Quantum Electromechanical Effects in Carbon Nanotubes and Nano-peapods I.V. Krive B. Verkin Institute for Low Temperature Physics and Engineering, NAS Ukraine V. Karazin Kharkov National University

Outline 1.Discovery of single-wall carbon nanotubes and carbon nano-peapods. 2.Dirac quasiparticles in carbon nanotubes. 3.Ballistic electron transport in metallic carbon nanotubes and Luttinger liquid properties. 4.Klein paradox and chiral tunneling. Giant thermopower. 5.Electron transport in single molecule transistors (Franck-Condon blockade). 6.Vibrational effects in suspended SWNT and carbon nano-peapods (experiments). 7.Nanoelectromechanical effects in nanotube-based Josephson junctions.

Graphene, Graphite, SWNT, Fullerene Nobel Prize in Chemistry for the Discovery of Fullerenes,1996: R. Curl, H. Kroto, R.Smalley; Nobel Prize in Physics (Graphene), 2010: A. Geim, K. Novoselov

Fullerene

Discovery of Multi-walled Carbon Nanotubes Carbon filaments and whiskers (coal industry, metallurgy) – XIX century Multi-walled carbon nanotube (MWNT): d~10-80 nm 1) Л.В. Радушкевич, В.М. Лукьянович, ЖФХ 26, 873 (1952) d≈50 nm (number of walls ~15-20) TEM measurments: 2) M. Hillert, N. Lange, “The structure of graphite filaments”, Z. Kristallogr. 111, 24 (1958) 3) S.Iijima, Nature 354, 56 (1991) Co-centric cylindrical MWNTs produced in electric arc discharge “reactor”. d~10-50 nm

Discovery: 1993 S. Ijima, T. Ichihashi, Nature 363, 603 (1993) NEC Labs, Japan D.S. Bethune et al., Nature 363, 605 (1993) IBM Labs, California, USA Nanotube radius R NT armchair, (n 1 =n 2 =n) → metallic zig-zag (n,0) chiral (n 1,n 2 ) d~1-2 nm SWNTs were discovered in failed attempts to fill MWNTs with pure transition metals (Ni, Co, etc.) → metallic or semiconducting Single-walled Carbon Nanotubes

SWCN is a hollow cylinder. The empty space inside a cylinder can be filled with molecules if their size is smaller d N ~(1-2)nm. It means that SWCNs can be used as a container for gases or liquids with molecular scale leads. An important problem arises → how can one fill and empty CNs? The first observation of peapod (i.e. SWCN filled with C 60 molecules) was announced in B.Smith, M.Monthioux, D.Luzzi Nature, 396, 323 (1998) First Observation and Structural Properties of Peapods

Filling SWNTs What for? (i)Physics of nano-world inside a nanotube (ii)1D crystals (iii)Container for catalysts, gases (H 2 ), medicine nanopills etc. Discovery: J. Sloan, J. Hammer, M. Zwiefka-Sibley, M.L.H. Green, Chem. Commun. (1998), Oxford Univ. RuCl 3 Halides: (KCl) x (UCl 4 ) y, AgCl x Br y, LnCl 3, KI, ZrCl 4 Oxides: Sb 2 O 3 1D KI crystal

How to fill? Filling procedure: acid treatment of SWNT → heating with the filler up to sublimation temperature → annealing (i) opening of SWNT SWNT opening has been demonstrated to be a side effect of the various acid- based purification procedures (HCl, HNO 3, H 2 SO 4 or oxidizing reactants H 2 O 2 ) (ii) gase phase method: (iii) another possibility – filling via liquid phase (molten state of the filling) low efficiency of filling! (iv) annealing High filling rates have not been achieved for solid phase materials with the exception of peapods. filling in a gas phase of the filler by heat treatment (for peapods T~ C)

SWNT: Electronic Spectrum Graphene (2D graphite sheet) is known to be a semimetal (the Fermi surface collapses to two points). The effective 2D Hamiltonian of electronic states around Fermi points is of the form of 2D massless Dirac Hamiltonian “Twisted” boundary conditions along the compactified direction (“y”) result in energy spectrum (Kane, Mele, 1997)

