Theory of Quantum Transport in Graphene and Nanotubes II
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1 CARGESE7B.OHP (August 26, 27) Theory of Quantum Transport in Graphene and Nanotubes II 1. Introduction Weyl s equation for neutrino 2. Berry s phase and topological anomaly Absence of backscattering in CN 3. Special time reversal symmetry Perfectly conducting channel in CN Symmetry crossover 4. Diamagnetic susceptibility 5. Bi-layer graphenes Susceptibility and conductivity Optical phonons Multi-layer graphenes 6. Summary Tsuneya ANDO Cargese, September 5 (Wed), 27) International School on Magnetic Fields for Science, Cargese Corsica, France, August 27 September 8, 27 [8:4 9:2 (3+1)] Page 1
2 Topological Anomaly and Berry s Phase CARGESE7B.OHP (August 26, 27) Weyl s equation : Neutrino Helicity (σ k) γ(σ ˆk) F sk (r) =ε s (k) F sk (r) F sk (r) = 1 L 2 R 1 [θ(k)] s ) R(θ±2π)= R(θ) R( π)= R(+π) ε s (k) =sγ k s = ±1 R(θ+2π) Pseudo spin Berry s phase =e iζ R(θ) T ζ = i dt sk(t) d θ sk(t) = π dt ζ Landau levels at ε= [J.W. McClure, PR 14, ζ π 666 (1956)] χ= g vg s γ 2 ( e ) 2 ( f(ε) ) δ(ε) dε 6π c h ε Absence of backscattering in metallic CNs Perfect conductor ε in the presence of scatterers T. Ando and T. Nakanishi, JPSJ 67, 174 (1998) T. Ando, R. Saito, and T. Nakanishi, JPSJ 67, 1857 (1998) Page 2
3 Conductance (units of 2e 2 /πh) 1. 2e 2 /π h.5 Length (units of L) L 5nm (1,1) CN ν = φ/φ =. εl/2πγ =. Λ/L = 1. u/2lγ =.1 Huge positive Magnetoresistance CARGESE7B.OHP (August 26, 27) Conductance of Finite-Length Nanotubes T. Ando and T. Nakanishi, JPSJ 67, 174 (1998) Nanotube radius R=L/2π Magnetic length l = c h/eb No scatterers with range smaller than lattice constant B 1 T Magnetic Field: (L/2πl) 2 (1,1) CN Perfect conductor in the presence of scatterers (B =) Page 3
4 CARGESE7B.OHP (August 26, 27) Special Time Reversal Symmetry and Universality Class Real time reversal (K K ): T FK T = σ zfk F K T = σ z FK T 2 =1 Special time reversal (within K and K ): S ( ) 1 F S = KF K = iσ y = K 1 2 = 1 Time reversal of P S 2 = 1 P S =K t PK 1 (Fα S,P S Fβ S)=(F β,pf α ) Time reversal Symmetry Matrix α Real T 2 =+1Orthogonal Real α Special S 2 = 1 Symplectic Quaternion β None Unitary Complex β Reflection coefficient: r βα =(F β,tf α )=(Fβ S,TF α) rᾱβ 1 T matrix: T = V +V E H +i V +V 1 E H +i V 1 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 4
5 CARGESE7B.OHP (August 26, 27) Metallic Nanotubes: Perfect Channel without Backscattering T. Ando and H. Suzuura, J. Phys. Soc. Jpn. 71, 2753 (22) 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 2e 2 /πh) Mean Free Path εl/2πγ ] W -1 =1. u/2γl = Odd channel number n c =1 Absence of backscattering Length (units of L) ] 3 5 Channel number n c W = n iu 2 4πγ 2 Page 5
