Theory of electronic transport in carbon nanotubes

Transcription

1 Theory of electronic transport in carbon nanotubes Reinhold Egger Institut für Theoretische Physik Heinrich-Heine Universität Düsseldorf Les Houches Seminar, July 4

2 Electronic transport in nanotubes Most mesoscopic effects have been observed (see seminar of C.Schönenberger) Disorder-related: MWNTs Strong-interaction effects Kondo and dot physics Superconductivity Spin transport allistic, localized, diffusive transport What has theory to say?

3 Overview Field theory of ballistic single-wall nanotubes: Luttinger liquid and beyond (A.O. Gogolin) Multi-terminal geometries Y junctions (S. Chen &. Trauzettel) Crossed nanotubes: Coulomb drag (A. Komnik) Multi-wall nanotubes: Nonperturbative Altshuler-Aronov effects (A.O. Gogolin) Superconductivity in ropes of nanotubes (A. De Martino)

4 Metallic SWNTs: Dispersion relation asis of graphite sheet contains two atoms: two sublattices p/-, equivalent to right/left movers r/- Two degenerate loch waves at each Fermi point K,K (α/-) φ pα ( x, y) E(k) p p K K r r- r r- E F k

5 SWNT: Ideal D quantum wire Transverse momentum quantization: k y is only relevant transverse mode, all others are far away D quantum wire with two spin-degenerate transport channels (bands) Massless D Dirac Hamiltonian Two different momenta for backscattering: r q E / v < k K F F F F

6 What about disorder? Experimentally observed mean free paths in high-quality metallic SWNTs l µ m allistic transport in not too long tubes No diffusive regime: Thouless argument gives localization length ξ N l l bands Origin of disorder largely unknown. Probably substrate inhomogeneities, defects, bends and kinks, adsorbed atoms or molecules, For now focus on ballistic regime

7 Field theory of interacting SWNTs Keep only two bands at Fermi energy Low-energy expansion of electron operator: Ψ σ ( x, y) ψ ( x) φ ( x y) p, α p ασ pα, ( ) r x y α, e i K φ pα πr D fermion operators: osonization applies Inserting expansion into full SWNT Hamiltonian gives D field theory Egger & Gogolin, PRL 997, EPJ 998 Kane, alents & Fisher, PRL 997 r r

8 Interaction potential (no gates ) Second-quantized interaction part: Unscreened potential on tube surface sin 4 ) ( / a z R y y R x x e U κ ( ) ( ) ( ) ( ) ( ) r r r r U r r d r d r H I r r r r r r r r σ σ σ σσ σ Ψ Ψ Ψ Ψ

9 D 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 ackscattering: Fast density modes, probes short-range properties of interaction ackscattering couplings scale as /R, sizeable only for ultrathin tubes

10 ackscattering couplings k F Momentum exchange q F K K p p K K p p with coupling constant b.e / R f.5e / R

11 osonized form of field theory Four bosonic fields, index a c, c, s, s Charge (c) and spin (s) Symmetric/antisymmetric K point combinations Luttinger liquid & nonlinear backscattering vf H f b dx dx dx a [ ( ) ] Π g ϕ [ cosϕ cosϕ cosϕ cosϕ cosϕ cosϕ ] c a s [ cosϕ ] s cosϑs cosϕc a x a c s s s

12 Luttinger parameters for SWNTs osonization gives g a c Logarithmic divergence for unscreened interaction, cut off by tube length g g c E c / 8e πκhv F. ln ( L ) πr / Pronounced non-fermi liquid correlations

13 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 T k f ( T b f / b) T De v b / b e R/ F R b T T b T f Luttinger liquid SDW CDW Supercond...5 g

14 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 from TDoS Explicit calculation gives ν ( x, E) Re Geometry dependence: iet dte Ψ( x, t) Ψ ( x,) η η bulk end E η ( g / g ) (/ g )/ / 4 > η bulk

15 Conductance probes tunneling DoS Conductance across kink: G T η end Universal scaling of nonlinear conductance: T η end di coth / dv ev k T ev sinh kt ImΨ η π Γ η end end iev πk T iev πk T Delft group

16 Evidence for Luttinger liquid gives g around. Yao et al., Nature 999

17 Multi-terminal circuits: Crossed tubes y chance Fusion: Electron beam welding (transmission electron microscope) Fuhrer et al., Science Terrones et al., PRL

