Introduction to Nanotube
Theory
Reinhold Egger
Institut für Theoretische Physik
Heinrich-Heine Universität Düsseldorf
Miraflores School, 27.9.-4.10.2003
Overview
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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):
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Ropes of SWNTs:
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One wrapped graphite sheet
Typical radius 1 nm, lengths up to several mm
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:
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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 sheet
Basis contains two atoms
a  3d , d  0.14nm
First Brillouin zone

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Hexagonal first Brillouin
zone
Exactly two independent 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

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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


E q   vF q
  
q  k  K , vF  8 105 m / sec

Graphite sheet is semimetallic
Nanotube = rolled graphite sheet
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(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
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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:
2v F
E 
 1eV
3R

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Metal if n=m: Armchair nanotubes
Small-gap semiconductor otherwise (curvatureinduced gap)
Experimentally observed: STM map plus
conductance measurement on same SWNT
In practice intrinsic doping, Fermi energy
typically 0.2 to 0.5 eV
Density of states
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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

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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=+/-)
 p ( x, y )
SWNT: Ideal 1D quantum wire
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Transverse momentum quantization: k y  0
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:

q F  EF / vF  k F  K
What about disorder?
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Experimentally observed mean free paths in
high-quality metallic SWNTs   1m
Ballistic transport in not too long tubes
No diffusive regime: Thouless argument
gives localization length   N bands  2
Origin of disorder largely unknown. Probably
substrate inhomogeneities, defects, bends
and kinks, adsorbed atoms or molecules,…
Here focus on ballistic regime
Conductance of ballistic SWNT
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Two spin-degenerate transport bands
Landauer formula: For good contact to
voltage reservoirs, conductance is
2
2e
2
G  N bands
 4e / h
h
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Experimentally (almost) reached recently
Ballistic transport is possible
What about interactions?
Breakdown of Fermi liquid in 1D
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Landau quasiparticles unstable in 1D because of
electron-electron interactions
Reduced phase space
Stable excitations: Plasmons (collective electronhole pair modes)
Often: Luttinger liquid
Luttinger, JMP 1963; Haldane, J. Phys. C 1981
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Physical realizations now emerging: Semiconductor
wires, nanotubes, FQH edge states, cold atoms,
long chain molecules,…
Some Luttinger liquid basics
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Gaussian field theory, exactly solvable
Plasmons: Bosonic displacement field
Without interactions: Harmonic chain problem

vF
2
2
H0 
dx

(
x
)

(


/

x
)
2 

Bosonization identities

x


R / L ( x)  exp  ik F x   ( x)   i  dx´ ( x´)


k F  k F
 ( x)  
 cos2k F x   ( x) 
 x 
Coulomb interaction

1D interaction potential externally screened by gate
e2
e2
U x  x´ 

x  x´
( x  x´)2  4d 2

Effectively short-ranged on large distance scales
Retain only k=0 Fourier component

Luttinger interaction parameter g
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1
g
1  U 0 / vF
Dimensionless parameter
Unscreened potential: only
multiplicative logarithmic corrections
Density-density interaction from  slow   / x
(forward scattering) gives Luttinger liquid
 2 1
vF
2
H   dx   2  / x  
2
g



Fast-density interactions (backscattering)
ignored here, often irrelevant
Luttinger liquid properties I
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Electron momentum
distribution function:
Smeared Fermi surface
at zero temperature
Power law scaling
nk  k F   k  k F

( g 1/ g  2 ) / 4
Similar power laws:
Tunneling DoS, with
geometry-dependent
exponents
Luttinger liquid properties II
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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
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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 Egger & Gogolin, PRL 1997
Kane, Balents & Fisher, PRL 1997
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Keep only the two bands at Fermi energy
Low-energy expansion of electron operator:
  x, y    p x  p x, y 
p ,
 p x, y  


 
1
iK r
e
2R
1D fermion operators: Bosonization applies
Inserting expansion into full SWNT
Hamiltonian gives 1D field theory
Interaction for insulating substrate

Second-quantized interaction part:
 
1
 
 
H I    dr dr ´ r  ´ r ´
2  ´
 


 U r  r ´ ´ r ´ r 

Unscreened potential on tube surface
U
e2 / 
 y  y´ 
2
( x  x´)  4 R sin 

a
z

2
R


2
2
2
1D fermion interactions
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Insert low-energy expansion
Momentum conservation allows only two
processes away from half-filling
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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
2k F
Momentum exchange
2qF
Coupling constant
b / a  0.1e 2 / R
f / a  0.05e 2 / R
Field theory for individual SWNT
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Four bosonic fields, index a  c, c, s, s 
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Charge (c) and spin (s)
Symmetric/antisymmetric K point combinations
Dual field:  a  a / x
H=Luttinger liquid + nonlinear backscattering


vF
2
2
2


H
dx


g




a
a
x a

2
a
 f  dx cos  c  cos  s   cos  c  cos  s   cos  s  cos  s   
 b  dxcos  s   cos s  cos  c 
Luttinger parameters for SWNTs
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Bosonization gives g a c   1
Logarithmic divergence for unscreened
interaction, cut off by tube length
1/ 2
2


