Relativistic magnetotransport in graphene

Relativistic magnetotransport in graphene Markus Müller in collaboration with Lars Fritz (Harvard) Subir Sachdev (Harvard) Jörg Schmalian (Iowa) Landau Memorial Conference June 6, 008

Outline Relativistic physics in graphene, quantum critical systems and conformal field theories Relativistic signatures in magnetotransport: el.+th. conductivity, Peltier, Nernst effect etc. Hydrodynamic description Collective, collision-broadened cyclotron resonance Boltzmann equation Recover and refine hydrodynamics with Boltzmann Describe relativistic-to-fermi liquid crossover Go beyond hydrodynamics

Dirac fermions in graphene Honeycomb lattice of C atoms (Semenoff 84, Haldane 88)

Dirac fermions in graphene (Semenoff 84, Haldane 88) Honeycomb lattice of C atoms Tight binding dispersion massless Dirac cones in the Brillouin zone: (Sublattice degree of freedom pseudospin) Fermi velocity (speed of light ) Close to the two Fermi points K, K : v F 1.1 10 m s 6 c 300 H v F ( p K) σ sublattice E k = v F k K

Dirac fermions in graphene (Semenoff 84, Haldane 88) Honeycomb lattice of C atoms Tight binding dispersion massless Dirac cones in the Brillouin zone: (Sublattice degree of freedom pseudospin) Fermi velocity (speed of light ) Close to the two Fermi points K, K : Coulomb interactions: Fine structure constant v F 1.1 10 m s 6 c 300 H v F ( p K) σ sublattice E k = v F k K e α ε h v F = O ( 1)

Relativistic fluid at the Dirac point Expect relativistic plasma physics of interacting particles and holes! Quantum critical n ( 1+ α ln[ Λ n] ) D. Sheehy, J. Schmalian, Phys. Rev. Lett. 99, 6803 (007).

Transport and phase diagram Expect relativistic plasma physics of interacting particles and holes!

Transport and phase diagram Expect relativistic plasma physics of interacting particles and holes! T = T imp ~ ρ imp

Transport and phase diagram Expect relativistic plasma physics of interacting particles and holes! T = Interaction limited transport Disorder limited transport T imp ~ ρ imp

Transport and phase diagram Expect relativistic plasma physics of interacting particles and holes! Hydrodynamics? ω << T Interaction limited transport T = Disorder limited transport T imp ~ ρ imp

Conductivity in and across the relativistic regime? J.-H. Chen et al. Nat. Phys. 4, 377 (008).

Conductivity in and across the relativistic regime? J.-H. Chen et al. Nat. Phys. 4, 377 (008). + Magnetotransport? e.g., Hall, Nernst effect?

Other relativistic fluids: Bismuth (3d Dirac fermions with very small mass) Effective theories close to quantum phase transitions Conformal field theories E.g.: strongly coupled Non-Abelian gauge theories (QCD): tretament via AdS-CFT

Low energy effective theory at quantum phase transitions Relativistic effective field theories z = 1; arise often due to particle-hole symmetry Example: Superconductor-insulator transition (SIT) Bhaseen, Green, Sondhi (PRL 07). Hartnoll, Kovtun, MM, Sachdev (PRB 07)

SI-transition: Bose Hubbard model Bose-Hubbard model + H = t b j bi + U ni ij i Coupling t g tunes the SI-transition U i n i QCP

SI-transition: Bose Hubbard model Bose-Hubbard model + H = t b j bi + U ni ij i i n i QCP Coupling t g U tunes the SI-transition ψ = 0 ψ 0 Effective action around g c ( = 0): e.g.

