Superconductivity in Carbon nanotube ropes

1
Superconductivity in Carbon nanotube ropes
A. De Martino Institut für Theoretische
Physik Heinrich-Heine Universität Düsseldorf,
Germany
R. Egger, Heinrich-Heine Universität,
Düsseldorf H. Bouchiat and M. Ferrier,
Laboratoire de Physique des Solides, Orsay, Paris
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Overview

• Introduction
• Superconductivity in CNs experimental results
• Attractive interactions in SWNTs ?
  Electron-phonon interaction
• Effective theory for superconductivity in ropes
• Comparison to experiments
• Summary and conclusions

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Carbon nanotube 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. Smalleys web image gallery)
(From Delaney et al., Science 1998)
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Luttinger liquid theory of SWNT
(Egger and Gogolin PRL1997, Kane et al. PRL 1997)

• Low-energy effective theory metallic NTs as
  2-channel strongly correlated Luttinger liquids
• Introduce four bosonic fields describing
  total/relative charge/spin densities
• effective Hamiltonian
• Coulomb interaction affects c mode
• with unscreened interactions
• for neutral modes and

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Superconductivityproximity effect
(Kasumov et al. Science 1999,Morpurgo et al.
Science 1999)

• Nanotubes suspended between superconducting
  contacts proximity induced superconductivity
  below 1K

S
S
S
S
Supercurrent
anomalously large critical current
(Kasumov et al. Science 1999)

• No detailed theory at present

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Intrinsic superconductivityisolated SWNTs

• Ultrathin (0.4 nm) SWNTs embedded in zeolite
  matrix
• NTs grown into the zeolite channels limit of
  single nanotube !
• strongly temperature-dependent anisotropic
  diamagnetism below 10 K
• I-V curves indicate opening of a gap at Fermi
  level
• no sharp transition, as expected in 1d
• no theory at present probably strong effects of
  curvature on electron and phonon dispersion
  relations

(Tang et al. Science 2001)
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Intrinsic superconductivityropes

• Ropes suspended between normal electrodes
• good contacts

(Kasumov et al. PRL 2001, PRB 2003)

• Occurrence of a superconducting transition
  depending on
• length
• number of metallic tubes
• normal state resistance

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Experimental results for resistance
(Kasumov et al. PRL 2001, PRB 2003)

• Cannot be understood in the framework of standard
  Luttinger liquid theory (purely electronic
  mechanism

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LL theory predictions for SWNTs
(Egger and Gogolin PRL1997, Kane et al. PRL 1997)

• 1D superconductivity is dominant instability only
  at with
  screened interactions ? purely electronic
  mechanism not sufficient
• What is physical mechanism for superconductivity
  ?
• Attractive interactions ?
• role of electron-phonon interactions

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Elastic model for phonons
(Mahan PRB 2002,Suzuura and Ando PRB 2002, Mahan
and Jeon PRB 2004)

• Detailed results for phonon spectra in SWNTs
  exist but difficult to use in analytical
  computations
• Simple model for low-energy and long-wavelength
  phonons as elastic vibrations ? elasticity theory
• 3D displacement field
• strain tensor
• elastic energy density
• resp. bulk and shear modulus, known
  for graphite

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1D phonon normal modes
(Suzuura and Ando PRB 2002)
Dispersion relations
breathing mode
twisting mode
stretching mode
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Electron-phonon coupling and effective action
(De Martino and Egger PRB 2003)

• Deformation potential ?
• local electron density
• large coupling constant
• Hopping amplitude modulation
• small coupling constant
• potentially relevant for Peierls distorsion
• Electronic effective action
• essentially exact integration over phonon modes
• in principle includes retardation effects, but
  unimportant
• renormalizes effective LL parameters

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Phonon-mediated interactions in SWNTs
(De Martino and Egger PRB 2003)

• In a SWNT with only screened Coulomb interactions
  
• Luttinger parameter
• Renormalization of LL parameter due to phonon
  exchange (breathing mode)
• additional contributions from optical modes

for (10,10) tube with screened Coulomb
interactions
attractive interactions !
(Gonzalez PRB 2003)
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Superconductivity in ropes

• Bundles of ballistic tubes in a
  closed packed triangular lattice with
  compositional disorder
• Electronic states localized on individual tubes
• single particle hopping exponentially suppressed
  due to Fermi momentum mismatch on adjacent tubes
• Intra- and inter-tube electrostatic interactions
  screened
• Dominant coupling mechanism Cooper pair hopping

(Maarouf, Kane and Mele PRB 2000)
(Gonzalez PRB 2003, Alvarez and Gonzalez PRB 2004)
(Gonzalez PRB 2003, De Martino and Egger PRB2004)
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Model for superconducting ropes
(De Martino and Egger, to appear in PRB 2004)

• Attractive electron-electron interactions within
  each metallic SWNT
• No single-particle hopping,no disorder
• Arbitrary Josephson coupling between singlet
  Cooper pair fields on adjacent tubes

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Quantum Ginzburg-Landau action
(De Martino and Egger PRB 2004)

• Superconducting order parameter fields
  decouple Josephson couplings via
  Hubbard-Stratonovich transformation
• Cumulant gradient expansion, expansion
  parameter
• coefficients are functions of temperature and
  microscopic parameters

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Mean field theory

• Mean field critical temperature
• Below , saddle point equation
• Finite amplitudes with a gap for fluctuations
• Transverse fluctuations negligible

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Phase-field action

• Low energy physics dominated by phase
  fluctuations
• amplitudes fixed at mean field value
• Intertube Josephson couplings lock phases
  together ? just a single phase
  for the full rope
• 1D phase action
• phase stiffness
• QLG gives but possibly affected by dissipation
  and disorder

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Quantum phase slips transition to normal state
(Zaikin et al. PRL 1997 De Martino and Egger PRB
2004)

• In 1D strong fluctuations of superconducting
  order parameter phase slips
• local vanishing of amplitude ? possibility of 2p
  shift in phase
• vortex in space-time
• Berezinsky-Kosterlitz-Thouless transition
  controlled by
• QPSs confined into neutral
  pairs ? superconductivity survives
• proliferation of QPSs ?
  normal metallic state
• depression of critical temperature

(Langer and Ambegaokar PR1967,McCumber and
Halperin PRB 1970, Zaikin et al. PRL 1997)
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Resistance well below Tc
(De Martino and Egger PRB 2004)

• QPS events ? dissipation via Josephson effect
• voltage related to the rate of phase slips
• Perturbative computation of induced resistance
• Not valid near transition
• Comparison to experimental data of Kasumov et. Al.

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Parameters in the theory

• LL interaction parameter, taken as
• Number of metallic tubes in the ropes,
  estimated from residual resistance (contact
  resistance)
• Josephson coupling, extracted from transition
  temperature
• Parameter used as a fit parameter, expected
  value

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Comparison to experiments
(De Martino Egger, PRB 2004, M. Ferrier et al.
Sol.State Comm. 2004)

• Rounding near transition not described, otherwise
  good agreement
• Evidence for QPSs

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Summary

• Sizeable phonon-mediated attractive interactions
  in SWNTs with screened Coulomb repulsion
• Josephson coupling between tubes in ropes
• Effective action for superconductivity in ropes
• Quantum phase slips finite resistance below
• Good agreement with experiment