Transcript Document

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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
Nanoelectromechanical Devices
Quantum ”bell”
A. Erbe et al., PRL 87, 96106 (2001);
Single-C60 transistor
H. Park et al., Nature 407, 57 (2000)
CNT-based nanoelectromechanical devices
B. J. LeRoy et al., Nature 432, 371 (2004)
V. Sazonova et al., Nature
431, 284 (2004)
Gorelik 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)
current
Nanomechanical Shuttling of Electrons
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:

E  E
J
C
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
H
nL
A supercurrent flows
between two leads kept at
a fixed phase difference
nR
Coherence between
isolated remote leads
created by “shuttling” of
Cooper pairs
I: Shuttling between coupled superconductors
H  HC  H J
e2 
Q( x) 
HC 
2n 

2C ( x) 
e 
2
ˆ)
H J    EJs ( x ) cos ( s  
s L,R
 x
EJL.R ( x)  E0 exp  

  
Dynamics: Louville-von Neumann equation

 i  H ,         0 ( H ) 
t
Relaxation suppresses the memory of initial conditions.
How does it work?
Between the leads Coulomb degeneracy is lifted producing
an additional "electrostatic" phase shift
    dt  E0 (1)  E0 (0)
Resulting Expression for the Current
Average current in units I 0  2ef as a function of
electrostatic,  , and superconducting, , phases
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 in
a screen)
Quantum nanomechanical
Interferometer (“quantum”
holes determined by
a wavefunction)
Interference
determines
the intensity
(Analogy applies for the elastic transport channel; need to add effects of inelastic scattering)
Model
H
H  H


 L, R
e
 Hm 
T


 L, R
  ie
3

H e   d r {  (r )  
2m
 r c
2
Hm  
L
2
L
2
H    a, a ,
2


A   (r )  U  y  u ( x, z )   (r )  (r )}

1 2
 2 u ( x) 
dx   ( x)  EI
2 
2


x


Renormalization of Electronic Tunneling
H  eiS HeiS
x


 (r ) eH


S  i  d r u ( x) (r )
i
(  dxu ( x)) (r ) (r ) 
y
c 0


3
TL
R
 eH

 TL exp i
R
c


L

0
2


dxu ( x) 


Coupling to the Fundamental Bending Mode
Only one vibration mode is taken into account
uˆ ( x)  Y0u0 ( x) (bˆ  bˆ)
2 ; Y0  

2 2
L

0  EI 
1
4
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
H    a , a , 
 ,
  gHLY0 / 20
 ˆ ˆ
ˆ
b b  (Teff ei (b
2
Y0 

bˆ )
Magnetic-flux dependent
tunneling
a



, 
2M 0  L
a
, R  , L
 h.c)
Teff 
TL TR*
Amplitude of quantum oscillations [about 0.01 nm]
E
Linear Conductance
(The vibrational subsystem is assumed to be in equilibrium)
2


G
2

 exp  4 
 ,

0

G0


G
4 2    2 h
 1
,
  
0  k BT
G0
3 
  gLY0 H .

 1
k BT

 1
k BT
0  hc / e
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 electron
dynamics
R.I. Shekhter et al., PRL 97 (2006)
Backscattering of Electrons due to the
Presence of Fullerene.
n
The probability of backscattering sums up all backscattering channels.
The result yields classical formula for non-movable target.
Wback  Wn  W
non movable
back
n
However the sum rule does not apply as Pauli principle puts
restrictions on allowed transitions .
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.
×
Magnetic Field Dependent Offset Current
I  I0 (V )  I ;
I
I 0 (V )
V0
I    



I 0 eV   0 
I
2
  
eV   


 0
 

eV0   
 0 
V
2
2
Different Types of NEM Coupling
C(x)
• Capacitive coupling
• Tunneling coupling
R(x)
• Shuttle coupling
C(x) R(x)
j
Lorentz force
for given j
• Inductive coupling
FL
H .
E
v
Electromotive
force at I = 0
for given v
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
Supercurrent-Driven Nanomechanics
Model: Driven, damped nonlinear oscillator
G. Sonne et al. arXiv:0806.4680
Compare:
NEM resonator as
part of a SQUID
mu  u  ku  HLJc sin( )

