1. An introduction to the theory of Carbon nanotubes A. De Martino Institut für Theoretische Physik Heinrich-Heine Universität Düsseldorf, Germany

2. Overview Introduction Band structure and electronic properties Low-energy effective theory for SWNTs Luttinger liquid physics in SWNTs Experimental evidence Summary and conclusions

3. Allotropic forms of Carbon Curl, Kroto, Smalley 1985 Iijima 1991 graphene (From R. Smalley´s web image gallery)

4. Classification of CNs: single layer Single-wall Carbon nanotubes (SWNTs,1993) - one graphite sheet seamlessly wrapped-up to form a cylinder - typical radius 1nm, length up to mm (From R. Smalley´s web image gallery) (From Dresselhaus et al., Physics World 1998) (10,10) tube

5. Classification of CNs: ropes Ropes: bundles of SWNTs - triangular array of individual SWNTs - ten to several hundreds tubes - typically, in a rope tubes of different diameters and chiralities (From R. Smalley´s web image gallery) (From Delaney et al., Science 1998)

6. Classification of CNs: many layers Multiwall nanotubes (Iijima 1991) - russian doll structure, several inner shells - typical radius of outermost shell > 10 nm (From Iijima, Nature 1991) (Copyright: A. Rochefort, Nano-CERCA, Univ. Montreal)

7. Why Carbon nanotubes so interesting ? Technological applications - conductive and high-strength composites - energy storage and conversion devices - sensors, field emission displays - nanometer-sized molecular electronic devices Basic research: most phenomena of mesoscopic physics observed in CNs - ballistic, diffusive and localized regimes in transport - disorder-related effects in MWNTs - strong interaction effects in SWNTs: Luttinger liquid - Coulomb blockade and Kondo physics - spin transport - superconductivity

8. Band structure of graphene Tight-binding model on hexagonal lattice - two atoms in unit cell - hexagonal Brillouin zone (Wallace PR 1947)

9. Band structure of graphene Tight-binding model - valence and conduction bands touch at E=0 - at half-filling Fermi energy is zero (particle-hole symmetry): no Fermi surface, six isolated points, only two inequivalent Near Fermi points relativistic dispersion relation Graphene: zero gap semiconductor (Wallace PR 1947)

10. Structure of SWNTs: folding graphene (n,m) nanotube specified by wrapping, i.e. superlattice vector: Tube axis direction

11. Structure of SWNTs (n,n) armchair (n,0) zig-zag chiral (4,0)

12. Electronic structure of SWNTs Periodic boundary conditions → quantization of - nanotube metallic if Fermi points allowed wave vectors, otherwise semiconducting ! - necessary condition: (2n+m)/3 = integer armchair zigzag metallic semiconducting

13. Electronic structure of SWNTs Band structure predicts three types: - semiconductor if (2n+m)/3 not integer; band gap: - metal if n=m (armchair nanotubes) - small-gap semiconductor otherwise (curvature-induced gap) Experimentally observed: STM map plus conductance measurement on same SWNT In practice intrinsic doping, Fermi energy typically 0.2 to 0.5 eV

14. Density of states Metallic tube: - constant DoS around E=0 - van Hove singularities at opening of new subbands Semiconducting tube: - gap around E=0 Energy scale ~1 eV - effective field theories valid for all relevant temperatures

15. Metallic SWNTs: 1D dispersion relation Only subband with relevant, all others more than 1 eV away - two degenerate Bloch waves, one for each Fermi point α=+/- - two sublattices p =+/-, equivalent to right/left movers r =+/- - electron states - typically doped:

16. SWNTs as ideal quantum wires Only one subband contributes to transport → two spin-degenerate channels Long mean free paths → ballistic transport in not too long tubes No Peierls instability SWNTs remain conducting at very low temperatures → model systems to study correlations in 1D metals

17. Conductance of ballistic SWNTs Landauer formula: for good contact to voltage reservoirs, conductance is Experimentally (almost) reached - clear signature of ballistic transport What about interactions?

18. Including electron-electron interactions in 1D In 1D metals dramatic effect of electron-electron interactions: breakdown of Landau´s Fermi liquid theory New universality class: Tomonaga-Luttinger liquid - Landau quasiparticles unstable excitations - stable excitations: bosonic collective charge and spin density fluctuations - power-law behaviour of correlations with interaction dependent exponents → suppression of tunneling DoS - spin-charge separation; fractional charge and statistics - exactly solvable by bosonization Experimental realizations: semiconductor quantum wires, FQHE edges states, long organic chain molecules, nanotubes,... (Tomonaga 1950, Luttinger 1963, Haldane 1981)

19. Luttinger liquid properties 1d electron system with dominant long-range interaction - charge and spin plasmon densities; - spin-charge separation ! - depends on interaction : Suppression of tunnelling density of states at Fermi surface : - exponent depends on geometry (bulk or edge) no interaction repulsive interaction

20. Field theory of SWNTs Low-energy expansion of electron field operator: - Two degenerate Bloch states at each Fermi point Keep only the two bands at Fermi energy, Inserting expansion in free Hamiltonian gives (Egger and Gogolin PRL1997, Kane et al. PRL 1997)

21. Coulomb interaction Second-quantized interaction part: Unscreened potential on tube surface

22. 1D fermion interactions Momentum conservation allows only two type of processes away from half-filling - Forward scattering: electrons remain at same Fermi point, probes long-range part of interaction - Backscattering: electrons scattered to opposite Fermi point, probes short-range properties of interaction - Backscattering couplings scale as 1/R, sizeable only for ultrathin tubes

23. Backscattering couplings Momentum exchange Coupling constant

24. Bosonization: four bosonic fields (linear combinations of ) - charge (c) and spin (s) - symmetric/antisymmetric K point combinations Bosonized Hamiltonian Effective low-energy Hamiltonian (Egger and Gogolin PRL1997, Kane et al. PRL 1997)

25. Luttinger parameters for SWNTs Bosonization gives Long-range part of interaction only affects total charge mode - logarithmic divergence for unscreened interaction, cut off by tube length very strong correlations !

26. Phase diagram (quasi long range order) Effective field theory can be solved in practically exact way Low temperature phases matter only for ultrathin tubes or in sub-mKelvin regime

27. Theoretical predictions Suppression of tunneling DoS: - geometry dependent exponent: Linear conductance across an impurity: Universal scaling of scaled non-linear conductance across an impurity as function of

28. Evidence for Luttinger liquid (Yao et al., Nature 1999) gives

29. Conclusions Effective field theory + bosonization for low-energy properties of SWNTs Very low-temperature : strong-coupling phases High-temperature : Luttinger liquid physics Clear experimental evidence from tunnelling conductance experiments