Abstract
In low-dimensional systems, the combination of reduced dimensionality, strong interactions and topology has led to a growing number of many-body quantum phenomena. Thermal transport, which is sensitive to all energy-carrying degrees of freedom, provides a discriminating probe of emergent excitations in quantum materials and devices. However, thermal transport measurements in low dimensions are dominated by the phonon contribution of the lattice, requiring an experimental approach to isolate the electronic thermal conductance. Here we measured non-local voltage fluctuations in a multi-terminal device to reveal the electronic heat transported across a mesoscopic bridge made of low-dimensional materials. Using two-dimensional graphene as a noise thermometer, we measured the quantitative electronic thermal conductance of graphene and carbon nanotubes up to 70 K, achieving a precision of ~1% of the thermal conductance quantum at 5 K. Employing linear and nonlinear thermal transport, we observed signatures of energy transport mediated by long-range interactions in one-dimensional electron systems, in agreement with a theoretical model.
This is a preview of subscription content, access via your institution
Access options
Access Nature and 54 other Nature Portfolio journals
Get Nature+, our best-value online-access subscription
$29.99 per month
cancel any time
Subscribe to this journal
Receive 12 print issues and online access
$259.00 per year
only $21.58 per issue
Rent or buy this article
Get just this article for as long as you need it
$39.95
Prices may be subject to local taxes which are calculated during checkout




Data availability
The data that support the findings of this study are available at the online depository Zenodo (https://doi.org/10.5281/zenodo.5500449).
References
Franz, R. & Wiedemann, G. Ueber die Wärme-Leitungsfähigkeit der Metalle. Ann. Phys. 165, 497–531 (1853).
Wakeham, N. et al. Gross violation of the Wiedemann–Franz law in a quasi-one-dimensional conductor. Nat. Commun. 2, 396 (2011).
Smith, R. P. et al. Marginal breakdown of the Fermi-liquid state on the border of metallic ferromagnetism. Nature 455, 1220–1223 (2008).
Tanatar, M. A., Paglione, J., Petrovic, C. & Taillefer, L. Anisotropic violation of the Wiedemann–Franz law at a quantum critical point. Science 316, 1320–1322 (2007).
Hill, R. W., Proust, C., Taillefer, L., Fournier, P. & Greene, R. L. Breakdown of Fermi-liquid theory in a copper-oxide superconductor. Nature 414, 711–715 (2001).
Crossno, J. et al. Observation of the Dirac fluid and the breakdown of the Wiedemann–Franz law in graphene. Science 351, 1058–1061 (2016).
Zhou, Y. et al. Bilayer Wigner crystals in a transition metal dichalcogenide heterostructure. Nature 595, 48–52 (2021).
Smoleński, T. et al. Signatures of Wigner crystal of electrons in a monolayer semiconductor. Nature 595, 53–57 (2021).
Shapir, I. et al. Imaging the electronic Wigner crystal in one dimension. Science 364, 870–875 (2019).
Andrei, E. Y. & MacDonald, A. H. Graphene bilayers with a twist. Nat. Mater. 19, 1265–1275 (2020).
Mak, K. F., Shan, J. & Ralph, D. C. Probing and controlling magnetic states in 2D layered magnetic materials. Nat. Rev. Phys. 1, 646–661 (2019).
Li, M. & Chen, G. Thermal transport for probing quantum materials. MRS Bull. 45, 348–356 (2020).
Tritt, T. M. Thermal Conductivity (Kluwer Academic and Plenum Press, 2004).
Jezouin, S. et al. Quantum limit of heat flow across a single electronic channel. Science 342, 601–604 (2013).
Banerjee, M. et al. Observed quantization of anyonic heat flow. Nature 545, 75–79 (2017).
Banerjee, M. et al. Observation of half-integer thermal Hall conductance. Nature https://doi.org/10.1038/s41586-018-0184-1 (2018).
Srivastav, S. K. et al. Universal quantized thermal conductance in graphene. Sci. Adv. 5, eaaw5798 (2019).
Dutta, B. et al. Thermal conductance of a single-electron transistor. Phys. Rev. Lett. 119, 077701 (2017).
Cui, L. et al. Quantized thermal transport in single-atom junctions. Science 355, 1192–1195 (2017).
Mosso, N. et al. Heat transport through atomic contacts. Nat. Nanotechnol. 12, 430–433 (2017).
Crossno, J., Liu, X., Ohki, T. A., Kim, P. & Fong, K. C. Development of high frequency and wide bandwidth Johnson noise thermometry. Appl. Phys. Lett. 106, 023121 (2015).
Fong, K. C. et al. Measurement of the electronic thermal conductance channels and heat capacity of graphene at low temperature. Phys. Rev. X 3, 041008 (2013).