18 Nanotube Y junctions Li et al., Nature 999

19 Landauer-üttiker type theory for Luttinger liquids? Standard scattering approach useless: Elementary excitations are fractionalized quasiparticles, not electrons No simple scattering of electrons, neither at junction nor at contact to reservoirs Generalization to Luttinger liquids Coupling to reservoirs via radiative boundary conditions (or g(x) approach) Junction: oundary condition plus impurities

20 Description of junction (node) Chen, Trauzettel & Egger, PRL Egger, Trauzettel, Chen & Siano, NJP 3 Landauer-üttiker: Incoming and outgoing states related via scattering matrix Ψ ) S Ψ ( ) Difficult to handle for correlated systems What to do? out ( in

21 Some recent proposals Perturbation theory in interactions Lal, Rao & Sen, PR Perturbation theory for almost no transmission Safi, Devillard & Martin, PRL Node as island Nayak, Fisher, Ludwig & Lin, PR 999 Node as ring Chamon, Oshikawa & Affleck, PRL 3 Node boundary condition for ideal symmetric junction (exactly solvable) additional impurities generate arbitrary S matrices, no conceptual problem Chen, Trauzettel & Egger, PRL

22 Ideal symmetric junctions N> branches, junction with S matrix z z S... z z z... z z z... z, N iλ λ implies wavefunction matching at node Ψ( ) Ψ ()... Ψ N () z Ψj ( ) Ψj in() Ψj,, out () Crossover from full to no transmission tuned byλ Texier & Montambaux, JP A

23 oundary conditions at the node Wavefunction matching implies density matching ρ ()... ρ N () can be handled for Luttinger liquid Additional constraints: Kirchhoff node rule Gauge invariance i I i Nonlinear conductance matrix can then be computed exactly for arbitrary parameters G ij e h I i µ j

24 Solution for Y junction withg/ Nonlinear conductance: with ( ) Ψ T V U ie T T ev i i i π / Im ( ) ) /( ) ( ), ( / λ λ λ N N N N N w w D T g i j j j i i ii U V U V G 9 9 8

25 Nonlinear conductance g/ µ ε F eu G.5 µ µ 3 ε F k T/ω c k T/ω c.5 k T/ω c. k T/ω c eu/ω c

26 Ideal junction: Fixed point Symmetric system breaks up into disconnected wires at low energies.. λ λ λ g/3 Only stable fixed point Typical Luttinger power law for all conductance coefficients -G e-6 e-8 e-.. k T/D

27 Asymmetric Y junction Add one impurity of strength W in tube close to node Exact solution possible for g3/8 (Toulouse limit in suitable rotated picture) Transition from truly insulating node to disconnected tube perfect wire 3

28 Asymmetric Y junction: g3/8 Full solution: Asymmetry contribution Strong asymmetry limit: /, 3, 3, I I I I I I δ δ [ ] D W W T e I I i W ew I / / / Im π π δ π πδ Ψ /,,3,3 I I I I

29 Crossed tubes: Theory vs. experiment Komnik & Egger, PRL 998, EPJ Gao, Komnik, Egger, Glattli, achtold, PRL 4 (a) (b) V V A A I A I A V I tube-tube Weakly coupled crossed nanotubes Single-electron tunneling between tubes irrelevant Electrostatic coupling relevant for strong interactions Without tunneling: Local Coulomb drag

30 Characterization: Tunneling DoS Tunneling conductance through crossing: Power law, consistent with Luttinger liquid Quantitative fit gives g.6 Evidence for Luttinger liquid beyond TDoS? G ) X ( S. (a) bulk-bulk.5 di / V ). T(K) V X (mv) X d X ( S (b) K bulk-bulk

31 Dependence on transverse current Experimental data show suppression of zero-bias anomaly when current flows through transverse tube Coulomb blockade or heating mechanisms can be ruled out -, - -,5-5,,5 5, Prediction of Luttinger 3 K. I liquid theory? ( ) -, - -,5-5,,5 5, di /dv ( S) A A di /dv ( S) A A G A ( S) 8K 4K V (mv) A V (mv) A I A (a) 8K 4K K I ( ). (b)

32 Hamiltonian for crossed tubes Without tunneling: Electrostatic coupling and crossing-induced backscattering Density operator: [ ] Π / ) ( () () () i x i i i A i i A A dx H H H H ϕ ρ λ ρ ρ λ [ ] ) ( 6 cos ) ( / / x g x A A ϕ π ρ

