Abstract
As high-quality single-crystal materials used in electronic devices approach the microscale and nanoscale, charge-transport phenomena in these devices result in inhomogeneous spatial signatures with strong implications for observable material properties. These signatures include spatially varying dissipation, which affects thermal management strategies in devices, and interface resistance between different materials, and are essential for the functional control of devices. In this Review, we investigate the spatially inhomogeneous signatures of charge flow in conductors, with particular emphasis on the recently rekindled field of electron hydrodynamics, a regime where electrons are strongly interacting and can flow collectively akin to fluids. We highlight recent experimental advances in transport measurements that enabled the observation of these signatures and review the theoretical frameworks used to interpret and predict these observations. We outline the new charge-transport phenomena introduced by crystal symmetry in materials, provide an outlook on future research opportunities and identify experimental and theoretical challenges in the study of hydrodynamic transport in materials.
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References
International Energy Agency. Digitalization and energy (IEA, 2017).
Shehabi, A., Smith, S. J., Masanet, E. & Koomey, J. Data center growth in the United States: decoupling the demand for services from electricity use. Env. Res. Lett. 13, 124030 (2018).
Masanet, E., Shehabi, A., Lei, N., Smith, S. & Koomey, J. Recalibrating global data center energy-use estimates. Science 367, 984–986 (2020).
International Energy Agency. Data centres and data transmission networks. IEA https://www.iea.org/energy-system/buildings/data-centres-and-data-transmission-networks (2022).
Lundstrom, M. S. & Alam, M. A. Moore’s law: the journey ahead. Science 378, 722–723 (2022).
Molenkamp, L. & de Jong, M. Observation of Knudsen and Gurzhi transport regimes in a two-dimensional wire. Solid State Electron. 37, 551–553 (1994).
de Jong, M. J. M. & Molenkamp, L. W. Hydrodynamic electron flow in high-mobility wires. Phys. Rev. B 51, 13389–13402 (1995). Early 2DEG hydrodynamic transport observations and dual-relaxation-time Boltzmann transport equation formalism.
Crossno, J. et al. Observation of the Dirac fluid and the breakdown of the Wiedemann–Franz law in graphene. Science 351, 1058–1061 (2016). One of three papers that were published back to back reporting transport measurements suggesting electron hydrodynamics.
Bandurin, D. A. et al. Negative local resistance caused by viscous electron backflow in graphene. Science 351, 1055–1058 (2016). Another of the three papers that were published back to back reporting transport measurements suggesting electron hydrodynamics.
Moll, P. J. W., Kushwaha, P., Nandi, N., Schmidt, B. & Mackenzie, A. P. Evidence for hydrodynamic electron flow in PdCoO2. Science 351, 1061–1064 (2016). The third of three papers that were published back to back reporting transport measurements suggesting electron hydrodynamics.
Levitov, L. & Falkovich, G. Electron viscosity, current vortices and negative nonlocal resistance in graphene. Nat. Phys. 12, 672–676 (2016).
Bandurin, D. A. et al. Fluidity onset in graphene. Nat. Commun. 9, 4533 (2018).
Gurzhi, R. Minimum of resistance in impurity-free conductors. Sov. Phys. JETP 44, 771 (1963).
Gurzhi, R. N. Hydrodynamic effects in solids at low temperature. Sov. Phys. Usp. 11, 255 (1968). Early review article by Gurzhi, who predicted electron hydrodynamics theoretically.
Kiselev, E. I. & Schmalian, J. Boundary conditions of viscous electron flow. Phys. Rev. B 99, 035430 (2019).
Moessner, R., Morales-Durán, N., Surówka, P. & Witkowski, P. Boundary-condition and geometry engineering in electronic hydrodynamics. Phys. Rev. B 100, 155115 (2019).
Aharon-Steinberg, A. et al. Direct observation of vortices in an electron fluid. Nature 607, 74–80 (2022). Direct observation of vortices in electron fluids in non-channel geometries.
Wolf, Y., Aharon-Steinberg, A., Yan, B. & Holder, T. Para-hydrodynamics from weak surface scattering in ultraclean thin flakes. Nat. Commun. 14, 2334 (2023).
