신상진 (한양대)2016.10.20@KPS

Holography of Dirac Fluid in Graphenewith two currents

Based on arXiv: 1609.03582서윤석, 송근호 (한양대)Philip Kim, Subir Sachdev (Harvard)

Effect of strong interaction

1

Quantum criticality. Plankian Dissipation (Rapid Thermalization), Strange Metal

Spectral function As coupling grows

Character of strongly correlated system

2

Loss of individuality: only characters as a whole remains.

Physics of StrCorrSys = Physics of Totalitarian

Lots of interesting phenomena like high Tc SC

Lots of such material!

Method: Holography

Black hole is a holographic image of Strange Metal

3

ClassicalWeak int4d

QuantumStrong int3d

Universality and Quantum criticality

4

Instead of individual material, want to study universality class.

Quantum criticality is classified by dynamical exponent z.

Problem: Usually str-corr-sys are complex material. Hard to analize

Science 4 March 2016

Graphene as as a strong correlated system

5

originally published online February 11, 2016 (6277), 1055-1058. [doi: 10.1126/science.aad0201]351Science

M. Polini (February 11, 2016) Novoselov, I. V. Grigorieva, L. A. Ponomarenko, A. K. Geim andTomadin, A. Principi, G. H. Auton, E. Khestanova, K. S. D. A. Bandurin, I. Torre, R. Krishna Kumar, M. Ben Shalom, A.grapheneNegative local resistance caused by viscous electron backflow in

Editor's Summary

, this issue p. 1061, 1055, 1058; see also p. 1026Sciencetransport in graphene, a signature of so-called Dirac fluids.

observed a huge increase of thermalet al.fluid flowing through a small opening. Finally, Crossno found evidence in graphene of electron whirlpools similar to those formed by viscouset al.Bandurin

had a major effect on the flow, much like what happens in regular fluids.2fluid in thin wires of PdCoO found that the viscosity of the electronet al.counterexamples (see the Perspective by Zaanen). Moll

flow rarely resembles anything like the familiar flow of water through a pipe, but three groups describe Electrons inside a conductor are often described as flowing in response to an electric field. This

Electrons that flow like a fluid

This copy is for your personal, non-commercial use only.

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(print ISSN 0036-8075; online ISSN 1095-9203) is published weekly, except the last weekScience

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Why graphene is Strongly interacting ?

6

1. g~1 +Marginal irrelevance (slow running)2. Near CNP: insufficient screening Dirac Fluid

QM criticality : z=1.3. StrCorSys: Such simple system in clean limit extraordinary guide for a theory.

One U(1) current model with Hydrodynamics

7

Hydrodynamic analysis with the effect of puddle (inhomogeneity)

Model: Two U(1) current with holography

8

Idea : Neutral current Enhance the heat conductivity

9

Results

Origin of two current : imbalance effect

• Due to Kinematics, Imbalance exist for a time electron and hole currents have independent chemical potential. (at equilibrium: )

suppressed kinematically due to E,p, conservation.

10

2

FIG. 1: K inematical constraints for imbalance relaxation.The left figure shows the Feynman diagram for the typi-cal two-particle decay process given by Eq. (1.1a); εi andpi respectively denote the energy and momentum of the i

t h

electron or hole. The right figure depicts momentum con-servation for this process. The length of the dashed pathL f ≡ |p2|+ |p3|+ |p4|traced out by the sum of “decay prod-uct” momenta is always greater than or equal to the lengthL i ≡ |p1|of the“parent” particlemomentum: L f ≥ L i . If theparticle energy spectrum takes the form ε(p) = |p|β , then thedepicted decay process is kinematically forbidden for β < 1.For β = 1, only forward-scattering is allowed.

h+ ↔ h+ + h+ + e−. (1.1b)

In graphene, however, these processes are kinematicallysuppressed by the conservation of energy ε and momen-tum p, because the spectrum ε(p) of thequasiparticles isnot decaying

d2ε(p)

d2p≤ 0.

(Negative curvature of the spectrum at T = 0K arises due to the logarithmic renormalization ofthe Fermi velocity vF attributed to electron-electroninteractions.)13,14,15 As explicated in Fig. 1, the linearspectrum allows only decay products with collinear mo-menta (i.e. pure forward scattering), but these processesmake a negligible contribution to the imbalance relax-ation. The sublinear spectrum forbids even this forwardscattering decay. In clean graphene, higher order (e.g.three particle collision) imbalance relaxation processesarealready allowed,whileimpurity-assistedcollisionswillcontribute in a disordered sample; τQ is therefore likelyfinite, although it may significantly exceed other relax-ation times in the system.In the limit of zero relaxation, a graphene monolayerprobed through thermally conducting, electrically insu-lating contacts would possess electron and hole popu-lations that are strictly conserved. In direct analogywith a single component, non-relativistic classical gas,16

the electron-hole plasma in clean graphene with vanish-ing imbalance relaxation would exhibit a finiteelectronicthermal conductivity κ for arbitrarily large system sizes.(The imbalance relaxation length lQ , introduced above,diverges as τQ →∞.) In this regime, interparticle colli-sions facilitate heat conduction without particle numberconvection. In this paper, we will demonstrate that thesamebehavior obtains for non-zero imbalance relaxation(1/lQ > 0) in the limit of short samples, L ≪ lQ . Bycomparison, prior work10 effectively assumed infinite re-

laxation of population imbalance (lQ → 0). We demon-strate that the results previously obtained in Ref. 10 forboth κ and the thermoelectric power α at non-zero dop-ing and temperature emerge in the limit of asymptoti-cally large system sizes L ≫ lQ for a finite rate of im-balance relaxation, (lQ > 0). We will show that whenthe ends of a graphene slab of length L lQ are held atdisparate temperaturesand noelectriccurrent ispermit-ted to flow, steady state particle convection does nev-ertheless occur; carrier flux is created or destroyed byimbalance relaxation processes near the terminals of thedevice. Thermopower measurements require the junc-tion of the graphene slab with metallic contacts. Weincorporate into our calculations carrier exchange withnon-ideal contacts, which also relax the imbalance, andwecarefully delimit the regimesin which deviationsfromthe infinite relaxation limit should be observable in ex-periments. The effects of weak quenched disorder areincluded in all of our computations.

