Electron transport and quantum criticality in disordered graphene. Alexander D. Mirlin

Transcription

1 Electron transport and quantum criticality in disordered graphene Alexander D. Mirlin Research Center Karslruhe & University Karlsruhe & PNPI St. Petersburg P. Ostrovsky, Research Center Karlsruhe & Landau Inst., Chernogolovka I. Gornyi, Research Center Karlsruhe & Ioffe Institute, St.Petersburg Phys. Rev. B 74, (2006) Phys. Rev. Lett 98, (2007) cond-mat/

2 Plan Experimental motivations Dirac fermions in disordered graphene Symmetry classification of disorder Concentration dependence of conductivity for different disorder types: Born vs. unitary limits Minimal conductivity (at the Dirac point) and quantum criticality Chiral disorder Long-range disorder Anomalous quantum Hall effect Summary

3 Graphene samples

4 Experiments on transport in graphene Novoselov, Geim et al; Zhang, Tan, Stormer, and Kim; Nature 2005 linear dependence of conductivity on electron density ( V g ) minimal conductivity σ 4e 2 /h ( e 2 /h per spin per valley) T independent in the range K

5 Graphene in transverse magnetic field Anomalous integer quantum Hall effect: σ xy = (n+ 1 2 ) (4e2 /h) 4 = 2 (spin) 2 (number of Dirac points) Center of the lowest Landau level, n e = 0: σ xx 2e2 h

6 nts Clean graphene: lattice structure and spectrum 2.46 Å K K Two sublattices: A and B (a) m K (b) k 0 K K K t k = t Hamiltonian in the sublattice representation: ( ) 0 tk H = t k 0 [ ] 1 + 2e i( 3/2)k y a cos(k x a/2) Spectrum ε 2 k = t k 2 The gap vanishes at 2 points, K, K = (±k 0, 0), where k 0 = 4π/3a. In the vicinity of K, K the spectrum is of massless Dirac-fermion type: H K = v 0 σk, H K = v 0 σ T k

7 Clean graphene: basis and notations Hamiltonian 4 4 matrix operating in: AB space of the two sublattices (σ Pauli matrices), K K space of the valleys (τ Pauli matrices). Four-component wave function is chosen as Ψ = {φ AK, φ BK, φ BK, φ AK } T. Hamiltonian in this basis has the form: Green function: H = v 0 τ 3 σk. G R(A) 0 (ε, k) = ε + v 0τ 3 σk (ε ± i0) 2 v 2 0 k2.

8 Clean graphene: symmetries Space of valleys K K : Isospin Λ x = σ 3 τ 1, Λ y = σ 3 τ 2, Λ z = σ 0 τ 3. Time inversion Chirality T 0 : H = σ 1 τ 1 H T σ 1 τ 1 C 0 : H = σ 3 τ 0 Hσ 3 τ 0 Combinations with Λ x,y,z T x : H = σ 2 τ 0 H T σ 2 τ 0 C x : H = σ 0 τ 1 Hσ 0 τ 1 T y : H = σ 2 τ 3 H T σ 2 τ 3 C y : H = σ 0 τ 2 Hσ 0 τ 2 T z : H = σ 1 τ 2 H T σ 1 τ 2 C z : H = σ 3 τ 3 Hσ 3 τ 3 Spatial isotropy T x,y and C x,y occur simultaneously T and C

9 Types of chiral disorder (i) bond disorder: randomness in hopping elements t ij (C z -symmetry) (ii) infinitely strong on-site impurities unitary limit: all bonds adjacent to the impurity are effectively cut = bond disorder (C z -symmetry) (iii) dislocations: random non-abelian gauge field (C 0 -symmetry), F. Guinea, A. Morpurgo, M.A.H.Vozmediano... (iv) random magnetic field, ripples (all four symmetries C 0,x,y,z )

