1. Introduction to Nanotube Theory Reinhold Egger Institut für Theoretische Physik Heinrich-Heine Universität Düsseldorf Miraflores School, 27.9.-4.10.2003

2. Overview Classification & band structure of nanotubes Interaction physics: Luttinger liquid, field theory of single-wall nanotubes Transport phenomena in single-wall tubes Screening effects in nanotubes Multi-wall nanotubes: Interplay of disorder and strong interactions Hot topics: Talk by Alessandro De Martino Experimental issues: Talk by Richard Deblock

3. Classification of carbon nanotubes Single-wall nanotubes (SWNTs):  One wrapped graphite sheet  Typical radius 1 nm, lengths up to several mm Ropes of SWNTs:  Triangular lattice of individual SWNTs (2…several 100) Multi-wall nanotubes (MWNTs):  Russian doll structure, several (typically 10) inner shells  Outermost shell radius about 5 nm

4. Transport in nanotubes Most mesoscopic effects have been observed:  Disorder-related: MWNTs  Strong-interaction effects  Kondo and dot physics  Superconductivity  Spin transport  Ballistic, localized, diffusive transport What has theory to say? Basel group

5. 2D graphite sheet Basis contains two atoms

6. First Brillouin zone Hexagonal first Brillouin zone Exactly two indepen- dent corner points K, K´ Band structure: Nearest-neighbor tight binding model Valence band and conduction band touch at E=0 (corner points!)

7. Dispersion relation: Graphite sheet For each C atom one π electron Fermi energy is zero, no closed Fermi surface, only isolated Fermi points Close to corner points, relativistic dispersion (light cone), up to eV energy scales Graphite sheet is semimetallic

8. Nanotube = rolled graphite sheet (n,m) nanotube specified by superlattice vector imposes transverse momentum quantization Chiral angle determined by (n,m) Important effect on electronic structure

9. Chiral angle and band structure Transverse momentum must be quantized Nanotube metallic only if K point (Fermi point) obeys this condition Necessary condition for metallic nanotubes: (2n+m)/3 = integer

10. Electronic structure 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

11. Density of states Metallic SWNT: constant DoS around E=0, van Hove singularities at opening of new subbands Semiconducting tube: gap around E=0 Energy scale in SWNTs is about 1 eV, effective field theories valid for all relevant temperatures

12. Metallic SWNTs: Dispersion relation Basis of graphite sheet contains two atoms: two sublattices p=+/-, equivalent to right/left movers r=+/- Two degenerate Bloch waves at each Fermi point K,K´ (α=+/-)

13. SWNT: Ideal 1D quantum wire Transverse momentum quantization: is only allowed mode, all others more than 1eV away (ignorable bands) 1D quantum wire with two spin-degenerate transport channels (bands) Massless 1D Dirac Hamiltonian Two different momenta for backscattering:

14. What about disorder? Experimentally observed mean free paths in high-quality metallic SWNTs Ballistic transport in not too long tubes No diffusive regime: Thouless argument gives localization length Origin of disorder largely unknown. Probably substrate inhomogeneities, defects, bends and kinks, adsorbed atoms or molecules,… Here focus on ballistic regime

15. Conductance of ballistic SWNT Two spin-degenerate transport bands Landauer formula: For good contact to voltage reservoirs, conductance is Experimentally (almost) reached recently Ballistic transport is possible What about interactions?

16. Breakdown of Fermi liquid in 1D Landau quasiparticles unstable in 1D because of electron-electron interactions Reduced phase space Stable excitations: Plasmons (collective electron- hole pair modes) Often: Luttinger liquid Luttinger, JMP 1963; Haldane, J. Phys. C 1981 Physical realizations now emerging: Semiconductor wires, nanotubes, FQH edge states, cold atoms, long chain molecules,…

17. Some Luttinger liquid basics Gaussian field theory, exactly solvable Plasmons: Bosonic displacement field Without interactions: Harmonic chain problem Bosonization identities

18. Coulomb interaction 1D interaction potential externally screened by gate Effectively short-ranged on large distance scales Retain only k=0 Fourier component

19. Luttinger interaction parameter g Dimensionless parameter Unscreened potential: only multiplicative logarithmic corrections Density-density interaction from (forward scattering) gives Luttinger liquid Fast-density interactions (backscattering) ignored here, often irrelevant

20. Luttinger liquid properties I Electron momentum distribution function: Smeared Fermi surface at zero temperature Power law scaling Similar power laws: Tunneling DoS, with geometry-dependent exponents

21. Luttinger liquid properties II Electron fractionalizes into spinons and holons (solitons of the Gaussian field theory) New Laughlin-type quasiparticles with fractional statistics and fractional charge Spin-charge separation: Additional electron decays into decoupled spin and charge wave packets  Different velocities for charge and spin  Spatial separation of this electrons´ spin and charge!  Could be probed in nanotubes by magnetotunneling, electron spin resonance, or spin transport

22. Field theory of SWNTs Keep only the two bands at Fermi energy Low-energy expansion of electron operator: 1D fermion operators: Bosonization applies Inserting expansion into full SWNT Hamiltonian gives 1D field theory Egger & Gogolin, PRL 1997 Kane, Balents & Fisher, PRL 1997

23. Interaction for insulating substrate Second-quantized interaction part: Unscreened potential on tube surface

24. 1D fermion interactions Insert low-energy expansion Momentum conservation allows only two processes away from half-filling  Forward scattering: „Slow“ density modes, probes long-range part of interaction  Backscattering: „Fast“ density modes, probes short-range properties of interaction  Backscattering couplings scale as 1/R, sizeable only for ultrathin tubes

