Hydrodynamic transport in the Dirac fluid in graphene. Andrew Lucas

Transcription

1 Hydrodynamic transport in the Dirac fluid in graphene Andrew Lucas Harvard Physics Condensed Matter Seminar, MIT November 4, 2015

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 Transport 3 Ohm s law the simplest experiment: I V = IR R 1 σ V

4 Transport 3 Ohm s law the simplest experiment: I V = IR R 1 σ V more generally, thermoelectric transport: ( J Q ) ( σ α = T ᾱ κ ) ( E T )

5 Fermi Liquids 4...are Analogous to a Classical Gas FL theory describes electronic fluid in iron, copper, etc. Fermi sea k B T µ

6 Fermi Liquids 4...are Analogous to a Classical Gas Fermi sea k B T FL theory describes electronic fluid in iron, copper, etc. long-lived quasiparticles; (quantum) kinetic theory µ

7 Fermi Liquids 4...are Analogous to a Classical Gas Fermi sea k B T µ FL theory describes electronic fluid in iron, copper, etc. long-lived quasiparticles; (quantum) kinetic theory interaction time: t ee µ (k B T ) s

8 Fermi Liquids 4...are Analogous to a Classical Gas Fermi sea k B T µ FL theory describes electronic fluid in iron, copper, etc. long-lived quasiparticles; (quantum) kinetic theory interaction time: t ee µ (k B T ) s QP interacts with impurity or phonon every s (exceptions are rare, e.g. GaAs, doped graphene)

9 Fermi Liquids 5 Wiedemann-Franz Law experimentally measured thermal conductivity: Q J=0 κ T, κ = κ T α2 σ.

10 Fermi Liquids 5 Wiedemann-Franz Law experimentally measured thermal conductivity: Q J=0 κ T, κ = κ T α2 σ. Wiedemann-Franz law in a Fermi liquid: L κ σt π2 kb 2 3e W Ω K 2. [Kumar, Prasad, Pohl, Journal of Materials Science (1993)]

11 Quantum Criticality 6 What is Quantum Criticality? quantum critical tee ~ 10 kb T 13 s (T = 100 K) T 0 g gc

12 Quantum Criticality 6 What is Quantum Criticality? quantum critical tee I ~ 10 kb T 13 s (T = 100 K) T quantum criticality in experiments: strange metal superfluid-insulator (cold atoms) 0 graphene g gc

13 Quantum Criticality 7 Emergent Lorentz Invariance l & l ee l a l. a hydrodynamics finite T QFT lattice model many quantum critical models have emergent Lorentz invariance

14 Quantum Criticality 7 Emergent Lorentz Invariance l & l ee l a l. a hydrodynamics finite T QFT lattice model many quantum critical models have emergent Lorentz invariance we re going to focus on l ξ hydrodynamics

15 Hydrodynamics 8 What is Hydrodynamics? l ee T (0) µ(0) u µ (0) T ( ) µ( ) u µ ( ) T (2 ) µ(2 ) u µ (2 ) T (3 ) µ(3 ) u µ (3 ) T (4 ) µ(4 ) u µ (4 ) effective description of relaxing to thermal equilibrium on long length scales

16 Hydrodynamics 8 What is Hydrodynamics? l ee T (0) µ(0) u µ (0) T ( ) µ( ) u µ ( ) T (2 ) µ(2 ) u µ (2 ) T (3 ) µ(3 ) u µ (3 ) T (4 ) µ(4 ) u µ (4 ) effective description of relaxing to thermal equilibrium on long length scales perturbative expansion in l ee /ξ opposite of usual QFT

17 Hydrodynamics 8 What is Hydrodynamics? l ee T (0) µ(0) u µ (0) T ( ) µ( ) u µ ( ) T (2 ) µ(2 ) u µ (2 ) T (3 ) µ(3 ) u µ (3 ) T (4 ) µ(4 ) u µ (4 ) effective description of relaxing to thermal equilibrium on long length scales perturbative expansion in l ee /ξ opposite of usual QFT classical equations of motion: conservation laws µ T µν = F µν J ν, µ J µ = 0