6 CARGESE7B.OHP (August 26, 27) Symmetry Breaking Effects: Symplectic Unitary Trigonal warping (S) H =α γa ( 4 (ˆk x +iˆk y ) 2 3 (ˆk x iˆk y ) 2 ) aky/2π Lattice distortion H PRB 65, (22)] = g 1 (u xx +u yy ) +g 2 [(u xx u yy )σ x 2u xy σ y ] Deformation potential : g 1 16 ev Bond-length (b) change: g 2 βγ /4 β = d ln γ d ln b, γ= 3γ a 3a, b = 2 2 u xx = u x x + u z R Curvature: H = p γa 4 3 [H. Ajiki & T. Ando, JPSJ 65, 55 (1996)] [H. Suzuura & T. Ando, ak /2π ( ux y + u ) y x ] u yy = u y y u xy = 1 2 [( 2 u z x 2 2 u ) z y 2 σ x 2 2 u z x y σ y Optical phonon: H = βγ b 2 σ [u A u B ] α 1 2 < β < 4 γ = ak x /2π K p=1 3 γ 8 γ 3 γ = 2 V ppa π 3 2 (V pp V σ pp)a π [T. Ando, JPSJ 69, 1757 (2)] [K. Ishikawa & T. Ando, JPSJ 75, (26)] Page 6
7 Symmetry Breaking Effects and Crossover CARGESE7B.OHP (August 26, 27) ε Short-range scatterers (d/a < 1) Symplectic Orthogonal Intervalley (K K ) Spin dependent potential Metallic nanotubes Absence of backscattering: Robust Perfect channel : Fragile T. Ando, JPSJ 73, 1273 (24) TA & K. Akimoto, JPSJ 73, 2895 (24) K. Akimoto & TA, JPSJ 73, 2194 (24) T. Ando, JPSJ 75, 5471 (26) Quantum correction to conductivity Orthogonal Symplectic Unitary Magnetoresistance Negative Positive No Inverse Localization Length (units of WL -1 ) W -1 = 1. u/2γl = ε(2πγ/l) Magnetic Flux (units of φ ) Crossover: H. Suzuura and T. Ando, PRL 81, (22) Experiments: S.V. Morozov et al., PRL 97, 1681 (26) X.-S. Wu et al., PRL 98, (27) Theory: E. McCann et al., PRL 97, (26) επγ Page 7
8 Diamagnetic Susceptibility: Disorder Effects Singular diamagnetism J.W. McClure, Phys. Rev. 14, 666 (1956) S.A. Safran & F.J. DiSalvo, PRB 2, 4889 (1979) χ= g vg s γ 2 ( e ) 2 δ(εf ) 6π c h Constant broadening Γ H. Fukuyama, JPSJ 76, (27) Γ δ(ε F ) π(ε 2 F +Γ2 ) Self-consistent Born approximation M. Koshino and T. Ando, PRB 75, (27) Susceptibility [(gvgsγ 2 /6πε)(e/ch) 2 ] δ(ε F ) W [ 2W ( ε F <Γ ) 2 ε F πγ Cutoff energy: Γ =ε c e 1/2W ] W W= CARGESE7B.OHP (August 26, 27) ε c /ε = 5. χ(ε) D(ε) Energy (units of ε ) Sharp peak and long tail ε F 1 Page 8 Density of States [gvgsε/πγ 2 ]
9 Density of states (units of ε c /2πγ2).4.2 CARGESE7B.OHP (August 26, 27) Weak Field Limit [M. Koshino & T. Ando, PRB 75, (27)] W=.2 1 =5 εc c : BandWidth width Energy (units of εc) Zero field limit Singular (ε c /hω B ) 2 = (ε c /hω B )2= 1 ε Density of states B 1 ε.1.2 Susceptibility (units of -e 2 γ 2 /h 2 /εc) Energy (units of εc) ε 1 Susceptibility ε Page 9
10 Bilayer Graphene CARGESE7B.OHP (August 26, 27) Quantum Hall effect in bilayer graphene K.S. Novoselov et al., Nature 438, 197 (25) K.S. Novoselov et al., Nat. Phys. 2, 177 (26) ARPES [T. Ohta et al., PRL 98, 2682 (27)] Effective Hamiltonian in bilayer graphene A 1 B 1 A 2 B 2 γˆk γˆk + Δ H= Δ γˆk γˆk + ) H h2 2m ( ˆk2 ˆk + 2 ε (k)=± h2 k 2 ± 2m D(ε)= g vg s m 2π h 2 m = h2 Δ 2γ 2.34m ε ˆk ± = ˆk x ±iˆk y Δ=γ 1.4eV ε ε ε γ γ γ E. McCann and V.I. Falko, PRL 96, 8685 (26) M. Koshino and T. Ando, PRB 73, (26) Tight-binding models S. Latil and L. Henrard, PRL 97, 3683 (26) F. Guinea et al., PRB 73, (26) Page 1