Driving Lorentz force
  (2eV / )  (2e[2HLu] / )
Induced el.motive force
jdcV   u 2 (t )
Energy balance in stationary regime
determines time-averaged dc supercurrent
Buks, Blencowe PRB 2006
Zhou, Mizel PRL 2006
Blencowe, Buks PRB 2007
Buks et al. EPL 2008
Giant Magnetoresistance
V
Alternating Josephson current
Alternating Lorentz force, FL
Mechanical resonances
For small amplitudes (u):
FL  HLJc sin(2eVt /   4eHLu(t ) / ) 
HLJc sin(2eVt / )  [4eH 2 L2 J c / ] u (t ) cos(2eVt / )
(I)
(II)
Force (I) leads to resonance at 2eV /   
Force (II) leads to parametric resonance at 2eV /   2
Accumulation and dissipation of a finite amount of energy during one
each nanowire oscillation period means that W  V j (t )  0 and
Therefore a nonzero average (dc) supercurrent on resonance
Giant Magnetoresistance
V
Alternating Josephson current
Alternating Lorentz force, FL
Mechanical resonances
FL  HLIc sin(2eVt /   4eHLu(t ) / ) 
HLIc sin(2eVt / )  [4eH 2 L2 I c / ] u (t ) cos(2eVt / )
(I)
(II)
Force (I) leads to resonance at 2eV /   
Force (II) leads to parametric resonance at 2eV /   2
Accumulation and dissipation of a finite amount of energy during
each nanowire oscillation period means that W  V I (t )  0 and
therefore a nonzero average (dc) supercurrent on resonance
Giant Magnetoresistance
The onset of the parametric resonance depends on magnetic field H.
By increasing H the resistance R  V / j(t ) jumps from R  
to a finite value.
Amplitude of wire oscillations
Resonance
Parametric resonance
”larger” H
”small” H
dc bias voltage
dc bias voltage
Superconductive Pumping of Nanovibrations
Mathematical formulation
Introduce dimensionless variables:
Y  (4eLH /  ) u (t )
~
~   / m;
V  2eV / 
  H / H 0 2 ; H 02  (m 2 ) /(8eL2 J c )
Equation of motion for the nanowire:
(Forced, damped, nonlinear oscillator)
~
Y  ~Y  Y   sin(Vt  Y )
Realistic numbers for a SWNT wire makes both parameters small:
~,   1 for L  1 μm, J c  100nA, H  20 mT,and1/ ~  Q  1000
Superconductive Pumping of Nanovibrations
Mathematical formulation
Introduce dimensionless variables:
Y  (4eLH /  ) u (t )
~
~   / m;
V  2eV / 
  H / H 0 2 ; H 02  (m 2 ) /(8eL2 J c )
Equation of motion for the nanowire:
(Forced, damped, nonlinear oscillator)
~
Y  ~Y  Y   sin(Vt  Y )
Realistic numbers for a SWNT wire makes both parameters small:
~,   1 for L  1 μm, J c  100nA, H  20 mT,and1/ ~  Q  1000
Superconductive Pumping of Nanovibrations
Resonance approximation
Assuming:
~
V  n    1;
the equation of motion:
  1;
~  1
~
~



Y   Y  Y   sin(Vt  Y )


~
Y
(
t
)

I
(
t
)
cos
V
t / n   n (t ) / n ;
by the Ansatz:
n
In (t ),  n (t )  1
Inserting the Ansatz in the equation of motion and integrating over
the fast oscillations one gets for the slowly varying variables:
 
   2n[dJ I / dI ] sin  
In  ~I n  2nJn I n1/ 2 cos n 
 n
Next: n=2, drop indices
n
1/ 2
n
n
n
Superconductive Pumping of Nanovibrations
Resonance approximation
Assuming:
~
V  n    1;
the equation of motion:
  1;
~  1
~
~



Y   Y  Y   sin(Vt  Y )


~
Y
(
t
)

I
(
t
)
cos
V
t / n   n (t ) / n ;
by the Ansatz:
n
In (t ),  n (t )  1
Inserting the Ansatz in the equation of motion and integrating over
the fast oscillations one gets for the slowly varying variables:
 
   2n[dJ I / dI ] sin  
In  ~I n  2nJn I n1/ 2 cos n 
 n
Next: n=2, drop indices
n
1/ 2
n
n
n
Multistability of the S-NEM Weak Link
Dynamics
4  J 2 (I) sin( )  0
 4  J ( I )  cos( )  ~ I
H2
  2;
H0
2

Pumping
 2 c 2 m 2 
 H 0 

2
8eL J c 

Dumping
I
~ I
 II
J 2 (I0 )  0
I0
I
4J 2 ( I )
0
 I  ~ / 2
I0
 II   I
I
I0
 9.6   I
2J 2 ( I0 )
0

2 
Onset of the dc Supercurrent on Resonance
c 2
jdc 
I ( )
2
2
64eL H
H  0  HcI 
H2
  2;
H0
 2 c 2 m 2 
 H 0 

2
8eL J c 

I (H )
I0
H  H cI  H cII 
H  H cII
jdc (H )
H cI ;
 I H0  HcI ;
H cII ;
H
 II H0  HcII ;
Dynamical Bistability
H  H cII
~
V  2
  increases
I
I0

0

I
I0
c
2
V
Current-Voltage Characteristics
I
If /2~1 GHz:
V0 ~ 5 V,
2c ~ 50 nV
2
1
If jdc ~ 100 nA
I1,2 ~ 5 nA
c
2
V0
c
V
V
V0
V
V   c
t
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.