Quantum Coherent Nanoelectromechanics

Robert Shekhter

Leonid Gorelik and Mats Jonson

*

University of Gothenburg / Heriot-Watt University / Chalmers Univ. of Technology

In collaboration with:

• Mechanically assisted superconductivity• NEM-induced electronic Aharonov-Bohm effect• Supercurrent-driven nanomechanics

H. Park et al., Nature 407, 57 (2000)

Quantum ”bell” Single-C60 transistor

A. Erbe et al., PRL 87, 96106 (2001);

Nanoelectromechanical Devices

V. Sazonova et al., Nature 431, 284 (2004)B. J. LeRoy et al., Nature 432, 371 (2004)

CNT-based nanoelectromechanical devices

Nanomechanical Shuttling of Electrons

bias voltage dissipation

curr

entGorelik et al, Phys Rev Lett 1998

Shekhter et al., J Comp Th Nanosc 2007 (review)

H.S.Kim, H.Qin, R.Blick, arXiv:0708.1646 (experiment)

How does mechanics contribute to tunneling of Cooper pairs?

Is it possible to maintain a mechanically-assisted supercurrent?

Gorelik et al. Nature 2001; Isacsson et al. PRL 89, 277002 (2002)

To preserve phase coherence only few degrees of freedom must be involved.

This can be achieved provided:

• No quasiparticles are produced

• Large fluctuations of the charge are suppressed by the Coulomb blockade:

CJ EE

Single-Cooper-Pair Box

Coherent superposition of two “nearby” charge states [2n and 2(n+1)] can be created by choosing a proper

gate voltage which lifts the Coulomb Blockade,

Nakamura et al., Nature 1999

Movable Single-Cooper-Pair Box

Josephson hybridization is produced at the trajectory turning points since near these points the Coulomb

blockade is lifted by the gates.

Shuttling of Superconducting Cooper Pairs

Possible setup configurations

A supercurrent flows between two leads kept at a fixed phase difference

H

LnRn

Coherence between isolated remote leads

created by “shuttling” of Cooper pairs

I: Shuttling between coupled superconductors

22

,

.0

( )2

2 ( )

ˆ( ) cos ( )

( ) exp

C J

C

sJ J s

s L R

L RJ

H H H

e Q xH n

C x e

H E x

xE x E

0

Louville-von Neumann equationDynamics:

, ( )i H Ht

Relaxation suppresses the memory of initial conditions.

How does it work?

0 0

Between the leads Coulomb degeneracy is lifted producing

an additional "electrostatic" phase shift

(1) (0)dt E E

Resulting Expression for the Current

0Average current in units as a function of

electrostatic, , and superconducting, , phases

2I ef

Black regions – no current. The current direction is indicated by signs

Mechanically Assisted Superconductive Coupling

Distribution of phase differences as a function of number of rotations. Suppression of quantum fluctuations of

phase difference

Electronic Transport through Vibrating CNT

Shekhter R.I. et al. PRL 97(15): Art.No.156801 (2006).

Quantum Nanomechanical Interferometer

Classical interferometer(two “classical” holes ina screen)

Quantum nanomechanicalInterferometer (“quantum”holes determined bya wavefunction)

Interferencedeterminesthe intensity

(Analogy applies for the elastic transport channel; need to add effects of inelastic scattering)

Model

, ,e m

L R L R

H H H H T

, ,H a a

22

3 { ( ) ( ) ( , ) ( ) ( )}2e

ieH d r r A r U y u x z r r

m r c

22 2

22

1 ( )( )

2

L

m L

u xH dx x EI

x

Renormalization of Electronic Tunneling

iS iSH e He

3

0

( )( ) ( ) ( ( )) ( ) ( )

xr eHS i d r u x r i dx u x r r

y c

2

0

exp ( )

L

L LR R

eHT T i dxu x

c

Coupling to the Fundamental Bending Mode

Only one vibration mode is taken into account

142 2

0 0 00

ˆ ˆˆ( ) ( ) ( ) 2 ; Lu x Y u x b b Y EI

CNT is considered as a complex scatterer for electrons tunneling from one metallic lead to the other

