Zero-Mode Anomalies and Related Physics in Graphene

MODANETALK.OHP (September 9, ) Zero-Mode Anomalies and Related Physics in Graphene. Introduction Weyl s equation for neutrino Berry s phase and topological anomaly. Zero-mode anomalies Diamagnetic response Conductivity 3. Time reversal and symmetry crossover Special vs real time reversal Symmetry crossover 4. Bilayer and multi-layer graphene Interlayer interaction Hamiltonian decomposition 5. Summary Aussois, September (Tues) GDR 46, Physique Quantique Meśoscopique Centre Paul Langevin, Aussois, France September 3, [5:45 6:3 (35+)] Tsuneya ANDO Collaborators M. Koshino (TITech) M. Noro (TITech) H. Suzuura (Hokkaido Univ) Page

MODANETALK.OHP (September 9, ) Effective-Mass Description: Neutrino or Massless Dirac Electron Graphene (Triangular antidot lattice) η a=.46 Å Weyl s equation for neutrino (K) γ(σ ˆk)F (r)=εf(r) Energy (units of γ) 4 3 - - -3 K Γ M K γ(σ xˆkx +σ yˆky )F (r)=εf(r) ( )( ) ( ) γ(ˆk x iˆk y ) F A (r) F Wave Vector γ(ˆk x +iˆk y ) F B = ε A (r) (r) F B (r) ˆk = i Massless (Dirac) v F c/3 (γ 3eV) Velocity: v F =γ/ h Constant velocity ( light, cannot stop) K : σ σ Topological anomaly γ = 3γ a/ (γ : Hopping integral) Page M Γ K K E F ε(k)=±γ k x +k y ε n ± n B

MODANETALK.OHP (September 9, ) Topological Anomaly and Berry s Phase ( e iθ/ ) Weyl s equation : Neutrino Helicity (σ k) R(θ)= e iθ/ γ(σ ˆk) F sk (r) =ε s (k) F sk (r) F sk (r) = eiϕ k exp(ik r) L R [θ(sk)] ) s R(θ±π)= R(θ) R( π)= R(+π) ε s (k)=sγ k s=± ψ(t )=e iζ ψ() Pseudo spin Berry s phase ( ) T ζ = i dt sk(t) d s θ sk(t) = π e iθ ζ dt Landau levels at ε= [J.W. McClure, PR 4, ζ π 666 (956)] χ= g vg s γ ( e ) ( f(ε) ) δ(ε) dε 6π c h ε ε Absence of backscattering Metallic CN with scatterers Perfect conductor T. Ando & T. Nakanishi, JPSJ 67, 74 (998) Backscattering Page 3

MODANETALK.OHP (September 9, ) Band-Gap Effect [M. Koshino & T. Ando, PRB 8, 9543 ()] Susceptibility (units of -gvgse γ /6πc h Δ).7.6.5.4.3.. Susceptibility Density of States Δ=. -3 - - 3 Energy (units of Δ) Graphene with a gap ( ) Δ γˆk χ(ε)= g vg s e γ γˆk + Δ 6πc γ 3 Density of States (units of g v g s Δ/πγ ) δ(ε) n 4 3 3 4 θ(δ ε ) Δ K K n 4 3 3 4 Hamiltonian at band edge H= hˆk m ± e h m c B m = h Δ ( γ e h ) D(ε) χ P (ε) =+ Pauli m c χ L (ε) = ( e h ) D(ε) Landau 3 m c D(ε) = g sg v m π h Page 4

MODANETALK.OHP (September 9, ) Diamagnetic Susceptibility: Disorder Effects Singular diamagnetism J.W. McClure, Phys. Rev. 4, 666 (956) S.A. Safran & F.J. DiSalvo, PRB, 4889 (979) χ= g sg v γ ( e ) δ(εf ) 6π c h Constant broadening Γ H. Fukuyama, JPSJ 76, 437 (7) Γ δ(ε F ) π(ε F +Γ ) Self-consistent Born approximation M. Koshino and T. Ando, PRB 75, 35333 (7) Susceptibility [(gvgsγ /6πε)(e/ch) ] [ πε ( ε F <ɛ ) δ(ε F ) W ε F Cutoff energy: ε =Wε c e /W ].5..5 W -... W= ε c /ε = 5. χ(ε) D(ε). -. -.5..5. Energy (units of ε ) Sharp peak and long tail 5 4 3 ε F Page 5 Density of States [gvgsε/πγ ]

