From Graphene to Nanotubes Zone Folding and Quantum Confinement at the Example of the Electronic Band Structure Christian Krumnow Freie Universit¨ at Berlin May 26, 2011 Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 26, 2011 1 / 21
Introduction Outline 1 Introduction 2 Going from Graphene to Nanotubes in k-space Real and k-space of Graphene Real and k-space of Nanotubes 3 Electronic Band Structure of Nanotubes Zone Folding Approximation Limits of Zone Folding 4 Summary Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 26, 2011 2 / 21
Introduction Motivation interesting electronic behavior about 1/3 of possible nanotubes are (quasi) metallic depending on geometric structure Hamada et al: Phys Rev Lett 68 1579 (1992) high degree of complexity ✽ -number of realizable tubes large number of atoms in unit cell possible Ñ efficient tool needed Reich et al: Carbon Nanotubes, Wiley (2004) Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 26, 2011 3 / 21
Going from Graphene to Nanotubes in k-space Outline 1 Introduction 2 Going from Graphene to Nanotubes in k-space Real and k-space of Graphene Real and k-space of Nanotubes 3 Electronic Band Structure of Nanotubes Zone Folding Approximation Limits of Zone Folding 4 Summary Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 26, 2011 4 / 21
Going from Graphene to Nanotubes in k-space Real and k-space of Graphene Real and k-space of Graphene real space 2 atomic unit cell lattice basis vectors a 1 and a 2 with ⑥ a 1 ⑥ ✏ ⑥ a 2 ⑥ ✏ a 0 and a 0 ✏ 2 . 461 ˚ A Reich et al: Carbon Nanotubes, Wiley (2004) Reich et al: Phys Rev B 66 035412 (2002) k-space reciprocal lattice basis vectors k 1 and k 2 K -point at 1 � 1 3 k 1 � 2 ✟ 3 ♣ k 1 ✁ k 2 q , 3 k 2 and � 2 3 k 1 � 1 ✟ 3 k 2 Reich et al: Carbon Nanotubes, Wiley (2004) Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 26, 2011 5 / 21
Going from Graphene to Nanotubes in k-space Real and k-space of Nanotubes Unitcell of Nanotubes in Real Space chiral vector c ✏ ♣ n 1 , n 2 q ✑ n 1 a 1 � n 2 a 2 lattice vector ✁ 2 n 2 � n 1 , 2 n 1 � n 2 � ✟ a ✏ nR nR where n greatest common divisor Reich et al: Carbon Nanotubes, Wiley (2004) of n 1 , n 2 ★ 3 if 3 ✗ n 1 ✁ n 2 n R ✏ 1 else ❜ diameter d ✏ ⑥ c ⑥ π ✏ a 0 n 2 1 � n 2 2 � n 1 n 2 Thomsen et al: Light Scat in Sol IX 108 (2007) π ⑥ a ⑥⑥ c ⑥ nR ♣ n 2 2 1 � n 2 # graphene unit cells: q ✏ ❄ 3 ④ 2 ✏ 2 � n 1 n 2 q a 0 Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 26, 2011 6 / 21
Going from Graphene to Nanotubes in k-space Real and k-space of Nanotubes Quantum Confinement and Boundary Conditions along tube axis one dimensional reciprocal lattice ✙ ⑥ k ⑤⑤ ⑥ ⑥ k ⑤⑤ ⑥ ✙ with k ⑤⑤ ✏ 2 π k t P ✁ ⑥ a ⑥ ˆ a 2 , 2 perpendicular to tube axis k 1 due to rolled up structure, periodic boundary conditions are exact k ❑ finite scale yields quantization of allowed k-values e i k ❑ c ✏ 1 yields k ❑ , m ✏ 2 π ⑥ c ⑥ m ˆ c k 2 k ⑤⑤ Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 26, 2011 7 / 21
Going from Graphene to Nanotubes in k-space Real and k-space of Nanotubes Number of Modes what is the maximal value of the quantum number m? projecting tube unitcell on c -axis ñ q equidistant intersections m is limited by c ✏ e i ♣ k ❑ � ∆ k ❑ q α ˆ e i k ❑ α ˆ c smallest physical period in real k 1 space given by α ✏ ⑥ c ⑥ q 2 π ñ ⑥ ∆ k ❑ ⑥ ✏ ⑥ c ⑥④ q k ❑ m ✏ ✁ q 2 � 1 , ✁ q 2 � 2 , . . . , 0 . . . , q 2 k 2 k ⑤⑤ Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 26, 2011 8 / 21
Going from Graphene to Nanotubes in k-space Real and k-space of Nanotubes Examples (10,10) tube (12,8) tube ❄ ❄ 10 2 � 10 2 � 10 2 12 2 � 8 2 � 12 ☎ 8 d ✏ a 0 d ✏ a 0 π π ✏ 1 . 357 nm ✏ 1 . 366 nm 10 ☎ 3 ♣ 10 2 � 10 2 � 10 2 q ✏ 20 4 ☎ 1 ♣ 12 2 � 8 2 � 12 ☎ 8 q ✏ 152 2 2 q ✏ q ✏ k 1 k 1 k ❑ k ⑤⑤ k ❑ k 2 k 2 k ⑤⑤ tube models created with: Blinder: Carbon Nanotubes, Wolfram Demonstrations Project Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 26, 2011 9 / 21
Electronic Band Structure of Nanotubes Outline 1 Introduction 2 Going from Graphene to Nanotubes in k-space Real and k-space of Graphene Real and k-space of Nanotubes 3 Electronic Band Structure of Nanotubes Zone Folding Approximation Limits of Zone Folding 4 Summary Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 26, 2011 10 / 21
