Introduction to Nanotube Theory

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 liquidLuttinger, 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 Egger & Gogolin, PRL 1997 Kane, Balents & Fisher, PRL 1997 • 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

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

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

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 Maarouf, Kane & Mele, PRB 2001 • 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

41. Interactions in MWNTs: Ballistic limit Egger, PRL 1999 • 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!

42. Experiment: TDOS of MWNT Bachtold et al., PRL 2001 (Basel group) • 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

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

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

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