References
Book: Andrew N. Cleland, Foundation of Nanomechanics
Springer,2003 (Chapter7,esp.7.1.4, Chapter 8,9);
Reviews: R.Shekhter et al. Low.Tepmp.Phys. 35, 662 (2009);
J.Phys. Cond.Mat. 15, R 441 (2003)
J. Comp.Theor.Nanosc., 4, 860 (2007)
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 2: Electronics and Mechanics on
the Nanometer Scale
Outline
 Electronics – Mesoscopic phenomena
 Mechanics - Classical dynamics of mechanical
deformations
Lecture 2: Electronics and Mechanics on the Nanometer Scale
Part 1
Electronics – Mesoscopic phenomena
4/48
Lecture 2: Electronics and Mechanics on the Nanometer Scale
5/48
Mesosopic phenomena
Persistent currents (in the ground state)
-Microscopic scale: Electrons move in atomic orbitals,
may generate net magnetization
-Macroscopic scale: No current in the ground state of bulk sample
-Mesoscopic scale: Persistent currents in the ground state
Coulomb blockade (due to discreteness of electronic charge)
-Microscopic scale: Electrons have finite charge e, Coulomb interactions
give rise to large ionization energies of atoms
-Macroscopic scale: Electron liquid, charge discreteness not important
-Mesoscopic scale: Coulomb blockade of tunneling through granular samples
Josephson effect (supercurrent passing through NS-region)
-A supercurrent may flow between two superconductors separated by a
non-superconducting region of mesoscopic size
Mesoscopic samples contain a large number of atoms but are small on the scale
of a temperature-dependent ”coherence length”. On such scales electronic and
mechanical phenomena coexist: Mesoscopic Nanoelectromechanics
Lecture 2: Electronics and Mechanics on the Nanometer Scale
Quantum Coherence of Electrons
•
•
•
•
Spatial quantization of electronic motion
Quantum tunneling of electrons
Resonance transmission phenomenon
Tunnel charge relaxation and tunnel resistance
6/48
Lecture 2: Electronics and Mechanics on the Nanometer Scale
Spatial quantization of orbital motion
• For a sample with symmetric shape the electronic spectrum is degenerate
• A distortion of the geometrical shape tends to lift degeneracies.
7/48
Lecture 2: Electronics and Mechanics on the Nanometer Scale
8/48
Quantum Level Spacing
Estimation of average level spacing, assuming all quantum states
are nondegenerate and homogeneously distributed in energy
n( E )

E  F
E
N
N – total number of
electrons
F
Lecture 2: Electronics and Mechanics on the Nanometer Scale
9/48
Quantum Tunneling
The classically moving electron is reflected by a
potential barrier and can not be “seen” in the
region x > 0. The quantum particle can
penetrate into such a forbidden region.
Under-the-barrier propagation:
x  x0
 i 2mE
  ( x)  exp 

x  x0
 2m(U 0  E )
  ( x)  c2 exp 

x


 1

 ( x)  c2 exp   dx 2m(U ( x)  E )  if
x0




| d  ( x)
dx

 2mE
x   c1 exp 



x

| 1
Under-the-barrier propagation is called tunneling. Wave function’s decay length l0 
is called the tunneling length.
2m(U  E )

x

Lecture 2: Electronics and Mechanics on the Nanometer Scale
10/48
Tunneling through a Barrier
Due to quantum tunneling a particle has a finite probability to penetrate
through a barrier of arbitrary height.
 ipx 
 ipx 
  r exp 

 
 h 
 ipx 
 ( x  )  t exp  
 
x2 ( E )
 1

 d
t  exp   dx 2m(U ( x)  E )   exp  
 l0 
x1 ( E )


| t |2  | r |2  1; t | t | exp i1 ; r | r | exp i 2 
 ( x  )  exp 
x


 1 2

t  c2 exp   dx 2m(U ( x)  E )  , d  ( x) | 1
dx
x1




t and r are probability amplitudes for the transmission and reflection of
the particle. These parameters characterise the barrier and can often
be considered to be only weakly energy dependent.
Lecture 2: Electronics and Mechanics on the Nanometer Scale
11/48
Tunneling Width of a Quantum Level
Let N be the number of ”tries” made
before the particle finally escapes the dot:
N 1 2
|t |
N | t |2  1
L
Escape time
  0N 
0
h  tunneling level width
;

