superconducting properties of carbon nanotubes
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Superconducting properties of carbon nanotubes. Reinhold Egger Institut für Theoretische Physik Heinrich-Heine Universität Düsseldorf A. De Martino, F. Siano. Overview. Superconductivity in ropes of nanotubes Attractive interactions via phonon exchange - PowerPoint PPT PresentationTRANSCRIPT
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Superconducting properties of carbon nanotubes
Reinhold Egger
Institut für Theoretische Physik
Heinrich-Heine Universität Düsseldorf
A. De Martino, F. Siano
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Overview
Superconductivity in ropes of nanotubes Attractive interactions via phonon exchange Effective low energy theory for superconductivity Quantum phase slips, finite resistance in the
superconducting state Josephson current through a short nanotube
Supercurrent through correlated quantum dot via Quantum Monte Carlo simulations
Kondo physics versus π-junction, universality
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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 (typically
up to a few 100) Multi-wall nanotubes (MWNTs):
Russian doll structure, several inner shells Outermost shell radius about 5 nm
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Superconductivity in ropes of SWNTs: Experimental results
Kasumov et al., PRB 2003
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Experimental results II
Kasumov et al., PRB 2003
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Continuum elastic theory of a SWNT: Acoustic phonons Displacement field: Strain tensor:
Elastic energy density:yxxyxy
zxxxx
yyyy
uuu
Ruuu
uu
2
/
222 422 xyyyxxyyxx uuuuu
BuU
),,(),( zyx uuuyxu
Suzuura & Ando, PRB 2002
De Martino & Egger, PRB 2003
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Normal mode analysis
Breathing mode
Stretch mode
Twist mode
AeV14.0
2 RMR
BB
sm102.1/ 4 MvT
sm102)(/4 4 BMBvS
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Electron-phonon coupling Main contribution from deformation potential
couples to electron density
Other electron-phonon couplings small, but
potentially responsible for Peierls distortion
Effective electron-electron interaction generated
via phonon exchange (integrate out phonons)
eV3020
VdxdyH phel
yyxx uuyxV ),(
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SWNT as Luttinger liquid
Low-energy theory of SWNT: Luttinger liquid
Coulomb interaction: Breathing-mode phonon exchange causes
attractive interaction:
Wentzel-Bardeen singularity: very thin SWNT
10 gg
nmBv
R
RRg
gg
FB
B
24.0)(
2
1
2
2
20
0
13.1 gFor (10,10) SWNT:
Egger & Gogolin; Kane et al., PRL 1997De Martino & Egger, PRB 2003
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Superconductivity in ropes
Model:
Attractive electron-electron interaction within each of the N metallic SWNTs
Arbitrary Josephson coupling matrix, keep only singlet on-tube Cooper pair field
Single-particle hopping negligible Maarouf, Kane & Mele, PRB
2003
jiij
ij
N
i
iLutt dyHH
*
1
)(
De Martino & Egger, cond-mat/0308162
,yi
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Order parameter for nanotube rope superconductivity Hubbard Stratonovich transformation:
complex order parameter field
to decouple Josephson terms Integration over Luttinger liquid fields gives
formally exact effective (Euclidean) action:
Lutt
Trjij
yiji eS
**
ln1
,
*
iiii ey ,
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Quantum Ginzburg Landau (QGL) theory 1D fluctuations suppress superconductivity Systematic cumulant & gradient expansion:
Expansion parameter QGL action, coefficients from full model
jijij
i
y
Tr
DCTr
BATrS
11
1*
22
4211
T2
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Amplitude of the order parameter Mean-field transition at
For lower T, amplitudes are finite, with gapped fluctuations
Transverse fluctuations irrelevant for
QGL accurate down to very low T
10 cTA
100N
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Low-energy theory: Phase action Fix amplitude at mean-field value: Low-
energy physics related to phase fluctuations
Rigidity
from QGL, but also influenced by dissipation or disorder
221
2 yss ccdydS
gg
cTTNT
2/)1(
01)(
1
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Quantum phase slips: Kosterlitz-Thouless transition to normal state Superconductivity can be destroyed by vortex
excitations: Quantum phase slips (QPS) Local destruction of superconducting order
allows phase to slip by 2π QPS proliferate for True transition temperature
2)( T
KN
TTgg
cc 5.0...1.02
1)1(2
0
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Resistance in superconducting state QPS-induced resistance Perturbative calculation, valid well below
transition:
4
2
0
4
23)(2
2
2/2
11
22/2
11
)(
cL
LT
cc TiuT
udu
TiuTu
du
TT
TR
TR
LcT s
L
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Comparison to experiment
Resistance below transition allows detailed comparison to Orsay experiments
Free parameters of the theory: Interaction parameter, taken as Number N of metallic SWNTs, known from
residual resistance (contact resistance) Josephson matrix (only largest eigenvalue
needed), known from transition temperature Only one fit parameter remains:
3.1g
1
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Comparison to experiment: Sample R2Nice agreement Fit parameter near 1 Rounding near
transition is not described by theory
Quantum phase slips → low-temperature resistance
Thinnest known superconductors
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Comparison to experiment: Sample R4 Again good agreement,
but more noise in experimental data
Fit parameter now smaller than 1, dissipative effects
Ropes of carbon nanotubes thus allow to observe quantum phase slips
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Josephson current through short tube
Short MWNT acts as (interacting) quantum dot Superconducting reservoirs: Josephson current,
Andreev conductance, proximity effect ? Tunable properties (backgate), study interplay
superconductivity ↔ dot correlations
Buitelaar, Schönenbergeret al., PRL 2002, 2003
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Model
Short MWNT at low T: only a single spin-degenerate dot level is relevant
Anderson model
(symmetric) Free parameters:
Superconducting gap ∆, phase difference across dot Φ Charging energy U, with gate voltage tuned to single
occupancy: Hybridization Γ between dot and BCS leads
nUnnnH
HHHH
dot
BCScoupdot
0
2/0 U
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Supercurrent through nanoscale dot How does correlated quantum dot affect the
DC Josephson current? Non-magnetic dot: Standard Josephson relation
Magnetic dot - Perturbation theory in Γ gives π-junction: Kulik, JETP
1965
Interplay Kondo effect – superconductivity? Universality? Does only ratio matter?
sincII
4/2.0 UK eUT
KT
0cI
Kondo temperature
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Quantum Monte Carlo approach: Hirsch-Fye algorithm for BCS leads Discretize imaginary time in stepsize Discrete Hubbard-Stratonovich transformation
→ Ising field decouples Hubbard-U Effective coupling strength: Trace out lead & dot fermions → self-energies
Now stochastic sampling of Ising field
1 ii ss
Js
ssZ
Ii
1
}{
Tr det1
2/cosh Ue
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QMC approach
Stochastic sampling of Ising paths Discretization error can be eliminated by
extrapolation Numerically exact results Check: Perturbative results are reproduced Low temperature, close to T=0 limit Computationally intensive
Siano & Egger
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Transition to π junction1/,1.0/,1 T
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Kondo regime to π junction crossover Universality: Instead of
Anderson parameters, everything controlled by ratio
Kondo regime has large Josephson current Glazman & Matveev, JETP 1989
Crossover to π junction at surprisingly large
KT/
18/ KT
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Conclusions
Ropes of nanotubes exhibit intrinsic superconductivity, thinnest superconducting wires known
Low-temperature resistance allows to detect quantum phase slips in a clear way
Josephson current through short nanotube: Interplay between Kondo effect, superconductivity, and π junction