Viscous electron fluids: Superballistic conduction; Odd-parity hydrodynamics

Transcription

1 Viscous electron fluids: Superballistic conduction; Odd-parity hydrodynamics Leonid Levitov (MIT) Frontiers in Many-Body Physics Memorial for Lev Petrovich Gor kov Tallahassee, 01/13/018

2 Hydrodynamics: theory of everything

3 Is hydrodynamics ever relevant in metals? credit: Jan Zaanen, Science 016 High-mobility electron systems (graphene, GaAs DES, PdCoO, etc): Non-Fermi liquids, high-tc superconductors, strange metals

4 Is hydrodynamics ever relevant in metals? credit: Jan Zaanen, Science 016 High-mobility electron systems (graphene, GaAs DES, PdCoO, etc): Non-Fermi liquids, high-tc superconductors, strange metals

5 Viscous electron fluids in graphene Strong interactions (enhanced in D, graphene) Weak electron-phonon scattering, no Umklapp ee scattering Fast p-conserving ee collisions, shear viscosity Sheehy and Schmalian, PRL 99, 6803 (007)

6 Viscous electron fluids in graphene Strong interactions (enhanced in D, graphene) Weak electron-phonon scattering, no Umklapp ee scattering Fast p-conserving ee collisions, shear viscosity Which geometries are sensitive to viscous effects? New collective phenomena? Sheehy and Schmalian, PRL 99, 6803 (007)

7 Fundamental bound for free particle conductance Landauer (1957): conduction=transmission Sharvin (1965): counting phase space 1/1/18

8 Experimental realization in quantum point contacts Disorder-free Width 1/1/18 B. J. van Wees et al. Phys. Rev. Lett. 60, 848 (1988); Phys. Rev. B 43, 1431 (1991).

9 Ballistic conduction Ballistic free fermion transport Conductance = # transmitting channels (Landauer) Phase space argument for Sharvin contact e G= N h w N = n t n=, 0< t n 1 λf w λ F

10 Is this really the highest conductance? No In a viscous flow electrons cooperate to overcome the ballistic conductance bound Experimental realization in graphene 1/1/18

11 Is this really the highest conductance? No In a viscous flow electrons cooperate to overcome the ballistic conductance bound Experimental realization in graphene l ee w Extreme viscous limit: conductance from Stokes eqn w 1/1/18 e v F l ee ν v = E ν= m 4 γ(k )m ρ=, γ(k )=ν k 1 ne k= w w Gvis G ball Gball l ee

12 Ballistic at T=0; Superballistic at T>0 Conductance enhanced as a result of cooperative behavior Electrons follow zig-zag paths to avoid boundaries 1/1/18 Phase space bottleneck relieved

13 Team Haoyu Guo MIT Patrick Ledwith MIT

14 Team Haoyu Guo MIT Andrey Shytov Exeter UK Patrick Ledwith MIT Gregory Falkovich WIS Israel

15 Team Haoyu Guo MIT Andrey Shytov Exeter UK Patrick Ledwith MIT Gregory Falkovich WIS Israel Roshan Krishna Kumar Denis Bandurin Manchester UK Andre Geim Manchester UK

16 Point contacts in high mobility graphene

17 Transport measurements Point contacts in high-mobility graphene Ballistic transport at low T 10 m 4 e w Gb = h λf Nat Phys 017

18 Transport measurements Viscous flow at elevated temperatures Ballistic transport at low T Semiconducting-like behavior R values below the ballistic limit 4 e w Gb = h λf Nat Phys 017

19 Transport measurements Viscous flow at elevated temperatures Hall bar Semiconducting-like behavior R values below the ballistic limit conventional behavior Nat Phys 017

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21 Different regimes: constriction width w vs. mean free path lee=vf/ ee Ballistic lee>>w Viscous lee<<w Crossover lee~w

22 Different regimes: constriction width w vs. mean free path lee=vf/ ee Ballistic lee>>w Viscous lee<<w Crossover lee~w

23 The viscous-to-ballistic crossover At lee~w solve quantum kinetic eqn ( t + v ) f =I ee (f )+ I b (f ) accounting for the zero modes (px,py,n) p-conserving -approximation for Iee I ee = γ ee (f P3 f ) I b= α( x) δ( y) P f Guo et al PNAS 017