8e
L
g  g c   1 
ln


2R
 vF


1

 0.2...0.3
1  2 Ec / 

Very strong correlations

Phase diagram (quasi long range
order)
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Effective field theory
can be solved in
practically exact way
Low temperature
phases matter only for
ultrathin tubes or in
sub-mKelvin regime
T f  ( f / b)Tb
k BTb  De
vF / b
e
 R / Rb
Tunneling DoS for nanotube
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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

n( x, E )  Re  dteiEt ( x, t )  ( x,0)  E
bulk  g  1 / g  2 / 4
Geometry dependence:
end  (1 / g  1) / 2  2bulk
0

Conductance probes tunneling DoS
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Conductance across
kink:
2 en d
G T

Universal scaling of
nonlinear conductance:
 eU  
ieU
 2 end
T
dI / dU  sinh 
 1   end 
2k BT
 2 k BT  

 eU  1

ieU 
 

 coth 
Im  1   end 
2k BT 
 2k BT  2


Delft
group



2
Transport theory

Simplest case: Single impurity, only charge
sector
Kane & Fisher, PRL 1992

H  H LL   cos 4  (0)
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
Conceptual difficulty: Coupling to reservoirs?
Landauer-Büttiker approach does not work
for correlated systems
First: Screening of charge in a Luttinger liquid
Electroneutrality in a Luttinger liquid
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On large lengthscales,
electroneutrality must
hold: No free uncompensated charges
Inject „impurity“ charge Q
Electroneutrality found
only when including
induced charges on gate
True long-range
interaction: g=0
Q  QLL  QGate  0
QGate   g 2Q


QLL   1  g Q
2
Coupling to voltage reservoirs

Two-terminal case,
applied voltage
eU   L   R

Left/right reservoir injects `bare´ density of
R/L moving charges
 R0 ( L / 2)   L / 2vF
 L0 ( L / 2)   R / 2vF

Screening: actual charge density is
 ( x)   R   L  g 2 (  R0   L0 )
Egger & Grabert, PRL 1997
Radiative boundary conditions
Egger & Grabert, PRB 1998
Safi, EPJB 1999


Difference of R/L currents unaffected by
screening:  R ( x)   L ( x)   R0 ( x)   L0 ( x)
Solve for injected densities
boundary conditions for chiral density
near adiabatic contacts
 1

 1

eU
 2  1  R ( L / 2)   2  1  L ( L / 2)  
2vF
g

g

Radiative boundary conditions …
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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?

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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
Egger & Grabert, PRL 1995
Leclair, Lesage & Saleur, PRB 1996

Bosonization gives Friedel oscillation as
( x) 

kF


cos2k F x   F  cos 4  ( x)
Asymptotics (large distance from barrier)
g



 x 
( x)  cos2k F x   F  
 x 
Very slow decay, inefficient screening in 1D
Singular backscattering at low energies
Friedel oscillation period in SWNTs

Several competing backscattering momenta:
2qF ,2(kF  qF ),4qF


Dominant wavelength and power law of
Friedel oscillations is interaction-dependent
Generally superposition of different
wavelengths, experimentally observed
Lemay et al., Nature 2001
Multi-wall nanotubes: Luttinger liquid?
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Russian doll structure, electronic transport in
MWNTs usually in outermost shell only
Energy scales one order smaller
Typically Nbands  20 due to doping
Inner shells can create `disorder´


Experiments indicate mean free path
Ballistic behavior on energy scales
E  1,   / vF
  R...10R
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
Interactions in MWNTs: Ballistic limit
Egger, PRL 1999

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
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Long-range tail of interaction unscreened
Luttinger liquid survives in ballistic limit, but
Luttinger exponents are close to Fermi liquid,
e.g.
 1
N bands
End/bulk tunneling exponents are at least
one order smaller than in SWNTs
Weak backscattering corrections to
conductance suppressed even more!
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
Tunneling DoS of MWNTs
Basel group, PRL 2001
Geometry dependence
end  2bulk
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

dt
modes

P ( E )  Re 
0


J (T  0, t )  
0
exp iEt  J t
d




I ( ) e it  1
Intrinsic Coulomb blockade

TDOS
Debye-Waller factor P(E):
 E / k BT
 (E)
1 e
  dPE   
 / k B T
0
1 e
For constant spectral density: Power law with
exponent   I (  0)
Here:
1/ 2
2
U0


* n
*
I ( ) 
Re    i / D 
 D D 
2
*
R
2 ( D  D)

n 





D / D  1   0U 0 , D  v  / 2
*
2
F
Field/charge diffusion constant
Dirty MWNT




High energies: E  EThouless  D /( 2R )
Summation can be converted to integral,
yields constant spectral density, hence power
R
law TDOS with

ln D* / D 
20 D
2
Tunneling into interacting diffusive 2D metal
Altshuler-Aronov law exponentiates into
power law. But: restricted to   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
  10 R,U 0 / 2vF  1, vF / R  1
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