Fluid dynamics of electrons in graphene. Andrew Lucas

Transcription

1 Fluid dynamics of electrons in graphene Andrew Lucas Stanford Physics Condensed Matter Seminar, Princeton October 17, 2016

2 Collaborators 2 Subir Sachdev Harvard Physics & Perimeter Institute Philip Kim Harvard Physics/SEAS Kin Chung Fong Raytheon BBN Jesse Crossno Harvard Physics/SEAS

3 Hydrodynamics 3 The Hydrodynamic Limit v 1

4 Hydrodynamics 3 The Hydrodynamic Limit v 1 complicated microscopic dynamics: t v 1 v 1 t ee +

5 Hydrodynamics 3 The Hydrodynamic Limit v 1 complicated microscopic dynamics: t v 1 v 1 t ee + hydrodynamic (t t ee ): N center t = flux out sides

6 Hydrodynamics 3 The Hydrodynamic Limit v 1 complicated microscopic dynamics: t v 1 v 1 t ee + hydrodynamic (t t ee ): N center = flux out sides t slow modes: locally conserved

7 Hydrodynamics 3 The Hydrodynamic Limit v 1 complicated microscopic dynamics: t v 1 v 1 t ee + hydrodynamic (t t ee ): N center = flux out sides t slow modes: locally conserved classical, universal phenomena

8 Hydrodynamics 4 Hydrodynamic Equations classical equations of motion: conservation laws t ρ + J = 0.

9 Hydrodynamics 4 Hydrodynamic Equations classical equations of motion: conservation laws t ρ + J = 0. gradient expansion (perturbative expansion in ): J = D(ρ) ρ

10 Hydrodynamics 4 Hydrodynamic Equations classical equations of motion: conservation laws t ρ + J = 0. gradient expansion (perturbative expansion in ): J = D(ρ) ρ diffusion (for lone conserved charge) t ρ = (D ρ)

11 Hydrodynamics 4 Hydrodynamic Equations classical equations of motion: conservation laws t ρ + J = 0. gradient expansion (perturbative expansion in ): J = D(ρ) ρ diffusion (for lone conserved charge) t ρ = (D ρ) local second law of thermodynamics: D 0, (dissipation only)

12 Hydrodynamics 5 Liquids and Gases gases: l mfp l l mfp l σ

13 Hydrodynamics 5 Liquids and Gases gases: l mfp l l mfp l σ reliable perturbative calculations

14 Hydrodynamics 5 Liquids and Gases gases: liquids: l mfp l mfp l l l mfp l σ reliable perturbative calculations l mfp l σ

15 Hydrodynamics 5 Liquids and Gases gases: liquids: l mfp l mfp l l l mfp l σ reliable perturbative calculations l mfp l σ breakdown of perturbation theory

16 Hydrodynamics 5 Liquids and Gases gases: liquids: l mfp l mfp l l l mfp l σ reliable perturbative calculations l mfp l σ breakdown of perturbation theory same hydrodynamic equations!

17 Hydrodynamics 6 Quantum Hydrodynamics quark-gluon plasma:

18 Hydrodynamics 6 Quantum Hydrodynamics quark-gluon plasma: cold atoms:

19 Hydrodynamics 6 Quantum Hydrodynamics quark-gluon plasma: cold atoms: what about correlated electrons (in metals)?

20 Fermi Liquids 7...are Analogous to a Classical Gas describes electrons in ordinary metals Fermi sea k B T µ

21 Fermi Liquids 7...are Analogous to a Classical Gas Fermi sea k B T µ describes electrons in ordinary metals interaction time constrained by near-fermi surface phase space: t ee µ (k B T ) 2

22 Fermi Liquids 7...are Analogous to a Classical Gas Fermi sea k B T µ describes electrons in ordinary metals interaction time constrained by near-fermi surface phase space: t ee µ (k B T ) 2 long-lived quasiparticles; (quantum) kinetic theory

