fluid mechanics? Why solid people need Falkovich WIS April 14, 2017 UVA

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1 Why solid people need fluid mechanics? Falkovich WIS 1. L Levitov & G Falkovich, Electron viscosity, current vortices and negative nonlocal resistance in graphene. Nature Physics 12 : (2016) 2. G Falkovich & L Levitov, Linking spatial distributions of potential and current in viscous electronics. Phys Rev. Lett. 2017, Arxiv: : 3. H Guo, E Ilseven, G Falkovich, L Levitov, Higherthan-ballistic conduction of viscous electron flows. PNAS 2017, Arxiv: , April 14, 2017 UVA

2 Electrons in a crystal la.ce Ohm s law: j=nev, v= F τ p /m = e φ τ p /m, j= n e 2 τ p /m φ

3 Is hydrodynamics ever relevant in metals? In one- component fluid or gas a hydrodynamic approach works because one has local conservahon of energy and momentum. Macroscopic hydrodynamic equahons describe propagahon of conserved quanhhes in space. Electron fluid in a solid can exchange energy and momentum with the laoce. Hydrodynamics not relevant? If the disorder scaqering Hme τ p exceeds the electron- electron scaqering Hme τ ee =l/ v F, then electrons behave as viscous liquid. High- mobility electron systems (GaAs 2DES, graphene). Non- Fermi liquids, high- Tc superconductors, strange metals. Fritz, L., Schmalian, J., Müller, M. & Sachdev, S. Quantum critical transport in clean graphene. Phys. Rev. B 78, (2008). Kashuba, A. B. Conductivity of defectless graphene. Phys. Rev. B 78, (2008).

4 KinemaHc viscosity - diffusivity of momentum ν v F 2 γ ee 1 v F l ee ν vl typical velocity mean free path water ν=0.01 cm 2 sec 1 air ν=0.15 cm 2 sec 1 electrons in graphene ν=1000 cm 2 sec 1

5 Dimensionless coupling constant In graphene (or any other Dirac material), the strength of electron-electron interactions is controlled by the dimensionless parameter, called fine structure constant (because of its analogy with the QED fine structure constant): This dimensionless number is: 1) not small, i.e. it is of order unity 2) not gate tunable (Fermi wave number drops out) 3) sensitive to dielectric environment (the epsilon factor ) V. N. Kotov et al., Rev. Mod. Phys. 84, 1067 (2012)

6 The main problem: WHAT IS THE MOST FUNDAMENTAL MANIFESTATION OF STRONGLY INTERACTING ELECTRON FLOW?

7 Hydrodynamic descriphon of transport Navier-Stokes equation t v+(v )v ν 2 v= P/mn

8 Life at low Reynolds numbers Re=vL/ν<<1 η v= P l Scallop theorem (Purcell 1977) to achieve pumping or propulsion at low Reynolds number one must deform in a way that is not invariant under time-reversal. l Berry phase & non-abelian gauge theory: Wilczek, Shapere (1989), Geometry of self-propulsion at low Re. Avron, Kenneth, Gat (2004). Swimming (pumping) consists in periodically changing shape to move relahve to the fluid

9 Life and death at low Reynolds number

10 Microfluidic flows and Ohmic- viscous currents Consider plane viscous flow between two plates separated by h. v(x,y,z)=6z(h z)v(x,y) h 1/h 0 h dz(η v P)=η v 12η/ h 2 v P=0 η v ρ (ne) 2 v ne φ=0 Ohmic resistance

11 Consider incompressible flow Purely Ohmic case Cauchy-Riemann conditions ψ= φ=0

12 Purely viscous case Current and potenhal in viscous electronics ω= 2 ψ= φ=0

13 Field expulsion from viscous current flow ω+iφ analytic function v 1/r v=0 ω=0, φ=const φ Conformal map metal r

14 The simplest non- trivial viscous current flow (non- metallic boundary)

15 NegaFve voltage IS THE MOST FUNDAMENTAL MANIFESTATION OF STRONGLY INTERACTING ELECTRON FLOW?

16 DC viscous flow Purely viscous case Mixed Ohmic-viscous case

17 DC viscous flow in a strip

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19 Comparing no- slip and no- stress cases

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21 Measured viscosity

22 Wiedemann- Franz law κ WF σ elec T = L 0 = π 2 3! # " k B e $ & % 2 Independent of density, mass, mean- free- path, scaqering Hme True in Drude model and Fermi liquid Kumar, Prasad, Pohl, J. of Materials Sci. 28, 4261 (1993) kc.fong@bbn.com 22

23 Experimental signature of Dirac fluid κ DL σ elec T > L 0 kc.fong@bbn.com 23

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25 How interachon between carriers of the same sign changes resistance? Does the resistance in the viscous regime always exceed that in the Ohmic regime? One must dishnguish between Ohmic resistance due to scaqering on a) phonons, b) impurihes or boundaries.

26 Exceeding ballishc conductance in viscous flows

27 Exceeding ballishc conductance in viscous flows viscous resistance

28 Exceeding ballishc conductance in viscous flows it is easier for interacfng electrons to go through the eye of a needle

29 Knudsen- Poiseuille transifon w Consider gas flowing through a tube with rough walls and define resistance as force divided by momentum: R 1/τ P/mnU, analog R V/I Knudsen regime w<l, τ~w/v T. Poiseuille regime w>l, τ~ w 2 /ν~ w 2 /v T l= w/v T w/l

30 Knudsen- Poiseuille transifon w Consider gas flowing through a tube with rough walls and define resistance as force divided by momentum: R 1/τ P/mnU, analog R V/I Knudsen regime w<l, τ~w/v T. Poiseuille regime w>l, τ~ w 2 /ν~ w 2 /v T l= w/v T w/l

31 Stokes Paradox, Back ReflecHons and InteracHon- Enhanced ConducHon H Guo, E Ilseven, G Falkovich and L Levitov, PNAS 2017, Arxiv:

32 Experiment R.K. Kumar, D.A. Bandurin, F.M.D. Pellegrino, Y Cao, A Principi, H Guo, G Auton, M Ben Shalom, L.A. Ponomarenko, G. Falkovich, I.V. Grigorieva, L.S. Levitov, M. Polini, and A.K. Geim, submitted to Nature Physics

33 R.K. Kumar, D.A. Bandurin, F.M.D. Pellegrino, Y. Cao, A. Principi, H. Guo, G.H. Auton, M. Ben Shalom, L.A. Ponomarenko, G. Falkovich, I.V. Grigorieva, L.S. Levitov, M. Polini, and A.K. Geim, submitted to Nature Physics

34 R.K. Kumar, D.A. Bandurin, F.M.D. Pellegrino, Y. Cao, A. Principi, H. Guo, G.H. Auton, M. Ben Shalom, L.A. Ponom G. Falkovich, I.V. Grigorieva, L.S. Levitov, M. Polini, and A.K. Geim, submitted to Nature Physics contact resistance (diffusive contribution) Sharvin (ballistic) e-e interaction contribution Guo,Ilseven,Falkovich,Levitov, PNAS March 2017 arxiv: ,

35 So where was the cheahng?

36 So where was the cheahng? If there is no cheafng, then it is not theorefcal physics but mathemafcs.

37 Moral: viscosity makes current- voltage relahon nonlocal opening new possibilihes; Strongly interachng electrons can flow like a laminar viscous flow and demonstrate negahve resistance, super- ballishc conductance and other wonders. Future viscous electronics needs fluid mechanics