10 Symmetries of various types of disorder in graphene Λ Λ z T 0 T T z C 0 C C z CT 0 CT CT z σ 0 τ 0 α σ {1,2} τ {1,2} β σ 1,2 τ 0 γ σ 0 τ 1,2 β z σ 3 τ 3 γ z σ 3 τ 1,2 β σ 0 τ 3 γ σ 1,2 τ 3 α σ 3 τ 0 α z Related works: S. Guruswamy, A. LeClair, and A.W.W. Ludwig, Nucl. Phys. B 583, 475 (2000) E. McCann, K. Kechedzhi, V.I. Fal ko, H. Suzuura, T. Ando, and B.L. Altshuler, PRL 97, (2006) I.L. Aleiner and K.B. Efetov, PRL 97, (2006)

11 Symmetries of various types of disorder in graphene Λ Λ z T 0 T T z C 0 C C z CT 0 CT CT z σ 0 τ 0 α σ {1,2} τ {1,2} β σ 1,2 τ 0 γ σ 0 τ 1,2 β z σ 3 τ 3 γ z σ 3 τ 1,2 β σ 0 τ 3 γ σ 1,2 τ 3 α σ 3 τ 0 α z Generic Gaussian disorder: thermodynamic and transport properties depend on α = α 0 + β 0 + γ 0 + α + β + γ + α z + β z + γ z α tr = 1 ( ) α0 + β 0 + γ 0 + α + β + γ + 3 ) αz + β z + γ z 2 2(

12 Self-Consistent Born Approximation: Drude conductivity = σ SCBA (ε) e2 π 2 [ ] ε α tr ε α Re Σ(ε) + 1 Exponentially small scale Γ 0 = e 1/α UV cut-off cf. P.A. Lee 93 (d-wave SC) High energies (ε Γ 0 ): ε-dependence only logarithmic σ SCBA (ε) e2 π 2 α tr [1 α2 log ] α tr ε +... Low energies (ε Γ 0 ): σ SCBA = 4 π e 2 h but makes no sense for generic disorder: localization effects.

13 2D disordered Dirac fermions: from SCBA to RG Self-consistent Born approximation: = + Σ Σ = = + + αε log ε α2 ε log 2 ε Non-SCBA contribution = α 2 ε log 2 ε Efficient tool to sum up logarithmic contributions in all orders: renormalization group

14 2D disordered Dirac fermions: ultraviolet RG (history) random bond Ising model (random mass) V.S. Dotsenko and V.S. Dotsenko 83 quantum Hall effect (random mass, scalar and vector potential) A.W.W. Ludwig, M.P.A. Fisher, R. Shankar, and G. Grinstein 94 disordered d-wave superconductors A. A. Nersesyan, A. M. Tsvelik, and F. Wenger 94, 95 M. Bocquet, D. Serban, and M.R. Zirnbauer 00 A. Altland, B.D. Simons, and M.R. Zirnbauer 02 Dirac fermions + C z chiral disorder (e.g., random-field XY model) S. Guruswamy, A. LeClair, and A.W.W. Ludwig 00 disordered graphene (T 0 invariance + some hierarchy of couplings) I.L. Aleiner and K.B. Efetov 06.

15 Ultraviolet (Ballistic) Renormalization Group v 0 =const, ε and disorder flow: (a) (b) (c) (d) RG stops at length L ε where ε reaches UV cutoff Logarithmic renormalization of disorder α tr σ(ε) = e 2 π 2 α tr (L ε )

16 2D disordered Dirac fermions: complete one-loop RG dα 0 d log L = 2α 0(α 0 + β 0 + γ 0 + α + β + γ + α z + β z + γ z ) + 2α α z + β β z + 2γ γ z, dα d log L = 2(2α 0α z + β 0 β z + 2γ 0 γ z ), dα z d log L = 2α z(α 0 + β 0 + γ 0 α β γ + α z + β z + γ z ) + 2α 0 α + β 0 β + 2γ 0 γ, dβ 0 d log L = 2[β 0(α 0 γ 0 + α + α z γ z ) + α β z + α z β + β γ 0 ], dβ d log L = 4(α 0β z + α z β 0 + β 0 γ 0 + β γ + β z γ z ), dβ z d log L = 2[ β z(α 0 γ 0 α + α z γ z ) + α 0 β + α β 0 + β γ z ], dγ 0 d log L = 2γ 0(α 0 β 0 + γ 0 + α β + γ + α z β z + γ z ) + 2α γ z + 2α z γ + β 0 β, dγ d log L = 4α 0γ z + 4α z γ 0 + β β2 + β2 z, dγ z d log L = 2γ z(α 0 α + α z β 0 + β β z + γ 0 γ + γ z ) + 2α 0 γ + 2α γ 0 + β β z. dε d log L = ε(1 + α 0 + β 0 + γ 0 + α + β + γ + α z + β z + γ z ).