SI-transition: Bose Hubbard model Bose-Hubbard model + H = t b j bi + U ni ij i i n i QCP Coupling t g U tunes the SI-transition ψ = 0 ψ 0 Effective action around g c ( = 0): e.g. Relativistic field theory in d=+1

Questions Transport characteristics of the relativistic plasma in lightly doped graphene and close to quantum criticality? Relativistic, hydrodynamic regime T = T imp ~ ρ imp

Questions Transport characteristics of the relativistic plasma in lightly doped graphene and close to quantum criticality? How does the relativistic regime connect to Fermi liquid behavior at large doping? Relativistic, hydrodynamic regime T = T imp ~ ρ imp

Questions Transport characteristics of the relativistic plasma in lightly doped graphene and close to quantum criticality? How does the relativistic regime connect to Fermi liquid behavior at large doping? What is the range of validity of relativistic magneto-hydrodynamics? Beyond hydrodynamics? Relativistic, hydrodynamic regime T = T imp ~ ρ imp

Model of graphene Graphene with Coulomb interactions and disorder H = H + 0 + H1 H dis Tight binding kinetic energy

Model of graphene Graphene with Coulomb interactions and disorder H = H + 0 + H1 H dis B-field: r r r i i ea/ c Tight binding kinetic energy

Model of graphene Graphene with Coulomb interactions and disorder H = H + 0 + H1 H dis B-field: r r r i i ea/ c Tight binding kinetic energy Coulomb interactions

Model of graphene Graphene with Coulomb interactions and disorder H = H + 0 + H1 H dis B-field: r r r i i ea/ c Tight binding kinetic energy Coulomb interactions Coulomb marginally irrelevant! e α ε hv r F = O ( 1) RG:

Time scales MM, L. Fritz, and S. Sachdev, cond-mat 0805.1413. 1. Inelastic scattering rate (Electron-electron interactions) τ 1 ee ~ α k B T 1 h max 1, [ T ] Relativistic regime ( < T): Relaxation rate set by temperature, like in quantum critical systems! τ 1 ~ α ee k B T h. Elastic scattering rate (Scattering from charged impurities) Subdominant at high T τ 1 imp ~ ( Ze ε ) h ρ imp 1 max [ T, ]

Regimes MM, L. Fritz, and S. Sachdev, cond-mat 0805.1413. 1. Hydrodynamic regime: (collision-dominated) τ 1 1 1 >> τ, τ, ω ee imp B

Regimes MM, L. Fritz, and S. Sachdev, cond-mat 0805.1413. 1. Hydrodynamic regime: (collision-dominated) τ 1 1 1 >> τ, τ, ω ee imp B. Ballistic magnetotransport (large field limit) 1 1 1 τ > τ >> τ, ω B ee imp 3. Disorder limited transport (inelastic scattering ineffective due to nearly conserved momentum) >> T τ < > τ 1 ee 1 imp

Hydrodynamic Approach

Hydrodynamics Hydrodynamic collision-dominated regime Long times, Large scales t >> τ ee τ 1 1 1 >> τ, τ, ω ee imp B

Hydrodynamics Hydrodynamic collision-dominated regime Long times, Large scales t >> τ ee τ 1 1 1 >> τ, τ, ω ee imp B Local equilibrium established: T loc ( r), loc ( r) ; u r loc ( r) Study relaxation towards global equilibrium Slow modes: Diffusion of the density of conserved quantities: Charge Momentum Energy

Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, 14450 (007). Energy-momentum tensor Current 3-vector u : Energy velocity: u = ( 1,0, 0) ν : Dissipative current ( heat current ) ρ ρux ρu y No energy current ε 0 0 0 P 0 0 0 P τ ν : Viscous stress tensor (Reynold s tensor)

Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, 14450 (007). Conservation laws (equations of motion): Charge conservation

Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, 14450 (007). Conservation laws (equations of motion): Charge conservation Energy/momentum conservation 1 τ imp 0 T ν δ 0 r E r π = ik ρ r Coulomb interaction k k

Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, 14450 (007). Heat current and viscous tensor? Landau-Lifschitz, Relat. plasma physics Heat current Q Entropy current = ( ε + P) u J S = Q T

Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, 14450 (007). Landau-Lifschitz, Relat. plasma physics Positivity of entropy production (Second law): Heat current and viscous tensor? Heat current S τ ν ν A α Q Entropy current ν α ν ( T,, F ) ν + Bαβ ( T,, F ) ν A ( T,, u; F ) = const. = = const. B ( ε + P) u J S ν = Q T + const. δ ν B α α τ αβ 0

Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, 14450 (007). Landau-Lifschitz, Relat. plasma physics Positivity of entropy production (Second law): Heat current and viscous tensor? Heat current Q Entropy current S τ ν ν A α = ( ε + P) u J S = Q ν α ν ( T,, F ) ν + Bαβ ( T,, F ) ν A ( T,, u; F ) = const. = const. B ν T + const. δ ν B α α τ αβ 0

Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, 14450 (007). Landau-Lifschitz, Relat. plasma physics Positivity of entropy production (Second law): Heat current and viscous tensor? Heat current Q Entropy current S τ ν ν A α = ( ε + P) u J S = Q ν α ν ( T,, F ) ν + Bαβ ( T,, F ) ν A ( T,, u; F ) = const. = const. B ν T + const. δ ν B α α τ αβ 0 B small!

Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, 14450 (007). Landau-Lifschitz, Relat. plasma physics Positivity of entropy production (Second law): Heat current and viscous tensor? Heat current Q Entropy current S τ ν ν A α = ( ε + P) u J S = Q ν α ν ( T,, F ) ν + Bαβ ( T,, F ) ν A ( T,, u; F ) = const. = const. B ν T + const. δ ν B α α τ αβ 0 Irrelevant for response at k 0

Relativistic Hydrodynamics S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, 14450 (007). Landau-Lifschitz, Relat. plasma physics Positivity of entropy production (Second law): Heat current and viscous tensor? Heat current Q Entropy current S τ ν ν A α = ( ε + P) u J S = Q ν α ν ( T,, F ) ν + Bαβ ( T,, F ) ν A ( T,, u; F ) = const. = const. B ν T + const. δ ν B α α τ αβ 0 Irrelevant for response at k 0 One single transport coefficient (instead of two)!

Meaning of σ Q? Dimension of electrical conductivity At zero doping (particle-hole symmetry): σ Q ( ) = σ ρ = 0 imp xx = Universal d.c. conductivity of the pure system σ xx ( ρ = 0 ) imp Why is finite??

Universal conductivity σ Q Particle-hole symmetry (ρ = 0) K. Damle, S. Sachdev, (1996). Key: Charge current without momentum (energy current)! (particle) (hole) J r 0, P r = 0 Finite quantum critical conductivity! Pair creation/annihilation leads to current decay

Universal conductivity σ Q Quantum critical situation: Particle-hole symmetry (ρ = 0) Key: Charge current without momentum (energy current) (particle) (hole) J r 0, = 0 Finite quantum critical conductivity! As in quantum criticality: Relaxation time set by temperature alone P r Pair creation/annihilation leads to current decay τ K. Damle, S. Sachdev, (1996). ee h α k B T Universal conductivity σ Drude e = ρτ m σ Q ~ k B e T v e ( kbt ) h 1 e ~ ( hv) α k T α h B

Universal conductivity σ Q Quantum critical situation: Particle-hole symmetry (ρ = 0) Key: Charge current without momentum (energy current) (particle) (hole) J r 0, Finite quantum critical conductivity! As in quantum criticality: Relaxation time set by temperature alone P r = 0 Pair creation/annihilation leads to current decay τ K. Damle, S. Sachdev, (1996). ee h α k B T σ Universal conductivity Drude e = ρτ σ m Q ~ k e T v B e ( kbt ) h 1 e ~ ( hv) α k T α h B Exact (Boltzmann) 0.76 e σ Q ( = 0) = α h

Thermoelectric response S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, 14450 (007). Charge and heat current: J Q = ρ ν = u ( ) ε + P u J Thermo-electric response etc.

Thermoelectric response S. Hartnoll, P. Kovton, MM, and S. Sachdev, Phys. Rev. B 76, 14450 (007). Charge and heat current: J Q = ρ ν = u ( ) ε + P u J Thermo-electric response etc. i) Solve linearized hydrodynamic equations ii) Read off the response functions (Kadanoff & Martin 1960)

Results from Hydrodynamics

Response functions at B=0 Symmetry z -z : σ xy = α xy = κ xy = 0 Longitudinal conductivity: Universal conductivity at the quantum critical point ρ = 0 Drude-like conductivity, divergent for Momentum conservation (ρ 0)! τ, ω 0, ρ 0

Response functions at B=0 Symmetry z -z : σ xy = α xy = κ xy = 0 Longitudinal conductivity: Coulomb correction g = π e ( ) +O( k )

Response functions at B=0 Symmetry z -z : σ xy = α xy = κ xy = 0 Longitudinal conductivity: Coulomb correction g = π e ( ) +O( k ) Thermal conductivity: Relativistic Wiedemann-Franz-like relations between σ and κ in the quantum critical window!