In this paper, we restrict our attention to the hy-drodynamic (or “interaction-limited”) transport regime,whereinelasticinterparticlecollisionsdominateover elas-tic impurity scattering. Prior work addressing ther-moelectric transport in the opposite, “disorder-limited”regime, in which real carrier-carrier scattering processesmay be neglected, includes that of Refs. 11,17,18. Inthe disorder-limited case, κ and α are determined bythe energy-dependenceof the electrical conductivity, viathe “generalized” Wiedemann-Franz law and Mott re-lation, respectively.19 The effect of the slow imbalancerelaxation upon the dc conductivity in graphene undernon-equilibrium interband photoexcitation hasalso beenaddressed.21

What essential new physics emerges through the in-corporation of imbalance relaxation effects into the de-scription of thermoelectric transport, and how can itbe extracted from experiments? The entirety of lineartransport phenomena in graphene within the hydrody-namic regime is essentially quantified by four intrinsicparameters. A finite rate of imbalance relaxation meansthat electrons and holes respond independently to ex-ternal forces; the single “quantum critical” conductiv-ity identified previously in Refs. 8,9,10 generalizes to a2x2 tensor of coefficients, with diagonal elements σee,σhh and off-diagonal elements σeh = σhe; all are me-diated entirely by inelastic interparticle collisions. Thedescription is similar to that of Coulomb drag:22 the di-agonal element σee (σhh ) characterizes the response ofthe conduction band electrons (valence band holes) to a(gedanken) electricfield that couplesonly to that carriertype, whereas σeh characterizes the “drag” exerted byone carrier species upon the other (due to electron-holecollisions) under the application of such a field. σee andσhh are related by particle-hole symmetry, leaving twoindependent parameters which we can take as σeh andσmin ≡ (σee+ σhh−2σeh ); the latter combination isequalto thebulk dcelectrical conductivity σ at theDiracpointin thehydrodynamicregime.12 The imbalance relaxation

Origin of two current :Imbalance by Kinematics

2

FIG. 1: K inematical constraints for imbalance relaxation.The left figure shows the Feynman diagram for the typi-cal two-particle decay process given by Eq. (1.1a); εi andp i respectively denote the energy and momentum of the i

t h

electron or hole. The right figure depicts momentum con-servation for this process. The length of the dashed pathL f ≡ |p2|+ |p3|+ |p4|traced out by the sum of “decay prod-uct” momenta is always greater than or equal to the lengthL i ≡ |p1|of the“parent” particlemomentum: L f ≥ L i . I f theparticle energy spectrum takes the form ε(p) = |p|β , then thedepicted decay process is kinematically forbidden for β < 1.For β = 1, only forward-scattering is allowed.

h+ ↔ h+ + h+ + e−. (1.1b)

In graphene, however, these processes are kinematicallysuppressed by the conservation of energy ε and momen-tum p, because the spectrum ε(p) of the quasiparticles isnot decaying

d2ε(p)

d2p≤ 0.

(Negative curvature of the spectrum at T = 0K arises due to the logarithmic renormalization ofthe Fermi velocity vF attributed to electron-electroninteractions.)13,14,15 As explicated in Fig. 1, the linearspectrum allows only decay products with collinear mo-menta (i.e. pure forward scattering), but these processesmake a negligible contribution to the imbalance relax-ation. The sublinear spectrum forbids even this forwardscattering decay. In clean graphene, higher order (e.g.three particle collision) imbalance relaxation processesarealready allowed,whileimpurity-assisted collisionswillcontribute in a disordered sample; τQ is therefore likelyfinite, although it may significantly exceed other relax-ation times in the system.In the limit of zero relaxation, a graphene monolayer

probed through thermally conducting, electrically insu-lating contacts would possess electron and hole popu-lations that are strictly conserved. In direct analogywith a single component, non-relativistic classical gas,16

the electron-hole plasma in clean graphene with vanish-ing imbalance relaxation would exhibit a finite electronicthermal conductivity κ for arbitrarily large system sizes.(The imbalance relaxation length lQ , introduced above,diverges as τQ → ∞.) In this regime, interparticle colli-sions facilitate heat conduction without particle numberconvection. In this paper, we will demonstrate that thesamebehavior obtains for non-zero imbalance relaxation(1/lQ > 0) in the limit of short samples, L ≪ lQ . Bycomparison, prior work10 effectively assumed infinite re-

laxation of population imbalance (lQ → 0). We demon-strate that the results previously obtained in Ref. 10 forboth κ and the thermoelectric power α at non-zero dop-ing and temperature emerge in the limit of asymptoti-cally large system sizes L ≫ lQ for a finite rate of im-balance relaxation, (lQ > 0). We will show that whenthe ends of a graphene slab of length L lQ are held atdisparate temperaturesand no electriccurrent ispermit-ted to flow, steady state particle convection does nev-ertheless occur; carrier flux is created or destroyed byimbalance relaxation processes near the terminals of thedevice. Thermopower measurements require the junc-tion of the graphene slab with metallic contacts. Weincorporate into our calculations carrier exchange withnon-ideal contacts, which also relax the imbalance, andwecarefully delimit the regimes in which deviations fromthe infinite relaxation limit should be observable in ex-periments. The effects of weak quenched disorder areincluded in all of our computations.

