Electrical Conduction in Carbon Nanotubes T. Nakanishi (AIST) - PowerPoint PPT Presentation

1 ISSP International Summer School for Young Researchers on “Quantum Transport in Mesoscopic Scale & Low Dimensions” Aug. 13 - 21, 2003. (My talk is given at 16 Aug. 2003.) ✓ ✏ Electrical Conduction in Carbon Nanotubes T. Nakanishi (AIST) ✒ ✑ 1. What is Carbon Nanotubes? Quasi-one dimensional system 2. Effective-Mass Scheme Electronic properties of carbon nanotubes 3. Impurity Scattering Ballistic transport (Absence of back-scattering for Slowly varying potential) ✓ ✏ 4. Point defects Collaborators 5. Topological defect Tsuneya Ando (TIT) Masatsura Igami (NISTEP) 6. Conclusion Riichiro Saito (Tohoku Univ.) ✒ ✑

You are free to use these slides, if correctly credited to the source. T. Nakanishi (http://staff.aist.go.jp/t.nakanishi/index-e.html) 2 ✓ ✏ Carbon Nanotubes ✒ ✑ Single-wall Multi-wall Quantum wire growing naturally Electron micrographs of CN Diameter ∼ 4 nm S. Iijima, Nature 354, 56 (1991) 1D level spacing ∼ 0 . 8 eV Length ∼ 1 µm ○ Graphene with periodic Diameter 2 ∼ 30 nm boundary condition

You are free to use these slides, if correctly credited to the source. T. Nakanishi (http://staff.aist.go.jp/t.nakanishi/index-e.html) 3 ✓ ✏ Graphite sheet (Graphene) ✒ ✑ First Brillouin Zone sp 2 covalent bonding A τ 1 single π band y’ τ 2 (m a ,m b ) τ 3 ✓ tight–binding model ✏ A B A x’ Nearest–neighbor Transfer Integral: γ 0 y (n a ,n b ) T x 3 − γ 0 l =1 ψ B ( R A − � τ l ) = εψ A ( R A ) , L � (0,0) η 3 − γ 0 l =1 ψ A ( R B + � τ l ) = εψ B ( R B ) . � b a ✒ ✑

You are free to use these slides, if correctly credited to the source. T. Nakanishi (http://staff.aist.go.jp/t.nakanishi/index-e.html) 4 ✓ ✏ Graphite and Chiral Vector ✒ ✑ k y k y’ A η τ 1 y’ τ 2 (m a ,m b ) τ 3 K A B A K’ Armchair ( η = π /6) x’ k x η y k x’ K (n a ,n b ) T K’ Zigzag ( η =0) x L (0,0) η K’ K b a Chiral Vector:L = n a a + n b b ≡ ( n a , n b ) , n a 2 + n b 2 − n a n b . � L = | L | = a ( n a , n b ) = (2 , 1) m : armchair CN ( n a , n b ) = (1 , 0) m : zigzag CN ✓ ✏ n a + n b = 3 N + ν ν = 0 metallic CN ν = ± 1 semiconducting CN ✒ ✑

You are free to use these slides, if correctly credited to the source. T. Nakanishi (http://staff.aist.go.jp/t.nakanishi/index-e.html) 5 ✓ ✏ Metallic and Semiconducting CN (Zigzag CN) ✒ ✑ k y’ =k y k y’ =k y k y’ =k y k x’ =k x k x’ =k x k x’ =k x K’ K Zigzag ( η =0) K’ K Zigzag ( η =0) K’ K Zigzag ( η =0) K 2 K 2 K 2 M M M K 1 K 1 K 1 Semiconductor Metal(Linear dispersion) Semiconductor 3 3 3 (n a ,n b )=(8,0) (n a ,n b )=(9,0) (n a ,n b )=(10,0) Energy (units of γ 0 ) 2 2 2 1 1 1 ✓ ✏ E F = 0 ✒ ✑ 0 0 0 -0.5 0.0 0.5 -0.5 0.0 0.5 -0.5 0.0 0.5 √ Wave Vector (units of 2 π/ 3 a )