Theoretical Model • Strong longitudinal quantization of

electrons on the CNT• Perturbative approach to resonant

tunneling though the quantized levels (only virtual localization of electrons

on the CNT is possible)

Effective Hamiltonian

ˆ ˆ( ), , , ,

, ,

ˆ ˆ ( . )2

i b beff R LH a a b b T e a a h c

0 0/ 2gHLY 0 02Y M L *

L Reff

T TT E

Amplitude of quantum oscillations [about 0.01 nm]

Magnetic-flux dependenttunneling

Linear Conductance(The vibrational subsystem is assumed to be in equilibrium)

1,4exp2

0

2

0

TkG

G

B

1,3

41

2

0

2

0

TkTk

h

G

G

BB

ehcHgLY /. 00

For L=1 m, = 108 Hz, T = 30 mK and H = 20-40 T we estimate G/G0 = 1-3%

The most striking feature is the temperature dependence. It comes from the dynamics of the entire nanotube, not from the electrondynamics R.I. Shekhter et al., PRL 97 (2006)

n

n

movablenonbacknback WWW

Backscattering of Electrons due to the Presence of Fullerene.

The probability of backscattering sums up all backscattering channels.The result yields classical formula for non-movable target.

However the sum rule does not apply as Pauli principle puts restrictions on allowed transitions .

×

The applied bias voltage selects the allowed inelastic transitions through vibrating nanowire as fermionic nature of electrons has to be considered.

Pauli Restrictions on Allowed Transitions Through Vibrating Nanowire

Magnetic Field Dependent Offset Current

0 ( ) ;I I V I 2

0

eV

I

V

I 2

00

eV)(0 VI

0V

2

00

eVI

I

Different Types of NEM Coupling

• Capacitive coupling

• Tunneling coupling

• Shuttle coupling

• Inductive coupling

C(x)

R(x)

C(x) R(x)

Lorentz forcefor given j

Electromotive force at I = 0for given v

j

FL

E

v

H .

Electronically Assisted NanomechanicsFrom the ”shuttle instability” we know that electronic and mechanicaldegrees of freedom couple strongly at the nanometre scale. So wemay ask....

Can a coherent flow of electrons drive nanomechanics?

• Does a Superconducting Nanoelectromechanical Single-Electron Transistor (NEM-SSET) have a shuttle instability?

- This is an open question

• Electronic Aharonov-Bohm effect induced by quantum vibrations: Can resonantly tunneling electrons in a B-field drive nanomechanics?

- This is an open question

• Can a supercurrent drive nanomechanics?

- Yes! Topic for the rest of this talk

Supercurrent-Driven Nanomechanics

)sin( cHLJkuuum

)/]2[2()/2( uHLeeV

)(2 tuVjdc

Model: Driven, damped nonlinear oscillator G. Sonne et al. arXiv:0806.4680

Driving Lorentz force

Induced el.motive force

Energy balance in stationary regimedetermines time-averaged dc supercurrent

Compare:

NEM resonator as part of a SQUID

Buks, Blencowe PRB 2006Zhou, Mizel PRL 2006 Blencowe, Buks PRB 2007Buks et al. EPL 2008

Giant Magnetoresistance

V

Alternating Josephson current

Mechanical resonances

Alternating Lorentz force, FL

)/2cos()( ]/4[)/2sin(

)/)(4/2sin(22

eVttuJLeHeVtHLJ

teHLueVtHLJF

cc

cL

Force (I) leads to resonance at

Force (II) leads to parametric resonance at

(I) (II)

/2eV

2/2 eV

Accumulation and dissipation of a finite amount of energy during oneeach nanowire oscillation period means that andTherefore a nonzero average (dc) supercurrent on resonance

0)( tjVW

For small amplitudes (u):

Giant Magnetoresistance

V

Alternating Josephson current

Mechanical resonances

Alternating Lorentz force, FL

)/2cos()( ]/4[)/2sin(

)/)(4/2sin(22

eVttuILeHeVtHLI

teHLueVtHLIF

cc

cL

Force (I) leads to resonance at

Force (II) leads to parametric resonance at

(I) (II)

/2eV

2/2 eV

Accumulation and dissipation of a finite amount of energy during each nanowire oscillation period means that andtherefore a nonzero average (dc) supercurrent on resonance

0)( tIVW

Giant Magnetoresistance

The onset of the parametric resonance depends on magnetic field H. By increasing H the resistance jumps from to a finite value.