Density of states: D(ε) = Zero-Mode Anomaly in Conductivity Boltzmann conductivity σ(ε F )=e D D(ε F )= e π h 4W Einstein relation D =vf τ = γ h τ W = n iu 4πγ τ = π h n iu D(ε F ) πw ε F / h τ D(ε F ) u Impurity strength n i Impurity density Independent of ε F (Metal!) σ() for D()= ( e /π h) MODANETALK.OHP (September 9, ) ε Zero-gap semiconductor πγ W (n s,ε F ) ε Singularity at the Dirac point (ε F =) Fermi energy scaling ) ( hωb σ xx (B) =σ xx hω e ε F ( hω ) B = γ/l (ω =) ) σ(ω) =σ = π hw ( hωb ε σ xy (B) =σ F xy e ε ( hω/ε F ) F Dynamical conductivity 4 h Magnetoconductivity Page 6

Density of States (units of gvgsε/πγ )..5 MODANETALK.OHP (September 9, ) Self-Consistent Born Approximation [N.H. Shon and T. Ando, J. Phys. Soc. Jpn. 67, 4 (998)] W = n iu 4πγ W - 5... Density of states ε ε c /ε =5. SCBA Boltzmann ε =Wε c e /W. - - Energy (units of ε ) Conductivity (units of g v g s e /π h) ε : Arbitrary energy ε c : Cutoff energy (width of π band) 35 3 5 5 5 Conductivity Long-Range Scatterers W - 5... σ = g vg s e π h (ε F =) - - Energy (units of ε ) ε c /ε =5. SCBA Boltzmann Page 7

Density of States (units of g s g v ε c /πγ ) Conductivity (units of g s g v e /π h).5..5..5. 5 5 Density of states Conductivity dk c5 W=.5 dk c5 W=.5 SCBA Ideal SCBA..5..5. Energy (units of ε c ) Boltzmann MODANETALK.OHP (September 9, ) Scatterers with Gaussian Potential (Self-Consistent Born Approx.) [M. Noro, M. Koshino & T. Ando JPSJ 79, 9473 ()] Conductivity (units of gsgve /π h) 3..5..5..5 Conductivity at Dirac point dk c. 5.... k c π a v i (r)= u πd exp ( r d )...5..5. W σ = g sg v e π h W = n iu 4πγ Page 8

MODANETALK.OHP (September 9, ) Charged Impurities (Self-Consistent Born Approximation) [M. Noro, M. Koshino & T. Ando, J. Phys. Soc. Jpn. 79, 9473 ()] Density of States (units of g s g v ε c /πγ ).5..5..5 Density of states Thomas-Fermi screening n i...5 σ = g sg v e π h SCBA Ideal...5..5..5 Energy (units of ε c ) Conductivity (units of g s g v e /π h) 5 4 3 σ min > σ n i...5 Conductivity vs n s n c = ε c 4πγ SCBA Boltzmann..5..5. Electron Concentration (units of n c ) Page 9

MODANETALK.OHP (September 9, ) Special Time Reversal Symmetry and Universality Class Real time reversal (K K ): T FK T = σ zfk F K T = σ z FK T = Special time reversal (within K and K ): S ( ) F S = KF K = iσ y = K = Time reversal of P S = P S =K t PK (Fα S,P S Fβ S)=(F β,pf α ) Time reversal Symmetry Matrix α Real T =+Orthogonal Real α Special S = Symplectic Quaternion β None Unitary Complex β Reflection coefficient: r βα =(F β,tf α )=(Fβ S,TF α) rᾱβ Tmatrix: T = V +V E H +i V +V E H +i V E H +i V + Real : rᾱβ =(Fα T,TF β )=(Fβ T,T(F α T ) T )=+(Fβ T,TF α)=+ r βα Special: rᾱβ =(Fα S,TF β )=(Fβ S,T(F α S ) S )= (Fβ S,TF α)= r βα Absence of backward scattering: rᾱα = ( Berry s phase) Presence of perfect channel (Odd channel numbers) Page

