From Graphene to Nanotubes

Transcription

1 From Graphene to Nanotubes Zone Folding and Quantum Confinement at the Example of the Electronic Band Structure Christian Krumnow Freie Universität Berlin May 6, 011 Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 6, / 1

2 Introduction Outline 1 Introduction 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 6, 011 / 1

3 Introduction Motivation interesting electronic behavior about 1/3 of possible nanotubes are (quasi) metallic depending on geometric structure high degree of complexity 8-number of realizable tubes large number of atoms in unit cell possible Ñ efficient tool needed Hamada et al: Phys Rev Lett (199) Reich et al: Carbon Nanotubes, Wiley (004) Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 6, / 1

4 Going from Graphene to Nanotubes in k-space Outline 1 Introduction 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 6, / 1

5 Going from Graphene to Nanotubes in k-space Real and k-space of Graphene Real and k-space of Graphene real space atomic unit cell lattice basis vectors a 1 and a with }a 1 } }a } a 0 and a Å Reich et al: Carbon Nanotubes, Wiley (004) k-space reciprocal lattice basis vectors k 1 and k K-point at 1 3 pk 1 1 k q, 3 k k 3 k 1 3 k and Reich et al: Phys Rev B (00) Reich et al: Carbon Nanotubes, Wiley (004) Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 6, / 1

6 Going from Graphene to Nanotubes in k-space Real and k-space of Nanotubes Unitcell of Nanotubes in Real Space where chiral vector c pn 1, n q n 1 a 1 n a lattice vector a n n 1 nr, n 1 n nr n greatest common divisor of n 1, n # 3 if 3 n 1 n R n 1 else Reich et al: Carbon Nanotubes, Wiley (004) diameter d }c} π a 0 π bn 1 n n 1n Thomsen et al: Light Scat in Sol IX 108 (007) # graphene unit cells: q }a}}c} a 0? 3{ nr pn 1 n n 1n q Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 6, / 1

7 Going from Graphene to Nanotubes in k-space Real and k-space of Nanotubes Quantum Confinement and Boundary Conditions along tube axis k 1 one dimensional reciprocal lattice k t P }k }, }k } with k }a}â π perpendicular to tube axis due to rolled up structure, periodic boundary conditions are exact k K k finite scale yields quantization of allowed k-values e ik Kc 1 yields k K,m π }c} mĉ k Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 6, / 1

8 Going from Graphene to Nanotubes in k-space Number of Modes Real and k-space of Nanotubes what is the maximal value of the quantum number m? projecting tube unitcell on c-axis ñ q equidistant intersections m is limited by e ik Kαĉ e ipk K k K qαĉ smallest physical period in real space given by α }c} ñ } k K } π }c}{q q k 1 k K m q 1, q,..., 0..., q k k Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 6, / 1

9 Going from Graphene to Nanotubes in k-space Examples Real and k-space of Nanotubes (10,10) tube d a? 0 π nm q 10 3 p q 0 (1,8) tube d a 0 π? nm q 4 1 p q 15 k 1 k 1 k k K k K k k k tube models created with: Blinder: Carbon Nanotubes, Wolfram Demonstrations Project Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 6, / 1

10 Electronic Band Structure of Nanotubes Outline 1 Introduction 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 6, / 1

11 Electronic Band Structure of Nanotubes Zone Folding Zone Folding Approximation 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 (003) Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 6, / 1

12 Electronic Band Structure of Nanotubes Zone Folding Zone Folding Approximation 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 (003) Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 6, / 1

13 Electronic Band Structure of Nanotubes Zone Folding Zone Folding Approximation 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 (003) Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 6, / 1

14 Electronic Band Structure of Nanotubes Zone Folding Approximation Band Structure of Graphene analytic expressions by tight binding model p z and p z -orbitals cross at K-point 3 1 a.u Reich et al: Phys Rev B (00) Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 6, / 1

15 Electronic Band Structure of Nanotubes Construction Zone Folding Approximation band structure of graphene 3 1 a.u Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 6, / 1

16 Electronic Band Structure of Nanotubes Construction Zone Folding Approximation band structure of graphene allowed k-states [(6,6) tube] k a.u k K k k Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 6, / 1

17 Electronic Band Structure of Nanotubes Construction Zone Folding Approximation band structure of graphene allowed k-states [(6,6) tube] k a.u k K k k band structure of nanotube Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 6, / 1