Sulpizio, J. A. et al. Visualizing Poiseuille flow of hydrodynamic electrons. Nature 576, 75–79 (2019). Direct visualization of electron hydrodynamics using spatially resolved transport measurements.
Ku, M. J. H. et al. Imaging viscous flow of the Dirac fluid in graphene. Nature 583, 537–541 (2020).
Gooth, J. et al. Thermal and electrical signatures of a hydrodynamic electron fluid in tungsten diphosphide. Nat. Commun. 9, 4093 (2018).
Vool, U. et al. Imaging phonon-mediated hydrodynamic flow in WTe2. Nat. Phys. 17, 1216–1220 (2021). Observation of electron hydrodynamics in bulk high-carrier-density materials.
Varnavides, G., Jermyn, A. S., Anikeeva, P., Felser, C. & Narang, P. Electron hydrodynamics in anisotropic materials. Nat. Commun. 11, 4710 (2020). Group-theoretical classification of viscosity tensor components in anisotropic materials beyond graphene.
Varnavides, G., Wang, Y., Moll, P. J. W., Anikeeva, P. & Narang, P. Mesoscopic finite-size effects of unconventional electron transport in pdcoo2. Phys. Rev. Mater. 6, 045002 (2022).
Wang, Y. et al. Generalized design principles for hydrodynamic electron transport in anisotropic metals. Phys. Rev. Mater. 6, 083802 (2022).
Lucas, A. & Fong, K. C. Hydrodynamics of electrons in graphene. J. Phys. Condens. Matter 30, 053001 (2018).
Narozhny, B. N. Hydrodynamic approach to two-dimensional electron systems. Riv. Nuovo Climento 45, 661–736 (2022).
Fritz, L. & Scaffidi, T. Hydrodynamic electronic transport. Preprint at https://doi.org/10.48550/arXiv.2303.14205 (2023).
Kumar, N. et al. Extremely high magnetoresistance and conductivity in the type-II Weyl semimetals WP2 and MoP2. Nat. Commun. 8, 1642 (2017).
Kumar, N. et al. Extremely high conductivity observed in the triple point topological metal MoP. Nat. Commun. 10, 2475 (2019).
Gooth, J. et al. Experimental signatures of the mixed axial–gravitational anomaly in the Weyl semimetal NbP. Nature 547, 324–327 (2017).
Kumar, N., Guin, S. N., Manna, K., Shekhar, C. & Felser, C. Topological quantum materials from the viewpoint of chemistry. Chem. Rev. 121, 2780–2815 (2021).
Wang, J., Yox, P. & Kovnir, K. Flux growth of phosphide and arsenide crystals. Front. Chem. 8, 186 (2020).
Takatsu, H. et al. Roles of high-frequency optical phonons in the physical properties of the conductive delafossite PdCoO2. J. Phys. Soc. Jpn 76, 104701 (2007).
Bachmann, M. D. et al. Directional ballistic transport in the two-dimensional metal PdCoO2. Nat. Phys. 18, 819–824 (2022).
Han, H. J. et al. Topological metal MoP nanowire for interconnect. Adv. Mater. 35, 2208965 (2023).
Callaway, J. Model for lattice thermal conductivity at low temperatures. Phys. Rev. 113, 1046–1051 (1959).
Jaoui, A. et al. Departure from the Wiedemann–Franz law in WP2 driven by mismatch in T-square resistivity prefactors. npj Quant. Mater. 3, 64 (2018).
Jaoui, A., Fauqué, B. & Behnia, K. Thermal resistivity and hydrodynamics of the degenerate electron fluid in antimony. Nat. Commun. 12, 195 (2021).
Lee, S. et al. Anomalously low electronic thermal conductivity in metallic vanadium dioxide. Science 355, 371–374 (2017).
Torre, I., Tomadin, A., Geim, A. K. & Polini, M. Nonlocal transport and the hydrodynamic shear viscosity in graphene. Phys. Rev. B 92, 165433 (2015).
Krishna Kumar, R. et al. Superballistic flow of viscous electron fluid through graphene constrictions. Nat. Phys. 13, 1182–1185 (2017).