23 Fermi Liquids 8 Metals are Disordered t ee t imp t ee t imp ultraclean metal + r J =0 ordinary metal (iron etc.) t ee t imp s =mess

24 Graphene 9 Crash Course in Graphene 1nm

25 Graphene 10 Crash Course in Graphene a = ~v F k + V int = e r T Dirac fluid hole FL electron FL 0 n

26 Graphene 10 Crash Course in Graphene a = ~v F k + V int = e r T Dirac fluid hole FL electron FL 0 n marginally irrelevant 1/r Coulomb interactions: α 0 α eff = 1 + (α 0 /4) log((10 5 K)/T ), α 0 1 c v F ɛ r

27 Graphene 10 Crash Course in Graphene a = ~v F k + V int = e r T Dirac fluid hole FL electron FL 0 n marginally irrelevant 1/r Coulomb interactions: α eff = α (α 0 /4) log((10 5 K)/T ), α thermalization length scale: l mfp max( µ, T ) T 70 K T 100 nm e.g. [Sheehy, Schmalian (2007); Müller, Fritz, Sachdev (2008)] c 0.5. v F ɛ r

28 Graphene 11 Graphene: an Ideal Experimental Platform gating farther than l ee possible!

29 Graphene 11 Graphene: an Ideal Experimental Platform gating farther than l ee possible! fabricating ultra pure monolayer graphene: [Dean et al (2010)] hbn monolayer graphene hbn

30 Figure 4 Spatial maps of the density of states of graphene on hbn and SiO2. a, Topography of graphene on hbn. b, Tip voltage at the Dirac point as a function of position for graphene on hbn. c, Tip voltage at the Dirac Graphene 11 Graphene: an Ideal Experimental Platform gating farther than l ee possible! fabricating ultra pure monolayer graphene: [Dean et al (2010)] b 100 nm hbn µ 30 K hbn monolayer graphene hbn c E d (mev) weak disorder: charge puddles [Xue et al (2011)] SiO 2 µ 300 K

31 Fermi Liquid Hydrodynamics 12 The Fermi Liquid Dirac fluid T hole FL electron FL 0 n

32 Fermi Liquid Hydrodynamics 12 The Fermi Liquid Dirac fluid T hole FL electron FL 0 n at high density, charge puddle disorder and interactions are weak: fluid dynamics of a gas of quasiparticles

33 Fermi Liquid Hydrodynamics 13 Linearized Hydrodynamics of a Disordered Fermi Liquid neglect thermal effects:

34 Fermi Liquid Hydrodynamics 13 Linearized Hydrodynamics of a Disordered Fermi Liquid neglect thermal effects: conservation of charge: (nv) n v = 0.

35 Fermi Liquid Hydrodynamics 13 Linearized Hydrodynamics of a Disordered Fermi Liquid neglect thermal effects: conservation of charge: (nv) n v = 0. Navier-Stokes equation: P η 2 v = n µ η 2 v Γ v

36 Fermi Liquid Hydrodynamics 13 Linearized Hydrodynamics of a Disordered Fermi Liquid neglect thermal effects: conservation of charge: (nv) n v = 0. Navier-Stokes equation: P η 2 v = n µ η 2 v Γ v Γ : rate of momentum relaxation (phonons/impurities)

37 Fermi Liquid Hydrodynamics 13 Linearized Hydrodynamics of a Disordered Fermi Liquid neglect thermal effects: conservation of charge: (nv) n v = 0. Navier-Stokes equation: P η 2 v = n µ η 2 v Γ v Γ : rate of momentum relaxation (phonons/impurities) Ohmic diffusion: η l Γ creep flow : η l Γ

38 Fermi Liquid Hydrodynamics 14 Flow Through Thin Opening [Levitov, Falkovich (2016); Torre, Tomadin, Geim, Polini (2015)]

