An introduction to the theory of Carbon nanotubes A. De Martino Institut für Theoretische Physik...
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Transcript of An introduction to the theory of Carbon nanotubes A. De Martino Institut für Theoretische Physik...
An introduction to the theory of Carbon nanotubes
A. De Martino
Institut für Theoretische Physik
Heinrich-Heine Universität
Düsseldorf, Germany
Overview
Introduction
Band structure and electronic properties
Low-energy effective theory for SWNTs
Luttinger liquid physics in SWNTs
Experimental evidence
Summary and conclusions
Allotropic forms of Carbon
Curl, Kroto, Smalley 1985
Iijima 1991graphene
(From R. Smalley´s web image gallery)
Classification of CNs:single layer
Single-wall Carbon nanotubes (SWNTs,1993)- one graphite sheet seamlessly wrapped-up to form a cylinder
- typical radius 1nm, length up to mm
(From R. Smalley´s web image gallery)
(From Dresselhaus et al., Physics World 1998)
(10,10) tube
Classification of CNs: ropes
Ropes: bundles of SWNTs- triangular array of individual SWNTs
- ten to several hundreds tubes
- typically, in a rope tubes of different diameters and chiralities
(From R. Smalley´s web image gallery)(From Delaney et al., Science 1998)
Classification of CNs: many layers
Multiwall nanotubes (Iijima 1991)
- russian doll structure, several inner shells
- typical radius of outermost shell > 10 nm
(From Iijima, Nature 1991) (Copyright: A. Rochefort, Nano-CERCA, Univ. Montreal)
Why Carbon nanotubes so interesting ?
Technological applications- conductive and high-strength composites
- energy storage and conversion devices
- sensors, field emission displays
- nanometer-sized molecular electronic devices
Basic research: most phenomena of mesoscopic physics observed in CNs- ballistic, diffusive and localized regimes in transport
- disorder-related effects in MWNTs
- strong interaction effects in SWNTs: Luttinger liquid
- Coulomb blockade and Kondo physics
- spin transport
- superconductivity
Band structure of graphene
Tight-binding model on hexagonal lattice
- two atoms in unit cell
- hexagonal Brillouin zone
(Wallace PR 1947)
iRRBRA
cHcctHi
,,,0 ..
nm 0.142d ; nm 246.0a
eV37.2
t1
23
K K
Band structure of graphene
Tight-binding model- valence and conduction bands touch at E=0
- at half-filling Fermi energy is zero (particle-hole symmetry): no Fermi surface, six isolated points, only two inequivalent
Near Fermi points
relativistic dispersion relation
Graphene: zero gap semiconductor
m/s108,
,5
FF
F
vKkq
qvqE
(Wallace PR 1947)
Structure of SWNTs: folding graphene
(n,m) nanotube specified by wrapping, i.e. superlattice vector:
Tube axis direction
21 amanC
2211 aatT
)2,2gcd(
/)2(
/)2(
2
1
nmmnd
dmnt
dnmt
R
R
R
Structure of SWNTs
(n,n)
armchair
(n,0)
zig-zag
chiral
(4,0)
Electronic structure of SWNTs
Periodic boundary conditions → quantization of
- nanotube metallic if Fermi points allowed wave vectors,
otherwise semiconducting !
- necessary condition: (2n+m)/3 = integer
k
armchair zigzag
metallic semiconducting
Electronic structure of SWNTs
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
eV13
2
R
vE F
Density of states
Metallic tube:
- constant DoS around E=0
- van Hove singularities at opening of new subbands
Semiconducting tube:
- gap around E=0
Energy scale ~1 eV
- effective field theories valid for all relevant temperatures
Metallic SWNTs: 1D dispersion relation
Only subband with relevant, all others more than 1 eV away
- two degenerate Bloch waves, one
for each Fermi point α=+/-
- two sublattices p =+/-, equivalent
to right/left movers r =+/-
- electron states
- typically doped:
0k
r
KkvEq FFFF
/
SWNTs as ideal quantum wires
Only one subband contributes to transport
→ two spin-degenerate channels
Long mean free paths
→ ballistic transport in not too long tubes
No Peierls instability
SWNTs remain conducting at very low temperatures → model systems to study correlations in 1D metals
m1
Conductance of ballistic SWNTs
Landauer formula: for good contact to voltage reservoirs, conductance is
Experimentally (almost) reached- clear signature of ballistic transport
What about interactions?
