Measurement of the Electronic Thermal Conductance Channels and Heat Capacity of Graphene at Low Temperature

The ability to transport energy is a fundamental property of the two-dimensional Dirac fermions in graphene. Electronic thermal transport in this system is relatively unexplored and is expected to show unique fundamental properties and to play an important role in future applications of graphene, including optoelectronics, plasmonics, and ultrasensitive bolometry. Here, we present measurements of bipolar thermal conductances due to electron diffusion and electron-phonon coupling and infer the electronic specific heat, with a minimum value of $10k_B$ (${10}^{-22}\mathrm{J}/\mathrm{K}$) per square micron. We test the validity of the Wiedemann-Franz law and find that the Lorenz number equals $1.32\times ({\pi }^2/3)(k_B/e{)}^2$. The electron-phonon thermal conductance has a temperature power law $T^2$ at high doping levels, and the coupling parameter is consistent with recent theory, indicating its enhancement by impurity scattering. We demonstrate control of the thermal conductance by electrical gating and by suppressing the diffusion channel using NbTiN superconducting electrodes, which sets the stage for future graphene-based single-microwave photon detection.

Popular Summary

Graphene is known to conduct heat well at ambient temperatures through lattice vibrations called phonons. At very low temperatures, the situation is more subtle: The coupling between electrons and phonons becomes very weak, and the electron gas can be remarkably isolated thermally from the phonons. In fact, thermal conduction through electron diffusion can dominate the transport of energy. Furthermore, recent theory of the electron-phonon coupling has predicted a large modification of the electron-phonon coupling due to the effects of disorder and scattering of the electrons. Experimental data on these effects have been very limited because of the high detection sensitivity needed for such measurements. In this work, we report a number of new fundamental findings based on high-sensitivity experimental observations and careful data analysis.

Our measurements of electron-based thermal transport in graphene not only span a wide temperature range from 100 K down to 400 mK but also cover a range of electron or hole density and different types of thermoelectrical contacts with the graphene sample. Our findings are very revealing. We have observed and quantified the theoretically predicted disorder-induced modifications to the electron-phonon coupling. For low temperatures (below 1 K), we have identified the thermal conductance that originated from electron diffusion and analyzed its ratio to the electrical conductance: The result reveals an intriguing deviation, at low electron densities, from the “robust” prediction of Fermi-liquid theory. From the thermal transport measurements, we have also been able to infer the heat capacity of the electron gas, which, at 300 mK, reaches an extremely low value of approximately ${10}^{22}\mathrm{J}/\mathrm{K}$ per micron square. This detection sensitivity is 10 times higher than what has been previously reported in nanobolometry.

These measurements are essential for the design of ultrasensitive graphene-based thermal sensors, which might find applications as bolometers, high-speed bolometric mixers, or ultrasensitive calorimeters. Furthermore, these techniques may be useful to illuminate physics where energy transport or thermodynamic measurements are more illuminating than electronic transport.

Figure 1

(a) Experimental setup for simultaneously measuring the thermal and electrical transport of a graphene device: LC, inductor capacitor matching network; LP, low-pass filter; LNA, low-noise amplifier; and BP, band-pass filter. (b) Graphene thermal model. Heat from the graphene electrons can flow out through two different channels: electronic diffusion to the electrodes $G_{\mathrm{WF}}$ and $ep$ coupling $G_{ep}$. (c) Optical micrograph of device $D2$. (d) dc graphene resistances. (e) Electrical mean-free path calculated using $l_e=\sigma E_F/n{e}^2{v}_F$.

Figure 2

Data from device $D1$ with normal metallic electrodes. (a) The bipolar $G_{\mathrm{WF}}$ as a function of charge-carrier density at various sample temperatures. (b) $G_{\mathrm{th}}$ data as a function of temperature at $n=-2.2\times {10}^{12}{\mathrm{cm}}^{-2}$. The solid line is the power-law fit to the $ep$ thermal conductance above 1.5 K, while the dashed line is the best linear $T$ fit to the WF thermal conductance. The size of the data points represents the measurement error. (c) Wiedemann-Franz law in graphene. Each data point represents a measurement at a different temperature and charge-carrier density. The fitted line is $G_{\mathrm{WF}}=\alpha {\mathcal{L}}_{\mathrm{meas}}T/R$ with a $y$ offset; ${\mathcal{L}}_{\mathrm{meas}}=3.25\pm 0.02\times {10}^{-8}\mathrm{W}\Omega {\mathrm{K}}^{-2}$. (d) The measured Lorenz number ${\mathcal{L}}_{\mathrm{meas}}$ as a function of density from fitting of the Wiedemann-Franz law. For electrons with $n\le -0.18\times {10}^{12}{\mathrm{cm}}^{-2}$, averaged ${\mathcal{L}}_{\mathrm{meas}}/\mathcal{L}=1.34\pm 0.06$, while for holes with $n\ge +0.18\times {10}^{12}{\mathrm{cm}}^{-2}$, averaged ${\mathcal{L}}_{\mathrm{meas}}/\mathcal{L}=1.26\pm 0.07$.

Figure 3

(a) The measured electron temperature versus dc heating power applied to the graphene devices at different phonon temperatures with $n=28\times {10}^{12}{\mathrm{cm}}^{-2}$ for device $D2$ and $n=-2.2\times {10}^{12}{\mathrm{cm}}^{-2}$ for device $D3$. The solid lines are fits to Eq. (2). The power law $T^3$ persists down to 420 mK for device $D3$ with NbTiN electrodes. (b),(c) Data from device $D3$ with NbTiN electrodes: (b) the bipolar electron-phonon thermal conductance as a function of charge-carrier density at various temperatures and (c) $G_{\mathrm{th}}$ data as a function of temperature at $n=-2.9\times {10}^{12}{\mathrm{cm}}^{-2}$. The solid line is the power-law fit to the $ep$ thermal conductance above 1.5 K, while the dashed line is the calculated $G_{\mathrm{WF}}=\alpha {\mathcal{L}}_{0}T/R$ using $R=1770\Omega$. The offset between high- and low-temperature data is due to thermal cycling of the device on two cryostats.

Figure 4

(a) The electron-phonon coupling parameter ${\Sigma }_{ep}$ and (b) the temperature power law $\delta$ extracted from the thermal conductance data above 1.5 K as a function of charge-carrier density in three measured devices. Inset: The values of the deformation potential $\mathcal{D}$ in graphene from the literature: Blue squares and orange diamonds represent data based on electrical- and thermal-related measurements, and orange circles represent the data from this report (see also Table ).