)(/ tjVR R

dc bias voltagedc bias voltage

Am

plitu

de o

f w

ire o

scill

atio

ns

Parametric resonanceResonance

”small” H

”larger” H

Superconductive Pumping of Nanovibrations

Mathematical formulation

)8/()( ;/

/2~

;/~)( )/4(

2220

20 cJeLmHHH

eVVm

tueLHY

Introduce dimensionless variables:

Equation of motion for the nanowire:

)~

sin(~ YtVYYY

(Forced, damped, nonlinear oscillator)

1000~/1 and mT, 20 ,nA 100 ,μm 1for 1,~ QHJL c

Realistic numbers for a SWNT wire makes both parameters small:

Superconductive Pumping of Nanovibrations

Mathematical formulation

)8/()( ;/

/2~

;/~)( )/4(

2220

20 cJeLmHHH

eVVm

tueLHY

Introduce dimensionless variables:

Equation of motion for the nanowire:

)~

sin(~ YtVYYY

(Forced, damped, nonlinear oscillator)

1000~/1 and mT, 20 ,nA 100 ,μm 1for 1,~ QHJL c

Realistic numbers for a SWNT wire makes both parameters small:

Superconductive Pumping of Nanovibrations

Resonance approximation

)~

sin(~ YtVYYY

1~ ;1 ;1~ nVAssuming:

the equation of motion:

by the Ansatz: 1)(),( ; /)(/~

cos)()( ttIntntVtItY nnnn

Inserting the Ansatz in the equation of motion and integrating overthe fast oscillations one gets for the slowly varying variables:

nnnnn

nnnnn

dIIdJn

InJII

sin]/[2

cos2~

2/1

2/1

Next: n=2, drop indices

Superconductive Pumping of Nanovibrations

Resonance approximation

)~

sin(~ YtVYYY

1~ ;1 ;1~ nVAssuming:

the equation of motion:

by the Ansatz: 1)(),( ; /)(/~

cos)()( ttIntntVtItY nnnn

Inserting the Ansatz in the equation of motion and integrating overthe fast oscillations one gets for the slowly varying variables:

nnnnn

nnnnn

dIIdJn

InJII

sin]/[2

cos2~

2/1

2/1

Next: n=2, drop indices

20

0I

I

II

I

0

2/~ I IIIIIJ

I 6.9)(2 02

0

0I I

I~

)(4 2 IJ

IIJ ~)cos()(4 2

)sin((I)4 2 J 0

0)(I02 J

Pumping Dumping

;20

2

H

H

c2

2220 J8eL

mcH

Multistability of the S-NEM Weak Link Dynamics

)(64 22

2

IHLe

cjdc

;0 cII HH

c2

2220 J8eL

mcH

)(Hjdc

;20

2

H

H

;cIIH;cIH

;0 cIIII HH

H

)(HI

cIHH 0

cIIcI HHH

cIIHH

0I

Onset of the dc Supercurrent on Resonance

0I

I

cIIHH 2~ V

0

I

2 V

0I

increases

c

Dynamical Bistability

I

V2c c0V

2

1

0VV

V

t

cV

Current-Voltage Characteristics

If ~1 GHz:

V0 ~ 5 V,

2c ~ 50 nV

If jdc ~ 100 nAI1,2 ~ 5 nA

• Phase coherence between remote superconductors can be supported by shuttling of Cooper pairs.

• Quantum nanovibrations cause Aharonov-Bohm interference determining finite magneto-resistance of suspended 1-D wire.

• Resonant pumping of nanovibrations modifies the dynamics of a NEM superconducting weak link and leads to a giant magnetoresistance effect (finite dc supercurrent at a dc driving voltage).

• Multistable nanovibration dynamics allow for a hysteretic I-V curve, sensitivity to initial conditions, and switching between different stable vibration regimes.

NEM-Assisted Quantum Coherence - Conclusions