17 Weak vs. strong impurities Born limit (Gaussian disorder): logarithmic dependence of σ on n e : σ(n e ) 2e2 π 2 α 0 ) (1 α 0 log 2 v0 2 n. e coincides with leading-order SCBA corrections; agrees with Aleiner, Efetov 06 Experiments: much stronger (approximately linear) dependence. Gaussian disorder irrelevant?! = Consider strong scatterers! Unitary limit (rare infinitely strong scatterers) = T -matrix New characteristic energy scale: η Γ η log(1/η). η dimensionless impurity concentration

18 From weak to strong impurities: SCTMA Self-consistent T -matrix approximation = full phase diagram : from weak (Born) to strong (unitary) scatterers Σ(ε, k) = n imp T (ε) T (ε) = m (d 2 k 1... d 2 k m 1 ) ˆV k k1 G(ε, k 1 ) ˆV k1 k 2... G(ε, k m 1 ) ˆV km 1 k =

19 Unitary limit: Conductivity Conductivity varies linearly with n e, with a logarithmic correction factor: σ(n e ) = e2 4π 2 n e n imp log 2 2 v 2 0 n e. Charged (Coulomb) impurities: crossover between unitary and Born limits, ε σ C e2 µ 2 n imp v 2 0 e2 n e n imp agrees with Nomura and MacDonald 06, Ando 06 Experiment: linear n e -dependence = impurities are strong! = either unitary-limit or Coulomb (long-range) scatterers!

20 Phase diagram ε energy, α disorder strength, η impurity concentration nts α Born unitary η η c (ε) PSfrag replacements α unitary Born α U (ε) 0 0 Γ ε 0 0 Γ η ε Phase boundaries : η η c (ε) αε2 2 log2 ε α α U (ε) η 2 2ε 2 log 2 ( / ε ) σ(n e ): crossover from linear to logarithmic dependence with lowering n e

21 Conductivity at µ = 0 For generic disorder, the Drude result σ = 4 e 2 /πh at µ = 0 does not make much sense: Anderson localization will drive σ 0. Experiment: σ 4 e 2 /h independent of T Quantum criticality? Can one have finite σ? Yes, if disorder either (i) preserves one of chiral symmetries or (ii) is of long-range character (does not mix the valleys)

22 Critical states of disordered graphene Disorder Symmetries Class Conductivity QHE Vacancies C z, T 0 BDI 4e 2 /πh normal Vacancies + RMF C z AIII 4e 2 /πh normal σ 3 τ 1,2 disorder C z, T z CII 4e 2 /πh normal Dislocations C 0, T 0 CI 4e 2 /πh chiral Dislocations + RMF C 0 AIII 4e 2 /πh chiral Ripples, RMF Λ z, C 0 2 AIII 4e 2 /πh odd-chiral Charged impurities Λ z, T 2 AII 4σSp random Dirac mass: σ 3 τ 0,3 Λ z, CT 2 D 4e 2 /πh odd Charged imp. + (RMF, ripples) Λ z 2 A 4σU odd

23 C 0 -chirality H = σ 3 Hσ 3 : symmetry consideration Disorder structures: σ 1,2 τ 3, σ 1,2 τ 1,2, σ 1,2 τ 0 Couplings: α (breaks T 0 ), β (preserves T 0 ), γ (preserves T 0 ) Symmetry classes: C 0 + no T 0 = AIII (ChUE); C 0 + T 0 = CI (BdG) Generic C 0 -disorder Non-Abelian vector potential problem Nersesyan, Tsvelik, and Wenger 94, 95; Fendley and Konik 00; Ludwig 00 RG equations (cf. Altland, Simons, and Zirnbauer 02): α log L = 0, β log L = 4β γ, γ log L = β2. DoS: ρ(ε) ε 1/7 (generic C 0 ); ρ(ε) ε (1 α )/(1+α ) (α only)