33 Renormalization group equations Lowest-order RG equations: Solution: Here: inter-tube coupling most relevant! ( ) ( ) A A A g dl d g dl d / / 4 8 λ λ λ λ λ λ [ ] ) ( () () () () ) ( () ) ( ) 8 ( ) 8 ( / ) 4 ( / A l g A l g A l g A e e l e l λ λ λ λ λ λ λ λ

34 Low-energy solution Keeping only inter-tube coupling, problem is exactly solvable by switching to symmetric and antisymmetric (±) boson fields For g3/6.875, particularly simple: ( ) Im 4 / / A A A V V V T k U V ie T k T k eu U U V h e I ± Ψ ± ± ± ± ± π

35 Comparison to experimental data.., G (4e /h) A (a).5. I A V A (mv) 8K 4K K G (4e /h) A.5,5 I A 8K 4K K. (c) V A (mv),,,,.., G (4e /h) A (b).5. K V A (mv) I ( A) Experimental data G (4e /h) A.5,5 I ( A).5.. K. (d) V A (mv),,,, Theory

36 New evidence for Luttinger liquid Gao, Komnik, Egger, Glattli & achtold, PRL 4 Rather good agreement, only one fit parameter: T. 6 K No alternative explanation works Agreement is taken as new evidence for Luttinger liquid in nanotubes, beyond previous tunneling experiments Additional evidence from photoemission experiments Ishii et al., Nature 3

37 Coulomb drag: Transconductance Strictly local coupling: Linear transconductance always vanishes Finite length: Couplings in /- sectors differ G [ ] ± ± ± ± g L L F A F T T D D T x k k dx L ) /( / /,, / ) ( cos λ λ λ λ Now nonzero linear transconductance, except at T!

38 Linear transconductance: g/4.5.5 T G c ± ± T ± / πt c c Ψ ( c ( c ' ± ± ± ± ' ± Ψ± ± / / ) )

39 Absolute Coulomb drag Averin & Nazarov, PRL 998 Flensberg, PRL 998 Komnik & Egger, PRL 998, EPJ For long contact & low temperature (but finite): Transconductance approaches maximal value G ( T, T / T ) e / h

40 Coulomb drag shot noise Trauzettel, Egger & Grabert, PRL Shot noise at T gives important information beyond conductance iωt P( ω ) dte δi ( t) δi () For two-terminal setup & one weak impurity: DC shot noise carries no information about fractional charge Crossed nanotubes: For A must be due to cross voltage (drag noise) P ei S Ponomarenko & Nagaosa, PR 999 V A, V P

41 Shot noise transmitted to other tube Mapping to decoupled two-terminal problems in ± channels implies δ I δ ( t ) ( ) I Consequence: Perfect shot noise locking P A P ( P P ) / Noise in tube A due to cross voltage is exactly equal to noise in tube Requires strong interactions, g</ Effect survives thermal fluctuations

42 Multi-wall nanotubes: The disorderinteraction problem Russian doll structure, electronic transport in MWNTs usually in outermost shell only Energy scales one order smaller Typically N bands due to doping Inner shells can also create `disorder Experiments indicate mean free path l allistic behavior on energy scales E τ >, τ l / v F R... R

43 Tunneling between shells ulk 3D graphite is a metal: and overlap, tunneling between sheets quantum coherent In MWNTs this effect is strongly suppressed Statistically /3 of all shells metallic (random chirality), since inner shells undoped For adjacent metallic tubes: Momentum mismatch, incommensurate structures Maarouf, Kane & Mele, PR Coulomb interactions suppress single-electron tunneling between shells

44 Interactions in MWNTs: allistic limit Egger, PRL 999 Long-range tail of interaction unscreened Luttinger liquid survives in ballistic limit, but Luttinger exponents are close to Fermi liquid, e.g. η N bands End/bulk tunneling exponents are at least one order smaller than in SWNTs Weak backscattering corrections to conductance suppressed even more!