Levin, A. D., Gusev, G. M., Levinson, E. V., Kvon, Z. D. & Bakarov, A. K. Vorticity-induced negative nonlocal resistance in a viscous two-dimensional electron system. Phys. Rev. B 97, 245308 (2018).
Braem, B. A. et al. Scanning gate microscopy in a viscous electron fluid. Phys. Rev. B 98, 241304 (2018).
Berdyugin, A. I. et al. Measuring Hall viscosity of graphene’s electron fluid. Science 364, 162–165 (2019).
Kim, M. et al. Control of electron–electron interaction in graphene by proximity screening. Nat. Commun. 11, 2339 (2020).
Gusev, G. M., Jaroshevich, A. S., Levin, A. D., Kvon, Z. D. & Bakarov, A. K. Stokes flow around an obstacle in viscous two-dimensional electron liquid. Sci. Rep. 10, 7860 (2020).
Ginzburg, L. V. et al. Superballistic electron flow through a point contact in a Ga[Al]As heterostructure. Phys. Rev. Res. 3, 023033 (2021).
Gupta, A. et al. Hydrodynamic and ballistic transport over large length scales in GaAs/AlGaAs. Phys. Rev. Lett. 126, 076803 (2021).
Pellegrino, F. M. D., Torre, I., Geim, A. K. & Polini, M. Electron hydrodynamics dilemma: whirlpools or no whirlpools. Phys. Rev. B 94, 155414 (2016).
Jenkins, A. et al. Imaging the breakdown of ohmic transport in graphene. Phys. Rev. Lett. 129, 087701 (2022).
Lorentz, H. A. Le mouvement des électrons dans les métaux [French]. Arch. Néerl. 10, 336 (1905).
Ledwith, P., Guo, H., Shytov, A. & Levitov, L. Tomographic dynamics and scale-dependent viscosity in 2D electron systems. Phys. Rev. Lett. 123, 116601 (2019).
Ledwith, P. J., Guo, H. & Levitov, L. The hierarchy of excitation lifetimes in two-dimensional Fermi gases. Ann. Phys. 411, 167913 (2019).
Okulov, V. & Ustinov, V. Boundary condition for the distribution function of conduction electrons scattered by a metal surface. Zh. Eksp. Teor. Fiz. 67, 1176 (1974).
Holder, T. et al. Ballistic and hydrodynamic magnetotransport in narrow channels. Phys. Rev. B 100, 245305 (2019).
Epstein, J. M. & Mandadapu, K. K. Time-reversal symmetry breaking in two-dimensional nonequilibrium viscous fluids. Phys. Rev. E 101, 052614 (2020).
Nye, J. F. Physical Properties of Crystals: Their Representation by Tensors and Matrices (Clarendon, 1984).
Neumann, F. Vorlesungen über die Theorie der Elastizität der Festen Körper und des Lichtäthers [German] (B. G. Teubner, 1885).
Principi, A., Vignale, G., Carrega, M. & Polini, M. Bulk and shear viscosities of the two-dimensional electron liquid in a doped graphene sheet. Phys. Rev. B 93, 125410 (2016).
Cook, C. Q. & Lucas, A. Viscometry of electron fluids from symmetry. Phys. Rev. Lett. 127, 176603 (2021).
Link, J. M., Narozhny, B. N., Kiselev, E. I. & Schmalian, J. Out-of-bounds hydrodynamics in anisotropic Dirac fluids. Phys. Rev. Lett. 120, 196801 (2018).
Avron, J. E. Odd viscosity. J. Stat. Phys. 92, 543–557 (1998).
Delacrétaz, L. V. & Gromov, A. Transport signatures of the Hall viscosity. Phys. Rev. Lett. 119, 226602 (2017).
Scaffidi, T., Nandi, N., Schmidt, B., Mackenzie, A. P. & Moore, J. E. Hydrodynamic electron flow and Hall viscosity. Phys. Rev. Lett. 118, 226601 (2017).
Gusev, G. M., Levin, A. D., Levinson, E. V. & Bakarov, A. K. Viscous transport and Hall viscosity in a two-dimensional electron system. Phys. Rev. B 98, 161303 (2018).
