore, F. D. Parmentier, A. Cavanna, U. Gennser, A. Ouerghi, Y. Jin, F. Pierre, 3192–3195 (1996). Heat Coulomb blockade of one ballistic channel. Nat. Phys. 14, 145–148 (2018). 10. L. G. C. Rego, G. Kirczenow, Fractional exclusion statistics and the universal quantum of 40. A. O. Slobodeniuk, I. P. Levkivskyi, E. V. Sukhorukov, Equilibration of quantum Hall edge thermal conductance: A unifying approach. Phys. Rev. B 59, 13080–13086 (1999). states by an Ohmic contact. Phys. Rev. B 88, 165307 (2013). 11. M. Banerjee, M. Heiblum, A. Rosenblatt, Y. Oreg, D. E. Feldman, A. Stern, V. Umansky, 41. L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, Observed quantization of anyonic heat flow. Nature 545,75–79 (2017). L. M. Campos, D. A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, C. R. Dean, 12. M. Banerjee, M. Heiblum, V. Umansky, D. E. Feldman, Y. Oreg, A. Stern, Observation of One-dimensional electrical contact to a two-dimensional material. Science 342, 614–617 half-integer thermal Hall conductance. Nature 559, 205–210 (2018). (2013). 13. D. E. Feldman, Comment on “Interpretation of thermal conductance of the n = 5/2 edge”. 42. A. V. Kretinin, Y. Cao, J. S. Tu, G. L. Yu, R. Jalil, K. S. Novoselov, S. J. Haigh, A. Gholinia, Phys. Rev. B 98, 167401 (2018). A. Mishchenko, M. Lozada, T. Georgiou, C. R. Woods, F. Withers, P. Blake, G. Eda, A. Wirsig, 14. S. H. Simon, Interpretation of thermal conductance of the n = 5/2 edge. Phys. Rev. B 97, C. Hucho, K. Watanabe, T. Taniguchi, A. K. Geim, R. V. Gorbachev, Electronic properties of 121406 (2018). graphene encapsulated with different two-dimensional atomic crystals. Nano Lett. 14, 15. C. Wang, A. Vishwanath, B. I. Halperin, Topological order from disorder and the quantized 3270–3276 (2014). Hall thermal metal: Possible applications to the n = 5/2 state. Phys. Rev. B 98, 045112 43. A. Venugopal, J. Chan, X. Li, C. W. Magnuson, W. P. Kirk, L. Colombo, R. S. Ruoff, E. M. Vogel, (2018). Effective mobility of single-layer graphene transistors as a function of channel dimensions. 16. D. F. Mross, Y. Oreg, A. Stern, G. Margalit, M. Heiblum, Theory of disorder-induced J. Appl. Phys. 109, 104511 (2011). half-integer thermal hall conductance. Phys. Rev. Lett. 121, 026801 (2018). 44. B.-R. Choi, A. E. Hansen, T. Kontos, C. Hoffmann, S. Oberholzer, W. Belzig, 17. D. T. Son, Is the composite fermion a dirac particle? Phys. Rev. X 5, 031027 (2015). C. Schönenberger, T. Akazaki, H. Takayanagi, Shot-noise and conductance measurements 18. P. T. Zucker, D. E. Feldman, Stabilization of the particle-hole Pfaffian order by of transparent superconductor/two-dimensional electron gas junctions. Phys. Rev. B 72, Landau-level mixing and impurities that break particle-hole symmetry. Phys. Rev. Lett. 024501 (2005). 117, 096802 (2016). 45. M. Büttiker, Absence of backscattering in the quantum Hall effect in multiprobe 19. M. Levin, B. I. Halperin, B. Rosenow, Particle-hole symmetry and the Pfaffian state. conductors. Phys. Rev. B 38, 9375–9389 (1988). Phys. Rev. Lett. 99, 236806 (2007). 46. N. Kumada, F. D. Parmentier, H. Hibino, D. C. Glattli, P. Roulleau, Shot noise generated by 20. S.