You are free to use these slides, if correctly credited to the source. T. Nakanishi (http://staff.aist.go.jp/t.nakanishi/index-e.html) 6 ✓ ✏ Effective-mass scheme ✒ ✑ √ K ′ =(2 π/a )(2 / 3 , 0) K =(2 π/a )(1 / 3 , 1 / 3) , A ( R A ) + e iη exp( i K ′ · R A ) F K ′  ψ A ( R A ) = exp( i K · R A ) F K A ( R A ) ,     ψ B ( R B ) = − ωe iη exp( i K · R B ) F K B ( R B ) + exp( i K ′ · R B ) F K ′ B ( R B ) ,     F K,K ′ A,B ( R A,B ) : Envelope Functions ω =exp(2 πi/ 3) ✓ tight–binding model ✏ 3 − γ 0 l =1 ψ B ( R A − � τ l ) = εψ A ( R A ) , � 3 − γ 0 l =1 ψ A ( R B + � τ l ) = εψ B ( R B ) . � ✒ ✑ τ l · ∂ F K,K ′ τ l ) = F K,K ′ F K,K ′ ( R A − � ( R A ) − � ( R A ) B B B ∂r l τ l · ∂ F K,K ′ τ l ) = F K,K ′ F K,K ′ ( R B − � ( R B ) − � ( R B ) A A A ∂r l k · p approximation

You are free to use these slides, if correctly credited to the source. T. Nakanishi (http://staff.aist.go.jp/t.nakanishi/index-e.html) 7 ✓ ✏ Effective–Mass Equation ✒ ✑ Envelope Function: F K ( r ) k · p Hamiltonian  F A   ✓ K point ✏ K F K ( r ) =    F B   K γ (ˆ k x − i ˆ    F A  F A     0 k y ) K K ✓ K’ point ✏  = ε         γ (ˆ k x + i ˆ  F B   F B    k y ) 0    K K ✒ ✑ γ ( σ x ˆ k x − σ y ˆ k y ) F ′ K ( r ) = ε F ′ K ( r ) γ ( σ x ˆ k x + σ y ˆ k y ) F K ( r ) = ε F K ( r ) ✒ ✑ Periodic Boundary Weyl’s equation for neutrinos Condition in x direction √ Band Parameter: γ = 3 aγ 0 / 2 Transfer Integral: γ 0 ∼ 2 . 6 [eV] ∇ + e k = − i� ˆ h A c ¯

You are free to use these slides, if correctly credited to the source. T. Nakanishi (http://staff.aist.go.jp/t.nakanishi/index-e.html) 8 ✓ ✏ Electronic States of CN’s ✒ ✑ Wave functions 1    b ν ( n, k y ) √ F K ( r ) =  exp [ iκ ν ( n ) x + ik y y ]     ± 1 2  b − ν ( n, k y ) ∗ 1   √ F K ′ ( r ) =  exp [ iκ − ν ( n ) x + ik y y ]     ± 1 2 with n a + n b = 3 N + ν b ν ( n, k y ) = κ ν ( n ) − ik y ✓ ✏ . κ ν ( n ) 2 + k 2 � y ν = 0 metallic CN Energy levels Linear dispersion κ ν ( n ) 2 + k 2 � ε ± ν ( n ) = ± γ y ε ± 0 (0) = ± γ | k y | Discritized wave number in ν = ± 1 semiconducting CN circumference direction k x = κ ν ( n ) = 2 π Band gap L ( n − ν/ 3) E g = 2 γ | κ ± 1 (0) | = 4 πγ Ajiki and Ando, J. Phys. Soc. Jpn.,62,1255 (1993) 3 L ✒ ✑

You are free to use these slides, if correctly credited to the source. T. Nakanishi (http://staff.aist.go.jp/t.nakanishi/index-e.html) 9 ✓ ✏ Band Gap ✒ ✑ E g = 4 πγ 3 L Band Gap of Zigzag Nanotubes M. S. Dresselhaus, G. Dresselhaus and R. Saito, Sol. State Com., 84 , 201 (1992). H. Ajiki and T. Ando, J. Phys. Soc. Jpn.,62,1255 (1993).