18 Hydrodynamics 9 The Gradient Expansion for a Lorentz-Invariant Fluid expand T µν, J µ in perturbative parameter l ee µ : ( T µν = P η µν + (ɛ + P )u µ u ν 2P µρ P νσ η (ρ u σ) P µν ζ 2η ) ρu ρ +, d J µ = nu µ σ qp µρ ( ρµ µ T ρt uν F ρν ) +, P µν η µν + u µ u ν, Q i = J i µt ti

19 Hydrodynamics 9 The Gradient Expansion for a Lorentz-Invariant Fluid expand T µν, J µ in perturbative parameter l ee µ : ( T µν = P η µν + (ɛ + P )u µ u ν 2P µρ P νσ η (ρ u σ) P µν ζ 2η ) ρu ρ +, d J µ = nu µ σ qp µρ ( ρµ µ T ρt uν F ρν ) +, P µν η µν + u µ u ν, Q i = J i µt ti quantum physics values of P, σ q, etc...

20 Hydrodynamics 9 The Gradient Expansion for a Lorentz-Invariant Fluid expand T µν, J µ in perturbative parameter l ee µ : ( T µν = P η µν + (ɛ + P )u µ u ν 2P µρ P νσ η (ρ u σ) P µν ζ 2η ) ρu ρ +, d J µ = nu µ σ qp µρ ( ρµ µ T ρt uν F ρν ) +, P µν η µν + u µ u ν, Q i = J i µt ti quantum physics values of P, σ q, etc... fluid has both electrons/holes; not separately conserved. distinct from usual plasma physics

21 Hydrodynamics 9 The Gradient Expansion for a Lorentz-Invariant Fluid expand T µν, J µ in perturbative parameter l ee µ : ( T µν = P η µν + (ɛ + P )u µ u ν 2P µρ P νσ η (ρ u σ) P µν ζ 2η ) ρu ρ +, d J µ = nu µ σ qp µρ ( ρµ µ T ρt uν F ρν ) +, P µν η µν + u µ u ν, Q i = J i µt ti quantum physics values of P, σ q, etc... fluid has both electrons/holes; not separately conserved. distinct from usual plasma physics transport equations well-posed at first order (dissipation allowed) [Hartnoll, Kovtun, Müller, Sachdev, Physical Review B (2007)]

22 Hydrodynamic Transport 10 Conductivity of a Clean Metal J =0 J = Qv and E =0

23 Hydrodynamic Transport J =0 10 Conductivity of a Clean Metal J =0 J = nv and E =0 J = Qv and E =0

24 Hydrodynamic Transport J =0 10 Conductivity of a Clean Metal J =0 J = nv and E =0 J = Qv and E =0 σ sensitive to how translational symmetry broken

25 Hydrodynamic Transport 11 Mean Field: Drude Model mean field treatment of translational symmetry breaking: [Hartnoll, Kovtun, Müller, Sachdev, Physical Review B (2007)] µ T µi = T it τ + ne i.

26 Hydrodynamic Transport 11 Mean Field: Drude Model mean field treatment of translational symmetry breaking: [Hartnoll, Kovtun, Müller, Sachdev, Physical Review B (2007)] constitutive relations : µ T µi = T it τ + ne i. T it = (ɛ + P )v i J i = σ q E i + nv i :

27 Hydrodynamic Transport 11 Mean Field: Drude Model mean field treatment of translational symmetry breaking: [Hartnoll, Kovtun, Müller, Sachdev, Physical Review B (2007)] constitutive relations : average over space: µ T µi = T it σ(ω) = σ q + τ + ne i. T it = (ɛ + P )v i J i = σ q E i + nv i : n 2 τ (ɛ + P )(1 iωτ).