-S. Lee, S. Ryu, C. Nayak, M. P. Fisher, Particle-hole symmetry and the n ¼ quantum Hall graphene p–n junctions in the quantum Hall effect regime. Nat. Commun. 6, 8068 (2015). state. Phys. Rev. Lett. 99, 236807 (2007). 47. S. Matsuo, S. Takeshita, T. Tanaka, S. Nakaharai, K. Tsukagoshi, T. Moriyama, T. Ono, 21. D. B. Chklovskii, B. I. Shklovskii, L. I. Glazman, Electrostatics of edge channels. Phys. Rev. B K. Kobayashi, Edge mixing dynamics in graphene p–n junctions in the quantum Hall 46, 4026–4034 (1992). regime. Nat. Commun. 6, 8066 (2015). 22. C. d. C. Chamon, X. Wen, Sharp and smooth boundaries of quantum Hall liquids. 48. U. Sivan, Y. Imry, Multichannel Landauer formula for thermoelectric transport with Phys. Rev. B 49,8227–8241 (1994). application to thermopower near the mobility edge. Phys. Rev. B 33, 551–558 (1986). 23. H. Inoue, A. Grivnin, Y. Ronen, M. Heiblum, V. Umansky, D. Mahalu, Proliferation of neutral 49. J.-H. Jiang, Y. Imry, Linear and nonlinear mesoscopic thermoelectric transport with modes in fractional quantum Hall states. Nat. Commun. 5, 4067 (2014). coupling with heat baths. C. R. Phys. 17, 1047–1059 (2016). 24. Y. Zhang, Y.-H. Wu, J. A. Hutasoit, J. K. Jain, Theoretical investigation of edge 50. C. W. J. Beenakker, M. Büttiker, Suppression of shot noise in metallic diffusive conductors. 5 7 reconstruction in the v ¼ and fractional quantum Hall states. Phys. Rev. B 90, 165104 Phys. Rev. B 46, 1889–1892 (1992). 2 3 (2014). 51. Y. M. Blanter, E. V. Sukhorukov, Semiclassical theory of conductance and noise in open 25. R. Sabo, I. Gurman, A. Rosenblatt, F. Lafont, D. Banitt, J. Park, M. Heiblum, Y. Gefen, chaotic cavities. Phys. Rev. Lett. 84, 1280–1283 (2000). V. Umansky, D. Mahalu, Edge reconstruction in fractional quantum Hall states. Nat. Phys. 52. A. Sergeev, V. Mitin, Electron-phonon interaction in disordered conductors: Static and 13, 491–496 (2017). vibrating scattering potentials. Phys. Rev. B 61, 6041–6047 (2000). 26. Z.-X. Hu, R. N. Bhatt, X. Wan, K. Yang, Realizing universal edge properties in graphene 53. S. S. Kubakaddi, Interaction of massless Dirac electrons with acoustic phonons in fractional quantum Hall liquids. Phys. Rev. Lett. 107, 236806 (2011). graphene at low temperatures. Phys. Rev. B 79, 075417 (2009). 27. G. Li, A. Luican-Mayer, D. Abanin, L. Levitov, E. Y. Andrei, Evolution of Landau levels into 54. F. Pierre, A. B. Gougam, A. Anthore, H. Pothier, D. Esteve, N. O. Birge, Dephasing of edge states in graphene. Nat. Commun. 4, 1744 (2013). electrons in mesoscopic metal wires. Phys. Rev. B 68, 085413 (2003). 28. F. Pientka, J. Waissman, P. Kim, B. I. Halperin, Thermal transport signatures of 55. P. W. Brouwer, M. Büttiker, Charge-relaxation and dwell time in the fluctuating broken-symmetry phases in graphene. Phys. Rev. Lett. 119, 027601 (2017). admittance of a chaotic cavity. Europhys. Lett. 37, 441–446 (1997). 