Dirac Quasiparticles. Energy Spectrum Mintmire et al., PRL, 68, 631 (1992) Saito et al., Appl.Phys.Lett., 60, 2204,(1992) Hamada et al., PRL, 68, 1579 (1992)

Ballistic Transport As a rule 1D metals are unstable with respect to Peierls phase transition 1D metal→electron-phonon inter.→Peierls dielectric (∆ P ) Conductivity: half-filling→ solitons ((CH) x ) otherwise→ charge density wave (CDW) Experiments showed that SWNTs are not Peierls-Fröhlich systems: “hard” phonon excitations tubular structures are not strictly 1D systems strong repulsive e-e interaction Nobel Prize in Chemistry for discovery of conducting polymers (2000)

1D Wigner Crystal LL correlation parameter : (experiments with SWNT)

Transport Properties and Kane-Fisher Effect Tunneling in Luttinger liquid (Kane, Fisher, 1991)

Tunneling into Luttinger Liquid α is different for tunneling to the bulk and to the end of quantum wire (QW). For SWNT: 4 independent channels in metallic SWNT (2 – spin degeneracy, 2 – valley degeneracy)

Experiments J.Nygård et al. “Electrical transport measurements on single-walled carbon nanotube”, Appl.Phys. A, 69, 297 (1999). L ~ 0,5 μmΔε ≈ 2 meV Low-T (T<<ε) dependence of G is in good agreement with “1/T-law” High-T (T>>ε) dependence of conductance G(T) ~ T α (α=0,7; α≈0,4) Contradicts G ≈ const valid for noninteracting multi-level QD

Non-relativistic particle, electrostatic potential barrier barrier transparency Ultra-relativistic (massless) particle (“Klein paradox”) helicity (finite backscattering) appears for non-normal particle incidence or for “magnetic scattering” Klein paradox

Hamiltonian (C.L. Kane et al., PRB 66, , 2002) - is the chiral angle – “armchair” nanotube – “zig-zag” nanotube Coulomb blockade oscillations in metallic SWNT Hopping transport in semiconducting SWNT Metallic Single-Wall Carbon Nanotube

General form of scattering potential – scalar potential produced by charged impurity or by nonuniform gate potential – pseudomagnetic potential produced by “strain engineering” It was predicted (T. Ando, 2002) that elastic strain induces an effective vector potential that arises from changes in the electron-hopping amplitude between carbon atoms – strain tensor N. Levy et al., Science 329, 544 (2010) “Strain-Induced Pseudo – Magnetic Fields Greater than 300 Tesla in Graphene Nanobubbles” Pseudomagnetic fields. Strain engineering

Local “chiral” scatterer in the limit Scattering problem is solved for transmission ( ) and reflection ( ) amplitudes by the standard “matching procedure” whereand (“Klein paradox”) A.V. Parafilo, I.V. Krive, E.N. Bogachek, U. Landman, R.I. Shekhter, M. Jonson, Phys. Rev. B83, (2011) Chiral Tunneling

Chiral tunneling is most pronounced at What is the physical meaning of oscillations and “quantization condition”? gap in the spectrum backscattering – Aharonov-Bohm – like phase inset (M. Katsnelson, K. Novoselov, A. Geim, Nature Physics 2, 620 (2006))

Energy dependence of transmission coefficient Smooth transmission coefficient – Mott formula for thermopower General formula for thermopower of noninteracting electrons (Sivan-Imry): electric conductance Thermoelectric Effects

Thermodynamic efficiency is described by “figure of merit” Ideal efficiency (Carnot efficiency) of heat engine Figure of Merit

Nature, 407, 57, (2000) Quantized nano-mechanical oscillations of the C 60 against the gold electrode (ω~1.2 THz; T≈1.5K) result in additional steps (hω~5 meV) in I-V curves. Single molecule transistors

“Toy” model (Holstein): Unitary transformation (Lang-Firsov): Vibron-assisted tunneling

