Hydrodynamics in the Dirac fluid in graphene. Andrew Lucas

Transcription

1 Hydrodynamics in the Dirac fluid in graphene Andrew Lucas Stanford Physics Fluid flows from graphene to planet atmospheres; Simons Center for Geometry and Physics March 20, 2017

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

3 Hydrodynamics 3 Interacting Quantum Systems quark-gluon plasma:

4 Hydrodynamics 3 Interacting Quantum Systems quark-gluon plasma: cold atoms:

5 Hydrodynamics 3 Interacting Quantum Systems quark-gluon plasma: cold atoms: electrons in solids:

6 Hydrodynamics 4 The Dirac Fluid graphene:

7 Hydrodynamics 4 The Dirac Fluid graphene: µ Fermi liquid

8 Hydrodynamics 4 The Dirac Fluid graphene: µ µ Fermi liquid Dirac fluid

9 Hydrodynamics 5 Collisionless-to-Hydrodynamic Crossover collisionless:!t ee 1!t ee 1 collisionless @t + r J =0

10 Hydrodynamics 5 Collisionless-to-Hydrodynamic Crossover collisionless: hydrodynamic:!t ee 1!t ee 1!t ee 1!t ee 1 collisionless collisionless hydrodynamic @t + =0+ r J =0

11 Hydrodynamics 6 Effective Field Theory: Hydrodynamics postulate: as t, all slow dynamics associated with local conservation laws

12 Hydrodynamics 6 Effective Field Theory: Hydrodynamics postulate: as t, all slow dynamics associated with local conservation laws energy, up to inelastic (substrate) phonon scattering

13 Hydrodynamics 6 Effective Field Theory: Hydrodynamics postulate: as t, all slow dynamics associated with local conservation laws energy, up to inelastic (substrate) phonon scattering momentum, up to impurities, umklapp, (substrate) phonons

14 Hydrodynamics 6 Effective Field Theory: Hydrodynamics postulate: as t, all slow dynamics associated with local conservation laws energy, up to inelastic (substrate) phonon scattering momentum, up to impurities, umklapp, (substrate) phonons charge, (substrate is insulating): Q = N electron N hole.

15 Hydrodynamics 6 Effective Field Theory: Hydrodynamics postulate: as t, all slow dynamics associated with local conservation laws energy, up to inelastic (substrate) phonon scattering momentum, up to impurities, umklapp, (substrate) phonons charge, (substrate is insulating): Q = N electron N hole. contrast with canonical (astrophysics) plasma: electrons, ions are often separate fluids

16 Hydrodynamics 7 The Relativistic Gradient Expansion hydrodynamics: classical effective theory

17 Hydrodynamics 7 The Relativistic Gradient Expansion hydrodynamics: classical effective theory equations are conservation laws: for relativistic fluid (to O(v/v F )): energy and momentum: µ T µν = 0: ( ) T µν ɛ (ɛ + P )v = i (ɛ + P )v i P δ ij η ( i v j + j v i δ ij k v k ) T µν must be symmetric, unlike in Galilean invariant fluid

18 Hydrodynamics 7 The Relativistic Gradient Expansion hydrodynamics: classical effective theory equations are conservation laws: for relativistic fluid (to O(v/v F )): energy and momentum: µ T µν = 0: ( ) T µν ɛ (ɛ + P )v = i (ɛ + P )v i P δ ij η ( i v j + j v i δ ij k v k ) T µν must be symmetric, unlike in Galilean invariant fluid charge: µ J µ = 0: ( J µ = n nv i σ q T i µ T ).

19 Hydrodynamics 7 The Relativistic Gradient Expansion hydrodynamics: classical effective theory equations are conservation laws: for relativistic fluid (to O(v/v F )): energy and momentum: µ T µν = 0: ( ) T µν ɛ (ɛ + P )v = i (ɛ + P )v i P δ ij η ( i v j + j v i δ ij k v k ) T µν must be symmetric, unlike in Galilean invariant fluid charge: µ J µ = 0: ( J µ = n nv i σ q T i µ T charge current momentum density ).

20 Hydrodynamics 8 Contrast with Fermi Liquid charge transport in Fermi liquid: i (nv i ) 0, }{{} i P = η j j v i. J

21 Hydrodynamics 8 Contrast with Fermi Liquid charge transport in Fermi liquid: i (nv i ) 0, }{{} i P = η j j v i. J diffusion at the neutrality point: 0 = i (σ q (E i i µ)). }{{} J

22 Hydrodynamics 8 Contrast with Fermi Liquid charge transport in Fermi liquid: i (nv i ) 0, }{{} i P = η j j v i. J diffusion at the neutrality point: 0 = i (σ q (E i i µ)). }{{} J hydrodynamic signatures for experiment will be qualitatively different

23 Microscopics 9 Dirac Fluid RG S = d 2 xdt i ψ ( ) γ t tψ + v Fγ i iψ dtd 2 xd 2 x α x x ψ (x)ψ(x)ψ(x ) ψ(x ).

24 Microscopics 9 Dirac Fluid RG S = d 2 xdt i ψ ( ) γ t tψ + v Fγ i iψ dtd 2 xd 2 x α x x ψ (x)ψ(x)ψ(x ) ψ(x ). strong coupling? α e2 ɛ v F α QED ɛ 0 ɛ c v F 2.5 ɛ 0 ɛ.