In this paper, we restrict our attention to the hy-drodynamic (or “interaction-limited”) transport regime,whereinelastic interparticlecollisionsdominateover elas-tic impurity scattering. Prior work addressing ther-moelectric transport in the opposite, “disorder-limited”regime, in which real carrier-carrier scattering processesmay be neglected, includes that of Refs. 11,17,18. Inthe disorder-limited case, κ and α are determined bythe energy-dependence of the electrical conductivity, viathe “generalized” Wiedemann-Franz law and Mott re-lation, respectively.19 The effect of the slow imbalancerelaxation upon the dc conductivity in graphene undernon-equilibrium interband photoexcitation hasalso beenaddressed.21

What essential new physics emerges through the in-corporation of imbalance relaxation effects into the de-scription of thermoelectric transport, and how can itbe extracted from experiments? The entirety of lineartransport phenomena in graphene within the hydrody-namic regime is essentially quantified by four intrinsicparameters. A finite rate of imbalance relaxation meansthat electrons and holes respond independently to ex-ternal forces; the single “quantum critical” conductiv-ity identified previously in Refs. 8,9,10 generalizes to a2x2 tensor of coefficients, with diagonal elements σee,σhh and off-diagonal elements σeh = σhe; all are me-diated entirely by inelastic interparticle collisions. Thedescription is similar to that of Coulomb drag:22 the di-agonal element σee (σhh ) characterizes the response ofthe conduction band electrons (valence band holes) to a(gedanken) electric field that couples only to that carriertype, whereas σeh characterizes the “drag” exerted byone carrier species upon the other (due to electron-holecollisions) under the application of such a field. σee andσhh are related by particle-hole symmetry, leaving twoindependent parameters which we can take as σeh andσmin ≡ (σee+ σhh−2σeh ); the latter combination is equalto thebulk dcelectrical conductivity σ at theDiracpointin the hydrodynamic regime.12 The imbalance relaxation

11

2

FIG. 1: K inematical constraints for imbalance relaxation.The left figure shows the Feynman diagram for the typi-cal two-particle decay process given by Eq. (1.1a); εi andpi respectively denote the energy and momentum of the i

t h

electron or hole. The right figure depicts momentum con-servation for this process. The length of the dashed pathL f ≡ |p2|+ |p3|+ |p4|traced out by the sum of “decay prod-uct” momenta is always greater than or equal to the lengthL i ≡ |p1|of the“parent” particlemomentum: L f ≥ L i . I f theparticleenergy spectrum takesthe form ε(p) = |p|β , then thedepicted decay process is kinematically forbidden for β < 1.For β = 1, only forward-scattering is allowed.

h+ ↔ h+ + h+ + e−. (1.1b)

In graphene, however, these processes are kinematicallysuppressed by the conservation of energy ε and momen-tum p, because thespectrum ε(p) of thequasiparticles isnot decaying

d2ε(p)

d2p≤ 0.

(Negative curvature of the spectrum at T = 0K arises due to the logarithmic renormalization ofthe Fermi velocity vF attributed to electron-electroninteractions.)13,14,15 As explicated in Fig. 1, the linearspectrum allows only decay products with collinear mo-menta (i.e. pure forward scattering), but theseprocessesmake a negligible contribution to the imbalance relax-ation. The sublinear spectrum forbids even this forwardscattering decay. In clean graphene, higher order (e.g.three particle collision) imbalance relaxation processesarealready allowed,whileimpurity-assistedcollisionswillcontribute in a disordered sample; τQ is therefore likelyfinite, although it may significantly exceed other relax-ation times in the system.In the limit of zero relaxation, a graphene monolayerprobed through thermally conducting, electrically insu-lating contacts would possess electron and hole popu-lations that are strictly conserved. In direct analogywith a single component, non-relativistic classical gas,16

the electron-hole plasma in clean graphenewith vanish-ing imbalancerelaxation would exhibit a finiteelectronicthermal conductivity κ for arbitrarily largesystem sizes.(The imbalance relaxation length lQ , introduced above,diverges as τQ →∞.) In this regime, interparticle colli-sions facilitate heat conduction without particle numberconvection. In this paper, we will demonstrate that thesamebehavior obtains for non-zero imbalance relaxation(1/lQ > 0) in the limit of short samples, L ≪ lQ . Bycomparison, prior work10 effectively assumed infinite re-

laxation of population imbalance (lQ → 0). We demon-strate that the results previously obtained in Ref. 10 forboth κ and the thermoelectric power α at non-zero dop-ing and temperature emerge in the limit of asymptoti-cally large system sizes L ≫ lQ for a finite rate of im-balance relaxation, (lQ > 0). We will show that whenthe ends of a grapheneslab of length L lQ are held atdisparatetemperaturesand noelectriccurrent ispermit-ted to flow, steady state particle convection does nev-ertheless occur; carrier flux is created or destroyed byimbalance relaxation processesnear the terminals of thedevice. Thermopower measurements require the junc-tion of the graphene slab with metallic contacts. Weincorporate into our calculations carrier exchange withnon-ideal contacts, which also relax the imbalance, andwecarefully delimit theregimesin which deviationsfromthe infinite relaxation limit should be observable in ex-periments. The effects of weak quenched disorder areincluded in all of our computations.