Response functions at B=0 Symmetry z -z : σ xy = α xy = κ xy = 0 Longitudinal conductivity: Coulomb correction g = π e ( ) +O( k ) Thermopower: 3e π 1 k T B α dσ xx xx d Relativistic fluid! Only valid in the Fermi liquid regime, but violated in the relativistic window. T

B > 0 : Cyclotron resonance E.g.: Longitudinal conductivity Poles in the response ω = ± ω QC c iγ i τ Collective cyclotron frequency of the relativistic plasma ω QC c ρ B = ω FL c ( ε + P) v m F e B = Intrinsic, interaction-induced broadening ( Galilean invariant systems: No broadening due to Kohn s theorem) γ = σ Q B ( ε + P) / v F

B > 0 : Cyclotron resonance Longitudinal conductivity Poles in the response ω = ± ω QC c iγ i τ

Can the resonance be observed? ω = ± ω iγ i c τ v F = 1.1 10 c / 300 6 m / s Conditions to observe collective cyclotron resonance Collison-dominated regime Small broadening Quantum critical regime } High T: no Landau quantization hω c << α k B T ρ ρ E 1 γ, τ < th = ω c ( kbt ) ( h ) v F eb hc LL = hv F << k B T Parameters: T 300K B 0.1T 11 ρ 10 cm 13 1 ω c 10 s

Does relativistic hydrodynamics apply? Do T and not break relativistic invariance? Validity at large chemical potential? Beyond linearization in magnetic field? Treatment of disorder?

Boltzmann Approach MM, L. Fritz, and S. Sachdev, cond-mat 0805.1413. Recover and refine the hydrodynamic description Describe relativistic-to-fermiliquid crossover Go beyond hydrodynamics

σ Q from Boltzmann Boltzmann equation in Born approximation k L. Fritz, J. Schmalian, MM, and S. Sachdev, condmat 080.489 Cb dis [ E + v B] f ( k t) = I [ k, t { f ( k, t) }] + I [ k, t { f ( k, t) }] t + e ±, α coll ± coll ± a.) +, i +, i +, i +, i +, i +, i +, i +, j +(N 1) +(N 1), i, i, i, i, j, j, j, i b.) +, i +, i +, i +, i +, i +, i 1 +(N 1) +, i +, i +, i +, i +, j +, j

σ Q from Boltzmann Boltzmann equation in Born approximation k L. Fritz, J. Schmalian, MM, and S. Sachdev, condmat 080.489 Cb dis [ E + v B] f ( k t) = I [ k, t { f ( k, t) }] + I [ k, t { f ( k, t) }] t + e ±, α coll ± coll ± eq Linearization: f (, t) = f ( k, t) δf ( k, t) ± k ± + ± Great simplification: Divergence of forward scattering amplitude in d [ Amp ] Equilibration along unidimensional spatial directions At p-h symmetry: ( ) ; = C( t) eq ( k, t) f k, + δ ( t) f ± = ± eq δ E k k

σ Q from Boltzmann Boltzmann equation in Born approximation k L. Fritz, J. Schmalian, MM, and S. Sachdev, condmat 080.489 Cb dis [ E + v B] f ( k t) = I [ k, t { f ( k, t) }] + I [ k, t { f ( k, t) }] t + e ±, α coll ± coll ± eq Linearization: f (, t) = f ( k, t) δf ( k, t) ± k ± + ± Great simplification: Divergence of forward scattering amplitude in d [ Amp ] Equilibration along unidimensional spatial directions At p-h symmetry: ( ) ; = C( t) eq ( k, t) f k, + δ ( t) f ± = ± eq δ E k k σ Q ( = 0) 0.76 α e h

