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Five-Lecture Course on the Basic Physics of
Nanoelectromechanical Devices
• Lecture 1: Introduction to nanoelectromechanical
systems (NEMS)
• Lecture 2: Electronics and mechanics on the
nanometer scale
• Lecture 3: Mechanically assisted single-electronics
• Lecture 4: Quantum nano-electro-mechanics
• Lecture 5: Superconducting NEM devices
Lecture 4: Quantum NanoElectromechanics
Outline
• Quantum coherence of electrons
• Quantum coherence of mechanical displacements
• Quantum nanomechanical operations:
a) shuttling of single electrons;
b) mechanically induced quantum
interferrence of electrons
Lecture 4: Quantum Nano-Electromechanics
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Quantum Coherence of Electrons
 p x 
r, p
Classical approach
2
Heisenberg principle in quantum
approach
Formalization of Heisenberg’s principle:
 Â
operators for physical variables
  
eigenfunctions – quantum states
 p (r )  Ce
i
pr
 (r )   a p p (r )
p
quantum state with definite momentum
In this state the momentum experiences
quantum fluctuations
Lecture 4: Quantum Nano-Electromechanics
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Stationary Quantum States
Hamiltonian of a single electron:
Stationary quantum states:
2
ˆ
p
Hˆ 
 U (r );
2m
Ĥ   E 

pˆ 
i r
Lecture 4: Quantum Nano-Electromechanics
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Second Quantization
• Spatial quantization  discrete quantum numbers
• Due to quantum tunneling the number of electrons in the body experiences
quantum fluctuations and is not an integer
• One therefore needs a description that treats the particle number N as a
quantum variable
Wave function for system of N electrons:
 N   (r )
n
Creation and annihilation operators
aˆ  N (rn )   N 1 (rn )
Nˆ   nˆ   a a
a , a    , fermions

aˆ N (rn )   N 1 (rn )
Hˆ   nˆ  
a , a    , bosons


 
 

Lecture 4: Quantum Nano-Electromechanics
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Field Operators
ˆ  (r )   a (r ); nˆ (r )  ˆ  (r )ˆ (r )

Hˆ   drˆ  (r ) Hˆ (r )ˆ (r )
ˆ (r1 ) ,ˆ  (r2 )    (r1  r2 )

Density Matrix
A  Sp{ˆ Aˆ };
ˆ
i
 [ Hˆ , ˆ ]
t
Louville – von Neumann
equation
Lecture 4: Quantum Nano-Electromechanics
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Zero-Point Oscillations
Consider a classical particle which oscillates in a quadratic potential well. Its equilibrium
position, X=0, corresponds to the potential minimum E=min{U(x)}.
A quantum particle can not be localized in space. Some “residual oscillations" are left
even in the ground states. Such oscillations are called zero point oscillations.
Classical motion:
d 2x
U
k
m 2 
 kx  
dt
x
m
Quantum motion:
1
U ( x)  U 0  kx 2
2
p
Amplitude of zero-point oscillations:
x0 
x  E ( x) 
2
2mx 2  kx 2 2
E( x0 )  min E( x)
m
Classical vs quantum description: the choice is determined by the parameter x0 d
where d is a typical length scale for the problem. “Quantum” when x0 d
1
Lecture 4: Quantum Nano-Electromechanics
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Quantum Nanoelectromechanics of
Shuttle Systems
 x p 
2
 x  2 x0 
M
If R( x   x)  R( x)  R( x) then quantum fluctuations of the
grain significantly affect nanoelectromechanics.
Lecture 4: Quantum Nano-Electromechanics
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Conditions for Quantum Shuttling
2 x0
x0 
1.
  Tunneling length
1
2M 

Fullerene based SET
X0

 0.1
Quasiclassical shuttle
vibrations.
  1 THz
2.
Suspended CNT
STM
L
  10
14
d 
Hz  
L
2
d  1 nm
  108 - 109 Hz for SWNT with L  1μm
9
Lecture 4: Quantum Nano-Electromechanics
10/29
Quantum Harmonic Oscillator
pˆ 2 1
H
 M  2 xˆ 2
2M 2
1
d
pˆ  i
dx
 1
En    n  
 2
1
 M  x2   M  
 1 2  M 4
| n   n  

 H n  x
 exp  
2
 2 n!    

