Metallic Nanotubes as a Perfect Conductor

Transcription

1 Metallic Nanotubes as a Perfect Conductor 1. Effective-mass description Neutrino on cylinder surface 2. Nanotube as a perfect conductor Absence of backward scattering Perfectly transmitting channel Some experiments 3. Effects of symmetry breaking Inelastic scattering Magnetic field and flux Short-range scatterers Trigonal warping 4. Inter-wall interaction Negligible inter-wall conductance 5. Summary and conclusion Nagano, June 23 (Fri), 2006 NT06: Seventh International Conference on the Science and Application of Nanotubes, Hotel Metropolitan Nagano, Nagano, Japan, June 18 23, 2006 [09:45-10:15 (25+5)] Tsuneya ANDO Sumio Iijima (1991) Collaborators T. Nakanishi (AIST) H. Suzuura (Hokkaido Univ) S. Uryu (Tokyo Tech) Page 1

2 Two-Dimensional Graphite and Carbon Nanotubes η a=2.46 Å Chiral vector : L=n a a+n b b Chiral angle : η Weyl s equation for neutrinos [ ][ ] [ ] 0 ˆkx iˆk γ y F A (r) F ˆk x +iˆk y 0 F B =ε A (r) (r) F B (r) γ(σ xˆkx +σ yˆky )F (r)=εf (r) γ( σ ˆk)F = εf ˆk = i Velocity: γ/ h Massless (Dirac) Constant velocity Energy (ev) light, cannot stop Topological anomaly M Γ K K E F σ bands (s, p x,p y ) π bands (p z ) -20 K Γ M K Wave Vector ε(k)=± k x 2 +k2 y Page 2

3 Periodic Boundary Conditions and Band Structure Periodic boundary conditions ψ(r) = F j (r)ψ jk (r) j=a,b Ψ jk (r+l) =Ψ jk (r) exp(ik L) exp(ik L) = exp(+2πνi/3) n a +n b =3N +ν (ν =0, ±1) ψ(r+l) =ψ(r) Wave functions F (r) exp[ iκ ν (n)x+iky ] κ ν (n) = 2π ( n ν ) L 3 Energy bands ε ν (±) (n, k)=±γ κ ν (n) 2 +k 2 Band gap (semiconductor) E g =4πγ/3L K : ν ν F (r+l)=f (r) exp( 2πνi/3) ε ν κ ε Fictitious AB flux ν k p scheme Global and essential properties of CN Page 3

4 Topological Anomaly Weyl s equation : Neutrino Helicity ( σ k) γ( σ ˆk) F sk (r) =ε s (k) F sk (r) F sk (r) = 1 exp(ik r) R 1 [θ(k)] s ) LA ε s (k) =sγ k s = ±1 R(θ±2π) = R(θ) R( π) = R(+π) Pseudo spin Berry s phase R(θ+2π) θ =e iζ T R(θ) ζ = i dt sk(t) d ζ sk(t) = π 0 dt ζ π Absence of backscattering in metallic CNs Perfect conductor in the presence of scatterers ε T. Ando and T. Nakanishi, JPSJ 67, 1704 (1998) Perfectly conducting channel T. Ando and H. Suzuura, JPSJ 71, 2753 (2002) Landau levels at ε=0 in 2D graphite Magnetic anisotropy of CN (Field alignment) H. Ajiki and T. Ando, JPSJ 62, 2470 (1993) Experiments: S. Zaric et al., Nano Lett. 4, 2219 (2004) Page 4

5 Conductance (units of 2e 2 /πh) 1.0 2e 2 /π h 0.5 Length (units of L) L 5nm (10,10) CN ν = 0 φ/φ 0 = 0.00 εl/2πγ = 0.0 Λ/L = 10.0 u/2lγ = 0.10 Huge positive Magnetoresistance Conductance of Finite-Length Nanotubes T. Ando and T. Nakanishi, JPSJ 67, 1704 (1998) Nanotube radius R=L/2π Magnetic length l = c h/eb No scatterers with range smaller than lattice constant B 100 T Magnetic Field: (L/2πl) 2 (10,10) CN Perfect conductor in the presence of scatterers (B =0) Page 5

6 Helicity, Spin-Rotation, and Berry s Phase T. Ando, T. Nakanishi, and R. Saito, J. Phys. Soc. Jpn. 67, 2857 (1998) Weyl s equation : Neutrino Helicity ( σ k) γ( σ ˆk) F sk (r) =ε s (k) F sk (r) F sk (r) = 1 exp(ik r) R 1 [θ(k)] s ) LA ε s (k) =sγ k s = ±1 R(θ±2π) = R(θ) R( π) = R(+π) R(θ+2π)=e iφ R(θ) φ: Berry s phase σ θ Time reversal (Pseudo-spin) path Backward Scattering Scattering matrix (T matrix) Cancellation: T + T = 0 T +π T π T θ σ' φ= i dt sk(t) d sk(t) = π 0 dt Absence of backward scattering in metallic nanotubes π Page 6

7 Multi-Channel Case: Presence of Perfectly Conducting Channel T. Ando and H. Suzuura, J. Phys. Soc. Jpn. 71, 2753 (2002) Time reversal processes: Reflection matrix det(r)=0 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 = u/2γl = Length (units of L) ] 3 5 Channel number n c W = n iu 2 4πγ 2 Odd channel number Page 7