MODANETALK.OHP (September 9, ) Metallic Nanotubes: Perfect Channel without Backscattering T. Ando and H. Suzuura, J. Phys. Soc. Jpn. 7, 753 () Time reversal processes: Reflection matrix det(r)= Perfect channel β Δθ β α π Δθ α ε πγ πγ πγ α β β ᾱ r βα = rᾱβ Conductance (units of e /πh) 5. 4. 3... Mean Free Path εl/πγ..5.9]..5.9 W - =. u/γl =.. 5 5 Odd channel number n c = Absence of backscattering Length (units of L) ] 3 5 Channel number n c W = n iu 4πγ Page

MODANETALK.OHP (September 9, ) Symmetry Breaking Effects: Symplectic Unitary [H. Ajiki & T. Ando, JPSJ 65, 55 (996)] Trigonal warping (S) H =α γa ( 4 (ˆk x +iˆk y ) 3 (ˆk x iˆk y ) ) [H. Suzuura & T. Ando, aky/π..5. Lattice distortion PRB 65, 354 ()] H = g (u xx +u yy ) -.5 +g [(u xx u yy )σ x u xy σ y ] -. Deformation potential : g 6 ev Bond-length (b) change: g βγ /4 -.5 β = d ln γ d ln b, γ= 3γ a 3a, b = u xx = u x x + u z R u yy = u y y u xy = ( ux φ Curvature: = a [( φ 4 γ (CN) 3R γ Optical phonon: H = βγ b σ [u A u B ] α < β < 4 ak /π.5...5. -.5 -. -.5..5. y + u y x )e 3iη + γ 8γ e 3iη] ak x /π ) η: Chiral angle R: Radius 3 γ = V ppa π 3 γ = (V pp V σ pp)a π [T. Ando, JPSJ 69, 757 ()] [K. Ishikawa & T. Ando, JPSJ 75, 8473 (6)] K Page

MODANETALK.OHP (September 9, ) Symmetry Breaking Effects: Symplectic Orthogonal Intervalley: K K Short-range scatterers (d/a<) Zone-boundary phonon H. Suzuura and T. Ando JPSJ 77, 4473 (8) Metallic nanotubes Absence of backscattering: Robust Perfect channel : Fragile Quantum corrections H. Suzuura & T. Ando, JPSJ75, 473 (6) Numerical study Localization length Fluctuations (UCF) ε επγ Inverse Localization Length (units of WL - ) - - W - =. u/γl =. φ -3 - - ε(πγ/l) -.5.5.5 3.5 Magnetic Flux (units of φ ) T. Ando, JPSJ 73, 73 (4) Figure T. Ando and K. Akimoto, JPSJ 73, 895 (4) K. Akimoto and T. Ando, JPSJ 73, 94 (4) T. Ando, JPSJ 75, 547 (6) Page 3

MODANETALK.OHP (September 9, ) Quantum Correction, Localization, and Conductance Fluctuations Symmetry Quantum correction Magnetoresistance UCF(α) Orthogonal Δσ< (weak localization) Negative Symplectic Δσ> (anti localization) Positive / Unitary Δσ = No / ΔG αe /h Theory H. Suzuura & T. Ando, PRL 8, 6663 () (6) D.V. Khveshchenko, PRL 97, 368 E. McCann et al., PRL 97, 4685 (6) Experiments S.V. Morozov et al., PRL 97, 68 (6) X.-S. Wu et al., PRL 98, 368 (7) F.V. Tikhonenko et al., PRL, 568 (8) D.-K. Kietal., PRB78, 549 (8) F.V. Tikhonenko et al., PRL 3, 68 (9) Page 4

