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

eVdeCECEF

LL 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

lzznznIlz

znzF2/cosh

122/exp21exp 2

22

zFGG mm

Experimental Results

Vibron-assisted tunneling

“Toy” model (Holstein) tunQDleads HHHH

)(,)(,

pkaaaaH jmRLj

pkkkjkleads mjjjj

0 0 int1( ) , ( ), ( )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 cx

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 LHkTG kT

G y LHkT

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

~ 20

1~~

p

p

12

1,

1,

arpRPAar GG 2/Im , tar

n

np n

AG

00

RL

pt

pRL ffG

Gd

heJ

2

2

2/1

4/22

00

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