Magnetopolaronic effects in single-molecule transistor

“Magnetopolaronic Effects in Electron Transport through a Single-Level Vibrating Quantum Dot” , Fizika Nizkikh Temperatur, Vol.37, 12, (December 2011), pp. 1295-1301. I.V.Krive, S.I.Kulinich, G.A.Skorobagatko M.Jonson and R.I.Shekhter - B.Verkin ILTPE of NAS of Ukraine, 47 Lenin Ave., Kharkov 61103, Ukraine -University of Gothenburg, SE-412 96 Gothenburg, Sweden

Plan. Single-molecule transistors (experiment). Vibrational effects: vibron-assisted tunneling, electron shuttling, polaronic blockade. Magnetic field-induced electromechanical coupling. Magnetopolaronic effects in sequential and resonant electron transport.

Single Molecule Transistor C60 in vacuum Low-T characteristics of SMT Coulomb blockade Conductance oscillations on VG (CBO)

Nature, 407, 57, (2000) Quantized nano-mechanical oscillations of the C60 against the gold electrode (ω~1.2 THz) result in additional steps (hω~5 μeV) in I-V curves.

Nano letters, 5(2), p.203, (2005)

Nanoelectromechanics of Suspended Carbon Nanotubes First experiment: S Sapmaz et al., PRL, 96, 026801 (2006), H.van der Zant group, Kavli Institute of Nanoscience, Delf Univ. of Technology Suspended SWNT<=>vibrating QD Low-T electron transport: T>>Г0 sequential electron tunneling T~Г0 resonant electron tunneling Electron tunneling in the presence of VG 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)

Nanoelectromechanical Coupling in Fullerene Peapods Theory: I.V. Krive, R. Ferone, R.I. Shekhter, M. Jonson, P. Utko, J. Nygard, New J. Phys. 10, 043043 (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 – Bose distribution function

Experimental Results

Vibron-assisted tunneling “Toy” model (Holstein) Unitary transformation: -polaronic shift

Sequantial electron tunneling and polaron tunneling approximation 1. Polaronic (Franck-Condon) “blockade” (strong coupling) sequential tunneling 2. Non-monotonic (anomalous) T-dependence of conductance at (strong coupling) 3. Vibron-assisted tunneling (weak or moderately strong coupling)

Electron Shuttling First publication: L.Y.Gorelik et al., PRL, 80, 4526, (1998) Single level quantum dot: D.Fedorets et al., Europhys. Lett., 58 (1), pp. 99-104, (2002) Nonlinear integral-differential equation for classical coordinate: At eV>hω0 xc=0 is unstable solution Cyclic (stable) solution

Nanomechanical Shuttling of Electrons Theory: Gorelik, Shekhter et al, Phys. Rev. Lett., 1998 Shekhter et al., J. Comp. Th. Nanosc., 2007 Experiment: H.S.Kim, H.Qin, R.Blick, arXiv:0708.1646 A.V.Moskalenko et al.,Phys.Rev B79 (2009) J. Kotthaus et al, Nature Nanotechnology 2008 bias voltage dissipation current

Quantum Fluctuation-Induced Aharonov-Bohm Effect R.I. Shekhter, L.Y. Gorelik, L.I. Glazman, M. Jonson, PRL 95(11), 156801 (2006) 14

Tunneling Transport in Magnetic Field. Hamiltonian Single-level QD with single vibrational mode (bending mode for SWNT) -is the tunneling length -is the “size” of quantum dot

Laplace and cohesive forces. Heisenberg equations of motion: 2 equations for fermionic operators : , Equation for coordinate operator Cohesive force: Laplace force:

Classical regime of vibrations: where: and with - Breit-Wigner transmission coefficient Fermi distribution function

Quantum regime of vibrations. Tunneling amplitude: - is the dimensionless strength of electron-vibron coupling I. Sequential tunneling: Spectral weights are defined by equation: -noninteracting vibrons! Equilibrium vibrons:

Magnetopolaronic Blockade; Anomalous Temperature Dependence ; Excess current. Conductance: Current: Frank-Condon factors: Excess current:

Polaronic Effects in Resonant Electron Tunneling Polaron tunneling approximation (PTA) electron dwell time characteristic time of polaron formation In this approximation polaron Green function By making use of the Meir-Wingreen formula for the average current through interaction QD we get In particular at low temperatures resonant conductance No polaronic effects at resonance condition

Conclusion In electron transport through a vibrating QD polaronic effects are the same for electric field or magnetic field-induced electromechanical coupling. The manifestations of polaronic (Franck-Condon) blockade are: (i) anomalous temperature dependence of conductance at , and (ii) the excess current in J-V curves at low temperatures. Magnetopolaronic effects are most pronounced in the regime of sequential electron tunneling. Resonant conductance is not renormalized by magnetic field in polaron tunneling approximation.