 

Ledder operators
M 
i
 xˆ 
2 
M
a
a 

pˆ 

M 
i
ˆ
x


2  M

pˆ 

a | n  n | n  1 
a  | n  n  1 | n  1 
a  a 
2M 

xˆ 
pˆ  i

H   a  a  1/ 2

a
2M 

a

[a, a ]  1
Probability densities |ψn(x)|2 for the
boundeigenstates, beginning with the
ground state (n = 0) at the bottom and
increasing in energy toward the top.
The horizontal axis shows the position
x,and brighter colors represent higher
probability densities.
Eigenstate of oscillator is a ideal gas of elementary excitations – vibrons,
which are a bose particles.
Lecture 4: Quantum Nano-Electromechanics
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Quantum Shuttle Instability
Quantum vibrations, generated by tunneling
electrons, remain undamped and accumulate
in a coherent “condensate” of phonons,
which is classical shuttle oscillations.

e
eV

   thr  
eE
d
2k
Shift in oscillator position
caused by charging it by a
single electron charge.
d

Phase space trajectory of
shuttling. From Ref. (3)
References:
(1) D. Fedorets et al. Phys. Rev. Lett. 92, 166801 (2004)
(2) D. Fedorets, Phys. Rev. B 68, 033106 (2003)
(3) T. Novotny et al. Phys. Rev. Lett. 90 256801 (2003)
Lecture 4: Quantum Nano-Electromechanics
12/29
Hamiltonian
Hˆ  Hˆ leads  Hˆ dot  Hˆ oscillator  Hˆ tunneling
Hˆ leads     k ak a k
 ,k
Hˆ dot    q cq cq  Ec Nˆ 2  e ( xˆ ) Nˆ
 ,k
Hˆ oscillator
Hˆ tunneling
pˆ 2 1

 M  2 xˆ 2
2M 2
  T kq ( xˆ ) ak cq  h.c.
 ,k ,q
Nˆ   cq cq
q
Lecture 4: Quantum Nano-Electromechanics
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Theory of Quantum Shuttle
x
The Hamiltonian:
Hˆ  Hˆ leads  Hˆ dot  Hˆ oscillator  Hˆ tunneling
Hˆ leads     k ak a k
 ,k
Dot
Lead
Lead
Time evolution in
Schrödinger picture:
 tˆ (t )  i[ H , ˆ (t )]
Total density operator
Reduced density operator
Hˆ dot    q cq cq  Ec Nˆ 2  e ( xˆ ) Nˆ
 ,k
Hˆ oscillator
Hˆ tunneling
pˆ 2 1

 M  2 xˆ 2
2M 2
  T kq ( xˆ )ak cq  H .c.
  L , R ;k , q
TL , R ( xˆ )  T0 e  xˆ / 
0 
 ˆ 0 (t )
ˆ (t )  Trleadsˆ  t   

ˆ
0

(
t
)

1

Lecture 4: Quantum Nano-Electromechanics
14/29
Generalized Master Equation
ˆ 0 :
density matrix operator of the uncharged shuttle
ˆ1 :
density matrix operator of the charged shuttle
At large voltages the equations for ˆ 0 , ˆ1 are local in time:
 t ˆ 0  i[ H osc  e ( xˆ ), ˆ 0 ]  { L ( xˆ ), ˆ 0 }   R ( xˆ ) ˆ1  R ( xˆ )  L ˆ 0
Free oscillator dynamics
Electron tunnelling
Dissipation
 t ˆ1  i[ H osc  eExˆ, ˆ1 ]  { R ( xˆ ), ˆ1}   L ( xˆ ) 0  L ( xˆ )  L ˆ1
L ˆ  
i
xˆ, pˆ , ˆ    xˆ, xˆ, ˆ 
2
2
Approximation:
x0   1,
ˆ   ˆ 0  ˆ1 : ˆ   ˆ 0  ˆ1 :
e '( x) M  2   1,
  
Lecture 4: Quantum Nano-Electromechanics
15/29
Shuttle Instability
After linearisation in x (using the small parameter xo/l) one finds:
x p/M
x(t )  Tr  ˆ  (t ) xˆ 
p   M  2 x   p   '( x  0)q
p(t )  Tr  ˆ  (t ) pˆ 
q   
 q   2e   x
q(t )  1  Tr  ˆ1 (t )
Result: an initial deviation from the equilibrium position grows exponentially if the
dissipation is small enough:
x(t )  exp( t );   ( thr   ) / 2;    thr 
 e '( x  0)
2M  2
Lecture 4: Quantum Nano-Electromechanics
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Quantum Shuttle Instability
Important conclusion: An electromechanical instability is possible even if the
initial displacement of the shuttle is smaller than its de Broglie wavelength and
quantum fluctuations of the shuttle position can not be neglected.
In this situation one speaks of a quantum shuttle instability.
Now once an instability occurs, how can one distinguish between quantum
shuttle vibrations and classical shuttle vibrations?
More generally: How can one detect quantum mechanical displacements?
This question is important since the detection sensitivity of modern devices is
approaching the quantum limit, where classical nanoelectromechanics is not
valid and the principle for detection should be changed.
Lecture 4: Quantum Nano-Electromechanics
17/29
How to Detect Nanomechanical
Vibrations in the Quantum Limit?
Try new principles for sensing the
quantum displacements!
Consider the transport of electrons through a suspended, vibrating carbon
nanotube beam in a transverse magnetic field H. What will the effect of H be
on the conductance?
Shekhter R.I. et al. PRL 97(15): Art.No.156801 (2006).
Lecture 4: Quantum Nano-Electromechanics
18/29
Aharonov-Bohm Effect
The particle wave, incidenting the device from the left splits at the left end of the device.
In accordance with the superposition principle the wave function at the right end will be
given by:
 (b)  1 (b)  2 (b)   A0 (b)exp i1 (1  exp i(2  1)
 (b)   i (b)   A0 (b) exp{i i }
i
i
 i ( x) 
 2  1  2