11 Landau Levels, Susceptibility, and Conductivity CARGESE7B.OHP (August 26, 27) Conductivity (units of e 2 /π 2 h) No Intervalley Scattering W =.4 Landau level Zero mode Monolayer hω n =sgn(n) hω B n (n=, ±1, ) 1 Bilayer hω n =± n(n+1) hω c (n=, 1, ) 2 ( )( ˆk SCBA Monolayer ˆk + φ Boltzmann ( )( ˆk2 Bilayer ˆk + 2 φ ( )( ˆk2.8 ˆk + 2 φ 1 ω c = eb m c Energy (units of ε ) ) = ) = ) = Susceptibility χ(ε)= g vg s e 2 γ 4π c 2 h 2 ln Δ ε [S.A. Safran, PRB 3, 421 (1984)] Conductivity σ min = g vg s e 2 2π 2 h [M. Koshino and T. Ando, PRB 73, (26)] Page 11
12 CARGESE7B.OHP (August 26, 27) Energy Dispersion and Density of States of Bilayer Graphene T. Ando, J. Phys. Soc. Jpn. 76, No. 1 (27) Energy (units of Δ) Allowed Allowed Monolayer Wave Vector (units of Δ/γ) Density of States (units of gvgsδ/2πγ 2 ) Density of States Electron Concetration Δ.4eV hω.2ev 1 2 Energy (units of Δ) 1 5 Electron Concentration (units of gvgsδ 2 /2πγ 2 ) Page 12
13 Optical Phonons in Bilayer Graphene T. Ando, J. Phys. Soc. Jpn. 76, No. 1 (27) CARGESE7B.OHP (August 26, 27) Frequency Shift and Broadening (units of λω ) Symmetric Δ/hω =2. δ/hω Shift Broadening Fermi Energy (units of hω ) Spectral Function (units of 1/hω ) Δ/hω =2. δ/hω =.1 Shift Symmetric Frequency (units of λω ) Fermi Energy (units of hω ) σ(ω) D.S.L. Abergel & V.I. Falko, PRB 75, (27) Page 13
14 CARGESE7B.OHP (August 26, 27) Magnetic Oscillation of Optical Phonons in Bilayer Graphene T. Ando, J. Phys. Soc. Jpn. 76, No. 1 (27) Energy (units of hω) ε F /hω =1. Δ/hω =2. Fermi Energy Landau Levels Magnetic Energy: hω c (units of hω ) Frequency Shift and Broadening (units of λω ) Δ/hω =2. ε F /hω =.25 Shift Broadening Symmetric -2 δ/hω = Magnetic Energy (units of hω ) σ(ω) D.S.L. Abergel & V.I. Falko, PRB 75, (27) Page 14
15 CARGESE7B.OHP (August 26, 27) Multi-Layer Graphene [M. Koshino & T. Ando, PRB 76, (27)] Exact decomposition of effective Hamiltonian 2M +1 Layers = 1 Monolayer + M Bilayers 2M Layers = Monolayer + M Bilayers Three parameters: γ,γ 1, γ 3 (trigonal warping) Diamagnetic susceptibility Page 15
16 CARGESE7B.OHP (August 26, 27) Summary: Theory of Quantum Transport in Graphene and Nanotubes II Collaborators 1. Introduction Weyl s equation for neutrino 2. Berry s phase and topological anomaly Absence of backscattering in CN 3. Special time reversal symmetry Perfectly conducting channel in CN Symmetry crossover 4. Diamagnetic susceptibility 5. Bi-layer graphenes Susceptibility and conductivity Optical phonons Multi-layer graphenes N.H. Shon (Vietnam) M. Koshino (Titech) Y. Zheng (China) T. Nakanishi (AIST) H. Suzuura (Hokkaido Univ) ando/reprint/graphene/reprints.htm Page 16
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