24 Conductivity at µ = 0: C 0 -chiral disorder Current operator j = ev 0 τ 3 σ relation between G R and G A & σ 3 j x = ij y, σ 3 j y = ij x, transform the conductivity at µ = 0 to RR + AA form: σ xx = 1 π [ ] d 2 (r r ) Tr j α G R (0; r, r )j α G R (0; r, r) σ RR. α=x,y Gauge invariance: p p + ea constant vector potential σ RR = 2 π 2 A 2 Tr ln GR σ RR = 0 (?!) But: contribution with no impurity lines anomaly: UV divergence shift of p is not legitimate (cf. Schwinger model 62).

25 Universal conductivity at µ = 0 for C 0 -chiral disorder Calculate explicitly (δ infinitesimal ImΣ) σ = 8e2 v 2 0 π d 2 k δ 2 (2π) 2 (δ 2 + v0 2k2 ) = 2e2 2 π 2 = 4 e 2 π h for C 0 -chiral disorder σ(µ = 0) does not depend on disorder strength Alternative derivation: use Ward identity ie(r r )G R (0; r, r ) = [G R jg R ](0; r, r ) and integrate by parts only surface contribution remains: σ = ev 0 4π 3 dk n Tr [ j G R (k) ] = e2 π 3 dkn k k 2 = 4 π e 2 h Related works: Ludwig, Fisher, Shankar, Grinstein 94; Tsvelik 95

26 C z chirality: H = σ 3 τ 3 Hσ 3 τ 3 Disorder structures: σ 1,2 τ 0, σ 0 τ 1,2, σ 1,2 τ 3, σ 3 τ 1,2 Couplings: γ (preserves T 0,z ), β z (breaks T z ), β 0 (breaks T 0 ), α (breaks T 0,z ) Symmetry classes: C z + no T 0,z = AIII (ChUE); C z + T 0 = BDI (ChOE); Related works: Gade, Wegner 91; Hikami, Shirai, Wegner 93; Guruswamy, LeClair, Ludwig 00; Ryu, Mudry, Furusaki, Ludwig 07 RG equations: C z + T z = CII (ChSE) α log L = 2β 0β z, β 0 log L = β z log L = 2α (β 0 + β z ), γ log L = β2 0 + β2 z DoS: ρ(ε) ε 1 exp( # log ε 2/3 ) Conductivity: σ = 4e2 πh [ ] 4 (β 0 β z ) 2 +

27 Long-range disorder Smooth random potential does not scatter between valleys Reduced Hamiltonian: H = v 0 σk + σ µ V µ (r) Disorder couplings: α 0 = V 2 Symmetries: 0 2πv 2 0 Ludwig, Fisher, Shankar, Grinstein 94, α = V 2 x + V 2 2πv 2 0 y, α z = V 2 z 2πv 2 0 α 0 disorder T -invariance H = σ y H T σ y AII (GSE) α disorder C-invariance H = σ z Hσ z AIII (ChUE) α z disorder CT -invariance H = σ x H T σ x D (BdG) generic long-range disorder A (GUE) Ultraviolet RG: α 0 ln L = 2(α 0+α z )(α +α 0 ), α ln L = 4α 0α z, α z ln L = 2(α 0+α z )(α α z )

28 Long-range disorder (cont d) Class D (random mass): Disorder is marginally irrelevant = diffusion never occurs DoS: ρ(ε) = ε πv 2 02α z log ε Conductivity: σ = 4e2 πh Class AIII (random vector potential): C 0 /C z chiral disorder; considered above DOS: ρ(ε) ε (1 α )/(1+α ) Conductivity: σ = 4e2 πh

29 Long-range disorder: unitary symmetry Generic long-range disorder breaks all symmetries = class A (GUE) Effective infrared theory is Pruisken s unitary sigma model with θ-term: S[Q] = 1 4 Str [ σ xx 2 ( Q)2 + ( σ xy + 1 ) ] Q x Q y Q 2 σ xx 8 Str( Q)2 +iπn[q] Compact (FF) sector of the model: Q FF M F = U(2n) U(n) U(n) Topological term takes values N[Q] π 2 (M F ) = Z Vacuum angle θ = π in the absence of magnetic field due to anomaly = Quantum Hall critical point σ = 4σ U 2e2 h d ln Σ d ln L 0 Σ U Θ Π Θ 0 ln Σ