You are free to use these slides, if correctly credited to the source. T. Nakanishi (http://staff.aist.go.jp/t.nakanishi/index-e.html) 10 ✓ ✏ Effective–Potential T. Ando and T. Nakanishi, J. Phys. Soc. Jpn. 67 ,1704 (1998) ✒ ✑ ✓ ✏ Effective–Mass Equation √ 3 a 2 ( H 0 + V ) F = ε F u A = u A ( R A ) , ˜ � 2 R A √ γ (ˆ k x − i ˆ 3 a 2    F K  0 k y ) 0 0 A ( r ) u B = u B ( R B ) , ˜ �     γ (ˆ k x + i ˆ F K     k y ) 0 0 0 B ( r ) 2     R B     H 0 = , F = √     F K ′ γ (ˆ k x + i ˆ    A ( r )  3 a 2 0 0 0 k y )     e i ( K ′ − K ) · R A ˜ u ′     A = u A ( R A ) , �    F K ′  γ (ˆ k x − i ˆ B ( r )  0 0 k y ) 0    2 R A √ e iη u ′ 3 a 2  u A ( r ) 0 A ( r ) 0  e i ( K ′ − K ) · R B ˜ u ′ B = u B ( R B ) , �   − ω − 1 e − iη u ′ 0 u B ( r ) 0 B ( r )   2   R B V =     e − iη u ′ A ( r ) ∗ 0 u A ( r ) 0 √     3 a 2 / 2: Area of a Unit Cell   − ωe iη u ′ B ( r ) ∗  0 0 u B ( r )  ✓ ✏ Slowly-varying Potential ✒ ✑ Potential Range d ≪ Circumference L = | L | Potential Range d ≫ a u A ( r ) = u B ( r ) u A ( r ) = u A δ ( r − r 0 ) , u B ( r ) = u B δ ( r − r 0 ) , u ′ A ( r ) = u ′ B ( r ) = 0 u ′ A ( r ) = u ′ u ′ B ( r ) = u ′ A δ ( r − r 0 ) , B δ ( r − r 0 ) . ✒ ✑ r 0 : Impurity Position

You are free to use these slides, if correctly credited to the source. T. Nakanishi (http://staff.aist.go.jp/t.nakanishi/index-e.html) 11 ✓ ✏ Right- and left-going channels ✒ ✑ ε ε (1) ✓ ✏ 1 � Solutions for V = 0 , | ε | < ε (1) = 2 πγ 0 ka/ π -1 -2/3 2/3 L (K’) (K)  F K 1     A ( r )  ∓ i F K ± = √  =  exp( iky ) ,      F K    B ( r ) 1 2 AL Metallic CN F K ′   1 A ( r )  ± i   F K ′ ± = √  =  exp( iky ) .   k y     F K ′   1   B ( r ) 2 AL  K Armchair ( η = π /6) A : Length of Nanotube k x + π /a M Energy: ε ( k )= ± γk +2 π /3a Group Velocity: v = ± γ/ ¯ h K’ Γ Right–going F K + , F K ′ + K 2     ±  -2 π /3a F K − , F K ′ − Left–going - π /a   K 1   ✒ ✑ ( n a , n b ) = (2 , 1) m armchair CN

You are free to use these slides, if correctly credited to the source. T. Nakanishi (http://staff.aist.go.jp/t.nakanishi/index-e.html) 12 ✓ ✏ Lowest Born Approximation ✒ ✑ � Inter-valley Scattering � ± i 1 � d r  e iη u ′ 1     A ( r ) 0  i � V K ± K ′ + =     − ω − 1 e − iη u ′     0 B ( r ) 1 2 AL   � d r 1 ∓ e iη u ′ A ( r ) − ω − 1 e − iη u ′ � � = B ( r ) 2 AL 1 2 AL ( ∓ u ′ A e iη − ω − 1 e − iη u ′ B ) = V ∗ = K ′ ± K + � Intra-valley Scattering � d r � ± i 1 1      u A ( r ) 0  − i � V K ± K + =         0 u B ( r ) 1 2 AL   � d r {± u A ( r ) + u B ( r ) } 1 = 2 AL 1 = 2 AL ( ± u A + u B ) = V K ′ ± K ′ + ✓ ✏ Absence of back-scattering for slowly varying potential V K − K ′ + = V ∗ K ′ − K + = 0 , V K − K + = V K ′ − K ′ + ∝ u B − u A = 0 ✒ ✑