45 Experiment: TDoS of MWNT achtold et al., PRL TDoS 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 a nm b 8nm

46 Tunneling DoS of MWNTs achtold et al., PRL a b Geometry dependence G (µs) 5 - V(mV) Scaled conductance c.35 K. K 3.5 K 9. K K G (µs) - ev/k T α.36 T(K) G (au) Voltage(mV) ηend η bulk

47 Interplay of disorder and interaction Egger & Gogolin, PRL Mishchenko, Andreev & Glazman, PRL Coulomb interaction enhanced by disorder Nonperturbative theory: Interacting Nonlinear σ Model Kamenev & Andreev, PR 999 Equivalent to Coulomb lockade: spectral density I(ω) of intrinsic electromagnetic modes dt P( E) J ( T Re, t) exp π dω I ( ω ) ω [ iet J ( t) ] ( iωt e )

48 Intrinsic Coulomb blockade TDoS Debye-Waller factor P(E): For constant spectral density: Power law with exponent Here: ( ) T k T k E e e E P d E / / ) ( ε ε ε ν ν ( ) ω α I [ ] ( ) /, / / Re ) ( ) ( * * / * * τ ν ω π ω F n v D U D D D D R n D i D D U I Field/particle diffusion constants

49 Dirty MWNT High energies: E > E D /( πr Thouless Summation can be converted to integral, yields constant spectral density, hence power law TDoS with R * α πν ( D / D ) Tunneling into interacting diffusive D metal Altshuler-Aronov law exponentiates into power law. ut: restricted to l < R D ln )

50 Numerical solution Egger & Gogolin, Chem.Phys. Power law well below Thouless scale Smaller exponent for weaker interactions, only weak dependence on mean free path D pseudogap at very low energies Mishchenko et al., PRL l R, U / πv, v / R F F

51 R(kW) Tc(K) R (kw) R(kW) Superconductivity in ropes of SWNTs. 6 H.4T R5..8 R..5. T(K) 4 HT..4 T(K) 5 T C (H) H T R T(K) field (T) Kasumov et al., PR 3

52 R/Rmax Experimental results for resistance R5 R R4 R T/T * Kasumov et al., PR 3

53 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 ( ) ( ) µ ( ) U u uxx uyy uxx uyy z / R x u y De Martino & Egger, PR 3 r u ( x, y) ( ux, u y, uz ) ( ) 4u xy Suzuura & Ando, PR

54 Normal mode analysis reathing mode ω µ.4 MR R Stretch mode v S 4µ / M( µ ) o ev A 4 m s Twist mode v T µ / M. 4 m s

55 Electron-phonon coupling Main contribution from deformation potential ( ) V ( x, y) α u xx u yy couples to electron density α 3 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)

56 SWNTs with phonon-induced interactions Luttinger parameter in one SWNT due to screened Coulomb interaction: g g Assume good screening (e.g. thick rope) reathing-mode phonon exchange causes attractive interaction: For (,) SWNT: g.3 > g R g g R R α π v ( µ ) F.4nm

57 Superconductivity in ropes Model: H N i ( ) H Lutt Λ ij i ij De Martino & Egger, PR 4 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 again negligible dy Θ * i Θ j Θ i ( y,τ )

58 Order parameter for nanotube rope superconductivity Hubbard Stratonovich transformation: complex order parameter field iφ y, τ e i ( ) i i to decouple Josephson terms Integration over Luttinger fields gives action: S ij, y τ * i Λ ij j ln e Tr ( * * Θ Θ ) Lutt

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

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

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

62 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 ) T T... 5K c c N ν

63 Resistance in superconducting state QPS-induced resistance Perturbative calculation, valid well below transition: ( ) ( ) ( ) ( ) ( ) Γ Γ Γ Γ ) ( / / ) ( 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 De Martino & Egger, PR 4

64 Comparison to experiment Ferrier, De Martino et al., Sol. State Comm. 4 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.3

65 Comparison to experiment: Sample R Nice agreement Fit parameter near Rounding near transition is not described by theory Quantum phase slips low-temperature resistance Thinnest known superconductors R(T)/R(T c ) ν.75 N 87 T c.55 K T/T c

66 Comparison to experiment: Sample R4 Again good agreement, but more noise in experimental data Fit parameter now smaller than, dissipative effects Ropes of carbon nanotubes thus allow to observe quantum phase slips R(T)/R(T c ) ν.6 N 43 T c. K T/T c

67 Conclusions Nanotubes allow for field-theory approach osonization & conformal field theory methods Disordered field theories Close connection to experiments Tunneling density of states Crossed nanotubes & local Coulomb drag Multiwall nanotubes Superconductivity