Chiatti, O. et al. Quantum thermal conductance of electrons in a one-dimensional wire. Phys. Rev. Lett. 97, 056601 (2006).
Molenkamp, L. W. et al. Peltier coefficient and thermal conductance of a quantum point contact. Phys. Rev. Lett. 68, 3765–3768 (1992).
Seol, J. H. et al. Two-dimensional phonon transport in supported graphene. Science 328, 213–216 (2010).
Kim, P., Shi, L., Majumdar, A. & McEuen, P. L. Thermal transport measurements of individual multiwalled nanotubes. Phys. Rev. Lett. 87, 215502 (2001).
Mosso, N. et al. Thermal transport through single-molecule junctions. Nano Lett. 19, 7614–7622 (2019).
Cui, L. et al. Thermal conductance of single-molecule junctions. Nature 572, 628–633 (2019).
Johnson, J. B. Thermal agitation of electricity in conductors. Phys. Rev. 32, 97–109 (1928).
Nyquist, H. Thermal agitation of electric charge in conductors. Phys. Rev. 32, 110–113 (1928).
Qu, J. F. et al. Johnson noise thermometry. Meas. Sci. Technol. 30, 112001 (2019).
Fong, K. C. & Schwab, K. C. Ultrasensitive and wide-bandwidth thermal measurements of graphene at low temperatures. Phys. Rev. X 2, 031006 (2012).
Yiǧen, S. & Champagne, A. R. Wiedemann–franz relation and thermal-transistor effect in suspended graphene. Nano Lett. 14, 289–293 (2014).
Sukhorukov, E. V. & Loss, D. Noise in multiterminal diffusive conductors: universality, nonlocality, and exchange effects. Phys. Rev. B 59, 13054–13066 (1999).
Talanov, A. V., Waissman, J., Taniguchi, T., Watanabe, K. & Kim, P. High-bandwidth, variable-resistance differential noise thermometry. Rev. Sci. Instrum. 92, 014904 (2021).
Pozderac, C. & Skinner, B. Relation between Johnson noise and heating power in a two-terminal conductor. Phys. Rev. B 104, L161403 (2021).
Principi, A. & Vignale, G. Violation of the Wiedemann–Franz law in hydrodynamic electron liquids. Phys. Rev. Lett. 115, 056603 (2015).
Lucas, A. & Das Sarma, S. Electronic hydrodynamics and the breakdown of the Wiedemann–Franz and Mott laws in interacting metals. Phys. Rev. B 97, 245128 (2018).
Li, S., Levchenko, A. & Andreev, A. V. Hydrodynamic electron transport near charge neutrality. Phys. Rev. B 102, 075305 (2020).
Xie, H.-Y. & Foster, M. S. Transport coefficients of graphene: interplay of impurity scattering, Coulomb interaction, and optical phonons. Phys. Rev. B 93, 195103 (2016).
Lucas, A., Davison, R. A. & Sachdev, S. Hydrodynamic theory of thermoelectric transport and negative magnetoresistance in Weyl semimetals. Proc. Natl Acad. Sci. USA 113, 9463–9468 (2016).
Zarenia, M., Principi, A. & Vignale, G. Disorder-enabled hydrodynamics of charge and heat transport in monolayer graphene. 2D Mater. 6, 035024 (2019).
Zarenia, M., Smith, T. B., Principi, A. & Vignale, G. Breakdown of the Wiedemann–Franz law in AB-stacked bilayer graphene. Phys. Rev. B 99, 161407 (2019).
Lucas, A. & Fong, K. C. Hydrodynamics of electrons in graphene. J. Phys. Condens. Matter 30, 53001 (2018).
Cui, X. et al. Multi-terminal transport measurements of MoS2 using a van der Waals heterostructure device platform. Nat. Nanotechnol. 10, 534–540 (2015).
Liu, Y. et al. Toward barrier free contact to molybdenum disulfide using graphene electrodes. Nano Lett. 15, 3030–3034 (2015).
El Abbassi, M. et al. Robust graphene-based molecular devices. Nat. Nanotechnol. 14, 957–961 (2019).
Yang, C., Qin, A., Tang, B. Z. & Guo, X. Fabrication and functions of graphene–molecule–graphene single-molecule junctions. J. Chem. Phys. 152, 120902 (2020).
Ilani, S. & McEuen, P. L. Electron transport in carbon nanotubes. Annu. Rev. Condens. Matter Phys. 1, 1–25 (2010).
Sfeir, M. Y. et al. Optical spectroscopy of individual single-walled carbon nanotubes of defined chiral structure. Science 312, 554–556 (2006).