Electronic Band Structure of Nanotubes Zone Folding Approximation Zone Folding calculate allowed k-states for corresponding tube approximate band structure of carbon nanotubes by using the band structure of graphene along allowed lines in k-space Samsonidze et al: J Nanosci Nanotech 3 (2003) Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 26, 2011 11 / 21
Electronic Band Structure of Nanotubes Zone Folding Approximation Zone Folding calculate allowed k-states for corresponding tube approximate band structure of carbon nanotubes by using the band structure of graphene along allowed lines in k-space why? Samsonidze et al: J Nanosci Nanotech 3 (2003) Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 26, 2011 11 / 21
Electronic Band Structure of Nanotubes Zone Folding Approximation Zone Folding calculate allowed k-states for corresponding tube approximate band structure of carbon nanotubes by using the band structure of graphene along allowed lines in k-space why? because it is simple! involves only once one calculation for graphene for all tubes Samsonidze et al: J Nanosci Nanotech 3 (2003) Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 26, 2011 11 / 21
Electronic Band Structure of Nanotubes Zone Folding Approximation Band Structure of Graphene analytic expressions by tight binding model 3 p z and p ✝ z -orbitals cross at K -point 2 1 a.u. 0 -1 -2 -3 Reich et al: Phys Rev B 66 035412 (2002) Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 26, 2011 12 / 21
❑ ⑤⑤ Electronic Band Structure of Nanotubes Zone Folding Approximation Construction band structure of graphene 3 2 1 a.u. 0 -1 -2 -3 Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 26, 2011 13 / 21
Electronic Band Structure of Nanotubes Zone Folding Approximation Construction allowed k-states [(6,6) tube] band structure of graphene k 1 3 2 1 k ❑ a.u. 0 -1 k 2 -2 k ⑤⑤ -3 Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 26, 2011 13 / 21
Electronic Band Structure of Nanotubes Zone Folding Approximation Construction allowed k-states [(6,6) tube] band structure of graphene k 1 3 2 1 k ❑ a.u. 0 -1 k 2 -2 k ⑤⑤ -3 band structure of nanotube Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 26, 2011 13 / 21
Electronic Band Structure of Nanotubes Zone Folding Approximation Closed Expression 2 (6,6) tube 1.5 1 0.5 a.u. 0 -0.5 -1 -1.5 general case -2 k ⑤⑤ Γ 2 k t ✓ ✂ 2 n 1 � n 2 ✡ ✂ 2 n 2 � n 1 ✡ 2 π m ✁ n 2 2 π m � n 1 E ♣ m , k t q ✾ 3 � 2 cos � 2 cos q 2 π k q 2 π k qnR qnR ✡✛ 1 ④ 2 ✂ n 1 ✁ n 2 2 π m ✁ n 1 � n 2 ✚ ✚ ✁ 1 2 , 1 � 2 cos where k P 2 π k qnR q 2 Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 26, 2011 14 / 21
Electronic Band Structure of Nanotubes Zone Folding Approximation Closed Expression 2 (6,6) tube 1.5 1 0.5 a.u. 0 -0.5 -1 -1.5 general case -2 k ⑤⑤ Γ 2 k t ✓ ✂ 2 n 1 � n 2 ✡ ✂ 2 n 2 � n 1 ✡ 2 π m ✁ n 2 2 π m � n 1 E ♣ m , k t q ✾ 3 � 2 cos � 2 cos q 2 π k q 2 π k qnR qnR ✡✛ 1 ④ 2 ✂ n 1 ✁ n 2 2 π m ✁ n 1 � n 2 ✚ ✚ ✁ 1 2 , 1 � 2 cos where k P 2 π k qnR q 2 Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 26, 2011 14 / 21
Electronic Band Structure of Nanotubes Zone Folding Approximation Closed Expression 2 (6,6) tube 1.5 1 0.5 a.u. 0 -0.5 -1 -1.5 general case -2 k ⑤⑤ Γ 2 k t ✓ ✂ 2 n 1 � n 2 ✡ ✂ 2 n 2 � n 1 ✡ 2 π m ✁ n 2 2 π m � n 1 E ♣ m , k t q ✾ 3 � 2 cos � 2 cos q 2 π k q 2 π k qnR qnR ✡✛ 1 ④ 2 ✂ n 1 ✁ n 2 2 π m ✁ n 1 � n 2 ✚ ✚ ✁ 1 2 , 1 � 2 cos where k P 2 π k qnR q 2 Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 26, 2011 14 / 21
Electronic Band Structure of Nanotubes Zone Folding Approximation Closed Expression 2 (6,6) tube 1.5 1 0.5 a.u. 0 -0.5 -1 -1.5 general case -2 k ⑤⑤ Γ 2 k t ✓ ✂ 2 n 1 � n 2 ✡ ✂ 2 n 2 � n 1 ✡ 2 π m ✁ n 2 2 π m � n 1 E ♣ m , k t q ✾ 3 � 2 cos � 2 cos q 2 π k q 2 π k qnR qnR ✡✛ 1 ④ 2 ✂ n 1 ✁ n 2 2 π m ✁ n 1 � n 2 ✚ ✚ ✁ 1 2 , 1 � 2 cos where k P 2 π k qnR q 2 Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 26, 2011 14 / 21
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