18 (6,6) tube Electronic Band Structure of Nanotubes Zone Folding Approximation Closed Expression a.u general case E pm, k t q 9 3 cos cos n1 n n πm qnr q πk n1 n qnr - Γ cos kt n n 1 k qnr πm n 1 q πk 1{ n1 n πm πk where k 1 P q, 1 Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 6, / 1

19 (6,6) tube Electronic Band Structure of Nanotubes Zone Folding Approximation Closed Expression a.u general case E pm, k t q 9 3 cos cos n1 n n πm qnr q πk n1 n qnr - Γ cos kt n n 1 k qnr πm n 1 q πk 1{ n1 n πm πk where k 1 P q, 1 Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 6, / 1

20 (6,6) tube Electronic Band Structure of Nanotubes Zone Folding Approximation Closed Expression a.u general case E pm, k t q 9 3 cos cos n1 n n πm qnr q πk n1 n qnr - Γ cos kt n n 1 k qnr πm n 1 q πk 1{ n1 n πm πk where k 1 P q, 1 Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 6, / 1

21 (6,6) tube Electronic Band Structure of Nanotubes Zone Folding Approximation Closed Expression a.u general case E pm, k t q 9 3 cos cos n1 n n πm qnr q πk n1 n qnr - Γ cos kt n n 1 k qnr πm n 1 q πk 1{ n1 n πm πk where k 1 P q, 1 Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 6, / 1

22 (6,6) tube Electronic Band Structure of Nanotubes Zone Folding Approximation Closed Expression a.u general case E pm, k t q 9 3 cos cos n1 n n πm qnr q πk n1 n qnr - Γ cos kt n n 1 k qnr πm n 1 q πk 1{ n1 n πm πk where k 1 P q, 1 Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 6, / 1

23 Electronic Band Structure of Nanotubes Zone Folding Approximation Examples: (1,0) and (13,0) Hamada et al: Phys Rev Lett (199) Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 6, / 1

24 Electronic Band Structure of Nanotubes Zone Folding Approximation Electronic Classification metallic tubes from quantization condition: K-point allowed if πz Kc 1 3 pk 1 k qc π 3 pn 1 n q nanotube band structure contains graphene K point if 3 pn 1 n q by courtesy of Prof. Dr. Reich semiconducting tubes all other /3 possible tubes are gapped Mintmire, White: Phys Rev Lett (1998) Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 6, / 1

25 Electronic Band Structure of Nanotubes Limits of Zone Folding Approximations within Zone Folding band structure of graphene approximation of graphene band structure directly implies deviations of obtained tube band structures Reich et al: Phys Rev B (00) Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 6, / 1

26 Electronic Band Structure of Nanotubes Limits of Zone Folding Approximations due to Zone Folding: Curvature geometric effects changes bond length in c-direction shifts point of crossing bands Ñ opens secondary gaps Kane, Mele: Phys Rev Lett (1997) rehybridization affects higher bands mixing of σ and π bands due to nonorthogonality Ouyang et al: Science 9 70 (001) Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 6, / 1

27 Electronic Band Structure of Nanotubes 4 Å tubes Limits of Zone Folding zone folding ab initio Zhaoming: University of Hong Kong, PhD thesis (004) Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 6, / 1

28 Summary Outline 1 Introduction 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 6, / 1

29 Summary Summary and Outlook summary knowledge about graphene allows approximations for nanotubes Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 6, / 1

30 Summary Summary and Outlook summary knowledge about graphene allows approximations for nanotubes in zone folding 1/3 of possible tubes are metallic Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 6, / 1

31 Summary Summary and Outlook summary knowledge about graphene allows approximations for nanotubes in zone folding 1/3 of possible tubes are metallic in reality 1/3 of possible tubes are metallic or small gapped quasi metallic Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 6, / 1

32 Summary Summary and Outlook summary knowledge about graphene allows approximations for nanotubes outlook bundles break symmetry and e.g. open pseudogaps in zone folding 1/3 of possible tubes are metallic in reality 1/3 of possible tubes are metallic or small gapped quasi metallic Ouyang et al: Science 9 70 (001) Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 6, / 1

33 Summary Summary and Outlook summary knowledge about graphene allows approximations for nanotubes outlook bundles break symmetry and e.g. open pseudogaps in zone folding 1/3 of possible tubes are metallic in reality 1/3 of possible tubes are metallic or small gapped quasi metallic Ouyang et al: Science 9 70 (001) multiwalled nanotubes correlations and Luttinger liquid applications? Christian Krumnow (FU Berlin) From Graphene to Nanotubes May 6, / 1