Introduction to Nanotube Theory

Reinhold Egger Institut für Theoretische PhysikHeinrich-Heine Universität DüsseldorfMiraflores School, 27.9.-4.10.2003

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

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

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

2D graphite sheetBasis contains two atoms nmdda 14.0,3

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!)

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

sec/108, 5mvKkq

qvqE

F

F

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

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

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

eVRvE F 132

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

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=+/-)

),( yxp

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:

0yk

KkvEq FFFF

/

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

m1

2 bandsN

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?

heheNG bands /42 22

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,…

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

Bosonization identities

220 )/()(

2xxdxvH F

x

FLR xdxixxkix ´)(´)(exp)(/

)(2cos)( xxkkx

kx FFF

Coulomb interaction

1D interaction potential externally screened by gate

Effectively short-ranged on large distance scales Retain only k=0 Fourier component

22

22

4´)(´´

dxx

exx

exxU

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

FvUg

/11

0

2

22 /1

2x

gdxvH F

xslow /

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

4/)2/1( ggFF kkkkn

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

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

rKip e

Ryx

21,

yxxyx pp

p ,,,

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

Interaction for insulating substrate Second-quantized interaction part:

Unscreened potential on tube surface

2222

2

2´sin4´)(

/

zaRyyRxx

eU

rrrrU

rrrdrdH I

´´

´´21

´

´´

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

Backscattering couplings

Fk2Momentum exchange

Fq2

Coupling constant

Reaf /05.0/ 2Reab /1.0/ 2

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

css

ssscsc

aaxaa

F

dxb

dxf

gdxvH

coscoscos

coscoscoscoscoscos

2222

sscca ,,,

xaa /

Luttinger parameters for SWNTs Bosonization gives Logarithmic divergence for unscreened

interaction, cut off by tube length

Very strong correlations

3.0...2.0/21

1

2ln812/12

c

Fc

E

RL

vegg

1cag

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

bF RRbvbB

bf

eDeTk

TbfT//

)/(

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:

ExtxdteExn iEt

)0,(),(Re),(0

bulkend

bulk

ggg

22/)1/1(4/2/1

Conductance probes tunneling DoS Conductance across

kink:

Universal scaling of nonlinear conductance:

endTG 2

TkieU

TkeÚ

TkieU

TkeUdUdIT

Bend

B

Bend

B

end

21Im

21

2coth

21

2sinh/

2

2

Delft group

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

)0(4cos LLHH

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 QgQ

QgQ

QQQ

LL

Gate

GateLL

2

2

1

0

Coupling to voltage reservoirs Two-terminal case,

applied voltage

Left/right reservoir injects `bare´ density of R/L moving charges

Screening: actual charge density isFRL

FLR

vL

vL

2/)2/(

2/)2/(0

0

RLeU

)()( 002LRLR gx

Egger & Grabert, PRL 1997

Radiative boundary conditions Difference of R/L currents unaffected by

screening: Solve for injected densities boundary conditions for chiral density

near adiabatic contacts

)()()()( 00 xxxx LRLR

FLR v

eULg

Lg

2

)2/(11)2/(1122

Egger & Grabert, PRB 1998Safi, EPJB 1999

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

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

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

)(4cos2cos)( xxkkx FFF

g

FF xxxkx

2cos)(

Egger & Grabert, PRL 1995Leclair, Lesage & Saleur, PRB 1996

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

FFFF qqkq 4),(2,2

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

RR 10...

FvE /,1

20bandsN

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

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

bandsN1

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)

Tunneling DoS of MWNTs

bulkend 2

Basel group, PRL 2001

Geometry dependence

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. 2002Rollbühler & Grabert, PRL 2001

1)(),0(

expRe)(

0

0

tieIdtTJ

tJiEtdtEP

Intrinsic Coulomb blockade TDOS Debye-Waller factor P(E):

For constant spectral density: Power law with exponent Here:

Tk

TkE

B

B

eeEPdE

/

/

0 11)(

)0( I 2/,1/

/Re)(2

)(

200

*

*2/1

22*

*0

F

n

vDUDD

DDR

nDiDD

UI

Field/charge diffusion constant

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

DDD

R /ln2

*

0

2)2/( RDEE Thouless

R

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

1/,12/,10 0 RvvUR FF

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

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?

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