In this paper, we restrict our attention to the hy-drodynamic (or “interaction-limited”) transport regime,whereinelasticinterparticlecollisionsdominateover elas-tic impurity scattering. Prior work addressing ther-moelectric transport in the opposite, “disorder-limited”regime, in which real carrier-carrier scattering processesmay be neglected, includes that of Refs. 11,17,18. Inthe disorder-limited case, κ and α are determined bythe energy-dependenceof the electrical conductivity, viathe “generalized” Wiedemann-Franz law and Mott re-lation, respectively.19 The effect of the slow imbalancerelaxation upon the dc conductivity in graphene undernon-equilibrium interband photoexcitation hasalsobeenaddressed.21

What essential new physics emerges through the in-corporation of imbalance relaxation effects into the de-scription of thermoelectric transport, and how can itbe extracted from experiments? The entirety of lineartransport phenomena in graphene within the hydrody-namic regime is essentially quantified by four intrinsicparameters. A finite rate of imbalance relaxation meansthat electrons and holes respond independently to ex-ternal forces; the single “quantum critical” conductiv-ity identified previously in Refs. 8,9,10 generalizes to a2x2 tensor of coefficients, with diagonal elements σee,σhh and off-diagonal elements σeh = σhe; all are me-diated entirely by inelastic interparticle collisions. Thedescription is similar to that of Coulomb drag:22 the di-agonal element σee (σhh ) characterizes the response ofthe conduction band electrons (valence band holes) to a(gedanken) electricfield that couplesonly to that carriertype, whereas σeh characterizes the “drag” exerted byone carrier species upon the other (due to electron-holecollisions) under the application of such a field. σee andσhh are related by particle-hole symmetry, leaving twoindependent parameters which we can take as σeh andσmin ≡ (σee+ σhh−2σeh ); the latter combination isequalto thebulk dcelectrical conductivity σ at theDiracpointin thehydrodynamicregime.12 The imbalancerelaxation

2

FIG. 1: K inematical constraints for imbalance relaxation.The left figure shows the Feynman diagram for the typi-cal two-particle decay process given by Eq. (1.1a); εi andpi respectively denote the energy and momentum of the i

t h

electron or hole. The right figure depicts momentum con-servation for this process. The length of the dashed pathL f ≡ |p2|+ |p3|+ |p4|traced out by the sum of “decay prod-uct” momenta is always greater than or equal to the lengthL i ≡ |p1|of the“parent” particlemomentum: L f ≥ L i . I f theparticle energy spectrum takes the form ε(p) = |p|β , then thedepicted decay process is kinematically forbidden for β < 1.For β = 1, only forward-scattering is allowed.

h+ ↔ h+ + h+ + e−. (1.1b)

In graphene, however, these processes are kinematicallysuppressed by the conservation of energy ε and momen-tum p, because thespectrum ε(p) of thequasiparticles isnot decaying

d2ε(p)

d2p≤ 0.

(Negative curvature of the spectrum at T = 0K arises due to the logarithmic renormalization ofthe Fermi velocity vF attributed to electron-electroninteractions.)13,14,15 As explicated in Fig. 1, the linearspectrum allows only decay products with collinear mo-menta (i.e. pure forward scattering), but these processesmake a negligible contribution to the imbalance relax-ation. The sublinear spectrum forbids even this forwardscattering decay. In clean graphene, higher order (e.g.three particle collision) imbalance relaxation processesarealready allowed,whileimpurity-assistedcollisionswillcontribute in a disordered sample; τQ is therefore likelyfinite, although it may significantly exceed other relax-ation times in the system.In the limit of zero relaxation, a graphene monolayerprobed through thermally conducting, electrically insu-lating contacts would possess electron and hole popu-lations that are strictly conserved. In direct analogywith a single component, non-relativistic classical gas,16

the electron-hole plasma in clean graphenewith vanish-ing imbalance relaxation would exhibit a finiteelectronicthermal conductivity κ for arbitrarily largesystem sizes.(The imbalance relaxation length lQ , introduced above,diverges as τQ →∞.) In this regime, interparticle colli-sions facilitate heat conduction without particle numberconvection. In this paper, we will demonstrate that thesamebehavior obtains for non-zero imbalance relaxation(1/lQ > 0) in the limit of short samples, L ≪ lQ . Bycomparison, prior work10 effectively assumed infinite re-

laxation of population imbalance (lQ → 0). We demon-strate that the results previously obtained in Ref. 10 forboth κ and the thermoelectric power α at non-zero dop-ing and temperature emerge in the limit of asymptoti-cally large system sizes L ≫ lQ for a finite rate of im-balance relaxation, (lQ > 0). We will show that whenthe ends of a grapheneslab of length L lQ are held atdisparatetemperaturesand noelectriccurrent ispermit-ted to flow, steady state particle convection does nev-ertheless occur; carrier flux is created or destroyed byimbalance relaxation processesnear the terminals of thedevice. Thermopower measurements require the junc-tion of the graphene slab with metallic contacts. Weincorporate into our calculations carrier exchange withnon-ideal contacts, which also relax the imbalance, andwecarefully delimit the regimesin which deviationsfromthe infinite relaxation limit should be observable in ex-periments. The effects of weak quenched disorder areincluded in all of our computations.