| t |2
 0  2L V
F
Lecture 2: Electronics and Mechanics on the Nanometer Scale
12/48
Resonant Tunneling
Electronic waves, like ordinary waves, experience a set of multiple reflections as they
move back and forth between two barriers. The total probability amplitude for the transfer
of a particle can be viewed as a sum of amplitudes, each corresponding to escape after
an increasing number of “bounces” between the barriers.
0
  0 ( x  d )  tt exp ipd /

1   1 ( x  d )  t  rr  t exp i3 pd

....
n
p  pn 
  n ( x  d )  t (rr ) n t exp i (2n  1) pd /

 n
 ipd 
t 2 exp 

 ipd   2
 i 2 pd  


2
T   t exp 
  | r | exp 
 



  1 | r |2 exp  2ipd 
n 0




d
If p = pn = nh/2d we have D=1 independently
of the barrier transparency! (Resonance)
D( E ) 
2
 E  En   
2
; En 
2
 2 2 n2
2m
;  | t |2 En ;
n
D | T | 
| t |4
2
2

2
2  2 pd  
4
2  2 pd 
   | r | sin 

1 | r | cos 





| E  En |
 1
En
Breit-Wigner formula
Lecture 2: Electronics and Mechanics on the Nanometer Scale
13/48
Tunneling Resistance
p
F  0 No acceleration of electrons
p
FS  p   scattering time
L
An electric field must be present in the vicinity of the barrier in order to
compensate for the ”scattering force” of the potential barrier and achieve a
stationary current flow
Fb  eE F  FS  Fb  0
The resulting voltage drop across the barrier, V = eEL , determines the
tunneling resistance, R = V/I
Lecture 2: Electronics and Mechanics on the Nanometer Scale
14/48
Quantization Effects in Electronic Tanspansport
Conductance of a quantum point
contact: G  I / V
Adiabatic point cointact
pn2 px2
EF 

; n  Nd   N F ;
2m 2m
pn 
n
d ( x)
 NF
d0
 2mEF
 quantized transverse momentum
Landauer formula
2
2
e
G  G0 N d | t |2 ; G0 
h
Lecture 2: Electronics and Mechanics on the Nanometer Scale
15/48
Charge Relaxation Due to Tunneling
Q
-Q
V Q ;
C
If one transfers a charge Q from one conductor to
the other, it will first accumulate in surface layers
on both sides of the tunnel barrier, and will then
relax due to tunneling of electrons .
dQ
1
 I V  Q
;
R
dt
RC
R 
1
RC
Lecture 2: Electronics and Mechanics on the Nanometer Scale
16/48
Characteristic Energy Scales
(summary)
Level spacing:
0.1-1 K
Level width:
0.01-0.1 K
Frequency of tunnel
charge relaxation :
0.01-0.1 K
d= 1-10nm
D=0.0001
At low enough temperatures all quantum coherent effects might be
experimentaly relevant.
Lecture 2: Electronics and Mechanics on the Nanometer Scale
17/48
Tunnel Transport of Discrete Charges
Charge transport in granular conductors is entirely due to tunneling of
electrons between small neighboring conducting grains.
• The electronic charge on each of the grains is quantized in units of the
elementary electronic charge.
• This results in quantization of the electrostatic energy, which may block the
intergrain tunneling of electrons.
Lecture 2: Electronics and Mechanics on the Nanometer Scale
18/48
Single Electron Transistor
V/2
-V/2
e
CS , CD , CG - Mutual capacitances
C  CS  CD  CG
Source
e
Q  ne
Drain
Q2
 C  CS V CG