1
; 
0
2
 dl A
L
 ds H ; 0 
S
e
c
 dl A
Li
2 c
 (flux quantum)
e
The probability for the particle transition through the device is given by:
2





  
2
2
2
| T | |  A0 | 1  cos  2
   sin  2


0 

  0  

Lecture 4: Quantum Nano-Electromechanics
19/29
Classical and Quantum Vibrations
In the classical regime the SWNT fluctuations u(x,t) follow well defined trajectories.
In the quantum regime the SWNT zero-point fluctuations (not drawn to scale) smear out
the position of the tube.
Lecture 4: Quantum Nano-Electromechanics
Quantum Nanomechanical
Interferometer
Classical interferometer
Quantum nanomechanical
interferometer
20/29
Electronic Propagation Through Swinging
Polaronic States
21
Lecture 4: Quantum Nano-Electromechanics
22/29
Model
H


 L, R
H  H e  H m 
  ie
3

H e   d r {  (r )  
2m
 r c
2
T


 L,R
H    a , a ,
2

A   (r )  U  y  u ( x, z )   (r )  (r )}

2
2




1
EI

u
(
x
)

2
2
H m   L dx  ˆ  
2  

2

2

x
2

 

L

  (r )   aqq (r )
q
ˆ  i
d
du
22
Lecture 4: Quantum Nano-Electromechanics
23/29
Coupling to the Fundamental Bending Mode
Only one vibration mode is taken into account
uˆ ( x)  x0u0 ( x) (bˆ  bˆ)
2 ; x0  

2 2
L

0  EI 
1
4
CNT is considered as a complex scatterer for electrons tunneling
from one metallic lead to the other.
Lecture 4: Quantum Nano-Electromechanics
24/29
Tunneling through Virtual Electronic States on CNT
•
•
Strong longitudinal quantization of electrons on CNT
No resonance tunneling though the quantized levels
(only virtual localization of electrons on CNT is possible)
Effective Hamiltonian
H     a , a ,

 ,
  gHLY0 / 20
 ˆ ˆ
i ( bˆ

b b  (Teff e

 bˆ )
2
Y0 
2M 0  L

a
  ,R a ,L  h.c)
 , 
Teff 
TLTR*
E
Lecture 4: Quantum Nano-Electromechanics
25/29
Calculation of the Electrical Current


I  G0   P (n) |  n |e
i ( bˆ  bˆ )
| n  l  |2 
n  0 l  n
 d  f ( ) 1  f (  l  )  f ( ) 1  f (  l
l
G
r

G0
r
  P ( n ) |  n | e i ( b
ˆ  bˆ )
l
 )  
| n |2 
n 0


 P(n) | n | e
n  0 l 1
i ( bˆ  bˆ )
| n  l  |2
2l 
kT
exp(l  )  1
kT
Lecture 4: Quantum Nano-Electromechanics
26/29
Vibron-Assisted Tunneling through
Suspended Nanowire
• Tunneling through vibrating nanowire is
performed in both elastic and inelastic
channels.
• Due to Pauli-principle, some of the
inelastic channels are excluded.
• Resonant tunneling at small energies of
electrons is reduced.
• Current reduction becomes independent
of both temperature and bias voltage.
Lecture 4: Quantum Nano-Electromechanics
27/29
Linear Conductance
Vibrational system is in equilibrium
2

G
    

 exp  
 ,
G0


  0 

2
G
1   
 1 
,

G0
6   0  kT
  4 gLx0 H ,

kT

kT
 1
 1
0  h e
For a 1 μm long SWNT at T = 30 mK and H ≈ 20 - 40 T a relative
conductance change is of about 1-3%, which corresponds to a magnetocurrent of 0.1-0.3 pA.
Lecture 4: Quantum Nano-Electromechanics
Quantum Nanomechanical
Magnetoresistor
R.I. Shekhter et al., PRL 97, 156801 (2006)
28/29
Lecture 4: Quantum Nano-Electromechanics
29/29
Magnetic Field Dependent Offset Current
I  I 0 (V )  I ;
I
I 0 (V )
I     