30 Long-range disorder: symplectic symmetry Diagonal disorder α 0 (charged impurities) preserves T -inversion symmetry = class AII (GSE) Partition function is real = Im S = 0 or π O(4n) Compact sector: Q FF M F = O(2n) O(2n) { Z Z, n = 1; = π 2 (M F ) = Z 2, n 2 At n = 1 M F = S 2 S 2 /Z 2 [Cooperons] [diffusons] = Im S = θ c N c [Q] + θ d N d [Q] T -invariance θ c = θ d = 0 or π Anomaly θ c,d = π At n 2 we use M F n=1 M F n 2 = Im S = πn[q] Symplectic sigma-model with θ = π term: S[Q] = σ xx 16 Str( Q)2 + iπn[q]

31 Long-range disorder: symplectic symmetry (cont d) Symplectic sigma-model with θ = π term: S[Q] = σ xx 16 Str( Q)2 + iπn[q] Possibility of θ = π topological term: Fendley 01 Similar to Pruisken sigma-model of QHE at criticality Instantons suppress localization = Novel critical point? Conductivity: σ = 4σ Sp d ln Σ d ln L 0 Θ Π Σ Sp Θ 0 Σ Sp ln Σ Alternative scenario: β function everywhere positive, σ Numerics needed! Related work: Ryu, Mudry, Obuse, Furusaki, cond-mat/

32 Long-range potential disorder: numerics Recent numerics confirm the absence of localization but the new critical point is not found. Bardarson, Tworzyd lo, Brouwer, Beenakker arxiv: Nomura, Koshino, Ryu arxiv:

33 Comments If an additional attractive fixed point exists, different microscopic models may flow into different fixed points Coulomb interaction (Altshuler-Aronov, Finkelstein) favors a novel fixed point Novel fixed point (if/when exists) is expected to describe the Quantum Spin Hall transition (in the presence of disorder)

34 Odd quantum Hall effect Decoupled valleys + magnetic field = unitary sigma model with anomalous topological term: S[Q] = 1 4 Str [ σ xx 2 ( Q)2 + ( σ xy + 1 ) ] Q x Q y Q 2 generic (valley-mixing) disorder = conventional IQHE = odd-integer QHE weakly valley-mixing disorder = even plateaus narrow, emerge at low T Decoupled valleys Weakly mixed valleys Σ 2e 2 h Σ 2e 2 h Ν Ν Estimate: T mix 10 50mK

35 Chiral quantum Hall effect C 0 -chiral disorder random vector potential Atiyah-Singer theorem: In magnetic field, zeroth Landau level remains degenerate!!! (Aharonov and Casher 79) Within zeroth Landau level Hall effect is classical Σ 2e 2 h Decoupled valleys (ripples) Weakly mixed valleys (dislocations) Σ 2e 2 h Ν Ν

36 Summary Graphene conductivity: type of disorder crucially important! Concentration dependence. Weak vs strong scatterers: log n e vs. n e Conductivity at µ = 0: generically localization. But quantum criticality for chiral and long-range disorder QHE: normal (generic disorder), odd (long-range disorder), chiral (C 0 disorder) Disorder Symmetries Class Conductivity QHE Vacancies C z, T 0 BDI 4e 2 /πh normal Vacancies + RMF C z AIII 4e 2 /πh normal σ 3 τ 1,2 disorder C z, T z CII 4e 2 /πh normal Dislocations C 0, T 0 CI 4e 2 /πh chiral Dislocations + RMF C 0 AIII 4e 2 /πh chiral Ripples, RMF Λ z, C 0 2 AIII 4e 2 /πh odd-chiral Charged impurities Λ z, T 2 AII 4σSp random Dirac mass: σ 3 τ 0,3 Λ z, CT 2 D 4e 2 /πh odd Charged imp. + (RMF, ripples) Λ z 2 A 4σU odd Exp.: linear σ(n e ), σ(µ = 0) 4e 2 /h, odd QHE = Coulomb impurities