En 
Q D

VG 
2C
2 C
 C

VG
Gate
En  En1  En  2EC n  N (V ,VG )
N (V , VG ) 
e2
EC 
2C
C D  CS
C
V  G VG
2e
e
Lecture 2: Electronics and Mechanics on the Nanometer Scale
As a result,
19/48
Lecture 2: Electronics and Mechanics on the Nanometer Scale
20/48
I-V curves: Coulomb staircase
How one can calculate the I-V curve?
g
e
-
c
+
(Master equation)
Lecture 2: Electronics and Mechanics on the Nanometer Scale
21/48
Stability Diagram for a Single-Electron
Transistor
Coulomb diamonds: all
transfer energies inside
are positive.
Conductance oscillates
as a function of gate
voltage – Coulomb
blockade oscillations.
Lecture 2: Electronics and Mechanics on the Nanometer Scale
22/48
Experimental test: Al-Al203 SET, temperature 30 mK
Coulomb blockade oscillations
V=10 μV
Lecture 2: Electronics and Mechanics on the Nanometer Scale
Coulomb Staircase
Thermal smearing
Coulomb
staircase
Calculations for different gate
potentials
Experiment:
STM of
surface
clusters
23/48
Lecture 2: Electronics and Mechanics on the Nanometer Scale
24/48
Single-Electron Transistor Device
SETs are promising for logical operations since they manipulate by
single electrons, and this is why have low power consumption per
bit.
The operation temperature is actually set by the relationship
between the charging energy, Ec=e2/2C, and the thermal smearing,
kΘ. At present time, room-temperature operation has been
demonstrated.
Coulomb blockade and single-electron effects are specifically
important for molecular electronics, where the size is intrinsically
small.
Negative feature of SETs is their sensitivity to fluctuations of the
background charges.
Lecture 2: Electronics and Mechanics on the Nanometer Scale
25/48
Submicron SET Sensors
• CB primary termometer (based on thermal smearing of the CB)
in the range 20 mK - 50 K (T~3%)
(J.Pekkola, J.Low Temp.Phys. 135, (2004), T. Bergsten et al. Appl.Phys.Lett. 78, 1264
(2001))
• Most sensitive electrometers (based on SET being sensitive to
the gate potential Vg): q ~ 10-6 eHz-1/2
(M.Devoret et al., Nature 406, 1039 (2000)).
• CB current meter (based on SET oscillations in the time domain)
(J.Bylander et al. Nature 434, 361 (2005) )
Lecture 2: Electronics and Mechanics on the Nanometer Scale
26/48
Quantum Fluctuation of Electric Charge
Qn
Charge fluctuations due to
quantum tunneling smear the
charge quantization . This
destroys the Coulomb Blockade.
  RC
 E 
e2
E 
C
R 
h
e2
Coulomb Blockade is destroyed by quantum fluctuations
of the charge
Coulomb Blockade is restored
Lecture 2: Electronics and Mechanics on the Nanometer Scale
Part 2.
Mechanics – Classical Dynamics of
Mechanical Deformations
27/48
Lecture 2: Electronics and Mechanics on the Nanometer Scale
28/48
Mechanical Dynamics of Nanostructures
Focus on spatial displacements of bodies and their parts
Examples
F
m
F (r )
F
F
Motion of a point-like mass
Rotational displacement +
center-of-mass motion
Elastic deformations
Displacements: Classical and Quantum
The discrete nature of solids can be ignored on the nanometer length scale
Lecture 2: Electronics and Mechanics on the Nanometer Scale
29/48
Classical Mechanics of a Point-Like Mass
Newton’s equation
d 2r
m 2  F (r )
dt
r (t )
In most cases we may consider F (r ) to be of elastic or electric origin
Classical harmonic oscillator:
 x
U ( x)  U 0  
 x0 
U
2
F ( x)  
dU
dx
d 2x
dx
m 2 
 kx  0
dt
dt
x   Cost   Sint
x
 ,  exp   t 
Lecture 2: Electronics and Mechanics on the Nanometer Scale
30/48
Euler-Bernoulli Equation
P(x)
U(x)
 2U ( x, t )
A
 P( x)  Pel ( x)
2
t
2 
2 
Pel ( x)  2  EI 2 U ( x, t )
x  x 
E – Young’s modulus – represents rigidity of the material
I – Second moment of crossection – represent influence of the crossectional