24 The viscous-to-ballistic crossover At lee~w solve quantum kinetic eqn ( t + v ) f =I ee (f )+ I b (f ) accounting for the zero modes (px,py,n) p-conserving -approximation for Iee I ee = γ ee (f P3 f ) Project on the 3D zero-mode subspace I b= α( x) δ( y) P f Guo et al PNAS 017

25 The viscous-to-ballistic crossover At lee~w solve quantum kinetic eqn ( t + v ) f =I ee (f )+ I b (f ) accounting for the zero modes (px,py,n) p-conserving -approximation for Iee I ee = γ ee (f P3 f ) Project on the 3D zero-mode subspace I b= α( x) δ( y) P f Yields an integral eqn of LippmannSchwinger type for 0-mode fields at the boundary (L 1 + α^ )(f 0 +δ f )=0 Guo et al PNAS 017

26 The viscous-to-ballistic crossover At lee~w solve quantum kinetic eqn ( t + v ) f =I ee (f )+ I b (f ) accounting for the zero modes (px,py,n) p-conserving -approximation for Iee I ee = γ ee (f P3 f ) Project on the 3D zero-mode subspace I b= α( x) δ( y) P f Yields an integral eqn of LippmannSchwinger type for 0-mode fields at the boundary (L 1 + α^ )(f 0 +δ f )=0 Solved at high ee by mapping to an auxiliary electrostatic problem Guo et al PNAS 017

27 The viscous-to-ballistic crossover At lee~w solve quantum kinetic eqn ( t + v ) f =I ee (f )+ I b (f ) accounting for the zero modes (px,py,n) p-conserving -approximation for Iee I ee = γ ee (f P3 f ) Project on the 3D zero-mode subspace I b= α( x) δ( y) P f Yields an integral eqn of LippmannSchwinger type for 0-mode fields at the boundary (L 1 + α^ )(f 0 +δ f )=0 Solved at high ee by mapping to an auxiliary electrostatic problem and at any ee numerically Guo et al PNAS 017

28 Potential distribution, high ee Guo et al PNAS 017 Agrees w exact solution (hydro) Negative E-field at the edge

29 Potential distribution, high ee Guo et al PNAS 017 Agrees w exact solution (hydro) Negative E-field at the edge

30 Potential distribution, low ee Guo et al PNAS 017 Positive E-field at the edge Current profile evolution: semicircle to a constant

31 Conductance of a VPC Solve quantum kinetic eqn w full account taken of zero modes (px,py,n) -apprx for the collision integral Project on the 3D zero-mode subspace Yields an integral eqn of LippmannSchwinger type for 0-mode fields at the boundary Solved at high by mapping to an auxiliary electrostatic problem Guo et al PNAS 017

32 Conductance of a VPC Solve quantum kinetic eqn w full Scaled dataofcollapse on one(pcurve account taken zero modes x,py,n) -apprx for the collision integral Project on the 3D zero-mode subspace Yields an integral eqn of LippmannSchwinger type for 0-mode fields at the boundary Solved at high by mapping to an auxiliary electrostatic problem For all ee, results expressed through a universal function of w/lee, with lee=v/ ee Guo et al PNAS 017

33 Conductance of a VPC Solve quantum kinetic w fullvs. w ee Conductance growseqn linearly account taken of zero modes (px,py,n) Guo et al PNAS 017 -apprx for the collision integral Project on the 3D zero-mode subspace Yields an integral eqn of LippmannSchwinger type for 0-mode fields at the boundary Solved by mapping to an auxiliary electrostatic problem at high For all, results expressed through a universal function of w/v With numerical precision find a simple linear dependence (exact solution?) π( ne) w 4e Gexact =G ball +G visc= N+ h 3 η

34 Conductance of a VPC Solve quantum kinetic w fullvs. w ee Conductance growseqn linearly account taken of zero modes (px,py,n) Guo et al PNAS 017 -apprx for the collision integral Project on the 3D zero-mode subspace Yields an integral eqn of LippmannSchwinger type for 0-mode fields at the boundary Solved by mapping to an auxiliary electrostatic problem at high For all, results expressed through a universal function of w/v With numerical precision find a simple linear dependence (exact solution?) π( ne) w 4e Gexact =G ball +G visc= N+ h 3 η

35 Measurements: extracting viscous conductance Unconventional scaling Gvisc~w π(ne ) w Gvisc = 3 η π(ne ) w Gvisc = 3 η Nat Phys 017

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37 A new transport regime in D Momentum conservation and fermion exclusion single out two types of collisions: a) head-on, and b) small-angle Momentum relaxation dominated by head-on collisions. Affects only the even-parity part of momentum distribution Very different relaxation rates for the angular harmonics of odd and even parity: / ~(T/TF), ~T/TF arxiv:

