Abstract
The universal quantization of thermal conductance provides information on a state's topological order. Recent measurements revealed that the observed value of thermal conductance of the
INTRODUCTION
Measurement of the quantization of thermal conductance at its quantum limit (
In this report, we carried out the thermal conductance measurement in the integer and FQHE of graphene devices using sensitive noise thermometry setup. We first establish the quantum limit of thermal conductance for integer plateaus of ν = 1, 2, and 6 in hBN-encapsulated monolayer graphene devices gated by an SiO2/Si back gate. We then further study the thermal conductance for fractional plateau of
We used two SiO2/Si back-gated devices and one graphite back-gated device for our measurements, where the hBN-encapsulated devices are fabricated using the standard dry transfer pickup technique (38) followed by the edge contacting method (see Materials and Methods). The schematic is shown in Fig. 1A, where the floating metallic reservoir in the middle connects both sides by edge contacts. The measurements are performed in a cryofree dilution refrigerator having a base temperature of ~12 mK. The thermal conductance was measured using noise thermometry based on LCR resonant circuit at resonance frequency of ~758 kHz, amplified by preamplifiers, and, lastly, measured by a spectrum analyzer (fig. S2). The conductance measured at the source contact in Fig. 1A for device 1 has been plotted as a function of back-gate voltage (VBG) at B = 9.8 T shown in Fig. 1B, where the clear plateaus at ν = 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 VBG in Fig. 1B, where the plateaus are also evident.
(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.
A DC current I, injected at the source contact (Fig. 1A), flows along the chiral edge toward the floating reservoir. The outgoing current from the floating reservoir splits into two equal parts, each propagating along the outgoing chiral edge from the floating reservoir to the cold grounds. The floating reservoir reaches a new equilibrium potential
Here,
RESULTS AND DISCUSSION
In our experiment, for an integer filling factor ν, the ν chiral edge modes impinge the current in the floating reservoir, and N = 2ν chiral edge modes leave the floating reservoir as shown in Fig. 1A. Figure 2 (A to C) shows the measured excess thermal noise SI for device 1 as a function of source current ISD for ν = 1, 2, and 6 at B = 9.8 T. The increment in the temperature of the floating reservoir as a function of ISD is exhibited in the increase of SI. The x and y axes of Fig. 2 (A to C) are converted to JQ and TM, respectively, and plotted in Fig. 2D for different ν, where each solid circle is generated after averaging nine consecutive data points (raw data in section S7). The T0 ~ 40 mK without DC current was determined from the thermal noise measurement and shown in section S3. As expected, the TM is higher for lower filling factor as less number of chiral edges are carrying the heat away from the floating reservoir. Thus, to maintain a constant TM, higher JQ is required for higher filling factor. In Fig. 2E, we plotted λ (= ΔJQ/(0.5κ0), where ΔJQ = JQ(νi, TM) − JQ(νj, TM), as a function of
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
To measure the thermal conductance for the FQHE state, we used a graphite back-gated device (device 3), where the graphene channel is isolated from the graphite gate by bottom hBN of thickness ~20 nm. For this device, the lower electron temperature T0~27 mK (section S3) was achieved by introducing extra low-pass filters at the mixing chamber. The conductance plateaus and the thermal noise as a function of VBG at B = 7 T are shown in Fig. 3A, where the ν = 1,
(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,
We would like to note that for device 3, the thermal conductance was obtained without varying the number of outgoing channels (ΔN). This may lead to the inaccuracy in the extracted thermal conductance values due to electron-phonon coupling and heat Coulomb blockade (39, 40). However, measuring the right value of the thermal conductance within 5% accuracy for device 3 corroborates the negligible contributions from the electron-phonon coupling and heat Coulomb blockade. The latter is discussed in more detail in section S10. The theoretical estimation (39, 40) of the heat Coulomb blockade for ν = 1 is shown by a dash curve in 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.
In conclusion, we measured the thermal conductance for three integer plateaus (1, 2, and 6) and one particle-like fractional plateau
MATERIALS AND METHODS
Device fabrication
Our encapsulated graphene devices were made using the following procedures similar to those used in previous reports (41, 42). First, an hBN/graphene/hBN stack was made using the “hot pickup” technique (38). This involved the mechanical exfoliation of graphite and bulk hBN crystal on the SiO2/Si wafer to obtain the single-layer graphene and thin hBN (~20 to 30 nm). Single-layer graphene and thin hBN (~20 to 30 nm) were identified using an optical microscope. Fabrication of this hetrostructure assembly involved four steps. Step 1: We used a poly-bisphenol-A-carbonate–coated polydimethylsiloxane block mounted on a glass slide attached to tip of a micromanipulator to pick up the exfoliated hBN flake. The exfoliated hBN flake was picked up at temperature of 90°C. Step 2: A previously picked-up hBN flake was aligned over a graphene. Now, this graphene was picked up at temperature of 90°C. Step 3: The bottom hBN flake was picked up using the previously picked-up hBN/graphene following step 2. Step 4: Last, this resulting hetrostructure (hBN/graphene/hBN) was dropped down on top of an oxidized silicon wafer (p++ doped silicon with SiO2 thickness of 285 nm) at temperature of 140°C, which served as a back gate (for the graphite back-gated device after step 3, the graphite flake was picked up 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 (CHCl3) followed by acetone and isopropyl alcohol (IPA). The next step involved electron-beam lithography (EBL) to define the contact region. Poly-methyl-methacrylate was coated on the resulting hetrostructure. Contact region was defined using EBL. Apart from conventional Hall probe geometry, we defined a region for floating reservoir of ~4- to 7-μm2 area. We used two SiO2/Si back-gated devices (device 1 and device 2) and one graphite back-gated device (device 3) for the thermal conductance measurement. The edge contacts were achieved by reactive ion etching (a mixture of CHF3 and O2 gas was used with a flow rate of 40 and 4 sccm, respectively, at 25°C with radio frequency power of 60 W), where the etching time has been varied from 100 to 50 s for the SiO2/Si and graphite back-gated devices, respectively, such that for the SiO2/Si device, the bottom hBN is being etched completely, whereas for the graphite back-gated device, the bottom hBN is partially etched to isolate the contacts from the bottom graphite back gate. Last, the thermal deposition of Cr/Pd/Au (5/15/60 nm) was performed to make the contacts in an evaporator chamber having base pressure of ~1 × 10−7 to 2 × 10−7 mbar and followed by lift-off procedure in acetone and IPA. The floating metallic reservoir in the middle was connected to both sides of the graphene part by the edge contacts. This procedure of making devices prevented contamination of exposed graphene edges with polymer residues, resulting in high-quality contacts.
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.
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REFERENCES AND NOTES
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