geometry
Why there is sensitivity to geometry of the beam crossection?
Easy to bend
Dificult to bend
Lecture 2: Electronics and Mechanics on the Nanometer Scale
Longitudinal and Flexural Vibrations
 2U ( x, t )
 2U
A
k 2
2
t
x
 2U ( x, t )
 4U
A
k 4
2
t
x
Londitudinal elastic vibrations
Flexural vibrations
Longitudinal deformation: Compression
across the whole crossection
Flexural deformation. Compression and
streching occur at different parts of the
crossection
31/48
Lecture 2: Electronics and Mechanics on the Nanometer Scale
32/48
Flexural Vibrations of a Strained Beam
 2u
 4u
 2u
 A 2  EI 4  T 2
t
x
x
APL 78 (2001) 162
Lecture 2: Electronics and Mechanics on the Nanometer Scale
33/48
Flexural Vibrations of a Doubly Clamped Beam
Ref: A.N. Cleland, Foundations of Nanoelectromechanics (Springer, 2003), Ch. 7
 2u
u 4
 A 2  EI 4  0
t
x
u (0)  u ( L)  0;
u´(0)  u´( L)  0
Nanotube:
L=100 nm, d=1.4 nm, f0=5 GHz
A: cross-section area (=HW)
ρ: mass density of the beam
E,I: assumed independent of position
The solution is:
un ( x)   an  cos  n x  cosh  n x   bn  sin  n x  sinh  n x   exp(int )
n  EI /  A n2
;
 n L  0,
Silicon: L=1mm, H=W=0.1 mm, f0=1 GHz
4.73004,
an / bn  1.01781,
7.8532,...
0.99923,
1.0000,...
Lecture 2: Electronics and Mechanics on the Nanometer Scale
34/48
Flexural Vibrations of a Cantilever
Ref: A.N. Cleland, Foundations of Nanoelectromechanics (Springer, 2003)
 2u
u 4
 A 2  EI 4  0
t
x
u (0)  u´(0)  u´´( L)  u´´´( L)  0
A: cross-section area (=HW)
ρ: mass density of the beam
E,I: assumed independent of position
The solution is:
wn ( x)  an cos  n x  cosh  n x   bn sin  n x  sinh  n x exp(int )
n  EI / A n2 ;
 n L  1.875,
an / bn  1.3622,
0.9819,
4.694,
1.008, ...
7.855,...
Lecture 2: Electronics and Mechanics on the Nanometer Scale
35/48
Lecture 2: Electronics and Mechanics on the Nanometer Scale
36/48
Lecture 2: Electronics and Mechanics on the Nanometer Scale
37/48
Damping of the Mechanical Motion
So far we have ignored any interaction of the mechanical vibrations with the many other
degrees of freedom present in the solid. Even though such interactions may be relatively
weak they could produce a significant effect on a large enough time scale. The
interactions cause dissipation of the mechanical energy and stochastic deviations from
the otherwise regular mechanical vibrations (noise).
Sources of dissipation and noise are the same and might come from:
a)
b)
c)
d)
Interaction with other mechanical modes
Interaction with electrons
(nonintrinsic source) motion of defects and ions due to imposed strain.
Interaction with a suface contaminations
Below we will present a phenomenological approach to describe these effects without
going into the microscopic theory for any particular mechanism.
Lecture 2: Electronics and Mechanics on the Nanometer Scale
38/48
Dissipation and Noise in Mechanical
Systems
Ref: A.N. Cleland, Foundations of Nanoelectromechanics (Springer, 2003), Ch. 8
Outline
- Langevin Equation (useful phenomenological approach)
- Dissipation and Quality Factor
- Dissipation in Nanoscale Mechanical Resonators
- Dissipation-Induced Amplitude Noise
Einstein (1905 – ”annus mirabulis”):
Friction and Brownian motion is connected;
where there is dissipation there is also noise
Lecture 2: Electronics and Mechanics on the Nanometer Scale
39/48
Langevin Equation
Consider a system of inertial mass m that interacts with its
environment through a conservative potential U(x)=kx2/2 +...
and in addition through a complex interaction term
characterized both by friction and noise.
Without friction the dynamic equation is Newton’s equation
which has a lossless solution x(t) where x0 and φ are
determined by the initial conditions:
2
m
d x
2