Bilayer Graphene MODANETALK.OHP (September 9, ) Quantum Hall effect in bilayer graphene K.S. Novoselov et al., Nature 438, 97 (5) K.S. Novoselov et al., Nat. Phys., 77 (6) ARPES [T. Ohta et al., PRL 98, 68 (7)] Effective Hamiltonian in bilayer graphene A B A B γˆk γˆk + Δ H= Δ γˆk γˆk + ( ) H h ˆk m ˆk + m = h Δ γ ˆk ± = ˆk x ±iˆk y Δ=γ.4eV γ γ γ γ γ E. McCann and V.I. Falko, PRL 96, 8685 (6) M. Koshino and T. Ando, PRB 73, 4543 (6) ε ε Tight-binding models S. Latil and L. Henrard, ε n =± hω c n(n+) (n=,,...) PRL 97, 3683 (6) ε F. Guinea et al., Two Landau levels at ε= Susceptibility χ(ε)= g vg s e γ 4π c h ln Δ PRB 73, 4546 (6), [S.A. Safran, PRB 3, 4 (984)] ε Page 5

MODANETALK.OHP (September 9, ) Energy Dispersion and Density of States of Bilayer Graphene T. Ando, J. Phys. Soc. Jpn. 76, 47 (7) Energy (units of Δ) - Δ Δ Monolayer Density of States (units of gvgsδ/πγ ) 4 3 Density of States Electron Density Δ.4eV hω.ev n s = g vg s Δ πγ (.5 3 cm ) 4 3 Electron Density (units of g v g s Δ /πγ ) - - - Wave Vector (units of Δ/γ) Energy (units of Δ) Page 6

MODANETALK.OHP (September 9, ) Multi-Layer Graphene [M. Koshino & T. Ando, PRB 76, 8545 (7)] Exact decomposition of effective Hamiltonian M + Layers = Monolayer + M Bilayers M Layers = Monolayer + M Bilayers Three parameters: γ,γ, γ 3 (trigonal warping) Diamagnetic susceptibility Page 7

MODANETALK.OHP (September 9, ) Dynamical Conductivity of Multi-Layer Graphene (Average ) M. Koshino and T. Ando, Phys. Rev. B 77, 533 (8).9.8 Δ B / γ =. (B ~.T) Re σ xx (units of g v g s e /h).6.7.8 (units of g v g s e /h) Re σ xx.7.6.5.9.8.7.6.5.9.8.7.6.5 Δ B / γ =. (B ~ 4.5T) Δ B / γ =.3 (B ~ T) Γ / Δ.. hω / Δ Δ B / Δ Δ B / Δ.4..4. κ = (L, L) κ = π/ κ = π/3 (L, H) κ = π/4 (L, H) κ = (L, H) hω / Δ Page 8 5 5 B(T) B(T)

Δ B / Δ.4. DOS difference (units of g v g s γ /πh v ) - 5 B(T) MODANETALK.OHP (September 9, ) Local Density of States of Multi-Layer Graphene (Average ) M. Koshino and T. Ando, Phys. Rev. B 77, 533 (8) Δ B / Δ Δ B / Δ.4..4...4.6.8 ε / Δ LDOS difference (units of g v g s γ /πh v ) -.. κ = π/ κ = π/3 κ =..4.6.8 ε / Δ 5 5 DOS (units of g v g s γ /πh v ) B(T) B(T) LDOS (units of g v g s γ /πh v ) 5.5 (κ = ) 5L+ 4L+ 3L+ L+ L+ + DOS + 3+ LDOS (Top layer) (κ = π/). 4+ Γ / Δ. Δ B / Δ =.3 (B ~ T)..4.6.8 ε / Δ Page 9

MODANETALK.OHP (September 9, ) Summary: Zero-Mode Anomalies and Related Physics in Graphene. Introduction Weyl s equation for neutrino Berry s phase and topological anomaly. Zero-mode anomalies Diamagnetic response Conductivity 3. Time reversal and symmetry crossover Special vs real time reversal Symmetry crossover 4. Bilayer and multi-layer graphene Interlayer interaction Hamiltonian decomposition 5. Summary Aussois, September (Tues) GDR 46, Physique Quantique Meśoscopique Centre Paul Langevin, Aussois, France September 3, [5:45 6:3 (35+)] Tsuneya ANDO Collaborators M. Koshino (TITech) M. Noro (TITech) H. Suzuura (Hokkaido Univ) Page