Research ArticlePHYSICS

Universal quantized thermal conductance in graphene

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Science Advances  12 Jul 2019:
Vol. 5, no. 7, eaaw5798
DOI: 10.1126/sciadv.aaw5798
  • 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 ν = 1, where one chiral edge channel (line with arrow) propagates along the edge of the sample. The current ISD is injected (green line) through the contact S, which is absorbed in the floating reservoir (red contact). Chiral edge channel (red line) at potential VM and temperature TM leave the floating reservoir and terminate into two cold grounds (CGs). The cold edges (without any current) at temperature T0 are shown by the blue lines. The resulting increase in the electron temperature TM of the floating reservoir is determined from the measured excess thermal noise at contact D. A resonant (LC) circuit, situated at contact D, with resonance frequency f0 = 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 VBG at f0 = 758 kHz (red line). The plateaus for ν = 1, 2, and 6 are visible in both measurements.

  • Fig. 2 Thermal conductance in integer QH.

    Excess thermal noise SI is measured as a function of source current ISD at ν = 1 (A), 2 (B), and 6 (C). (D) The increased temperatures TM of the floating reservoir are plotted (solid circles) as a function of dissipated power JQ for ν = 1 (N = 2), 2 (N = 4), and 6 (N = 12), respectively, where N = 2ν is the total outgoing channels from the floating reservoir. (E) The λ = ΔJQ/(0.5κ0) is plotted as a function of TM2 for ΔN = 2 (between ν = 1 and 2) and ΔN = 8 (between ν = 2 and 6), respectively, in red and black solid circles, where ΔJQ = JQi, TM) − JQj, TM). The solid lines are the linear fittings to extract the thermal conductance values. Slope of these linear fits are 1.92 and 7.92 for ΔN = 2 and 8, respectively, which gives the gQ = 0.96κ0T and 0.99κ0T for the single edge mode, respectively.

  • 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 VBG at B = 7 T. The plateaus for ν = 1, 43, and 2 are visible in both the measurements. (B) Similar to the previous plots (Fig. 2), the excess thermal noise SI is measured as a function source current ISD, and the TM is shown as a function of the dissipated power JQ in figs. S15 and S16, from which we have extracted the JQ (solid circles) as a function of TM2T02 for ν = 1, 43, and 2 and shown up to TM ~ 60 to 70 mK. The solid lines are the linear fits to extract the slopes, which give the thermal conductance values of 1.02, 2.08, and 2.02κ0T for ν = 1, 43, and 2, respectively. One can see that the thermal conductance values are quantized for ν = 1 and 2, and the values are the same for both the ν = 43 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 ν = 1, showing its negligible contribution to the net thermal current.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/7/eaaw5798/DC1

    Section S1. Device characterization and measurement setup

    Section S2. Gain of the amplification chain

    Section S3. Electron temperature (T0) determination

    Section S4. Partition of current and contact resistance

    Section S5. Dissipated power in the floating reservoir

    Section S6. Determination of the temperature (TM) of floating reservoir

    Section S7. Extended excess thermal noise data

    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.

    Fig. S2. Experimental setup for noise measurement.

    Fig. S3. Schematic used to derive the gain in section S2.

    Fig. S4. Gain of amplification chain: Output voltage from a known input signal in QH state at resonance frequency.

    Fig. S5. Gain of amplification chain: From the temperature-dependent thermal noise.

    Fig. S6. Gain of amplification chain during measurement of device 3 (graphite back-gated device).

    Fig. S7. RC filter assembly and thermal anchoring on the cold finger.

    Fig. S8. Electron temperature (T0) determination.

    Fig. S9. Electron temperature (T0) determination: From shot noise measurement in a p-n junction of graphene device.

    Fig. S10. Equipartition of current in left and right moving chiral states.

    Fig. S11. Determination of contact resistance and source noise.

    Fig. S12. Extended excess thermal noise raw data.

    Fig. S13. Extended data of device 1 at B = 6 T.

    Fig. S14. Extended data of device 2 at B = 6 T.

    Fig. S15. Extended data of device 3 (graphite back gate) at B = 7 T.

    Fig. S16. Extended data of device 3 (graphite back gate) at B = 7 T.

    Fig. S17. Heat loss by electron-phonon coupling.

    Table S1. Gain of amplification chain.

    Table S2. Electron temperature (T0).

    Table S3. Contact resistance and the source noise.

    Table S4. Contact resistance and the source noise of device 3 (graphite back-gated device).

    Table S5. Change in thermal conductance for different electron temperature T0.

    References (4355)

  • Supplementary Materials

    This PDF file includes:

    • Section S1. Device characterization and measurement setup
    • Section S2. Gain of the amplification chain
    • Section S3. Electron temperature (T0) determination
    • Section S4. Partition of current and contact resistance
    • Section S5. Dissipated power in the floating reservoir
    • Section S6. Determination of the temperature (TM) of floating reservoir
    • Section S7. Extended excess thermal noise data
    • 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.
    • Fig. S2. Experimental setup for noise measurement.
    • Fig. S3. Schematic used to derive the gain in section S2.
    • Fig. S4. Gain of amplification chain: Output voltage from a known input signal in QH state at resonance frequency.
    • Fig. S5. Gain of amplification chain: From the temperature-dependent thermal noise.
    • Fig. S6. Gain of amplification chain during measurement of device 3 (graphite back-gated device).
    • Fig. S7. RC filter assembly and thermal anchoring on the cold finger.
    • Fig. S8. Electron temperature (T0) determination.
    • Fig. S9. Electron temperature (T0) determination: From shot noise measurement in a p-n junction of graphene device.
    • Fig. S10. Equipartition of current in left and right moving chiral states.
    • Fig. S11. Determination of contact resistance and source noise.
    • Fig. S12. Extended excess thermal noise raw data.
    • Fig. S13. Extended data of device 1 at B = 6 T.
    • Fig. S14. Extended data of device 2 at B = 6 T.
    • Fig. S15. Extended data of device 3 (graphite back gate) at B = 7 T.
    • Fig. S16. Extended data of device 3 (graphite back gate) at B = 7 T.
    • Fig. S17. Heat loss by electron-phonon coupling.
    • Table S1. Gain of amplification chain.
    • Table S2. Electron temperature (T0).
    • Table S3. Contact resistance and the source noise.
    • Table S4. Contact resistance and the source noise of device 3 (graphite back-gated device).
    • Table S5. Change in thermal conductance for different electron temperature T0.
    • References (4355)

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