28 Hydrodynamic Transport 11 Mean Field: Drude Model mean field treatment of translational symmetry breaking: [Hartnoll, Kovtun, Müller, Sachdev, Physical Review B (2007)] constitutive relations : average over space: µ T µi = T it σ(ω) = σ q + τ + ne i. T it = (ɛ + P )v i J i = σ q E i + nv i : n 2 τ (ɛ + P )(1 iωτ). but missing sharp predictions for τ...

29 Hydrodynamic Transport 12 Wiedemann-Franz Revisited L 0 = L = κ σt L 0 (1 + (n/n 0 ) 2 ) 2 (ɛ + P )τ σ q T 2, n 2 0 = (ɛ + P )σ q τ

30 Hydrodynamic Transport 12 Wiedemann-Franz Revisited L 0 = L = κ σt L 0 (1 + (n/n 0 ) 2 ) 2 (ɛ + P )τ σ q T 2, n 2 0 = (ɛ + P )σ q τ L/LWF heat/charge currents nearly same hydro FL decoupled heat/charge n

31 Hydrodynamic Transport 13 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

32 Hydrodynamic Transport 13 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)

33 Hydrodynamic Transport 13 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

34 Hydrodynamic Transport 13 Non-Perturbative Approach l ee n(x) > 0 s(x) > 0 n(x) < 0 n x non-perturbative hydrodynamic transport: disorder on scale ξ l ee static fluid in an inhomogeneous chemical potential µ 0 (x) transport from linearized hydrodynamic equations mean-field formalism with specific τ as µ 0 0: 1 τ ( µ 0) 2 ( ) n 2 [ µ 1 σ q (ɛ + P ) + 4ηµ 2 ξ 2 (ɛ + P ) 3 [Lucas, New Journal of Physics (2015)]; [Lucas et al, arxiv: ] similar ideas in FL: [Andreev, Kivelson, Spivak, PRL (2011)] ]

35 Hydrodynamic Transport 14 Exact Statements Onsager reciprocity

36 Hydrodynamic Transport 14 Exact Statements Onsager reciprocity variational (Thomson s) principle for DC transport: [Lucas, New Journal of Physics (2015)] resistances power dissipated on arbitrary unit flows analogous to resistor network theory

37 Hydrodynamic Transport 14 Exact Statements Onsager reciprocity variational (Thomson s) principle for DC transport: [Lucas, New Journal of Physics (2015)] resistances power dissipated on arbitrary unit flows analogous to resistor network theory application: ruling out disorder-driven metal-insulator transition in some AdS/CMT models [Grozdanov, Lucas, Sachdev, Schalm, arxiv: , PRL (to appear)]

38 Dirac Fluid in Graphene 15 Crash Course in Graphene 1nm

39 Dirac Fluid in Graphene 16 Crash Course in Graphene a = ~v F k + V int = e r T Dirac fluid hole FL electron FL 0 n

40 Dirac Fluid in Graphene 16 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

41 Dirac Fluid in Graphene 16 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 effective quantum critical description at room T! c 0.5. v F ɛ r

42 Dirac Fluid in Graphene 16 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 effective quantum critical description at room T! thermalization length scale 100 nm. c 0.5. v F ɛ r e.g. [Sheehy, Schmalian, Physical Review Letters (2007)] [Müller, Fritz, Sachdev, Physical Review B (2008)]

43 Dirac Fluid in Graphene 17 Graphene: an Ideal Experimental Platform gating farther than l ee possible!

44 Dirac Fluid in Graphene 17 Graphene: an Ideal Experimental Platform gating farther than l ee possible! fabricating ultra pure monolayer graphene: [Dean et al, Nature Nanotechnology (2010)] hbn monolayer graphene hbn

45 Dirac Fluid in Graphene 17 Graphene: an Ideal Experimental Platform gating farther than l ee possible! fabricating ultra pure monolayer graphene: [Dean et al, Nature Nanotechnology (2010)] b 100 nm hbn µ 30 K hbn monolayer graphene c E d (mev) hbn SiO 2 µ 300 K weak disorder: charge puddles [Xue et al, Nature Materials (2011)] 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 point as a function of position for graphene on SiO2. The colour scale is the