m

x  0;
0
dt 2
x(t )  x0 exp(i0t   )
Paul Langevin
(1872-1946)
Friction and noise in the system is due to the interaction of the
mass m with a large number of degrees of freedom in the environment. It can be included by adding a time-dependent environmental
force term to Newton’s equation
d 2x
m 2  m02 x  Fenv (t )
dt
Lecture 2: Electronics and Mechanics on the Nanometer Scale
40/48
Dissipation and Noise are Due to the
Environment
In many dissipative systems the environmental force can be separated into a
dissipation (or loss) term proportional to the ensemble average velocity and
a noise term due to a random force
d 2x
dx
m 2  m02 x  m
 FN (t );
dt
dt
FN (t )  0
Equations of this form are known as Langevin equations.
The dissipative term in the Langevin equation causes energy to be transferred
from the harmonic oscillator to the environment.
Thermal equilibrium in a system controlled by the Langevin equation is achieved
through the second moment of the noise force, which must satisfy:
2mkBT  



FN (0) FN (t ) dt
Lecture 2: Electronics and Mechanics on the Nanometer Scale
Dissipation and Environmental Noise Drives the
System to Equilibrium and Maintains Equilibrium
The mean energy of a harmonic oscillator is
1
1
2
E  m x  m02 x 2
2
2
The energy of an undriven harmonic oscillator described by our Langevin
equation will equilibriate to the energy of the environment by losing any initial
excess energy to the environment by the velocity-dependent dissipation term
and then, gaining and losing energy stochastically through the noise term the
noise force will produce this equilibrium.
Without proof we state that:
E  kBT
1
1
1
2
2
2
m x  m0 x  k BT
2
2
2
41/48
Lecture 2: Electronics and Mechanics on the Nanometer Scale
42/48
Fundamental Relation between Environmental Noise,
Dissipation and Temperature (Einstein 1905)
If we assume that the noise force is uncorrelated for time scales over which
the harmonic oscillator responds, we have so called white noise, and
1
S () 
2


FN (t ) FN (t   ) ei d

We can define a spectral density for the (noise) force-force correlation function
as:
FN (t )FN (t´)  2m kBT (t  t´)
Noise
Dissipation
Temperature
For white noise the spectral density is constant (independent of frequency):
1
S ( ) 
2

 2m k T ( )e
B

i
d 
m kBT

Lecture 2: Electronics and Mechanics on the Nanometer Scale
43/48
Dissipation and Quality Factor (Q)
In the absence of the noise term the solution to the Langevin equation
d 2x
dx
2
m 2  m0 x  m
 FN (t )
dt
dt
is x(t)=x0exp(-iωt+φ), where the complex-valued frequency is given by
 2  i  02  0
The frequency ω has both real and imaginary parts, ω = ωR + i ωI:
 R  02   2 / 4 ,  I   / 2
The quality factor Q is defined as:
I

1/ Q  2

  / 0
2
2
R
0   / 4
(if
  0 )
Lecture 2: Electronics and Mechanics on the Nanometer Scale
Damping of Mechanical Oscillations
Now, since
x(t )  x0 exp(it   )  x0 exp(iRt   ) exp(I t )
the oscillation amplitude damps as
x(t )  exp(t / 2)  exp(0t / 2Q)
and the energy damps as
x (t )  exp(0t / Q)
2
44/48
Lecture 2: Electronics and Mechanics on the Nanometer Scale
45/48
Dissipation in Nanoscale Mechanical
Resonators
Recall the Euler-Bernoulli equation:
 2u
u 4
 A 2  E ( ) I 4  0
t
x
u (0)  u ( L)  0;
u´(0)  u´( L)  0
A: cross-section area (=HW)
ρ: mass density of the beam
And its solution
un ( x)   an  cos  n x  cosh  n x   bn  sin  n x  sinh  n x   exp(int )
Different with dissipation!
 'n  1  i / 2Q  EI /  A n2  1  i / 2Q  n ;
an / bn  1.01781,
 n L  0,
4.73004,...
0.99923, 1.0000,...
The imaginary part of ’n indicates that the n:th eigenmode will decay in
amplitude as exp(-n/2Q), similar to the damped harmonic oscillator
Lecture 2: Electronics and Mechanics on the Nanometer Scale
46/48
Driven Damped Beams
We add a harmonic driving force F(x,t)=f(x)exp(-ict), where f(x) is a positiondependent force per unit length and c is the drive – or carrier – frequency. The
equation of motion is now:
 2u
u 4
 A 2  E ( ) I 4  f ( x) exp(it )
t
x
Solve this for times longer than the damping time for the beam by expansion
in terms of eigenfunctions:
L

