Date post: | 26-Dec-2015 |
Category: |
Documents |
Upload: | oliver-ball |
View: | 224 times |
Download: | 0 times |
Magnetopolaronic effects Magnetopolaronic effects in single-molecule in single-molecule transistortransistor
““Magnetopolaronic Effects in Electron Transport Magnetopolaronic Effects in Electron Transport through a Single-Level Vibrating Quantum Dot” , through a Single-Level Vibrating Quantum Dot” , FizikaFizika Nizkikh Temperatur, Vol.37, 12, (December 2011), Nizkikh Temperatur, Vol.37, 12, (December 2011), pp. 1295-1301.pp. 1295-1301.
I.V.Krive, S.I.Kulinich, I.V.Krive, S.I.Kulinich, G.A.SkorobagatkoG.A.Skorobagatko
M.Jonson and M.Jonson and R.I.ShekhterR.I.Shekhter
- B.Verkin ILTPE of NAS - B.Verkin ILTPE of NAS of Ukraine, 47 Lenin of Ukraine, 47 Lenin Ave., Kharkov 61103, Ave., Kharkov 61103, UkraineUkraine
-University of -University of Gothenburg, SE-412 96 Gothenburg, SE-412 96 Gothenburg, SwedenGothenburg, Sweden
Plan.Plan.
Single-molecule transistors (experiment).Single-molecule transistors (experiment).
Vibrational effects: vibron-assisted tunneling, electron Vibrational effects: vibron-assisted tunneling, electron shuttling, polaronic blockade.shuttling, polaronic blockade.
Magnetic field-induced electromechanical coupling.Magnetic field-induced electromechanical coupling.
Magnetopolaronic effects in sequential and resonant Magnetopolaronic effects in sequential and resonant electron transport.electron transport.
Single Molecule Transistor
C60 in vacuum
eV 4.76-LUMO
eV 6.40-HOMO
eVEE HL 6.1
eVCEL 8.1)( 60
eVd
eCECE
FLL 3)()(
2
6060
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
Low-T electron transport:(i) T>>Г0 sequential electron tunneling(ii) T~Г0 resonant electron tunneling
Suspended SWNT<=>vibrating QD
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
,, gBWg VGfdTVG T TG m 1~
Tz /0 0
n
l
l
lz
znznIlzznzF
2/cosh
122/exp21exp
2
22
zFGG mm
Experimental Results
Vibron-assisted tunneling
“Toy” model (Holstein) tunQDleads HHHH
)(,)(,
pkaaaaH jmRLj
pkkkjkleads mjjjj
0 0 int
1( ) , ( ), ( )
2 2QD
iH c c b b b b c c x b b p b b
(0)
,
H.c., [ , ] 1, [ , ] 1jtun j k
k j
H t a c b b c c
Unitary transformation: ˆ ˆ ˆexp( ),H UHU U i pn n c c
bbccH pQD 0
~
H.c.~ )(
0)(
k
pik
jjt ecatH
j
20 0
int 02 /
p
-polaronic shift
Sequantial electron tunneling and polaron tunneling approximation
2. Non-monotonic (anomalous) T-dependence of conductance at (strong coupling)
0T
TTGe
2ch, 2-
00
2
0 0T
sequential tunneling
3. Vibron-assisted tunneling (weak or moderately strong coupling)
1. Polaronic (Franck-Condon) “blockade” (strong coupling)
ˆ ( )cx x t
Nonlinear integral-differential equation for classical coordinate:
)(20 txFxx ccc
At eV>hω0 xc=0 is unstable solution
Electron Shuttling
0exp( / ), ( , ) ( , )j tt t jx j L R
,
( ) ( 1) Rej
jc j k
k j
HF n x t t a c
x
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)
Cyclic (stable) solution ( ) sin( )cx t A t
Nanomechanical Shuttling of Electrons
bias voltage dissipation
current
Theory:Gorelik, Shekhter et al, Phys. Rev. Lett., 1998Shekhter et al., J. Comp. Th. Nanosc., 2007
Experiment: H.S.Kim, H.Qin, R.Blick, arXiv:0708.1646A.V.Moskalenko et al.,Phys.Rev B79 (2009)J. Kotthaus et al, Nature Nanotechnology 2008
Quantum Fluctuation-Induced Aharonov-Bohm Effect
B
2
0
0
20 0
0
411 , 1,6
41 1exp ,2
y LH
kTG kTG y LH
kT
R.I. Shekhter, L.Y. Gorelik, L.I. Glazman, M. Jonson, PRL 95(11), 156801 (2006)
Tunneling Transport in Magnetic Field.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.Laplace and cohesive forces.
Heisenberg equations of motion:Heisenberg equations of motion: 2 equations for fermionic operators : ,2 equations for fermionic operators : , Equation for coordinate operator Equation for coordinate operator
Cohesive force:
Laplace force:
Classical regime of vibrations: Classical regime of vibrations:
where:
and
with - Breit-Wigner transmission coefficient
- Fermi distribution function
Quantum regime of vibrations.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 Magnetopolaronic Blockade; Anomalous Temperature Dependence ; Excess Temperature Dependence ; Excess
current.current.
Conductance:
Current:
Frank-Condon factors:
Excess current:
Polaronic Effects in Resonant Electron Tunneling
Polaron tunneling approximation (PTA)
20t
te
~ 2
0
1~~
pp
12
1
,1
,
arpRPAar GG 2/Im , tar
n
np n
AG
00
RL
pt
pRL ffG
Gd
h
eJ
2
2
2/1
4/22
0
0
RLF
RLGG
RLj e ,
2
0 F
electron dwell time characteristic time of polaron formation
In this approximation
By making use of the Meir-Wingreen formula for the average current through interaction QD we get
No polaronic effects at resonance condition
polaron Green function
In particular at low temperatures resonant conductance
ConclusionConclusion
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.
T