In this paper, we restrict our attention to the hy-drodynamic (or “interaction-limited”) transport regime,whereinelasticinterparticlecollisionsdominateover elas-tic impurity scattering. Prior work addressing ther-moelectric transport in the opposite, “disorder-limited”regime, in which real carrier-carrier scattering processesmay be neglected, includes that of Refs. 11,17,18. Inthe disorder-limited case, κ and α are determined bythe energy-dependenceof the electrical conductivity, viathe “generalized” Wiedemann-Franz law and Mott re-lation, respectively.19 The effect of the slow imbalancerelaxation upon the dc conductivity in graphene undernon-equilibrium interband photoexcitation hasalsobeenaddressed.21

What essential new physics emerges through the in-corporation of imbalance relaxation effects into the de-scription of thermoelectric transport, and how can itbe extracted from experiments? The entirety of lineartransport phenomena in graphene within the hydrody-namic regime is essentially quantified by four intrinsicparameters. A finite rate of imbalance relaxation meansthat electrons and holes respond independently to ex-ternal forces; the single “quantum critical” conductiv-ity identified previously in Refs. 8,9,10 generalizes to a2x2 tensor of coefficients, with diagonal elements σee,σhh and off-diagonal elements σeh = σhe; all are me-diated entirely by inelastic interparticle collisions. Thedescription is similar to that of Coulomb drag:22 the di-agonal element σee (σhh ) characterizes the response ofthe conduction band electrons (valence band holes) to a(gedanken) electricfield that couplesonly to that carriertype, whereas σeh characterizes the “drag” exerted byone carrier species upon the other (due to electron-holecollisions) under the application of such a field. σee andσhh are related by particle-hole symmetry, leaving twoindependent parameters which we can take as σeh andσmin ≡ (σee+ σhh−2σeh ); the latter combination isequalto thebulk dcelectrical conductivity σ at theDiracpointin thehydrodynamicregime.12 The imbalance relaxation

2

FIG. 1: K inematical constraints for imbalance relaxation.The left figure shows the Feynman diagram for the typi-cal two-particle decay process given by Eq. (1.1a); εi andpi respectively denote the energy and momentum of the i

t h

electron or hole. The right figure depicts momentum con-servation for this process. The length of the dashed pathL f ≡ |p2|+ |p3|+ |p4|traced out by the sum of “decay prod-uct” momenta is always greater than or equal to the lengthL i ≡ |p1|of the“parent” particlemomentum: L f ≥ L i . I f theparticleenergy spectrum takesthe form ε(p) = |p|β , then thedepicted decay process is kinematically forbidden for β < 1.For β = 1, only forward-scattering is allowed.

h+ ↔ h+ + h+ + e−. (1.1b)

In graphene, however, these processes are kinematicallysuppressed by the conservation of energy ε and momen-tum p, because thespectrum ε(p) of thequasiparticles isnot decaying

d2ε(p)

d2p≤ 0.

(Negative curvature of the spectrum at T = 0K arises due to the logarithmic renormalization ofthe Fermi velocity vF attributed to electron-electroninteractions.)13,14,15 As explicated in Fig. 1, the linearspectrum allows only decay products with collinear mo-menta (i.e. pure forward scattering), but theseprocessesmake a negligible contribution to the imbalance relax-ation. The sublinear spectrum forbids even this forwardscattering decay. In clean graphene, higher order (e.g.three particle collision) imbalance relaxation processesarealready allowed,whileimpurity-assistedcollisionswillcontribute in a disordered sample; τQ is therefore likelyfinite, although it may significantly exceed other relax-ation times in the system.In the limit of zero relaxation, a graphene monolayerprobed through thermally conducting, electrically insu-lating contacts would possess electron and hole popu-lations that are strictly conserved. In direct analogywith a single component, non-relativistic classical gas,16

the electron-hole plasma in clean graphenewith vanish-ing imbalancerelaxation would exhibit a finiteelectronicthermal conductivity κ for arbitrarily largesystem sizes.(The imbalance relaxation length lQ , introduced above,diverges as τQ →∞.) In this regime, interparticle colli-sions facilitate heat conduction without particle numberconvection. In this paper, we will demonstrate that thesamebehavior obtains for non-zero imbalance relaxation(1/lQ > 0) in the limit of short samples, L ≪ lQ . Bycomparison, prior work10 effectively assumed infinite re-

laxation of population imbalance (lQ → 0). We demon-strate that the results previously obtained in Ref. 10 forboth κ and the thermoelectric power α at non-zero dop-ing and temperature emerge in the limit of asymptoti-cally large system sizes L ≫ lQ for a finite rate of im-balance relaxation, (lQ > 0). We will show that whenthe ends of a grapheneslab of length L lQ are held atdisparatetemperaturesand noelectriccurrent ispermit-ted to flow, steady state particle convection does nev-ertheless occur; carrier flux is created or destroyed byimbalance relaxation processesnear the terminals of thedevice. Thermopower measurements require the junc-tion of the graphene slab with metallic contacts. Weincorporate into our calculations carrier exchange withnon-ideal contacts, which also relax the imbalance, andwecarefully delimit theregimesin which deviationsfromthe infinite relaxation limit should be observable in ex-periments. The effects of weak quenched disorder areincluded in all of our computations.

In this paper, we restrict our attention to the hy-drodynamic (or “interaction-limited”) transport regime,whereinelasticinterparticlecollisionsdominateover elas-tic impurity scattering. Prior work addressing ther-moelectric transport in the opposite, “disorder-limited”regime, in which real carrier-carrier scattering processesmay be neglected, includes that of Refs. 11,17,18. Inthe disorder-limited case, κ and α are determined bythe energy-dependenceof the electrical conductivity, viathe “generalized” Wiedemann-Franz law and Mott re-lation, respectively.19 The effect of the slow imbalancerelaxation upon the dc conductivity in graphene undernon-equilibrium interband photoexcitation hasalsobeenaddressed.21