u ( x, t )   anun ( x) exp(it )
3
u
(
x
)
u
(
x
)
dx

L
 m,n
 n m
n 1
0
The equation for the expansion coefficients an is
 4un ( x)
  A anun ( x)  E ( ) I  an
 f ( x)
4
x
n 1
n 1

2

Lecture 2: Electronics and Mechanics on the Nanometer Scale
47/48
Using the definitions of the eigenfunctions and their properties, and the
definition of the complex-valued eigenfrequencies ’n this can be written as:
L
1
2
2
n    an   AL3  un ( x) f ( x)dx
0
For  close to 1, only the n=1 term has a significant amplitude,
given by:
L
1
1
a1 
u1 ( x) f ( x)dx
3
2
2
2

 AL 1    i1 / Q 0
For a uniform force distribution, f(x)=f0 , the integral is evaluated to 1L2,
1=0.8309 and we have, since ’n=(1-i/Q)n:
1
f0
a1  2
1   2  i12 / Q M
Lecture 2: Electronics and Mechanics on the Nanometer Scale
48/48
Dissipation-Induced Amplitude Noise
The displacement of a forced damped beam driven near its fundamental
frequence is – as we have seen – given by
1
f0
u ( x, t )  2
u1 ( x) exp(i t )
2
2
1    i1 / Q M
In the absence of noise the motion is purely harmonic at the carrier frequency
. But if there is dissipation (finite Q), there is also necessarily noise and a
noise force fN(t) that can be expanded in terms of the eigenfunctions un(x):

1
f N (t )   f N ,n (t )un ( x)
L n1
As we discussed already dissipation drives the beam to equilibrium with its
environment at temperature T and the stochastic noise force maintains the
equilibrium.
Lecture 2: Electronics and Mechanics on the Nanometer Scale
49/48
Without driving force the mean total energy for each mode is kBT. This requires
the spectral density of the noise force fN,n(t) to be:
2k BTMn
S f N ,n ( ) 
QL2
Force per length, hence the
term L2, which is not there
for a simple harmonic osc.
Using this result we can calculate the spectral density for the thermally driven
amplitude as
2k BTMn
San ( ) 
2
2
2
2 2
2
QL
n    n / Q

n
 

Lecture 2: Electronics and Mechanics on the Nanometer Scale
Speaker: Professor Robert Shekhter, Gothenburg University 2009
51/52
Comments
to the
next slide Scale
Lecture 2: Electronics and Mechanics
on the
Nanometer
This equation can be used to find the vibrational spectrum
of a double clamped beam. Inserting an inertion term and
extractind an external force we find thye equation. Note
that it differs from the wave equation due to fourth order
spacial derivative instead second one is present. The
boundary conditions just demand that discplacement and
deformation of a beam material are equal to zero if end of
the beam are spacialy fixed.
Discrete sets of different solutions(modes) are presented
here. Notice that frequency is inversely proportional to the
square of the beam length.
(This is in contrast to the bulk elastic vibrations which
lowers phjononic frequency is inversely proportionasl to the
lewngth of the sample not to the length squared.)
Speaker: Professor Robert Shekhter, Gothenburg University 2009
52/52
Coments to the next slide
The same for the beam clamped only from one side. The boundary condition for
the free side express an absence of the tension and share tension (correct ?) at the
free end. Ther same properties of thye solutions
53
Coments to the next slide
An estimation of the frequency of the nanovibrations.
What is the meaning of the note ”not harmonic”?
54
Coments to the next slide
Would be nice to get comments to ”W” and ”G” which appear on the slide
55