25 Microscopics 9 Dirac Fluid RG S = d 2 xdt i ψ ( ) γ t tψ + v Fγ i iψ dtd 2 xd 2 x α x x ψ (x)ψ(x)ψ(x ) ψ(x ). strong coupling? α e2 ɛ v F α QED ɛ 0 ɛ c v F 2.5 ɛ 0 ɛ. RG flow: α(t ) α α 0 4 log Λ T, v F (T ) = v F0 α 0 α(t ).

26 Microscopics 9 Dirac Fluid RG S = d 2 xdt i ψ ( ) γ t tψ + v Fγ i iψ dtd 2 xd 2 x α x x ψ (x)ψ(x)ψ(x ) ψ(x ). strong coupling? α e2 ɛ v F α QED ɛ 0 ɛ c v F 2.5 ɛ 0 ɛ. RG flow: α(t ) α α 0 4 log Λ T, v F (T ) = v F0 α 0 α(t ). Λ 10 5 K, T 10 2 K, = α 0.3.

27 Microscopics 10 Kinetic Theory at (exponentially) low T, α(t ) 0: l ee v F α 2 k B T 100 nm (T = 100 K)

28 Microscopics 10 Kinetic Theory at (exponentially) low T, α(t ) 0: l ee v F α 2 k B T 100 nm (T = 100 K) small dissipative coefficients: σ q 0.76e2 α 2, η 0.45(k BT ) 2 v 2 F0 α2 0 [Müller et al; PRB, ], [Müller et al; PRL, ]

29 Microscopics 10 Kinetic Theory at (exponentially) low T, α(t ) 0: l ee v F α 2 k B T 100 nm (T = 100 K) small dissipative coefficients: σ q 0.76e2 α 2, η 0.45(k BT ) 2 v 2 F0 α2 0 [Müller et al; PRB, ], [Müller et al; PRL, ] at O(α 2 ): N electron, N hole separately conserved. (broken at higher orders in α)

30 Disorder 11 Metals are Disordered t ee t imp e?) ordinary metal (iron etc.) t ee s t imp =mess

31 Disorder 11 Metals are Disordered t ee t imp t ee t imp e?) ordinary metal (iron etc.) ultraclean metal (GaAs, graphene?) t ee t imp r J =0 t ee

32 Disorder 12 Charge Puddles in Graphene fabricating ultra pure monolayer graphene: [Dean et al, (2010)] hbn monolayer graphene hbn

33 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 Disorder 12 Charge Puddles in Graphene fabricating ultra pure monolayer graphene: [Dean et al, (2010)] hbn monolayer graphene hbn b c 100 nm E d (mev) hbn µ 30 K weak disorder: charge puddles [Xue et al, (2011)] SiO 2 µ 300 K

34 Transport 13 Thermal and Electrical Conductivity at Charge Neutrality rt E

35 Transport 13 Thermal and Electrical Conductivity at Charge Neutrality rt E in a clean charge neutral metal: κ =, σ = finite.

36 Transport 14 Model I: Mean Field hydrodynamic constitutive relations : J i = σ q (E i + µ ) T it + nv i, Q i = (ɛ + P )v i µj i, with Newton s law: [Hartnoll et al; PRB, ] ɛ + P τ v i = ne i s i T.

37 Transport 14 Model I: Mean Field hydrodynamic constitutive relations : J i = σ q (E i + µ ) T it + nv i, Q i = (ɛ + P )v i µj i, with Newton s law: [Hartnoll et al; PRB, ] ɛ + P v i = ne i s i T. τ electrical vs. thermal conductivity σ = σ q + n2 τ ɛ + P κ = σ q σ(n) T s2 τ ɛ + P κ/σ heat/charge currents nearly same hydro W-F decoupled heat/charge n

38 Transport 15 Wiedemann-Franz Law Violations in Experiment Tbath (K) phonon-limited disorder-limited n (10 9 cm -2 ) L / L0 [Crossno et al; Science, ]

39 Transport 16 Model II: Charge Puddles hydrodynamic transport: disorder on scale ξ l ee l ee n(x) > 0 s(x) > 0 n(x) < 0 n x

40 Transport 16 Model II: Charge Puddles hydrodynamic transport: disorder on scale ξ l ee l ee n(x) > 0 s(x) > 0 n(x) < 0 n x solve partial differential equations ( ( i n(µ 0 (x))δv i + σ q E i i δµ + µ(x) )) T iδt = 0, etc.