What essential new physics emerges through the in-corporation of imbalance relaxation effects into the de-scription of thermoelectric transport, and how can itbe extracted from experiments? The entirety of lineartransport phenomena in graphene within the hydrody-namic regime is essentially quantified by four intrinsicparameters. A finite rate of imbalance relaxation meansthat electrons and holes respond independently to ex-ternal forces; the single “quantum critical” conductiv-ity identified previously in Refs. 8,9,10 generalizes to a2x2 tensor of coefficients, with diagonal elements σee,σhh and off-diagonal elements σeh = σhe; all are me-diated entirely by inelastic interparticle collisions. Thedescription is similar to that of Coulomb drag:22 the di-agonal element σee (σhh ) characterizes the response ofthe conduction band electrons (valence band holes) to a(gedanken) electricfield that couplesonly to that carriertype, whereas σeh characterizes the “drag” exerted byone carrier species upon the other (due to electron-holecollisions) under the application of such a field. σee andσhh are related by particle-hole symmetry, leaving twoindependent parameters which we can take as σeh andσmin ≡ (σee+ σhh−2σeh ); the latter combination isequalto thebulk dcelectrical conductivity σ at theDiracpointin thehydrodynamicregime.12 The imbalancerelaxation

2

FIG. 1: Kinematical constraints for imbalance relaxation.The left figure shows the Feynman diagram for the typi-cal two-particle decay process given by Eq. (1.1a); εi andpi respectively denote the energy and momentum of the i

t h

electron or hole. The right figure depicts momentum con-servation for this process. The length of the dashed pathL f ≡ |p2|+ |p3|+ |p4|traced out by thesum of “decay prod-uct” momenta is always greater than or equal to the lengthL i ≡ |p1|of the“parent” particlemomentum: L f ≥ L i . If theparticleenergy spectrum takestheform ε(p) = |p|β , then thedepicted decay process is kinematically forbidden for β < 1.For β = 1, only forward-scattering is allowed.

h+ ↔ h+ + h+ + e−. (1.1b)

In graphene, however, these processes are kinematicallysuppressed by the conservation of energy ε and momen-tum p, becausethespectrum ε(p) of thequasiparticles isnot decaying

d2ε(p)

d2p≤0.

(Negative curvature of the spectrum at T = 0K arises due to the logarithmic renormalization ofthe Fermi velocity vF attributed to electron-electroninteractions.)13,14,15 As explicated in Fig. 1, the linearspectrum allowsonly decay productswith collinear mo-menta (i.e. pure forward scattering), but theseprocessesmake a negligible contribution to the imbalance relax-ation. The sublinear spectrum forbids even this forwardscattering decay. In clean graphene, higher order (e.g.three particle collision) imbalance relaxation processesarealready allowed,whileimpurity-assistedcollisionswillcontribute in a disordered sample; τQ is therefore likelyfinite, although it may significantly exceed other relax-ation times in the system.In the limit of zero relaxation, a graphenemonolayerprobed through thermally conducting, electrically insu-lating contacts would possess electron and hole popu-lations that are strictly conserved. In direct analogywith a single component, non-relativistic classical gas,16

the electron-holeplasma in clean graphenewith vanish-ing imbalancerelaxation would exhibit a finiteelectronicthermal conductivity κ for arbitrarily largesystem sizes.(The imbalance relaxation length lQ , introduced above,divergesas τQ →∞.) In this regime, interparticle colli-sions facilitate heat conduction without particle numberconvection. In this paper, wewill demonstrate that thesamebehavior obtainsfor non-zero imbalancerelaxation(1/lQ > 0) in the limit of short samples, L ≪ lQ. Bycomparison, prior work10 effectively assumed infinite re-

laxation of population imbalance (lQ → 0). We demon-strate that the results previously obtained in Ref. 10 forboth κ and the thermoelectricpower α at non-zerodop-ing and temperature emerge in the limit of asymptoti-cally large system sizes L ≫ lQ for a finite rate of im-balance relaxation, (lQ > 0). We will show that whenthe endsof a grapheneslab of length L lQ areheld atdisparatetemperaturesand noelectriccurrent ispermit-ted to flow, steady state particle convection does nev-ertheless occur; carrier flux is created or destroyed byimbalance relaxation processesnear the terminalsof thedevice. Thermopower measurements require the junc-tion of the graphene slab with metallic contacts. Weincorporate into our calculations carrier exchange withnon-ideal contacts, which also relax the imbalance, andwecarefully delimit theregimesin which deviationsfromthe infinite relaxation limit should be observable in ex-periments. The effects of weak quenched disorder areincluded in all of our computations.

In this paper, we restrict our attention to the hy-drodynamic (or “interaction-limited”) transport regime,whereinelasticinterparticlecollisionsdominateover elas-tic impurity scattering. Prior work addressing ther-moelectric transport in the opposite, “disorder-limited”regime, in which real carrier-carrier scattering processesmay be neglected, includes that of Refs. 11,17,18. Inthe disorder-limited case, κ and α are determined bythe energy-dependenceof the electrical conductivity, viathe “generalized” Wiedemann-Franz law and Mott re-lation, respectively.19 The effect of the slow imbalancerelaxation upon the dc conductivity in graphene undernon-equilibrium interband photoexcitation hasalsobeenaddressed.21

What essential new physics emerges through the in-corporation of imbalance relaxation effects into the de-scription of thermoelectric transport, and how can itbe extracted from experiments? The entirety of lineartransport phenomena in graphene within the hydrody-namic regime is essentially quantified by four intrinsicparameters. A finite rateof imbalance relaxation meansthat electrons and holes respond independently to ex-ternal forces; the single “quantum critical” conductiv-ity identified previously in Refs. 8,9,10 generalizes to a2x2 tensor of coefficients, with diagonal elements σee,σhh and off-diagonal elements σeh = σhe; all are me-diated entirely by inelastic interparticle collisions. Thedescription is similar to that of Coulomb drag:22 the di-agonal element σee (σhh) characterizes the response ofthe conduction band electrons (valence band holes) to a(gedanken) electricfield that couplesonly to that carriertype, whereas σeh characterizes the “drag” exerted byone carrier species upon the other (due to electron-holecollisions) under the application of such a field. σee andσhh are related by particle-hole symmetry, leaving twoindependent parameters which we can take as σeh andσmin ≡ (σee+ σhh−2σeh); thelatter combination isequaltothebulk dcelectrical conductivity σ at theDiracpointin thehydrodynamicregime.12 Theimbalancerelaxation