46 Dirac Fluid in Graphene 18 Early Evidence: Fast Interaction Time time-resovled ARPES: electron dynamics with t ee s (higher T ) [Johannsen et al, Physical Review Letters (2013)]

47 Dirac Fluid in Graphene 18 Early Evidence: Fast Interaction Time time-resovled ARPES: electron dynamics with t ee s (higher T ) electrons reached thermal equilibrium on time scale s [Johannsen et al, Physical Review Letters (2013)]

48 Dirac Fluid in Graphene 18 Early Evidence: Fast Interaction Time time-resovled ARPES: electron dynamics with t ee s (higher T ) electrons reached thermal equilibrium on time scale s but electrons coupled to optical phonons [Johannsen et al, Physical Review Letters (2013)]

49 Dirac Fluid in Graphene 18 Early Evidence: Fast Interaction Time time-resovled ARPES: electron dynamics with t ee s (higher T ) electrons reached thermal equilibrium on time scale s but electrons coupled to optical phonons can we see evidence of strong e-e interaction at lower T? [Johannsen et al, Physical Review Letters (2013)]

50 Dirac Fluid in Graphene 19 Johnson Noise Thermometry measuring heat flow Q in graphene impossible

51 Dirac Fluid in Graphene 19 Johnson Noise Thermometry measuring heat flow Q in graphene impossible Johnson noise thermometer: V 2 rms = 4k BT R tot B

52 Dirac Fluid in Graphene 19 Johnson Noise Thermometry measuring heat flow Q in graphene impossible Johnson noise thermometer: V 2 rms = 4k BT R tot B Joule heating: h T i = J 2 L 2 12apple DT (x) DT ave I s-d x [Fong, Schwab, Physical Review X (2012)] [Crossno et al., arxiv: ]

53 Dirac Fluid in Graphene 20 Wiedemann-Franz Law Violations in Experiment Tbath (K) phonon-limited disorder-limited n (10 9 cm -2 ) L / L0 [Crossno et al, arxiv: ]

54 n min (cm -2 ) Dirac Fluid in Graphene K 150 K 250 K Wiedemann-Franz Law Violations in Experiment T bath (K) Thermal Conductivity (nw/k) 8 C 75 K 6 4 D T dis κe T el-ph σtl0 2 0 V 0 E K 10 mm K V Vg (V) Tbath (K) L/L 0 S3 Disorder Thermally Limited Limited Temperature (K) C H +V e h h e C -V L / L Temperature (K) H (ev/µm 2 ) T (K) n (10 10 cm 2 ) compare to perturbative prediction L = L DF (1 + (n/n 0 ) 2 ) 2 [Crossno et al, arxiv: ]

55 Dirac Fluid in Graphene 22 3 Experimental Puzzles experimental puzzles: σ(n) is nearly constant

56 Dirac Fluid in Graphene 22 3 Experimental Puzzles experimental puzzles: σ(n) is nearly constant what is controlling the T -dependence of κ near n = 0?

57 Dirac Fluid in Graphene 22 3 Experimental Puzzles experimental puzzles: σ(n) is nearly constant what is controlling the T -dependence of κ near n = 0? non-monotonic growth in κ(t )?

58 Dirac Fluid in Graphene 22 3 Experimental Puzzles experimental puzzles: σ(n) is nearly constant what is controlling the T -dependence of κ near n = 0? non-monotonic growth in κ(t )? proposed resolution: density/viscosity dependence neglected (can be addressed in perturbation theory!)

