Electron Hydrodynamics and Hall Viscosity Thomas Scaffidi UC

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Electron Hydrodynamics and Hall Viscosity Thomas Scaffidi UC Berkeley HYDRODYNAMIC MODELS FOR TRANSPORT IN

Electron Hydrodynamics and Hall Viscosity Thomas Scaffidi UC Berkeley HYDRODYNAMIC MODELS FOR TRANSPORT IN 2 D MATERIALS University of Minnesota [TS, Nandi, Schmidt, Mackenzie, and Moore, Phys. Rev. Lett. 118, 226601] [Holder, Queiroz, TS, Silberstein, Rozen, Sulpizio, Ella, Ilani, Stern, ar. Xiv: 1901. 08546] [J. A. Sulpizio 1†, L. Ella 1†, A. Rozen 1†, J. Birkbeck 2, 3, D. J. Perello 2, 3, D. Dutta 1, M. Ben. Shalom 2, 3, 4, T. Taniguchi 5, K. Watanabe 5, T. Holder 1, R. Queiroz 1, A. Stern 1, T. Scaffidi 6, A. K. Geim 2, 3, and S. Ilani, ar. Xiv: 1905. 11662 ]

Electric conduction versus water flow Metal Water • Resistance arises through external scattering due

Electric conduction versus water flow Metal Water • Resistance arises through external scattering due to the lattice (impurities, phonons, …) • Theory: Drude • Resistance arises through internal scattering (viscosity) • Theory: Hydrodynamics

Hydrodynamics • Universal description of fluids based on conserved quantities: momentum, energy, charge, …

Hydrodynamics • Universal description of fluids based on conserved quantities: momentum, energy, charge, … • Works at length/time scales much larger than the microscopic ones • The more interacting, the better! mean free path (internal scattering) Wavelength of perturbation

Why look for hydro in solids? Novel regime of transport: Access to “exotic” hydrodynamics:

Why look for hydro in solids? Novel regime of transport: Access to “exotic” hydrodynamics: Dirac fluid Integrable hydro in spin chains Available experiments: 2 DEGs Graphene Pd. Co. O 2 … Crossno et al, Science 351, 1058 (2016) 300 μm De Jong, Molenkamp (1995), … Bandurin et al, Science 351, 1055 -1058 (2016) Crossno et al, Science 351, 1058 (2016), … Moll et al, Science (2016) [Bertini et al] [Bulchandani et al]

Viscous fluid Microscopics Hydrodynamics Navier-Stokes

Viscous fluid Microscopics Hydrodynamics Navier-Stokes

Hydrodynamic flow of electrons possible if… Momentum-conserving mean free path (“internal scattering”): • e-e

Hydrodynamic flow of electrons possible if… Momentum-conserving mean free path (“internal scattering”): • e-e scattering Sample size W First proposed by Gurzhi in 1963. Many papers since then. . . Momentum-relaxing mean free path (“external scattering”): • Impurities • Phonons

Three regimes Ohmic Viscous Classical-Ballistic “Knudsen flow”

Three regimes Ohmic Viscous Classical-Ballistic “Knudsen flow”

Viscous fluid Microscopics Hydrodynamics Navier-Stokes

Viscous fluid Microscopics Hydrodynamics Navier-Stokes

Viscous electronic fluid Microscopics Hydrodynamics

Viscous electronic fluid Microscopics Hydrodynamics

Two regimes Viscous Non-local relation between current and electric field! m tes r e

Two regimes Viscous Non-local relation between current and electric field! m tes r e a If t min do If do term m ina te s Ohmic

Where did the convection term go? Today:

Where did the convection term go? Today:

Outline • Electron hydrodynamics – What and why? • Electron hydrodynamics in good metals

Outline • Electron hydrodynamics – What and why? • Electron hydrodynamics in good metals • Using magnetic fields to detect hydro effects • Conclusion and outlook

How to identify hydro effects? • Idea: Look at finite size corrections to transport

How to identify hydro effects? • Idea: Look at finite size corrections to transport coefficients • Problem: how to distinguish from ballistic effects? • Solution: use magnetic field W 300 μm