2

FIG. 1: K inematical constraints for imbalance relaxation.The left figure shows the Feynman diagram for the typi-cal two-particle decay process given by Eq. (1.1a); εi andpi respectively denote the energy and momentum of the i

t h

electron or hole. The right figure depicts momentum con-servation for this process. The length of the dashed pathL f ≡ |p2|+ |p3|+ |p4|traced out by thesum of “decay prod-uct” momenta is always greater than or equal to the lengthL i ≡ |p1|of the“parent” particlemomentum: L f ≥ L i . If theparticleenergy spectrum takestheform ε(p) = |p|β , then thedepicted decay process is kinematically forbidden for β < 1.For β = 1, only forward-scattering is allowed.

h+ ↔ h+ + h+ + e−. (1.1b)

In graphene, however, these processes are kinematicallysuppressed by the conservation of energy ε and momen-tum p, becausethespectrum ε(p) of thequasiparticles isnot decaying

d2ε(p)

d2p≤0.

(Negative curvature of the spectrum at T = 0K arises due to the logarithmic renormalization ofthe Fermi velocity vF attributed to electron-electroninteractions.)13,14,15 As explicated in Fig. 1, the linearspectrum allowsonly decay productswith collinear mo-menta (i.e. pure forward scattering), but theseprocessesmake a negligible contribution to the imbalance relax-ation. The sublinear spectrum forbids even this forwardscattering decay. In clean graphene, higher order (e.g.three particle collision) imbalance relaxation processesarealready allowed,whileimpurity-assistedcollisionswillcontribute in a disordered sample; τQ is therefore likelyfinite, although it may significantly exceed other relax-ation times in the system.In the limit of zero relaxation, a graphenemonolayerprobed through thermally conducting, electrically insu-lating contacts would possess electron and hole popu-lations that are strictly conserved. In direct analogywith a single component, non-relativistic classical gas,16

the electron-hole plasma in clean graphenewith vanish-ing imbalancerelaxationwould exhibit a finiteelectronicthermal conductivity κ for arbitrarily largesystem sizes.(The imbalance relaxation length lQ , introduced above,diverges as τQ →∞.) In this regime, interparticle colli-sions facilitate heat conduction without particle numberconvection. In this paper, wewill demonstrate that thesamebehavior obtains for non-zero imbalancerelaxation(1/lQ > 0) in the limit of short samples, L ≪ lQ. Bycomparison, prior work10 effectively assumed infinite re-

laxation of population imbalance (lQ → 0). We demon-strate that the results previously obtained in Ref. 10 forboth κ and the thermoelectricpower α at non-zero dop-ing and temperature emerge in the limit of asymptoti-cally large system sizes L ≫ lQ for a finite rate of im-balance relaxation, (lQ > 0). We will show that whenthe endsof a grapheneslab of length L lQ areheld atdisparatetemperaturesand noelectriccurrent ispermit-ted to flow, steady state particle convection does nev-ertheless occur; carrier flux is created or destroyed byimbalance relaxation processesnear the terminals of thedevice. Thermopower measurements require the junc-tion of the graphene slab with metallic contacts. Weincorporate into our calculations carrier exchange withnon-ideal contacts, which also relax the imbalance, andwecarefully delimit theregimesin which deviationsfromthe infinite relaxation limit should be observable in ex-periments. The effects of weak quenched disorder areincluded in all of our computations.

In this paper, we restrict our attention to the hy-drodynamic (or “interaction-limited”) transport regime,whereinelasticinterparticlecollisionsdominateover elas-tic impurity scattering. Prior work addressing ther-moelectric transport in the opposite, “disorder-limited”regime, in which real carrier-carrier scattering processesmay be neglected, includes that of Refs. 11,17,18. Inthe disorder-limited case, κ and α are determined bythe energy-dependenceof the electrical conductivity, viathe “generalized” Wiedemann-Franz law and Mott re-lation, respectively.19 The effect of the slow imbalancerelaxation upon the dc conductivity in graphene undernon-equilibrium interband photoexcitation hasalsobeenaddressed.21

What essential new physics emerges through the in-corporation of imbalance relaxation effects into the de-scription of thermoelectric transport, and how can itbe extracted from experiments? The entirety of lineartransport phenomena in graphene within the hydrody-namic regime is essentially quantified by four intrinsicparameters. A finite rateof imbalance relaxation meansthat electrons and holes respond independently to ex-ternal forces; the single “quantum critical” conductiv-ity identified previously in Refs. 8,9,10 generalizes to a2x2 tensor of coefficients, with diagonal elements σee,σhh and off-diagonal elements σeh = σhe; all are me-diated entirely by inelastic interparticle collisions. Thedescription is similar to that of Coulomb drag:22 the di-agonal element σee (σhh) characterizes the response ofthe conduction band electrons (valence band holes) to a(gedanken) electricfield that couplesonly to that carriertype, whereas σeh characterizes the “drag” exerted byone carrier species upon the other (due to electron-holecollisions) under the application of such a field. σee andσhh are related by particle-hole symmetry, leaving twoindependent parameters which we can take as σeh andσmin ≡ (σee+ σhh−2σeh); thelatter combination isequaltothebulk dcelectrical conductivity σ at theDiracpointin thehydrodynamicregime.12 Theimbalancerelaxation