41 Transport 16 Model II: Charge Puddles hydrodynamic transport: disorder on scale ξ l ee l ee n(x) > 0 s(x) > 0 n(x) < 0 n x solve partial differential equations ( ( i n(µ 0 (x))δv i + σ q E i i δµ + µ(x) )) T iδt = 0, etc. weak disorder: µ 0 (x) = µ + µ 0 (x): 1 τ µ2 0 ( ) n 2 [ 1 4η µ 2 ] 2( ɛ + P + ) µ σ q ( ɛ + P ) 2 ξ 2.

42 Transport 16 Model II: Charge Puddles hydrodynamic transport: disorder on scale ξ l ee l ee n(x) > 0 s(x) > 0 n(x) < 0 n x solve partial differential equations ( ( i n(µ 0 (x))δv i + σ q E i i δµ + µ(x) )) T iδt = 0, etc. weak disorder: µ 0 (x) = µ + µ 0 (x): 1 τ µ2 0 ( ) n 2 [ 1 4η µ 2 ] 2( ɛ + P + ) µ σ q ( ɛ + P ) 2 ξ 2. at strong disorder: numerical solution

43 Transport 17 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 [Lucas, Crossno, Fong, Kim, Sachdev; PRB, ]

44 Sound Waves 18 Speed of Sound gapless modes of relativistic fluid: sound: ω = v s k iη 6P k2 + charge diffusion: ω = i σq µn k2

45 Sound Waves 18 Speed of Sound gapless modes of relativistic fluid: sound: ω = v s k iη 6P k2 + charge diffusion: ω = i σq µn k2 when µ, T only energy scales: ɛ = 2P

46 Sound Waves 18 Speed of Sound gapless modes of relativistic fluid: sound: ω = v s k iη 6P k2 + charge diffusion: ω = i σq µn k2 when µ, T only energy scales: ɛ = 2P dissipationless hydro: t δɛ + x δπ = 0, t δπ + x δɛ 2 = 0.

47 Sound Waves 18 Speed of Sound gapless modes of relativistic fluid: sound: ω = v s k iη 6P k2 + charge diffusion: ω = i σq µn k2 when µ, T only energy scales: dissipationless hydro: speed of sound: t δɛ + x δπ = 0, ɛ = 2P v s = v F 2. t δπ + x δɛ 2 = 0.

48 Sound Waves 19 Sound Waves in Disordered Media in clean sample of size L: [Lucas; PRB, ] η s k B 1 10n 2 k B T L v F (if n resonances observed)

49 Sound Waves 19 Sound Waves in Disordered Media in clean sample of size L: [Lucas; PRB, ] η s k B 1 10n 2 k B T L v F (if n resonances observed) electronic sound resonances at ω 30 GHz

50 Sound Waves 19 Sound Waves in Disordered Media in clean sample of size L: [Lucas; PRB, ] η s k B 1 10n 2 k B T L v F (if n resonances observed) electronic sound resonances at ω 30 GHz momentum relaxation time: J γ = 0.01 γ = 0.02 γ = 0.04 γ = 0.1 γ = ω

51 Sound Waves 19 Sound Waves in Disordered Media in clean sample of size L: [Lucas; PRB, ] η s k B 1 10n 2 k B T L v F (if n resonances observed) J electronic sound resonances at ω 30 GHz momentum relaxation time: J γ = 0.01 γ = 0.02 γ = 0.04 γ = 0.1 γ = 0 J charge puddles: ! ω !

52 Sound Waves 19 Sound Waves in Disordered Media in clean sample of size L: [Lucas; PRB, ] η s k B 1 10n 2 k B T L v F (if n resonances observed) J electronic sound resonances at ω 30 GHz 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!

53 Outlook 20 Dirac fluid to do list: measure viscosity (sound waves, other techniques?)

54 Outlook 20 Dirac fluid to do list: measure viscosity (sound waves, other techniques?) spatial imaging of hydrodynamic flows?

55 Outlook 20 Dirac fluid to do list: measure viscosity (sound waves, other techniques?) spatial imaging of hydrodynamic flows? time-resolved dynamics?

56 Outlook 20 Dirac fluid to do list: measure viscosity (sound waves, other techniques?) spatial imaging of hydrodynamic flows? time-resolved dynamics? ballistic-to-hydrodynamic crossover? (and other problems about fast dynamics)