Holder, T., Queiroz, R. & Stern, A. Unified description of the classical Hall viscosity. Phys. Rev. Lett. 123, 106801 (2019).
Rao, P. & Bradlyn, B. Hall viscosity in quantum systems with discrete symmetry: point group and lattice anisotropy. Phys. Rev. X 10, 021005 (2020).
Cook, C. Q. & Lucas, A. Electron hydrodynamics with a polygonal Fermi surface. Phys. Rev. B 99, 235148 (2019).
Varnavides, G., Jermyn, A. S., Anikeeva, P. & Narang, P. Probing carrier interactions using electron hydrodynamics. Preprint at https://arxiv.org/abs/2204.06004 (2022).
Coulter, J., Sundararaman, R. & Narang, P. Microscopic origins of hydrodynamic transport in the type-II Weyl semimetal WP2. Phys. Rev. B 98, 115130 (2018).
Allen, P. B. New method for solving Boltzmann’s equation for electrons in metals. Phys. Rev. B 17, 3725–3734 (1978).
Jermyn, A. S. et al. Transport of hot carriers in plasmonic nanostructures. Phys. Rev. Mater. 3, 075201 (2019).
Varnavides, G., Jermyn, A. S., Anikeeva, P. & Narang, P. Nonequilibrium phonon transport across nanoscale interfaces. Phys. Rev. B 100, 115402 (2019).
Kumar, C. et al. Imaging hydrodynamic electrons flowing without Landauer–Sharvin resistance. Nature 609, 276–281 (2022).
Levchenko, A. & Schmalian, J. Transport properties of strongly coupled electron–phonon liquids. Ann. Phys. 419, 168218 (2020).
Yang, H.-Y. et al. Evidence of a coupled electron–phonon liquid in NbGe2. Nat. Commun. 12, 5292 (2021).
Huang, X. & Lucas, A. Electron–phonon hydrodynamics. Phys. Rev. B 103, 155128 (2021).
Krebs, Z. J. et al. Imaging the breaking of electrostatic dams in graphene for ballistic and viscous fluids. Science 379, 671–676 (2023).
Mendoza, M., Herrmann, H. J. & Succi, S. Preturbulent regimes in graphene flow. Phys. Rev. Lett. 106, 156601 (2011).
Galitski, V., Kargarian, M. & Syzranov, S. Dynamo effect and turbulence in hydrodynamic Weyl metals. Phys. Rev. Lett. 121, 176603 (2018).
Di Sante, D. et al. Turbulent hydrodynamics in strongly correlated kagome metals. Nat. Commun. 11, 3997 (2020).
Morimoto, T., Zhong, S., Orenstein, J. & Moore, J. E. Semiclassical theory of nonlinear magneto-optical responses with applications to topological Dirac/Weyl semimetals. Phys. Rev. B 94, 245121 (2016).
Sano, R., Toshio, R. & Kawakami, N. Nonreciprocal electron hydrodynamics under magnetic fields: applications to nonreciprocal surface magnetoplasmons. Phys. Rev. B 104, L241106 (2021).
Acknowledgements
The authors thank A. Jermyn for discussions. G.V. acknowledges support from the Miller Institute for Basic Research in Science. A.Y. acknowledges support from the Army Research Office under grant no. W911NF-22-1-0248, the Gordon and Betty Moore Foundation through grant GBMF 9468, and the Quantum Science Center (QSC), a National Quantum Information Science Research Center of the US Department of Energy. P.N. acknowledges support from the QSC, a National Quantum Information Science Research Center of the US Department of Energy, the Office of Naval Research under grant no. 13672292 and the Gordon and Betty Moore Foundation through grant no. 8048.
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All authors contributed to literature review, manuscript writing, figure composition and article integration. G.V. and P.N. conceived the topical focus of this Review. P.N. supervised the project.
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Varnavides, G., Yacoby, A., Felser, C. et al. Charge transport and hydrodynamics in materials. Nat Rev Mater 8, 726–741 (2023). https://doi.org/10.1038/s41578-023-00597-3
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DOI: https://doi.org/10.1038/s41578-023-00597-3