1. Magnetopolaronic effects in single-molecule transistor

2. “Magnetopolaronic Effects in Electron Transport through a Single-Level Vibrating Quantum Dot” , Fizika Nizkikh Temperatur, Vol.37, 12, (December 2011), pp
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 Gothenburg, Sweden

3. 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.

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

5. 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.

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

7. Nanoelectromechanics of Suspended Carbon Nanotubes
First experiment: S Sapmaz et al., PRL, 96, (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)

8. Nanoelectromechanical Coupling in Fullerene Peapods
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
– Bose distribution function

9. Experimental Results

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

11. 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)

12. 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 , (2002)
Nonlinear integral-differential equation for classical coordinate:
At eV>hω0 xc=0 is unstable solution
Cyclic (stable) solution

13. 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:
A.V.Moskalenko et al.,Phys.Rev B79 (2009)
J. Kotthaus et al, Nature Nanotechnology 2008
bias voltage
dissipation
current

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

15. 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

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

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

18. 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:

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

20. 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

21. 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.