Superconducting properties of carbon nanotubes

Transcription

1 Superconducting properties of carbon nanotubes Reinhold Egger Institut für Theoretische Physik Heinrich-Heine Universität Düsseldorf A. De Martino, F. Siano

2 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 versusπ-junction, universality

3 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

4 R(kW) Tc(K) R (kw) R(kW) Superconductivity in ropes of SWNTs: Experimental results 1. 6 H=1.4T R R T(K) 4 H=0T T(K) T C (H) H= T R T(K) field (T) Kasumov et al., PRB 003

5 R/Rmax Experimental results II R5 R R4 R T/T * Kasumov et al., PRB 003

6 Continuum elastic theory of a SWNT: Acoustic phonons Displacement field: Strain tensor: u = u u yy xx u xy = y x = u y y x u + u + x Elastic energy density: r B ( ) ( ) µ ( ) U u = uxx uyy + uxx uyy + z / R x u y De Martino & Egger, PRB 003 r u ( x, y) = ( ux, u y, uz ) ( ) 4u xy Suzuura & Ando, PRB 00

7 Normal mode analysis Breathing mode ω B B + µ 0.14 = MR R Stretch mode v S = 4Bµ / M( B + µ ) o ev A 10 4 m s Twist mode v T = µ / M m s

8 Electron-phonon coupling Main contribution from deformation potential ( ) V ( x, y) = α u xx + u yy couples to electron density α 0 30 ev H el ph = dxdy Vρ Other electron-phonon couplings small, but potentially responsible for Peierls distortion Effective electron-electron interaction generated via phonon exchange (integrate out phonons)

9 SWNT as Luttinger liquid Low-energy theory of SWNT: Luttinger liquid Coulomb interaction: = g 0 1 Breathing-mode phonon exchange causes attractive interaction: g0 For (10,10) SWNT: g 1.3 > 1 Wentzel-Bardeen singularity: very thin SWNT g Egger & Gogolin; Kane et al., PRL 1997 De Martino & Egger, PRB 003 g R B = = 1 g R R α π v ( B + µ ) F 0 B 0.4nm

10 Superconductivity in ropes Model: H De Martino & Egger, cond-mat/ N = ( i ) Λ ij i = 1 ij H Lutt dy Θ * i Θ j 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 Θ i ( y,τ ) Maarouf, Kane & Mele, PRB 003

11 Order parameter for nanotube rope superconductivity Hubbard Stratonovich transformation: complex order parameter field iφ y, τ = e i ( ) i to decouple Josephson terms i Integration over Luttinger liquid fields gives formally exact effective (Euclidean) action: S = ij, yτ * i Λ 1 ij j ln e Tr ( * * Θ + Θ ) Lutt

12 Quantum Ginzburg Landau (QGL) theory 1D fluctuations suppress superconductivity Systematic cumulant & gradient expansion: Expansion parameter πt QGL action, coefficients from full model S = + Tr + Tr Tr {( ) } 1 4 Λ A + B { } C + D ij y * i 1 ( 1 1 Λ ) ij Λ 1 j τ + +

13 Amplitude of the order parameter Mean-field transition at A ( 0 T ) c = Λ 1 For lower T, amplitudes are finite, with gapped fluctuations Transverse fluctuations irrelevant for N 100 QGL accurate down to very low T ο/πτ T/T c

14 Low-energy theory: Phase action Fix amplitude at mean-field value: Lowenergy physics related to phase fluctuations S Rigidity = µ π [ ( Φ) + ( Φ) ] 1 dydτ c s c s ν 1 from QGL, but also influenced by dissipation or disorder τ µ( T ) = Nν 1 T 0 T c y ( g 1)/ g

15 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 π QPS proliferate for µ ( T ) True transition temperature g ( g 1) = T T K c c N ν

16 Resistance in superconducting state QPS-induced resistance Perturbative calculation, valid well below transition: ( ) ( ) ( ) ( ) ( ) Γ + Γ + Γ + Γ + = ) ( / 1 1 / 1 1 ) ( µ µ µ µ µ c L L T c c T iut u du T iut u du T T R T T R L c T s L π =

17 Comparison to experiment Resistance below transition allows detailed comparison to Orsay experiments Free parameters of the theory: Interaction parameter, taken as g 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: ν =1.3 1

18 Comparison to experiment: Sample R Nice agreement Fit parameter near 1 Rounding near transition is not described by theory Quantum phase slips low-temperature resistance Thinnest known superconductors R(T)/R(T c ) ν = 0.75 N = 87 T c = 0.55 K T/T c

19 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 R(T)/R(T c ) ν = 0.16 N = 43 T c = 0.1 K T/T c

20 Josephson current through short tube Buitelaar, Schönenberger et al., PRL 00, 003 Short MWNT acts as (interacting) quantum dot Superconducting reservoirs: Josephson current, Andreev conductance, proximity effect? Tunable properties (backgate), study interplay superconductivity dot correlations

21 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: ε 0 = U / HybridizationΓbetween dot and BCS leads H H = dot H = dot ε 0 + H coup + H BCS ( n + n ) + Un n

22 Supercurrent through nanoscale dot How does correlated quantum dot affect the DC Josephson current? Non-magnetic dot: Standard Josephson relation ( φ) sinφ I = I c Magnetic dot - Perturbation theory in Γ gives π-junction: I < 0 Kulik, JETP 1965 c Interplay Kondo effect superconductivity? Universality? Does only ratio matter? T K 0. ΓU e πu / 4Γ Kondo temperature T K

23 Quantum Monte Carlo approach: Hirsch-Fye algorithm for BCS leads Discretize imaginary time in stepsize Discrete Hubbard-Stratonovich transformation Ising field s sτ =± decouples Hubbard-U = i ( ) 1 Effective coupling strength: i Trace out lead & dot fermions self-energies I 1 ( φ) = det[ + Σ + λs] Tr [ + Σ + λs] Z { s i } τ cosh Now stochastic sampling of Ising field ( ) 1 Σ τ λ δ τ = e Uδ / τ J

24 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

25 Transition to junction = 1, T / = 0.1, Γ / = 1

26 Kondo regime to π junction crossover Universality: Instead of Anderson parameters, everything controlled by ratio /T K Kondo regime has large Josephson current Glazman & Matveev, JETP 1989 Crossover to π junction at surprisingly large / 18 T K

27 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