39 Fermi Liquid Hydrodynamics 15 Experimental Evidence experimental geometry: [Bandurin et al, (2016)]

40 Fermi Liquid Hydrodynamics 15 Experimental Evidence experimental geometry: [Bandurin et al, (2016)]

41 Fermi Liquid Hydrodynamics 15 Experimental Evidence experimental geometry: [Bandurin et al, (2016)] but no signal when n = 0! (will explain shortly)

42 Fermi Liquid Hydrodynamics 16 Viscosity of the Fermi Liquid on general grounds, we expect η ɛ l ee v F µ4 T 2

43 Fermi Liquid Hydrodynamics 16 Viscosity of the Fermi Liquid on general grounds, we expect η ɛ l ee v F µ4 T 2 η is a qualitative measure of l ee, interaction strength: large l ee : fast momentum di usion small l ee : slow momentum di usion v v

44 Fermi Liquid Hydrodynamics 16 Viscosity of the Fermi Liquid on general grounds, we expect η ɛ l ee v F µ4 T 2 η is a qualitative measure of l ee, interaction strength: large l ee : fast momentum di usion small l ee : slow momentum di usion experiment suggests v η m2 0.1 mn s (2 orders of magnitude larger than water) v

45 Dirac Fluid Hydrodynamics 17 The Dirac Fluid Dirac fluid T hole FL electron FL 0 n

46 Dirac Fluid Hydrodynamics 17 The Dirac Fluid Dirac fluid T hole FL electron FL 0 n need a sample with µ avg = 0, T > µ dis

47 Dirac Fluid Hydrodynamics 17 The Dirac Fluid Dirac fluid T hole FL electron FL 0 n need a sample with µ avg = 0, T > µ dis fluid dynamics of relativistic plasma

48 Dirac Fluid Hydrodynamics 17 The Dirac Fluid Dirac fluid T hole FL electron FL 0 n need a sample with µ avg = 0, T > µ dis fluid dynamics of relativistic plasma very different from ordinary plasmas, e.g. astrophysics (not separate fluid for positive/negative charges)

49 Dirac Fluid Hydrodynamics 18 Linearized Hydrodynamics of the Disordered Dirac Fluid conservation of charge: ( (nv + σ q E µ + µ )) 0 T = 0. T 0

50 Dirac Fluid Hydrodynamics 18 Linearized Hydrodynamics of the Disordered Dirac Fluid conservation of charge: ( (nv + σ q E µ + µ )) 0 T = 0. T 0 conservation of heat: ( (T 0 sv µ 0 σ q E µ + µ )) 0 T = 0. T 0

51 Dirac Fluid Hydrodynamics 18 Linearized Hydrodynamics of the Disordered Dirac Fluid conservation of charge: ( (nv + σ q E µ + µ )) 0 T = 0. T 0 conservation of heat: ( (T 0 sv µ 0 σ q E µ + µ )) 0 T = 0. T 0 Navier-Stokes equation n( µ E) + s T = (η( v + v T )) (η v)

52 Dirac Fluid Hydrodynamics 18 Linearized Hydrodynamics of the Disordered Dirac Fluid conservation of charge: ( (nv + σ q E µ + µ )) 0 T = 0. T 0 conservation of heat: ( (T 0 sv µ 0 σ q E µ + µ )) 0 T = 0. T 0 Navier-Stokes equation n( µ E) + s T = (η( v + v T )) (η v) note: Coulomb interactions just re-define µ (at ω = 0) [Lucas (2015); [Lucas, Crossno, Fong, Kim, Sachdev (2016)]

53 Dirac Fluid Hydrodynamics 19 Fermi Liquid Transport: Wiedemann-Franz Law thermal conductivity κ; electrical conductivity σ: Q J=0 κ T, J T =0 = σe.