59 Dirac Fluid in Graphene 22 3 Experimental Puzzles experimental puzzles: σ(n) is nearly constant what is controlling the T -dependence of κ near n = 0? non-monotonic growth in κ(t )? proposed resolution: density/viscosity dependence neglected (can be addressed in perturbation theory!) perturbative: universal κ T 3 from charge puddles

60 Dirac Fluid in Graphene 22 3 Experimental Puzzles experimental puzzles: σ(n) is nearly constant what is controlling the T -dependence of κ near n = 0? non-monotonic growth in κ(t )? proposed resolution: density/viscosity dependence neglected (can be addressed in perturbation theory!) perturbative: universal κ T 3 from charge puddles momentum relaxation from disorder vs. acoustic phonons? [Lucas, Crossno, Fong, Kim, Sachdev, arxiv: ]

61 Dirac Fluid in Graphene 23 A Hydrodynamic Model for Graphene s = C 0 T0 2 + C 2 2 µ2 0 C 4µ 4 0 4T0 2 +, l ee n(x) > 0 s(x) > 0 n(x) < 0 n x n = C 2 µ 0 T 0 + C 4 T 0 µ 3 0 +, η = η 0 T 2 0, ζ = 0, σ q = σ 0 numerically solve linearized hydrodynamic equations: ( ( 0 = i nv i + σ q E i i µ µ 0 ζ i + µ )) 0 i T, T 0 0 = i ( T 0 sv i µ 0 σ q ( E i i µ µ 0 ζ i + µ 0 T 0 i T )), 0 = n( i µ E i ) + s ( i T T 0 ζ i ) j (η ( i v j + j v i δ ij k v k )). [Lucas, Crossno, Fong, Kim, Sachdev, arxiv: ]

62 Dirac Fluid in Graphene 24 Comparing Theory to Experiment comparison to experiment at T = 75 K fit parameters: (k 1 ) hole FL Dirac fluid elec. FL apple (nw/k) hole FL Dirac fluid elec. FL n (µm 2 ) n (µm 2 ) Figure 1: testing

63 Dirac Fluid in Graphene 24 Comparing Theory to Experiment comparison to experiment at T = 75 K fit parameters: (k 1 ) hole FL Dirac fluid elec. FL apple (nw/k) hole FL Dirac fluid elec. FL n (µm 2 ) n (µm 2 ) η s 10 k B Figure 1: testing (though possibly substantial overestimate) [Lucas, Crossno, Fong, Kim, Sachdev, arxiv: ]

64 Dirac Fluid in Graphene 25 Comparing Theory to Experiment comparison to experiment at n = 0 (no new fit parameters): (k 1 ) puddle FLs Dirac fluid phonons apple (nw/k) puddle FLs Dirac fluid phonons T (K) T (K) Figure 1: testing satisfactory quantitative agreement in Dirac fluid! [Lucas, Crossno, Fong, Kim, Sachdev, arxiv: ]

65 Hydrodynamics of the Fermi Liquid in Graphene 26 nonlocal negative voltage measurement: signature of viscous electronic flow in FL:

66 Hydrodynamics of the Fermi Liquid in Graphene 26 nonlocal negative voltage measurement: signature of viscous electronic flow in FL: theory: [Torre et al, arxiv: ]; [Levitov, Falkovich, arxiv: ] experiment: [Bandurin et al, arxiv: ]

67 Outlook 27 strong evidence emerging for hydrodynamics in electronic fluids!

68 Outlook 27 strong evidence emerging for hydrodynamics in electronic fluids! first quantitative contact between AdS/CMT-inspired model and condensed matter experiment

69 Outlook 27 strong evidence emerging for hydrodynamics in electronic fluids! first quantitative contact between AdS/CMT-inspired model and condensed matter experiment direct imaging of hydrodynamic flows?

70 Outlook 27 strong evidence emerging for hydrodynamics in electronic fluids! first quantitative contact between AdS/CMT-inspired model and condensed matter experiment direct imaging of hydrodynamic flows? accurate determination of thermo/hydro coefficients of Dirac fluid?

71 Outlook 27 strong evidence emerging for hydrodynamics in electronic fluids! first quantitative contact between AdS/CMT-inspired model and condensed matter experiment direct imaging of hydrodynamic flows? accurate determination of thermo/hydro coefficients of Dirac fluid? better understanding of electronic hydrodynamics w/o translation symmetry; prediction of new phenomena