2

FIG. 1: K inematical constraints for imbalance relaxation.The left figure shows the Feynman diagram for the typi-cal two-particle decay process given by Eq. (1.1a); εi andpi respectively denote the energy and momentum of the i

t h

electron or hole. The right figure depicts momentum con-servation for this process. The length of the dashed pathL f ≡ |p2|+ |p3|+ |p4|traced out by thesum of “decay prod-uct” momenta is always greater than or equal to the lengthL i ≡ |p1|of the“parent” particlemomentum: L f ≥ L i . If theparticleenergy spectrum takes the form ε(p) = |p|β , then thedepicted decay process is kinematically forbidden for β < 1.For β = 1, only forward-scattering is allowed.

h+ ↔ h+ + h+ + e−. (1.1b)

In graphene, however, these processes are kinematicallysuppressed by the conservation of energy ε and momen-tum p, because thespectrum ε(p) of thequasiparticles isnot decaying

d2ε(p)

d2p≤ 0.

(Negative curvature of the spectrum at T = 0K arises due to the logarithmic renormalization ofthe Fermi velocity vF attributed to electron-electroninteractions.)13,14,15 As explicated in Fig. 1, the linearspectrum allows only decay productswith collinear mo-menta (i.e. pure forward scattering), but theseprocessesmake a negligible contribution to the imbalance relax-ation. The sublinear spectrum forbids even this forwardscattering decay. In clean graphene, higher order (e.g.three particle collision) imbalance relaxation processesarealready allowed,whileimpurity-assistedcollisionswillcontribute in a disordered sample; τQ is therefore likelyfinite, although it may significantly exceed other relax-ation times in the system.In the limit of zero relaxation, a graphene monolayerprobed through thermally conducting, electrically insu-lating contacts would possess electron and hole popu-lations that are strictly conserved. In direct analogywith a single component, non-relativistic classical gas,16

the electron-hole plasma in clean graphenewith vanish-ing imbalancerelaxation would exhibit a finiteelectronicthermal conductivity κ for arbitrarily largesystem sizes.(The imbalance relaxation length lQ , introduced above,diverges as τQ →∞.) In this regime, interparticle colli-sions facilitate heat conduction without particle numberconvection. In this paper, we will demonstrate that thesamebehavior obtains for non-zero imbalance relaxation(1/lQ > 0) in the limit of short samples, L ≪ lQ . Bycomparison, prior work10 effectively assumed infinite re-

laxation of population imbalance (lQ → 0). We demon-strate that the results previously obtained in Ref. 10 forboth κ and the thermoelectric power α at non-zero dop-ing and temperature emerge in the limit of asymptoti-cally large system sizes L ≫ lQ for a finite rate of im-balance relaxation, (lQ > 0). We will show that whenthe ends of a grapheneslab of length L lQ are held atdisparatetemperaturesand noelectriccurrent ispermit-ted to flow, steady state particle convection does nev-ertheless occur; carrier flux is created or destroyed byimbalance relaxation processesnear the terminals of thedevice. Thermopower measurements require the junc-tion of the graphene slab with metallic contacts. Weincorporate into our calculations carrier exchange withnon-ideal contacts, which also relax the imbalance, andwecarefully delimit theregimesin which deviationsfromthe infinite relaxation limit should be observable in ex-periments. The effects of weak quenched disorder areincluded in all of our computations.

In this paper, we restrict our attention to the hy-drodynamic (or “interaction-limited”) transport regime,whereinelasticinterparticlecollisionsdominateover elas-tic impurity scattering. Prior work addressing ther-moelectric transport in the opposite, “disorder-limited”regime, in which real carrier-carrier scattering processesmay be neglected, includes that of Refs. 11,17,18. Inthe disorder-limited case, κ and α are determined bythe energy-dependenceof the electrical conductivity, viathe “generalized” Wiedemann-Franz law and Mott re-lation, respectively.19 The effect of the slow imbalancerelaxation upon the dc conductivity in graphene undernon-equilibrium interband photoexcitation hasalsobeenaddressed.21

What essential new physics emerges through the in-corporation of imbalance relaxation effects into the de-scription of thermoelectric transport, and how can itbe extracted from experiments? The entirety of lineartransport phenomena in graphene within the hydrody-namic regime is essentially quantified by four intrinsicparameters. A finite rate of imbalance relaxation meansthat electrons and holes respond independently to ex-ternal forces; the single “quantum critical” conductiv-ity identified previously in Refs. 8,9,10 generalizes to a2x2 tensor of coefficients, with diagonal elements σee,σhh and off-diagonal elements σeh = σhe; all are me-diated entirely by inelastic interparticle collisions. Thedescription is similar to that of Coulomb drag:22 the di-agonal element σee (σhh) characterizes the response ofthe conduction band electrons (valence band holes) to a(gedanken) electricfield that couplesonly to that carriertype, whereas σeh characterizes the “drag” exerted byone carrier species upon the other (due to electron-holecollisions) under the application of such a field. σee andσhh are related by particle-hole symmetry, leaving twoindependent parameters which we can take as σeh andσmin ≡ (σee+ σhh−2σeh); the latter combination isequalto thebulk dcelectrical conductivity σ at theDiracpointin thehydrodynamicregime.12 The imbalancerelaxation

For beta=2, three momentum 2,3,4 on the sphere of diameter p1.

In graphenebeta<1 due to logarithmic renormalization of fermi velocity v*: E=v* p

Foster+Almeiner (0810.4342)

Conclusion

We have calculated Transports by holography.They fit data very well.Appearance of new field theory for many body theorydue to the help of simple system.