54 Dirac Fluid Hydrodynamics 19 Fermi Liquid Transport: Wiedemann-Franz Law thermal conductivity κ; electrical conductivity σ: Q J=0 κ T, J T =0 = σe. Wiedemann-Franz law in a Fermi liquid: L κ σt π2 kb 2 3e W Ω K 2. [Kumar, Prasad, Pohl (1993)]

55 Dirac Fluid Hydrodynamics 20 Thermal and Electrical Conductivity at Charge Neutrality rt E

56 Dirac Fluid Hydrodynamics 20 Thermal and Electrical Conductivity at Charge Neutrality rt E in a clean charge neutral metal boost to a moving reference frame; finite heat current at T > 0; hence κ =

57 Dirac Fluid Hydrodynamics 20 Thermal and Electrical Conductivity at Charge Neutrality rt E in a clean charge neutral metal boost to a moving reference frame; finite heat current at T > 0; hence κ = at charge neutrality, σ = σ q finite!

58 Dirac Fluid Hydrodynamics 21 Momentum Relaxation Time Approximation momentum conservation: ne s T = ɛ + P τ v

59 Dirac Fluid Hydrodynamics 21 Momentum Relaxation Time Approximation momentum conservation: ne s T = ɛ + P τ use constitutive relations J = nv + σ q ( E µ T T ), Q = (ɛ + P )v µj v

60 Dirac Fluid Hydrodynamics 21 Momentum Relaxation Time Approximation momentum conservation: ne s T = ɛ + P τ use constitutive relations J = nv + σ q ( E µ T T ), Q = (ɛ + P )v µj v transport coefficients: [Hartnoll, Kovtun, Müller, Sachdev (2007)] σ = σ q + n2 τ ɛ + P, κ = ɛ + P T τ σ q σ(n).

61 Dirac Fluid Hydrodynamics 21 Momentum Relaxation Time Approximation momentum conservation: ne s T = ɛ + P τ use constitutive relations J = nv + σ q ( E µ T T ), Q = (ɛ + P )v µj v transport coefficients: [Hartnoll, Kovtun, Müller, Sachdev (2007)] σ = σ q + n2 τ ɛ + P, κ = ɛ + P T τ σ q σ(n). generalization of Drude peak: transport dominated by slow momentum relaxation

62 Dirac Fluid Hydrodynamics 22 Wiedemann-Franz Law Violations in Experiment Tbath (K) phonon-limited disorder-limited n (10 9 cm -2 ) L / L0 [Crossno et al (2016)]

63 Dirac Fluid Hydrodynamics 23 Non-Perturbative Approach s(x) > 0 l ee non-perturbative hydrodynamic transport: disorder on scale ξ l ee n(x) > 0 n(x) < 0 n x

64 Dirac Fluid Hydrodynamics 23 Non-Perturbative Approach non-perturbative hydrodynamic transport: disorder on scale ξ l ee l ee n(x) > 0 s(x) > 0 n(x) < 0 n x static fluid in an inhomogeneous chemical potential µ 0 (x)

65 Dirac Fluid Hydrodynamics 23 Non-Perturbative Approach non-perturbative hydrodynamic transport: disorder on scale ξ l ee l ee n(x) > 0 s(x) > 0 n(x) < 0 n x static fluid in an inhomogeneous chemical potential µ 0 (x) transport from linearized hydrodynamic equations

66 Dirac Fluid Hydrodynamics 23 Non-Perturbative Approach non-perturbative hydrodynamic transport: disorder on scale ξ l ee l ee n(x) > 0 s(x) > 0 n(x) < 0 n x static fluid in an inhomogeneous chemical potential µ 0 (x) transport from linearized hydrodynamic equations charge puddle disorder suggests: 1 τ u2 ( ) n 2 ( µ 1 σ q (ɛ + P ) + 4ηµ 2 ξ 2 (ɛ + P ) 3 [Lucas (2015); [Lucas, Crossno, Fong, Kim, Sachdev (2016)] similar ideas in Fermi liquid: [Andreev, Kivelson, Spivak (2011)] ).