29. K. Yang, S. D. Sarma, A. H. MacDonald, Collective modes and skyrmion excitations in graphene SU(4) quantum Hall ferromagnets. Phys. Rev. B 74, 075423 (2006). Acknowledgments: We would like to thank J. Jain for the critical inputs and help in writing 30. I. Sodemann, A. H. MacDonald, Broken SU(4) symmetry and the fractional quantum Hall the manuscript. We would also like to thank M. Heiblum for the fruitful discussion. We also effect in graphene. Phys. Rev. Lett. 112, 126804 (2014). thank R. Pandit, H. Choi, Y. Ronen, and V. Singh for the useful discussions. Fundings: The 31. M. Kharitonov, Phase diagram for the n = 0 quantum Hall state in monolayer graphene. authors acknowledge device fabrication and characterization facilities in CeNSE, IISc, Phys. Rev. B 85, 155439 (2012). Bangalore. A.D. thanks the Department of Science and Technology (DST), government of India, 32. B. E. Feldman, J. Martin, A. Yacoby, Broken-symmetry states and divergent resistance in under grant nos. DSTO1470 and DSTO1597. K.W. and T.T. acknowledge support from the suspended bilayer graphene. Nat. Phys. 5, 889–893 (2009). Elemental Strategy Initiative conducted by the MEXT, Japan and the CREST (JPMJCR15F3), JST. 33. R. T. Weitz, M. T. Allen, B. E. Feldman, J. Martin, A. Yacoby, Broken-symmetry states in Author contributions: S.K.S. contributed to the device fabrication, data acquisition, and doubly gated suspended bilayer graphene. Science 330, 812–816 (2010). analysis. M.R.S. contributed in the noise setup, data acquisition, and analysis. A.D. contributed 34. A. F. Young, C. R. Dean, L. Wang, H. Ren, P. Cadden-Zimansky, K. Watanabe, T. Taniguchi, in conceiving the idea and designing the experiment, data interpretation, and analysis. J. Hone,K. L. Shepard,P.Kim,Spin and valleyquantum Hall ferromagnetismin graphene. S.B. contributed in the data interpretation and theoretical understanding of the manuscript. Nat. Phys. 8,550–556 (2012). K.W. and T.T. synthesized the hBN single crystals. All authors contributed in writing the 35. P. Maher, C. R. Dean, A. F. Young, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, P. Kim, manuscript. Competing interests: The authors declare that they have no competing interests. Evidence for a spin phase transition at charge neutrality in bilayer graphene. Nat. Phys. 9, Data and materials availability: All data needed to evaluate the conclusions in the paper 154–158 (2013). are present in the paper and/or the Supplementary Materials. Additional data related to 36. A. F. Young, J. D. Sanchez-Yamagishi, B. Hunt, S. H. Choi, K. Watanabe, T. Taniguchi, this paper may be requested from the authors. R. C. Ashoori, P. Jarillo-Herrero, Tunable symmetry breaking and helical edge transport in a graphene quantum spin Hall state. Nature 505, 528–532 (2014). Submitted 7 January 2019 37. J. I. A. Li, C. Tan, S. Chen, Y. Zeng, T. Taniguchi, K. Watanabe, J. Hone, C. R. Dean, Accepted 31 May 2019 Even-denominator fractional quantum Hall states in bilayer graphene. Science 358, Published 12 July 2019 648–652 (2017). 10.1126/sciadv.aaw5798 38. F. Pizzocchero, L. Gammelgaard, B. S. Jessen, J. M. Caridad, L. Wang, J. Hone, P. Bøggild, T. J. Booth, The hot pick-up technique for batch assembly of van der Waals Citation: S. K. Srivastav, M. R. Sahu, K. Watanabe, T. Taniguchi, S. Banerjee, A. Das, Universal heterostructures. Nat. Commun. 7, 11894 (2016). quantized thermal conductance in graphene. Sci. Adv. 5, eaaw5798 (2019). Srivastav et al., Sci. Adv. 2019; 5 : eaaw5798 12 July 2019 5of5

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