Transport problem can be solved analytically in perturbation theory on Г 0 ~|t 0 | 2 (bare level width) 2. Non-monotonic (anomalous) T-dependence of conductance at (strong coupling) sequential tunneling 3. Vibron-assisted tunneling (weak or moderately strong coupling) 1. Polaronic (Franck-Condon) “blockade” (strong coupling) satellites

Nonlinear integral-differential equation for classical coordinate: At eV>hω 0 x c =0 is unstable solution First publication: L.Y.Gorelik et al., PRL, 80, 4526, (1998) Single level quantum dot: D.Fedorets et al., Europhys. Lett., 58 (1), pp , (2002) Cyclic (stable) solution Electron Shuttling

A.V.Moskalenko et al., University of Bath (UK), Phys.Rev.B 79 (2009) AFM images of shuttle device (20 nm gold nanoparticle) Electron Shuttle. Experiments.

D.R.Koenig, and E.M.Weig, Center for Nanoscience, Munchen, Germany (2012) Electron Shuttle. Experiments.

First experiment: S Sapmaz et al., PRL, 96, (2006), H.van der Zant group, Kavli Institute of Nanoscience, Delft Univ. of Technology Low-T electron transport: (i)T>>Г 0 sequential electron tunneling (ii) T≤Г 0 resonant electron tunneling Suspended SWNT vibrating QD Electron tunneling in the presence of V G is accompanied by the shift of c.m.c. of the nanotube towards back gate (tunneling induces mechanical vibrations of the nanotube) I-V curve of nanotube-based SET (L~0.1-1 μm) revealed vibrational effects induced by stretching mode (~0.6 meV) Nanoelectromechanics of Suspended Carbon Nanotubes

Theory: I.V. Krive, R. Ferone, R.I. Shekhter, M. Jonson, P. Utko, J. Nygard, New J. Phys. 10, (2008) Experiment: P. Utko, R. Ferone, I.V. Krive, R.I. Shekhter, M. Jonson, M. Monthioux, L. Noe, J. Nygard, Nature Com. 1, 37 (2010) Empty SWNT “peapod” – mechanical frequency of cluster oscillations – dimensionless electromechanical coupling Nanoelectromechanical Coupling in Fullerene Peapods

Experimental Results Empty SWNT Nanopeapod scale-bar 5nm

First publication: A. Kasumov et al., Science (1999) E. Pallecchi et al., Appl. Phys. Lett. (2008) First experiment with suspended nanowire between superconducting electrodes: A. Kretinin et al. Cond. Mat. March 2013 Nanowire-based S-Quantum Dot-S junctions

Additional energy scale for superconducting junctions: Δ 0 – superconducting gap 1) “hard” vibrons Franck-Condon blockade of n=0 level dc Josephson current is exponentially suppressed Flensberg et al., PRB (2005) What is the signature of vibrational effects in dc Josephson current? 2) “soft” vibrons Zazunov, Egger, PRB (2010) Numerical calculations in the “dissipative” regime Weak influence of vibrational effects on Josephson current Influence of vibrational subsystem on dc Josephson current

In SNS junction Josephson current is supported by Andreev levels. Short SNS junction: Special case: SINIS (“I” stands for insulator) is equivalent S-QD-S junction Supercurrent is supported by resonant (±ε 0 ) Andreev levels high-T scaling: Model: Josephson Current through a Resonant Level

Sun et al., PRB 61 (2000) Meir, Wingreen PRL, 1992 Analogous formula for superconducting junction retarded GF of interacting electrons in QD (retarded GF’s in Nambu representation) I n (z) is modified Bessel function In perturbation theory on Γ 0 the averages over fermionic and bosonic operators are factorized Normal transport: Anomalous Temperature Dependence of Critical Current

low-Thigh-T enhanced critical current (numerically effect is small) anomalous temperature dependence as compared to tanh(ε 0 /2T) Experiments with suspended SWNT allows one to estimate polaronic energy electron-vibron coupling (stretching mode) (bending mode) A.V. Parafilo, I.V. Krive, R.I. Shekhter, Y.W. Park, M. Jonson, Phys. Rev. B 89, (2014)