Navier-Stokes under magnetic field Add B field Lorentz force Hall viscosity

Navier-Stokes under magnetic field Add B field Lorentz force Hall viscosity

A tale of two viscosities Shear viscosity Probed by Hall viscosity Probed by

A tale of two viscosities Shear viscosity Probed by Hall viscosity Probed by

A tale of two viscosities Shear viscosity Probed by Hall viscosity Probed by

A tale of two viscosities Shear viscosity Probed by Hall viscosity Probed by

Hydrodynamic solution Result from Boltzmann theory for a charged Fermi liquid:

Hydrodynamic solution Result from Boltzmann theory for a charged Fermi liquid:

Navier-Stokes is not enough In theory: Navier-Stokes In practice: Kinetic theory : quasi-particle density

Navier-Stokes is not enough In theory: Navier-Stokes In practice: Kinetic theory : quasi-particle density

Boltzmann equation for quasi-particle density Non-uniform current density Lorentz force Momentumrelaxing scattering Collision integral:

Boltzmann equation for quasi-particle density Non-uniform current density Lorentz force Momentumrelaxing scattering Collision integral: simplest form which conserves momentum

Results from kinetic theory: magnetoresistance

Results from kinetic theory: magnetoresistance

A tale of two viscosities Shear viscosity Probed by Hall viscosity Probed by

A tale of two viscosities Shear viscosity Probed by Hall viscosity Probed by

Historical perspective on Hall effects Classical Hall Effect [Hall 1879] Quantum Hall Effect [von

Historical perspective on Hall effects Classical Hall Effect [Hall 1879] Quantum Hall Effect [von Klitzing 1980] Classical Viscous Hall Effect Quantum Viscous Hall Effect

Hall resistivity => Hall viscosity W Hall viscosity could be measured by looking at

Hall resistivity => Hall viscosity W Hall viscosity could be measured by looking at finite-size effects in Hall resistivity Nicer probe than , because bulk value is “universal”

Results of kinetic theory: Hall resistivity (normalized by bulk value)

Results of kinetic theory: Hall resistivity (normalized by bulk value)

Hall voltage 0 B

Hall voltage 0 B

More on this tomorrow: ar. Xiv: 1806. 01606

More on this tomorrow: ar. Xiv: 1806. 01606

Last part of the talk: local properties • So far we’ve only discussed quantities

Last part of the talk: local properties • So far we’ve only discussed quantities that are averaged over the cross section of the channel • What about their spatial dependence? Any smoking gun features of hydro? • Naïvely, yes: Ohmic Hydro

Last part of the talk: local properties • So far we’ve only discussed quantities

Last part of the talk: local properties • So far we’ve only discussed quantities that are averaged over the cross section of the channel • What about their spatial dependence? Any smoking gun features of hydro? • Naïvely, yes: But: Ohmic Hydro Ballistic

Magnetic fields are useful again: Ballistic Current density Hydro

Magnetic fields are useful again: Ballistic Current density Hydro

Magnetic fields are useful again: Ballistic Current density Hall electric field Hydro

Magnetic fields are useful again: Ballistic Current density Hall electric field Hydro

Magnetic fields are useful again: Ballistic Current density Hall electric field Hydro

Magnetic fields are useful again: Ballistic Current density Hall electric field Hydro

“Phase diagram” based on profile curvature Hall electric field Current

“Phase diagram” based on profile curvature Hall electric field Current

Measurements Single-electron transistor [J. A. Sulpizio 1†, L. Ella 1†, A. Rozen 1†, J.

Measurements Single-electron transistor [J. A. Sulpizio 1†, L. Ella 1†, A. Rozen 1†, J. Birkbeck 2, 3, D. J. Perello 2, 3, D. Dutta 1, M. Ben. Shalom 2, 3, 4, T. Taniguchi 5, K. Watanabe 5, T. Holder 1, R. Queiroz 1, A. Stern 1, T. Scaffidi 6, A. K. Geim 2, 3, and S. Ilani, ar. Xiv: 1905. 11662 ]

Acknowledgements • UC Berkeley: Joel Moore • MPI Dresden: Nabhanila Nandi, Burkhard Schmidt, Philip

Acknowledgements • UC Berkeley: Joel Moore • MPI Dresden: Nabhanila Nandi, Burkhard Schmidt, Philip Moll, Pallavi Kushwaha, Andrew Mackenzie • Weizmann institute: Tobias Holder, Raquel Queiroz, Navot Silberstein, Asaf Rozen, Joseph A. Sulpizio, Lior Ella, Shahal Ilani, Ady Stern Thanks for your attention!