Quantum Coherent Nanoelectromechanics

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Quantum Coherent Nanoelectromechanics
Robert Shekhter
In collaboration with
Leonid Gorelik and Mats Jonson
University of Gothenburg / Heriot-Watt University
/ Chalmers Univ. of Technology

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

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Nanoelectromechanical Devices
Quantum bell
Single-C60 transistor
A. Erbe et al., PRL 87, 96106 (2001)
H. Park et al., Nature 407, 57 (2000)
CNT-based nanoelectromechanical devices
V. Sazonova et al., Nature 431, 284 (2004)
B. J. LeRoy et al., Nature 432, 371 (2004)
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Nanomechanical Shuttling of Electrons
bias voltage
dissipation
Gorelik et al, Phys Rev Lett 1998 Shekhter et
al., J Comp Th Nanosc 2007
(review) H.S.Kim, H.Qin, R.Blick,
arXiv0708.1646 (experiment)
current
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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)
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• 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

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Single-Cooper-Pair Box
Coherent superposition of two nearby charge
states 2n and 2(n1) can be created by choosing
a proper gate voltage which lifts the Coulomb
Blockade, Nakamura et al., Nature 1999
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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.
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Shuttling of Superconducting Cooper Pairs
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Possible setup configurations
A supercurrent flows between two leads kept at a
fixed phase difference
Coherence between isolated remote leads created
by shuttling of Cooper pairs
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I Shuttling between coupled superconductors
Relaxation suppresses the memory of initial
conditions.
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How does it work?
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Resulting Expression for the Current
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Black regions no current. The current direction
is indicated by signs
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Mechanically Assisted Superconductive Coupling
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Distribution of phase differences as a function
of number of rotations. Suppression of quantum
fluctuations of phase difference
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Electronic Transport through Vibrating CNT
Shekhter R.I. et al. PRL 97(15) Art.No.156801
(2006).
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Quantum Nanomechanical Interferometer
Classical interferometer (two classical holes
in a screen)
Interference determines the intensity
Quantum nanomechanical Interferometer
(quantum holes determined by a wavefunction)
(Analogy applies for the elastic transport
channel need to add effects of inelastic
scattering)
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Model
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Renormalization of Electronic Tunneling
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Coupling to the Fundamental Bending Mode
Only one vibration mode is taken into account
CNT is considered as a complex scatterer for
electrons tunneling from one metallic lead to
the other
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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
Magnetic-flux dependent tunneling
Amplitude of quantum oscillations about 0.01 nm
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Linear Conductance
(The vibrational subsystem is assumed to be in
equilibrium)
For L1 mm, ? 108 Hz, T 30 mK and H 20-40 T
we estimate DG/G0 1-3 The most striking
feature is the temperature dependence. It comes
from the dynamics of the entire nanotube, not
from the electron dynamics
R.I. Shekhter et al., PRL 97 (2006)
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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 .
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Pauli Restrictions on Allowed Transitions
Through Vibrating Nanowire
The applied bias voltage selects the allowed
inelastic transitions through vibrating nanowire
as fermionic nature of electrons has to be
considered.

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Magnetic Field Dependent Offset Current
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Different Types of NEM Coupling
C(x)

• Capacitive coupling
• Tunneling coupling
• Shuttle coupling
• Inductive coupling

R(x)
C(x)
R(x)
j
Lorentz force for given j
FL
H .
E
Electromotive force at I 0 for given v
v
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Electronically Assisted Nanomechanics
From the shuttle instability we know that
electronic and mechanical degrees of freedom
couple strongly at the nanometre scale. So we may
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
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Supercurrent-Driven Nanomechanics
Model Driven, damped nonlinear oscillator
G. Sonne et al. arXiv0806.4680
Compare NEM resonator as part of a SQUID
Driving Lorentz force
Buks, Blencowe PRB 2006 Zhou, Mizel PRL 2006
Blencowe, Buks PRB 2007 Buks et al. EPL 2008
Induced el.motive force
Energy balance in stationary
regime determines time-averaged dc supercurrent
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Giant Magnetoresistance
V
Alternating Josephson current
Alternating Lorentz force, FL
Mechanical resonances
For small amplitudes (u)
(I)
(II)
Force (I) leads to resonance at
Force (II) leads to parametric resonance at
Accumulation and dissipation of a finite amount
of energy during one each nanowire oscillation
period means that
and Therefore a nonzero average (dc) supercurrent
on resonance
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Giant Magnetoresistance
V
Alternating Josephson current
Alternating Lorentz force, FL
Mechanical resonances
(I)
(II)
Force (I) leads to resonance at
Force (II) leads to parametric resonance at
Accumulation and dissipation of a finite amount
of energy during each nanowire oscillation
period means that
and therefore a nonzero average (dc) supercurrent
on resonance
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Giant Magnetoresistance
The onset of the parametric resonance depends on
magnetic field H. By increasing H the resistance
jumps from to a finite
value.
Parametric resonance
Resonance
larger H
small H
Amplitude of wire oscillations
dc bias voltage
dc bias voltage
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Superconductive Pumping of Nanovibrations
Mathematical formulation
Introduce dimensionless variables
Equation of motion for the nanowire
(Forced, damped, nonlinear oscillator)
Realistic numbers for a SWNT wire makes both
parameters small
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Superconductive Pumping of Nanovibrations
Mathematical formulation
Introduce dimensionless variables
Equation of motion for the nanowire
(Forced, damped, nonlinear oscillator)
Realistic numbers for a SWNT wire makes both
parameters small
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Superconductive Pumping of Nanovibrations
Resonance approximation
Assuming
the equation of motion
by the Ansatz
Inserting the Ansatz in the equation of motion
and integrating over the fast oscillations one
gets for the slowly varying variables
Next n2, drop indices
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Superconductive Pumping of Nanovibrations
Resonance approximation
Assuming
the equation of motion
by the Ansatz
Inserting the Ansatz in the equation of motion
and integrating over the fast oscillations one
gets for the slowly varying variables
Next n2, drop indices
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Multistability of the S-NEM Weak Link Dynamics
Pumping
Dumping
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Onset of the dc Supercurrent on Resonance
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Dynamical Bistability
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Current-Voltage Characteristics
If w/2p1 GHz V0 5 mV, 2dc 50 nV If jdc
100 nA DI1,2 5 nA
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NEM-Assisted Quantum Coherence - Conclusions

• 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.