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Introduction to Nanotube Theory
Reinhold Egger Institut für Theoretische PhysikHeinrich-Heine Universität DüsseldorfMiraflores School, 27.9.-4.10.2003
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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
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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
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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
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2D graphite sheet
Basis contains two atoms nmdda 14.0,3
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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!)
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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
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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
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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
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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
eVR
vE F 1
3
2
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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
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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
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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
/
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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
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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?
heh
eNG bands /4
2 22
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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,…
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Some Luttinger liquid basics Gaussian field theory, exactly solvable Plasmons: Bosonic displacement field Without interactions: Harmonic chain problem
Bosonization identities
220 )/()(
2xxdx
vH F
x
FLR xdxixxkix ´)(´)(exp)(/
)(2cos)( xxkk
x
kx F
FF
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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
e
xx
exxU
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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
/1
1
0
2
22 /
1
2x
gdx
vH F
xslow /
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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( gg
FF kkkkn
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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
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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
2
1,
yxxyx pp
p ,,,
Egger & Gogolin, PRL 1997Kane, Balents & Fisher, PRL 1997
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Interaction for insulating substrate Second-quantized interaction part:
Unscreened potential on tube surface
2222
2
2´
sin4´)(
/
zaRyy
Rxx
eU
rrrrU
rrrdrdH I
´´
´´2
1
´
´´
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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
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Backscattering couplings
Fk2Momentum exchange
Fq2
Coupling constant
Reaf /05.0/ 2Reab /1.0/ 2
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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
gdxv
H
coscoscos
coscoscoscoscoscos
2222
sscca ,,,
xaa /
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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
2ln8
12/12
c
Fc
E
RL
v
egg
1cag
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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//
)/(
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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
g
gg
22/)1/1(
4/2/1
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Conductance probes tunneling DoS Conductance across
kink:
Universal scaling of nonlinear conductance:
endTG 2
Tk
ieU
Tk
eU
Tk
ieU
Tk
eUdUdIT
Bend
B
Bend
B
end
21Im
2
1
2coth
21
2sinh/
2
2
Delft group
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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
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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
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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
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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
eUL
gL
g
2)2/(1
1)2/(1
122
Egger & Grabert, PRB 1998Safi, EPJB 1999
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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
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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
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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)( xxkk
x FFF
g
FF x
xxkx
2cos)(
Egger & Grabert, PRL 1995Leclair, Lesage & Saleur, PRB 1996
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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
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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
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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
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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
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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)
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Tunneling DoS of MWNTs
bulkend 2
Basel group, PRL 2001
Geometry dependence
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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
tieId
tTJ
tJiEtdt
EP
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Intrinsic Coulomb blockade TDOS Debye-Waller factor P(E):
For constant spectral density: Power law with exponent Here:
Tk
TkE
B
B
e
eEPd
E/
/
0 1
1)(
)0( I
2/,1/
/Re)(2
)(
200
*
*2/1
2
2**0
F
n
vDUDD
DDR
nDiDD
UI
Field/charge diffusion constant
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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/ln
2*
0
2)2/( RDEE Thouless
R
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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
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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
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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?
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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