25. Backscattering couplings Momentum exchange Coupling constant

26. Field theory for individual SWNT Four bosonic fields, index  Charge (c) and spin (s)  Symmetric/antisymmetric K point combinations Dual field: H=Luttinger liquid + nonlinear backscattering

27. Luttinger parameters for SWNTs Bosonization gives Logarithmic divergence for unscreened interaction, cut off by tube length Very strong correlations

28. 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

29. Tunneling DoS for nanotube Power-law suppression of tunneling DoS reflects orthogonality catastrophe: Electron has to decompose into true quasiparticles Experimental evidence for Luttinger liquid in tubes available Explicit calculation gives Geometry dependence:

30. Conductance probes tunneling DoS Conductance across kink: Universal scaling of nonlinear conductance: Delft group

31. Transport theory Simplest case: Single impurity, only charge sector Kane & Fisher, PRL 1992 Conceptual difficulty: Coupling to reservoirs? Landauer-Büttiker approach does not work for correlated systems First: Screening of charge in a Luttinger liquid

32. Electroneutrality in a Luttinger liquid On large lengthscales, electroneutrality must hold: No free uncompen- sated charges Inject „impurity“ charge Q Electroneutrality found only when including induced charges on gate True long-range interaction: g=0

33. Coupling to voltage reservoirs Two-terminal case, applied voltage Left/right reservoir injects `bare´ density of R/L moving charges Screening: actual charge density is Egger & Grabert, PRL 1997

34. Radiative boundary conditions Difference of R/L currents unaffected by screening: Solve for injected densities boundary conditions for chiral density near adiabatic contacts Egger & Grabert, PRB 1998 Safi, EPJB 1999

35. Radiative boundary conditions … hold for arbitrary correlations and disorder in Luttinger liquid imposed in stationary state apply also to multi-terminal geometries preserve integrability, full two-terminal transport problem solvable by thermodynamic Bethe ansatz Egger, Grabert, Koutouza, Saleur & Siano, PRL 2000

36. Friedel oscillation Why zero conductance at T=0?  Barrier generates oscillatory charge disturbance (Friedel oscillation)  Incoming electron is backscattered by Hartree potential of Friedel oscillation (in addition to bare impurity potential)  Energy dependence linked to Friedel oscillation asymptotics

37. Screening in a Luttinger liquid Bosonization gives Friedel oscillation as Asymptotics (large distance from barrier) Very slow decay, inefficient screening in 1D Singular backscattering at low energies Egger & Grabert, PRL 1995 Leclair, Lesage & Saleur, PRB 1996

38. Friedel oscillation period in SWNTs Several competing backscattering momenta: Dominant wavelength and power law of Friedel oscillations is interaction-dependent Generally superposition of different wavelengths, experimentally observed Lemay et al., Nature 2001

39. Multi-wall nanotubes: Luttinger liquid? Russian doll structure, electronic transport in MWNTs usually in outermost shell only Energy scales one order smaller Typically due to doping Inner shells can create `disorder´  Experiments indicate mean free path  Ballistic behavior on energy scales

40. Tunneling between shells Bulk 3D graphite is a metal: Band overlap, tunneling between sheets quantum coherent In MWNTs this effect is strongly suppressed  Statistically 1/3 of all shells metallic (random chirality), since inner shells undoped  For adjacent metallic tubes: Momentum mismatch, incommensurate structures  Coulomb interactions suppress single-electron tunneling between shells Maarouf, Kane & Mele, PRB 2001

41. Interactions in MWNTs: Ballistic limit Long-range tail of interaction unscreened Luttinger liquid survives in ballistic limit, but Luttinger exponents are close to Fermi liquid, e.g. End/bulk tunneling exponents are at least one order smaller than in SWNTs Weak backscattering corrections to conductance suppressed even more! Egger, PRL 1999

42. Experiment: TDOS of MWNT DOS observed from conductance through tunnel contact Power law zero-bias anomalies Scaling properties similar to a Luttinger liquid, but: exponent larger than expected from Luttinger theory Bachtold et al., PRL 2001 (Basel group)

43. Tunneling DoS of MWNTs Basel group, PRL 2001 Geometry dependence

44. Interplay of disorder and interaction Coulomb interaction enhanced by disorder Microscopic nonperturbative theory: Interacting Nonlinear σ Model Equivalent to Coulomb Blockade: spectral density I(ω) of intrinsic electromagnetic modes Egger & Gogolin, PRL 2001, Chem. Phys. 2002 Rollbühler & Grabert, PRL 2001

45. Intrinsic Coulomb blockade TDOS Debye-Waller factor P(E): For constant spectral density: Power law with exponent Here: Field/charge diffusion constant

46. Dirty MWNT High energies: Summation can be converted to integral, yields constant spectral density, hence power law TDOS with Tunneling into interacting diffusive 2D metal Altshuler-Aronov law exponentiates into power law. But: restricted to

47. Numerical solution Power law well below Thouless scale Smaller exponent for weaker interactions, only weak dependence on mean free path 1D pseudogap at very low energies

48. Multi-wall nanotubes… are strongly interacting but disordered conductors Mesoscopic effects for disordered electrons show up in a strongly interacting situation again Many open questions remain

49. Conclusions Nanotubes allow for sophisticated field-theory approaches, e.g.  Bosonization & conformal field theory methods  Disordered field theories (Wegner-Finkelstein type) Close connection to experiments Looking for open problems to work on?

50. Some hot topics in nanotube theory Intrinsic superconductivity: Nanotube arrays and ropes. Superconducting properties in the ultimate 1D limit? Resonant tunneling in nanotubes Optical properties (e.g. Raman spectra) Transport in MWNTs from field theory of disordered electrons Physics linked to interactions, low dimensions, and possibly disorder