h
e
h
eNG bands
22 42
Including electron-electron interactions in 1D
In 1D metals dramatic effect of electron-electron interactions: breakdown of Landau´s Fermi liquid theory
New universality class: Tomonaga-Luttinger liquid
- Landau quasiparticles unstable excitations
- stable excitations: bosonic collective charge and spin density fluctuations
- power-law behaviour of correlations with interaction dependent exponents → suppression of tunneling DoS
- spin-charge separation; fractional charge and statistics
- exactly solvable by bosonization
Experimental realizations: semiconductor quantum wires, FQHE edges states, long organic chain molecules, nanotubes, ...
(Tomonaga 1950, Luttinger 1963, Haldane 1981)
Luttinger liquid properties
1d electron system with dominant long-range interaction
- charge and spin plasmon densities; - spin-charge separation !
- depends on interaction :
Suppression of tunnelling density of states at Fermi surface :
- exponent depends on geometry (bulk or edge)
ExtxedtEx iEt
)0,(),(Re1
),(0
22222LL 22 sxs
Fcxcc
F dxv
Kdxv
H
sxcx ,
cK1
1
c
c
K
K no interaction
repulsive interaction
Field theory of SWNTs
Low-energy expansion of electron field operator:
- Two degenerate Bloch states
at each Fermi point
Keep only the two bands at Fermi energy,
Inserting expansion in free Hamiltonian gives
)()()( ''0 xxdxivH pppxxpF
1
0;
0
1
2,
BA
p
rKi
pR
eyx
p
pp yxFyxyx ),(,,
(Egger and Gogolin PRL1997, Kane et al. PRL 1997)
0k
)(),( xyxF pp
Coulomb interaction
Second-quantized interaction part:
Unscreened potential on tube surface
2222
2
2
´sin4´)(
/
zaR
yyRxx
eU
rrrrUrrrdrdH I
´´´´
2
1´´
´
1D fermion interactions
Momentum conservation allows only two type of processes away from half-filling
- Forward scattering: electrons remain at same Fermi point, probes long-range part of interaction
- Backscattering: electrons scattered to opposite Fermi point, probes short-range properties of interaction
- Backscattering couplings scale as 1/R, sizeable only for ultrathin tubes
Backscattering couplings
Fk2
Momentum exchange
Fq2
Coupling constant
Reaf /05.0/ 2Reab /1.0/ 2
Bosonization:
four bosonic fields (linear combinations of )- charge (c) and spin (s)
- symmetric/antisymmetric K point combinations
Bosonized Hamiltonian
Effective low-energy Hamiltonian
termsnonlinear2
222 aaaa
F Kv
dxH
),,,( ssccaa
(Egger and Gogolin PRL1997, Kane et al. PRL 1997)
rFF ixirqxki
r e 4~
r
Luttinger parameters for SWNTs
Bosonization gives
Long-range part of interaction only affects total charge mode
- logarithmic divergence for unscreened interaction, cut off by tube length
very strong correlations !
3.02.02ln8
12/12
RL
v
eKg
Fc
1caK
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//
)/(
Theoretical predictions
Suppression of tunneling DoS:
- geometry dependent exponent:
Linear conductance across an impurity:
Universal scaling of scaled non-linear conductance across an impurity as function of
EEx ),(
4/)1/1(
8/2/1
g
gg
end
bulk
endTTG 2)(
TkeV B/
Evidence for Luttinger liquid
(Yao et al., Nature 1999)
gives 22.0g
end2
bulk
Conclusions
Effective field theory + bosonization for low-energy properties of SWNTs
Very low-temperature : strong-coupling phases
High-temperature : Luttinger liquid physics
Clear experimental evidence from tunnelling conductance experiments