Cheng, A., Taniguchi, T., Watanabe, K., Kim, P. & Pillet, J. D. Guiding dirac fermions in graphene with a carbon nanotube. Phys. Rev. Lett. 123, 216804 (2019).
McEuen, P., Bockrath, M., Cobden, D., Yoon, Y.-G. & Louie, S. Disorder, pseudospins, and backscattering in carbon nanotubes. Phys. Rev. Lett. 83, 5098–5101 (1999).
Garg, A., Rasch, D., Shimshoni, E. & Rosch, A. Large violation of the Wiedemann–Franz Law in Luttinger liquids. Phys. Rev. Lett. 103, 096402 (2009).
Kane, C. L. & Fisher, M. P. A. Thermal transport in a Luttinger liquid. Phys. Rev. Lett. 76, 3192–3195 (1996).
Li, M.-R. & Orignac, E. Heat conduction and Wiedemann–Franz law in disordered Luttinger liquids. Europhys. Lett. 60, 432–438 (2002).
Pecker, S. et al. Observation and spectroscopy of a two-electron Wigner molecule in an ultraclean carbon nanotube. Nat. Phys. 9, 576–581 (2013).
Shi, Z. et al. Observation of a Luttinger-liquid plasmon in metallic single-walled carbon nanotubes. Nat. Photonics 9, 515–519 (2015).
Fazio, R., Hekking, F. W. J. & Khmelnitskii, D. E. Anomalous thermal transport in quantum wires. Phys. Rev. Lett. 80, 5611–5614 (1998).
Wang, L. et al. One-dimensional electrical contact to a two-dimensional material. Science 342, 614–617 (2013).
Müller, M., Schmalian, J. & Fritz, L. Graphene: a nearly perfect fluid. Phys. Rev. Lett. 103, 025301 (2009).
Tielrooij, K. J. et al. Photoexcitation cascade and multiple hot-carrier generation in graphene. Nat. Phys. 9, 248–252 (2013).
Walsh, E. D. et al. Graphene-based Josephson-junction single-photon detector. Phys. Rev. Appl. 8, 024022 (2017).
Lee, G.-H. et al. Graphene-based Josephson junction microwave bolometer. Nature 586, 42–46 (2020).
Kokkoniemi, R. et al. Bolometer operating at the threshold for circuit quantum electrodynamics. Nature 586, 47–51 (2020).
Acknowledgements
We thank B. Halperin, A. Lucas, S. D. Sarma, C. Mousatov, K. C. Fong and J. Crossno for helpful discussions, J. MacArthur for assistance with electronics design and construction, and M. Arino and H. Bartolomei for their assistance in the early stages of this work. This work was supported by ARO (W911NF-17-1-0574) for developing RF technology and characterization, and DOE (DE-SC0012260) for device fabrication and measurements. A.V.T. acknowledges support from the DoD through the NDSEG Fellowship Program. J.W. and P.K. acknowledge support from NSF (DMR-1922172) for data analysis. D.G.N. acknowledges support by the Office of Basic Energy Sciences of the DOE (DE-SC0017619). K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan (grant number JPMXP0112101001), JSPS KAKENHI (grant number JP20H00354) and CREST (JPMJCR15F3), JST. Work by K.A.M. at the Argonne National Laboratory was supported by the DOE, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division.
Author information
Authors and Affiliations
Contributions
J.W. and P.K. conceived the experiments. J.W. performed the experiments and analysed the data. L.E.A. fabricated the NT–graphene devices. J.W., Y.J.S., and D.H.N. fabricated the graphene devices. M.R. fabricated the α-RuCl3 devices. X.F. and D.G.N. synthesized the bulk α-RuCl3 crystals. T.T. and K.W. synthesized the bulk hBN crystals. B.S. performed non-local noise calculations. K.A.M. developed the plasmon hopping theory. J.W., L.E.A., A.V.T., Z.Y., M.R., B.S., K.A.M. and P.K. discussed the results and interpretations. J.W. and P.K. wrote the manuscript in consultation with the other authors.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Peer review information Nature Nanotechnology thanks Pramod Reddy and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Supplementary Information
Supplementary Discussion and Figs. 1–11.
Rights and permissions
About this article
Cite this article
Waissman, J., Anderson, L.E., Talanov, A.V. et al. Electronic thermal transport measurement in low-dimensional materials with graphene non-local noise thermometry. Nat. Nanotechnol. 17, 166–173 (2022). https://doi.org/10.1038/s41565-021-01015-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/s41565-021-01015-x
This article is cited by
-
Direct determination of the topological thermal conductance via local power measurement
Nature Physics (2023)
-
Cascading electron transfer and photophysics in a donor-π-acceptor graphene nanoconjugate
Nano Research (2022)