I 0 eV   0 
I
2
  
eV   


 0
2


eV0   

 0
V0
V
2
30
Lecture 4: Quantum Nano-electromechanics
Quantum Suppression of Electronic Backscattering in
a 1-D Channel
G.Sonne, L.Y.Gorelik, R.I.Shekhter, M.Jonson,
Europhys.Lett. (2008), (in press);
Speaker: Professor Robert Shekhter, Gothenburg University 2009
31/41
31
Lecture 4: Quantum Nano-electromechanics
CNT with an Enclosed Fullerene
Shallow harmonic potential for the
fullerene position vibrations along CNT
Strong quantum fluctuations of the
fullerene position, comparable to
Fermi wave vector of electrons.
x0 
2M 
Speaker: Professor Robert Shekhter, Gothenburg University 2009
32/41
32
Lecture 4: Quantum Nano-electromechanics
Electronic Transport through 1-D
Channel
•
•
•
Weak interaction with the impurity
Only a small fraction of electrons
scatter back.
Elastic and inelastic
backscattering channels
I.
 Left
y
 Right
No backscattering
Left and right side reservoirs inject electrons with energy distribution according to
Hibbs with different chemical potentials
 Left
V
I0 
;
RQ
4e
R  G0 
h
1
Q
2
 Right
py
Speaker: Professor Robert Shekhter, Gothenburg University 2009
33/41
33
Lecture 4: Quantum Nano-electromechanics
II. Backflow Current
I  I 0  I back
I back
 dN 
 e

 dt back
r
 dN 

   f Right ( p ) 1  f Left (q )  P(n) 
 dt back h p ,q
n ,l
n Vback n  l
2
Backscattering potential
 R† ( y )   a †p e ipy
p
 L† ( y )   aq†e iqy
Hˆ back  r R† ( y ) L† ( y ) ( y  X )  h.c.
2 ikF Xˆ
†
ˆ
H back  r  a p aq e
 h.c.
p ,q
q
Vback  re
2 ikF Xˆ
 h.c.
Speaker: Professor Robert Shekhter, Gothenburg University 2009
34/41
34
Lecture 4: Quantum Nano-electromechanics
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.
non movable
Wback  Wn  Wback
n
However the sum rule does not apply as Pauli principle puts restrictions on
allowed transitions .
Speaker: Professor Robert Shekhter, Gothenburg University 2009
35/41
35
Lecture 4: Quantum Nano-electromechanics
Pauli Restrictions on Allowed
Transitions through Oscillator
Bias potential across tube selects allowed transitions through oscillator as
fermionic nature of electrons has to be considered.
×
Speaker: Professor Robert Shekhter, Gothenburg University 2009
36/41
36
Lecture 4: Quantum Nano-electromechanics
Excess Current
I excess  I  G0V (1  r )  G0 r
lim
eV  
 
e
αħω=4εFm/M, excess current independent of confining potential, voltage
and temperature. Scales as ratio of masses in the system.
Speaker: Professor Robert Shekhter, Gothenburg University 2009
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37
Model
H


 L, R
H  H e  H m 
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


38
Backscattering of Electrons on Vibrating
Nanowire.
n
The probability of backscattering sums up all backscattering channels.
The result yields classical formula for non-movable target.
W  W  W
back
n
non movable
back
n
However the sum rule does not apply as Pauli principle puts
restrictions on allowed transitions .
39
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.
×
40
4-5
41
4-6
42
Lecture 4: Quantum Nano-electromechanics
Speaker: Professor Robert Shekhter, Gothenburg University 2009
43/41
43
Lecture 4: Quantum Nano-electromechanics
General conclusion from the course:
Mesoscopic effects in electronic system and quantum dynamics of
mechanical displacements qualitatively modify principles of NEM
operations bringing new functionality which is now determined by
quantum mechnaical phase and discrete charges of electrons.
Speaker: Professor Robert Shekhter, Gothenburg University 2009
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44
Lecture 4: Quantum Nano-Electromechanics
45/29
Magnetic Field Dependent Tunneling
iSˆ ˆ  iSˆ
e
He
In order to proceed it is convenient to make the unitary transformation
x



 
ˆ


(
r
)
eH

3

ˆ
S  i  d r uˆ ( x)ˆ (r )
i
  dxuˆ ( x) ˆ (r )ˆ (r ) 
y
c 0



2
2



Hˆ el   dxˆ ( x)  


ˆ ( x)
2
 2m x

2
x
2
 1

ˆ


eH
EI

u
(
x
)

2
Hˆ mech   dx  [ˆ ( x) 
dxˆ ( x)ˆ ( x)] 

 
2

2

c
2

x

 
0



 eH
Tˆl ( H )  T exp  i
r
 c


L
2

0


dxuˆ ( x)  Tˆl (0)  h.c.
 r