8 Boltzmann Conductivity and Mean Free Path Transport equation ē h E g mk k = m k W m k mk(g m k g mk ) f(ε mk ) g mk =f(ε mk )+eev mk τ mk ε mk (K m m + K m+m +)Λ m (ε)=1 m ε Λ m (ε) = v mk τ mk πγ K νμ = A V νμ 2 πγ h 2 v μ v ν ( σ = dε f ) σ(ε) ε σ(ε) = 4e2 Λ m (ε) π h m Conductivity σ Mean free path Infinite for ε <2πγ/L Finite for ε >2πγ/L Conductivity (units of 2e 2 L/πhW) Conductivity Density of States n c : Energy (units of 2πγ/L) Page Density of States (units of 1/2πγ)

9 Symmetry Breaking Effects π φφ Magnetic field: (L/2πl) 2 Magnetic flux: φ/φ 0 (φ 0 =ch/e) Short-range scatterers Intervalley (K K ) Spin dependent potential Trigonal warping (β 1) H = γ ak y /2π ak 0 /2π Range d Long : d/a > 1 Short: d/a<1 0 ˆkx iˆk y + βa 4 3 e3iη (ˆk x +iˆk y ) 2 ˆk x +iˆk y + βa 4 3 e 3iη (ˆk x iˆk y ) ak x /2π [H. Ajiki and T. Ando, JPSJ 65, 505 (1996)] TB model β =1 K Page 9

10 Conductance in Weak Magnetic Fields T. Ando, J. Phys. Soc. Jpn. 73, 1273 (2004) Conductance (units of 2e 2 /πh) (L/2πl) <G> exp<ln(g)> Localization Length W -1 = u/2γl = 0.01 ε(2πγ/l) -1 = Length (units of LW -1 ) ε επγ (L/2πl) <G> exp<ln(g)> Localization Length W -1 = u/2γl = 0.01 ε(2πγ/l) -1 = Length (units of LW -1 ) Page 10

11 Inverse Localization Length α vs Magnetic Field and Flux T. Ando, J. Phys. Soc. Jpn. 73, 1273 (2004) Inverse Localization Length (units of WL -1 ) W -1 = u/2γl = 0.01 G exp( αa) A : Length Magnetic Field: (L/2πl) 2 ε(2πγ/l) W -1 = u/2γl = 0.01 NΛ 0 NΛ 0 ε επγ W = n iu 2 4πγ ε(2πγ/l) Magnetic Flux (units of φ 0 ) Page 11

12 Effective Magnetic Flux Curvature: φ = 2π φ Lattice distortion: a L p cos 3η [T. Ando, JPSJ 69, 1757 (2000)] p=1 3 γ γ 1 γ = 2 V ppa π γ = 2 (V pp V σ pp)a π cf. C.L. Kane and E.J. Mele, PRL 78, 1932 (1997) φ = Lg 2 φ 0 2πγ [(u xx u yy ) cos 3η 2u xy sin 3η] u xx = u x x +u z R L : Circumference R : Radius L/2π a : Lattice constant η : Chiral angle Zigzag : η =0 Armchair: η =π/6 g 2 : Coupling constant ( Vpp ) π [H. Suzuura and T. Ando, PRB 65, (2002)] u yy = u y y u xy = 1 2 ( ux ) y + u y x Page 12

13 Inverse Localization Length (units of WL -1 ) W -1 = u/2γl = 0.01 NΛ 0 ε(2πγ/l) Ratio of Short-Range Scatterers: δ Effects of Short-Range Scatterers T. Ando and K. Akimoto, JPSJ 73, 1895 (2004) Scattering strength W =W L +W S δ =W S /W ε επγ Page 13

14 Inverse Localization Length (units of WL -1 ) W -1 = u/2γl = 0.01 η(π/6) -1 = 0.00 Zigzag βa/l Typical single wall nanotube ε(2πγ/l) ak y /2π Effects of Trigonal Warping K. Akimoto & T. Ando, JPSJ 73, 2194 (2004) ak 0 /2π ε ak x /2π επγ K Page 14

15 π Incommensurate metallic tubes Δ G e2 π h 10 4 Length-independent average Small, Irregular oscillation Inter-Wall Conductance of Double-Wall Nanotube S. Uryu and T. Ando Phys. Rev. B (2005) Multi-wall nanotubes Incommensurate lattices Large inter-wall distance 3.6 Å (3.35 Å Graphite) Four-terminal geometry L 1 /a= 21 Page 15

16 Inter-Wall Coupling [S. Uryu and T. Ando, PRB 72, (2005)] Inter-tube conductance G= e2 a 2 1 h 4L 1 L 2 γ0 2 t(r) 2 R Series: +1, 1, +1, 1, t(r) 2 typical t(r) 2 R γ t(r) < 0.1 γ 0 G 10 4 e2 (4, 16)/(7, 4) π h Length independent (up to 1 mm) Some related works ω ω Y.-G. Yoon et al., PRB 66, (2002) ω ω K.-H. Ahn et al., PRB 90, (2003) ω ω F. Triozon et al., PRB 69, (2004) J. Cumings and A. Zettl, PRL 93, (2004) Page 16

17 Summary: Metallic Nanotubes as a Perfect Conductor 1. Effective-mass description Neutrino on cylinder surface 2. Nanotube as a perfect conductor Absence of backward scattering Perfectly transmitting channel Symmetry and channel number Some experiments 3. Effects of symmetry breaking Inelastic scattering Magnetic field and flux Short-range scatterers Trigonal warping Absence of backscattering: Robust Perfect channel : Fragile 4. Inter-wall interaction Negligible inter-wall conductance Tsuneya ANDO Sumio Iijima (1991) Collaborators T. Nakanishi (AIST) H. Suzuura (Hokkaido Univ) S. Uryu (Tokyo Tech) Page 17