Download Magnetopolaronic effects in single

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
LUMO
- 4.76 eV
HOMO - 6.40 eV
  EL  EH  1.6eV

EL (C60
)  1.8eV
2
e

  EL (C60
)  EL (C60 ) 
 3eV
dF
Low-T characteristics of SMT
(i) Coulomb blockade
(ii) 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:
(i) T>>Г0 sequential electron tunneling
(ii) 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)
G V g , T    d  f  G BW V g ,  
Empty SWNT
 T  
G m  ~ 1 T
Gm   G m  F z 
“peapod”

2


exp

lz
/
2
I
2

nz 1  nz 
2
l
F z   exp   1  2nz  
cosh 2 lz / 2
l  

z  0 / T



 0 – mechanical frequency of cluster
oscillations
– dimensionless electromechanical
coupling
n
– Bose distribution function

Experimental Results
Vibron-assisted tunneling
H  H leads  H QD  H tun
“Toy” model (Holstein)
H leads 
H QD

(



)
a
 k j j k j ak j
j  L,R
a


,
a
kj
pm   jm ( k  p )
1 
i
  0c c  0b b   int (b  b)c c, x 
(b  b), p 
(b   b)
2
2







H tun   t (0)
a
c

H.c.,
[
b
,
b
]

1,
[
c
,
c
] 1
j
kj
k, j
Unitary transformation:
ˆ ˆ ), nˆ  c c
H  UHU  U  exp(i pn
~
H QD   p c  c  0b b
~
H t( j )   t0( j ) ak j c  eip  H.c.
k
 p   0   2 0
   2 int / 0
-polaronic shift
Sequantial electron tunneling and polaron
tunneling approximation
1. Polaronic (Franck-Condon) “blockade” (strong coupling)
0    e
 2
0 , G 
2. Non-monotonic (anomalous)
T-dependence of conductance
at T  0 (strong coupling)
3. Vibron-assisted tunneling (weak
or moderately strong coupling)


ch -2
T
2T
0  T  0
sequential tunneling
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)
 xˆ  xc (t )
Nonlinear integral-differential equation for classical coordinate:
xc  02 xc  F xc (t )
F  
t j  t0 exp( jx / t ), j  ( L, R)  (, )
H
  n  xc (t )   (1) j t j Re  akj c 
x
k, j
At eV>hω0 xc=0 is unstable solution
Cyclic (stable) solution
xc (t )  Asin(t   )
Nanomechanical Shuttling of Electrons
Theory:
Gorelik, Shekhter et al, Phys. Rev. Lett., 1998
Shekhter et al., J. Comp. Th. Nanosc., 2007
current
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
Quantum Fluctuation-Induced Aharonov-Bohm Effect
B
2



1  4 y0 LH

 1

 ,
 1,
6 kT   0 
G 
kT

 1  4 y LH 2 

G0 
0
 1
  , kT
exp  2  
0
 
 

R.I. Shekhter, L.Y. Gorelik,
L.I. Glazman, M. Jonson, PRL
95(11), 156801 (2006)
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.
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

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)
e ~

t
  p ~
electron dwell time

p
polaron Green function

1
GrRPA
, a  G p     r , a
G p   
By making use of the Meir-Wingreen
formula for the average current
through interaction QD we get
In particular at low
temperatures
resonant conductance
G  G0

  
n  
1
0
2 1
2
characteristic time of polaron formation
t   0 2
In this approximation
~

Im  r ,a   t / 2
1
An
0  n 0
L R G p2  
e
 f L    f R  
J   d
2
h
1  t G p   / 2


L  R  
 F   0 2  L    R  2 / 4
No polaronic effects at resonance condition
F  0
 j    e  L , R
2
Conclusion

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In electron transport through a vibrating QD polaronic
effects are the same for electric field or magnetic fieldinduced electromechanical coupling.
The manifestations of polaronic (Franck-Condon) blockade
are: (i) anomalous temperature dependence of conductance
at T   , 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.