38 Kinematics of twobody collisions: 1 1 arxiv:

39 Small scattering angles: Δ θ T / T F 1 1 arxiv:

40 Related publications R. Shankar, RMP. 66, 19 (1994). B. Laikhtman, PRB 45, 159 (199). R. N. Gurzhi, A. N. Kalinenko, and A. I. Kopeliovich, PRL 74, 387 (1995) H. Buhmann, L. W. Molenkamp, Physica E 1, (00) K. E. Nagaev and O. S. Ayvazyan, PRL 101, (008). K. E. Nagaev and T. V. Kostyuchenko, PRB 81, (010) P J Ledwith, H Guo and LL, arxiv: arxiv:

41 Slow momentum relaxation and its role in transport: a simple model Eigenmodes and eigenvalues of a linearized collision operator: imθ ( t + v I )δ f p ( x,t )=0, δ f p = m δ f m e, p I ei m θ =γ m e i m θ, γ m even =γ (1 δ m,0 ), γ m odd γ ' γ =γ ' (1 δ m,1 δm, 1 ) arxiv:

42 Slow momentum relaxation and its role in transport: a simple model Eigenmodes and eigenvalues of a linearized collision operator: imθ ( t + v I )δ f p ( x,t )=0, δ f p = m δ f m e, p I ei m θ =γ m e i m θ, γ m even =γ (1 δ m,0 ), γ m odd γ ' γ =γ ' (1 δ m,1 δm, 1 ) Plane-wave solns f~eikx-i t: a tight-binding model in fm basis kv kv (γ m i ω)δ f m= δ f m 1 + δ f m +1 arxiv:

43 Slow momentum relaxation and its role in transport: a simple model Eigenmodes and eigenvalues of a linearized collision operator: imθ ( t + v I )δ f p ( x,t )=0, δ f p = m δ f m e, p I ei m θ =γ m e i m θ, γ m even =γ (1 δ m,0 ), γ m odd γ ' γ =γ ' (1 δ m,1 δm, 1 ) Plane-wave solns f~eikx-i t: a tight-binding model in fm basis kv kv (γ m i ω)δ f m= δ f m 1 + δ f m +1 m=0 on every other site in the limit infinitely long lived dark state: p δ f m= p +1=( 1), δ f m = p=0 arxiv:

44 Slow momentum relaxation and its role in transport: a simple model Eigenmodes and eigenvalues of a linearized collision operator: imθ ( t + v I )δ f p ( x,t )=0, δ f p = m δ f m e, p I ei m θ =γ m e i m θ, γ m even =γ (1 δ m,0 ), γ m odd γ ' γ =γ ' (1 δ m,1 δm, 1 ) Plane-wave solns f~eikx-i t: a tight-binding model in fm basis kv kv (γ m i ω)δ f m= δ f m 1 + δ f m +1 m=0 on every other site in the limit an infinitely long-lived dark state: p δ f m= p +1=( 1), δ f m = p=0 A family of soft modes originating from the dark state: overlap w/ current modes m=1,-1, hence impact conduction & viscosity FDT: slow relaxation implies strong current fluctuations & reduced dissipation; k-dependent couplings, scale dependence

45 The odd-parity transport regime Occurs between the ballistic and conventional hydro regimes, at the length scales l ee <r < ξ= v wavenumbers ξ 1 <k < 1/l ee γ γ' Scale-dependent viscosity η(k ) 1/ k Scale-dependent conductivity: σ (k ) 1/k 3/ Conductance of a strip of width l ee < w< ξ G(w) w 5 / Exponent 5/ between the Gurzhi value 3 and ballistic value Multiple particle/hole switchbacks on straight paths Velocities collimated within Δ θ w /ξ, stronger than in the collisionless ballistic regime arxiv:

46 The odd-parity transport regime Occurs between the ballistic and conventional hydro regimes, at the length scales l ee <r < ξ= v wavenumbers ξ 1 <k < 1/l ee γ γ' Scale-dependent viscosity η(k ) 1/ k Scale-dependent conductivity: σ (k ) 1/k 3/ Conductance of a strip of width l ee < w< ξ G(w) w 5 / Exponent 5/ between the Gurzhi value 3 and ballistic value Multiple particle/hole switchbacks on straight paths Velocities collimated within Δ θ w /ξ, stronger than in the collisionless ballistic regime arxiv:

47 The odd-parity transport regime Occurs between the ballistic and conventional hydro regimes, at the length scales l ee <r < ξ= v wavenumbers ξ 1 <k < 1/l ee γ γ' Scale-dependent viscosity η(k ) 1/ k Scale-dependent conductivity: σ (k ) 1/k 3/ Conductance of a strip of width l ee < w< ξ G(w) w 5 / Exponent 5/ between the Gurzhi value 3 and ballistic value Beamlets: multiple particle/hole switchbacks on straight paths Velocities collimated within Δ θ w /ξ, stronger than in the collisionless ballistic regime arxiv:

48 A microscopic approach Integrate out the even-m modes, derive a closed-form master equation for the odd-m modes, valid at the length scales r > lee=v/ ( t I + )δ f ( x, t)+ v δ f p ( x, t)=0, + (0 ) ( t I )δ f p ( x,t )+ v δ f p ( x, t)= e E ( x) p f p + p arxiv:

49 A microscopic approach Integrate out the even-m modes, derive a closed-form master equation for the odd-m modes, valid at the length scales r > lee=v/ ( t I + )δ f ( x, t)+ v δ f p ( x, t)=0, + (0 ) ( t I )δ f p ( x,t )+ v δ f p ( x, t)= e E ( x) p f p + p Fast spatial diffusion along the velocity v direction, plus a slow angular diffusion that gradually randomizes the orientation of v: (0) ) γ ' ) δ f ( x,t )= e ( t D( v^ E ( x) f, D=v /γ θ p p p arxiv:

50 A microscopic approach Integrate out the even-m modes, derive a closed-form master equation for the odd-m modes, valid at the length scales r > lee=v/ ( t I + )δ f ( x, t)+ v δ f p ( x, t)=0, + (0 ) ( t I )δ f p ( x,t )+ v δ f p ( x, t)= e E ( x) p f p + p Fast spatial diffusion along the velocity v direction, plus a slow angular diffusion that gradually randomizes the orientation of v: (0) ) γ ' ) δ f ( x,t )= e ( t D( v^ E ( x) f, D=v /γ θ p p p Analyze response to a transverse field E ( x)= E cos k x, E k =0 obtain current, momentum flux & viscosity L=i ω+ Dk sin θ + γ ' Invert Mathieu operator θ Quantum harmonic oscillator at the oddparity length scales l ee <r < ξ arxiv:

51 A microscopic approach Integrate out the even-m modes, derive a closed-form master equation for the odd-m modes, valid at the length scales r > lee=v/ ( t I + )δ f ( x, t)+ v δ f p ( x, t)=0, + (0 ) ( t I )δ f p ( x,t )+ v δ f p ( x, t)= e E ( x) p f p + p Fast spatial diffusion along the velocity v direction, plus a slow angular diffusion that gradually randomizes the orientation of v: (0) ) γ ' ) δ f ( x,t )= e ( t D( v^ E ( x) f, D=v /γ θ p p p Analyze response to a transverse field E ( x)= E cos k x, E k =0 obtain current, momentum flux & viscosity L=i ω+ Dk sin θ + γ ' Invert Mathieu operator θ Quantum harmonic oscillator at the oddparity length scales l ee <r < ξ γ3 / 4 σ (k ) 1 /4 3 / γ' k

52 Summary & future A new collective effect: hydrodynamic screening of disorder, conductance enhancement: Gvisc > Gball At crossover: ballistic and viscous conductances additive (no Mathiessen rule): G=Gball+Gvisc

53 Summary & future A new collective effect: hydrodynamic screening of disorder, conductance enhancement: Gvisc > Gball At crossover: ballistic and viscous conductances additive (no Mathiessen rule): G=Gball+Gvisc Probe ballistic-to-viscous crossover as a function of temperature: good agreement with experiment Odd-parity hydrodynamics: head-on collisions, multiple slow modes, scale-dependent viscosity

54 Summary & future A new collective effect: hydrodynamic screening of disorder, conductance enhancement: Gvisc > Gball At crossover: ballistic and viscous conductances additive (no Mathiessen rule): G=Gball+Gvisc Probe ballistic-to-viscous crossover as a function of temperature: good agreement with experiment Odd-parity hydrodynamics: head-on collisions, multiple slow modes, scale-dependent viscosity Low dissipation in the viscous limit. Limits on dissipation? Thermal transport? Noise? How general is this behavior?

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