67 Dirac Fluid Hydrodynamics 24 Comparing Theory to Experiment (k 1 ) hole FL Dirac fluid elec. FL apple (nw/k) hole FL Dirac fluid elec. FL (k 1 ) n (µm 2 ) puddle FLs Dirac fluid 10 Figure 1: testing phonons 8 apple (nw/k) n (µm 2 ) puddle FLs Dirac fluid phonons T (K) T (K) Figure 1: testing [Crossno et al, (2016); Lucas, Crossno, Fong, Kim, Sachdev (2016)]

68 Dirac Fluid Hydrodynamics 25 Viscosity of Dirac Fluid? strongly coupled fluid: [Kovtun, Son, Starinets, (2005)] η s k B 1 4π.

69 Dirac Fluid Hydrodynamics 25 Viscosity of Dirac Fluid? strongly coupled fluid: [Kovtun, Son, Starinets, (2005)] η s k B 1 4π. in graphene, kinetic theory gives [Müller, Schmalian, Fritz (2009)] η s 0.1 ( ) 1 k B αeff 2 + O αeff 2 log α. eff

70 Dirac Fluid Hydrodynamics 25 Viscosity of Dirac Fluid? strongly coupled fluid: [Kovtun, Son, Starinets, (2005)] η s k B 1 4π. in graphene, kinetic theory gives [Müller, Schmalian, Fritz (2009)] η s 0.1 ( ) 1 k B αeff 2 + O αeff 2 log α. eff transport data suggests [Lucas, Crossno, Fong, Kim, Sachdev (2016)] 2 k B η s 10 k B.

71 Dirac Fluid Hydrodynamics 26 Sound Waves and Resonances observe resonances of electronic sound waves? in clean sample of size L: [Lucas (2016)] η s k B 1 10n 2 k B T L v F (if n resonances observed)

72 Dirac Fluid Hydrodynamics 26 Sound Waves and Resonances observe resonances of electronic sound waves? in clean sample of size L: [Lucas (2016)] η s k B 1 10n 2 k B T L v F (if n resonances observed) electronic sound resonances at ω 30 GHz (challenging)

73 Dirac Fluid Hydrodynamics 26 Sound Waves and Resonances observe resonances of electronic sound waves? in clean sample of size L: [Lucas (2016)] η s k B 1 10n 2 k B T L v F (if n resonances observed) electronic sound resonances at ω 30 GHz (challenging) momentum relaxation time: J γ = 0.01 γ = 0.02 γ = 0.04 γ = 0.1 γ = ω

74 Dirac Fluid Hydrodynamics 26 Sound Waves and Resonances J observe resonances of electronic sound waves? in clean sample of size L: [Lucas (2016)] η s k B 1 10n 2 k B T L v F (if n resonances observed) electronic sound resonances at ω 30 GHz (challenging) momentum relaxation time: J γ = 0.01 γ = 0.02 γ = 0.04 γ = 0.1 γ = 0 J charge puddles: ! ω !

75 Dirac Fluid Hydrodynamics 26 Sound Waves and Resonances J observe resonances of electronic sound waves? in clean sample of size L: [Lucas (2016)] η s k B 1 10n 2 k B T L v F (if n resonances observed) electronic sound resonances at ω 30 GHz (challenging) momentum relaxation time: J γ = 0.01 γ = 0.02 γ = 0.04 γ = 0.1 γ = ! ω J interplay of diffusion and (classically) localized waves 10 charge puddles: ! momentum relaxation time approx. fails!

76 Outlook 27 emerging field of electronic hydrodynamics

77 Outlook 27 emerging field of electronic hydrodynamics direct measurement of viscosity?

78 Outlook 27 emerging field of electronic hydrodynamics direct measurement of viscosity? which phenomena uniquely hydro? (not ballistic or Ohmic)

79 Outlook 27 emerging field of electronic hydrodynamics direct measurement of viscosity? which phenomena uniquely hydro? (not ballistic or Ohmic) other materials?

80 Outlook 27 emerging field of electronic hydrodynamics direct measurement of viscosity? which phenomena uniquely hydro? (not ballistic or Ohmic) other materials? practical applications? good conductors/thermoelectrics?