σ Q from Boltzmann Boltzmann equation in Born approximation k L. Fritz, J. Schmalian, MM, and S. Sachdev, condmat 080.489 Cb dis [ E + v B] f ( k t) = I [ k, t { f ( k, t) }] + I [ k, t { f ( k, t) }] t + e ±, α coll ± coll ± General analysis in linear response: Central element of analysis: Choose appropriate basis λ=± = ( k, t) a n φ ( λ k) g, Momentum or energy-current mode Charge current mode n n

σ Q from Boltzmann Boltzmann equation in Born approximation k L. Fritz, J. Schmalian, MM, and S. Sachdev, condmat 080.489 Cb dis [ E + v B] f ( k t) = I [ k, t { f ( k, t) }] + I [ k, t { f ( k, t) }] t + e ±, α coll ± coll ± General analysis in linear response: = Central element of analysis: Choose appropriate basis g ( k, t) a n φ ( λ, k) λ=± Momentum or energy-current mode Charge current mode n Relativistic dispersion ensures that ϕ 0 only couples to ϕ 1 for clean systems! n

Conductivity: σ Q L. Fritz, J. Schmalian, MM, and S. Sachdev, condmat 080.489 General doping: Clean system: Precise expression for σ Q!

Conductivity: σ Q L. Fritz, J. Schmalian, MM, and S. Sachdev, condmat 080.489 General doping: Clean system: Precise expression for σ Q! Gradual disappearance of relativistic physics σ Q ( = 0) σ Q 0.76 α T ( ) e h Will appear in all Boltzmann formulae below!

Conductivity: crossover General doping: L. Fritz, J. Schmalian, MM, and S. Sachdev, condmat 080.489 Lightly disordered system: Correction to hydrodynamics

Conductivity: crossover General doping: L. Fritz, J. Schmalian, MM, and S. Sachdev, condmat 080.489 Lightly disordered system: Correction to hydrodynamics Fermi liquid regime: Α Σ xx Ρ 4 3 Impurity limited conductivity 1 1.0 0.5 0.5 1.0 Universal Cb limited conductivity Ρ Ρ imp

Conductivity: crossover General doping: L. Fritz, J. Schmalian, MM, and S. Sachdev, condmat 080.489 Lightly disordered system: Correction to hydrodynamics Fermi liquid regime: J.-H. Chen et al. Nat. Phys. 4, 377 (008). Maryland group ( 08): Graphene Α Σ xx Ρ 4 3 1 1.0 0.5 0.5 1.0 Universal Cb limited conductivity Ρ Ρ imp Impurity limited conductivity

Magnetotransport Strategy: describe the slow dynamics of the momentum mode ϕ 0 in very weak disorder and moderate magnetic field

Magnetotransport Strategy: describe the slow dynamics of the momentum mode ϕ 0 in very weak disorder and moderate magnetic field Result: Full thermoelectric response (for general B) obtained in terms of thermodynamic quantities + only independent transport coefficients (collision matrix elements)! At small B, one transport coefficient is subdominant Relativistic hydrodynamics with only one transport coefficient σ Q is recovered! 1 1 >>τ τ ee B Corrections to hydrodynamics

Cyclotron resonance revisited Crossover to Fermi liquid regime: Semiclassical ω c recovered at >> T Broadening goes to zero - Kohn s theorem recovered: Non-broadening of the resonance for a single parabolic band. γ σ Q >> T ( ) 0

Cyclotron resonance revisited Beyond hydrodynamics: Towards ballistic magnetotransport = T Large fields 1 1 1 τ > τ >> τ, ω B ee imp Resonance Damping

Strongly coupled liquids Same trends as in exact (AdS-CFT) results for strongly coupled relativistic fluids! Graphene Resonance S. Hartnoll, C. Herzog (007) Damping N = 4 SUSY SU(N) gauge theory [flows to CFT at low energy] Exact Exact MHD B MHD B

Summary Relativistic physics in graphene and quantum critical systems Hydrodynamic description: collective cyclotron resonance in the relativistic regime covariance: 6 frequency dependent response functions given by thermodynamics and only one parameter σ Q. Boltzmann approach Confirmed and refined hydrodynamic description Understood relativistic-to-fermi liquid crossover: From universal Coulomb-limited to disorder-limited linear conductivity in graphene From collective-broadened to semiclasscial sharp cyclotron resonance Beyond hydrodynamics: describe large fields and disorder