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
Electronics
–
Mesoscopic
phenomena
Mechanics

Classical dynamics of mechanical
deformations
Outline
Part
1
Electronics
–
Mesoscopic
phenomena
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
4
/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
5
/48
Quantum Coherence of Electrons
•
Spatial quantization of electronic motion
•
Quantum tunneling of electrons
•
Resonance transmission phenomenon
•
Tunnel charge relaxation and tunnel resistance
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
6
/48
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.
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
7
/48
Quantum Level Spacing
Estimation
of
average
level
spacing,
assuming
all
quantum
states
are
nondegenerate
and
homogeneously
distributed
in
energy
N
–
total number of
electrons
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
8
/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:
Under

the

barrier propagation is called
tunneling
. Wave function’s decay length
is called the
tunneling length
.
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
9
/48
Tunneling through a Barrier
Due to quantum tunneling a particle has a finite
probability
to
penetrate
through a
barrier
of arbitrary height.
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
10
/48
Tunneling Width of a Quantum Level
Let
N
be the number of ”tries” made
before the particle finally escapes the dot:
Escape
time
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
11
/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.
If
p = p
n
= nh/2d
we have D=1
independently
of the barrier transparency! (Resonance)
Breit

Wigner formula
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
12
/48
Tunneling Resistance
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
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
13
/48
L
Quantization Effects in Electronic Tanspansport
Conductance of a quantum point
contact:
Adiabatic point cointact
Landauer formula
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
14
/48
Charge Relaxation Due to Tunneling
Q

Q
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 .
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
15
/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
16
/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
17
/48
Single Electron Transistor
e
e
Gate
Source
Drain
V/2

V/2

Mutual capacitances
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
18
/48
As a result,
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
19
/48
I

V curves: Coulomb staircase
How one can calculate the I

V curve?
e
g
c
+

(Master equation)
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
20
/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
21
/48
Experimental test: Al

Al
2
0
3
SET, temperature 30 mK
V=10
μ
V
Coulomb blockade oscillations
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
22
/48
Calculations for different gate
potentials
Thermal smearing
Coulomb
staircase
Experiment:
STM
of
surface
clusters
Coulomb Staircase
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
23
/48
Single

Electron Transistor Device
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
24
/48
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,
E
c
=e
2
/2C
, and the thermal smearing,
k
Θ
. 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.
Submicron SET Sensors
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
25
/48
•
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 V
g
):
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) )
Quantum Fluctuation of Electric Charge
Charge
fluctuations
due to
quantum tunneling smear the
charge quantization . This
destroys
the Coulomb Blockade.
Coulomb Blockade is
destroyed
by quantum fluctuations
of the
charge
Coulomb Blockade is
restored
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
26
/48
Part
2.
Mechanics
–
Classical Dynamics of
Mechanical Deformations
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
27
/48
Mechanical Dynamics of Nanostructures
Focus on spatial displacements of bodies and their parts
Examples
m
Motion of a point

like mass
Rotational displacement
+
center

of

mass motion
F
F
F
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
28
/48
Classical Mechanics of a Point

Like Mass
Newton’s
equation
In most cases we may consider to be of elastic or electric origin
Classical harmonic oscillator:
U
x
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
29
/48
Euler

Bernoulli Equation
P(x)
U(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
30
/48
Longitudinal and Flexural Vibrations
Londitudinal
elastic
vibrations
Flexural
vibrations
Longitudinal
deformation: Compression
across the whole crossection
Flexural
deformation. Compression and
streching occur at different parts of the
crossection
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
31
/48
Flexural Vibrations of a Strained Beam
APL 78 (2001) 162
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
32
/48
Flexural Vibrations of a Doubly Clamped Beam
Ref: A.N. Cleland, Foundations of Nanoelectromechanics (Springer, 2003), Ch. 7
A
: cross

section area (
=HW
)
ρ
: mass density of the beam
E
,
I
: assumed independent of position
The solution is:
Silicon:
L=1
m
m
,
H=W=0.1
m
m
,
f0=1 GHz
Nanotube:
L=100
湭
,
d=1.4 nm
,
f
0
=5 GHz
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
33
/48
A
: cross

section area (
=HW
)
ρ
: mass density of the beam
E
,
I
: assumed independent of position
The
solution
is:
Flexural Vibrations of a Cantilever
Ref: A.N. Cleland, Foundations of Nanoelectromechanics (Springer, 2003)
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
34
/48
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
35
/48
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
36
/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)
Interaction with other
mechanical modes
b)
Interaction
with
electrons
c)
(nonintrinsic source)
motion of
defects
and
ions
due to imposed strain.
d)
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
37
/48
Dissipation and Noise in Mechanical
Systems
Ref: A.N. Cleland, Foundations of Nanoelectromechanics (Springer, 2003), Ch. 8

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
Outline
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
38
/48
Langevin Equation
Consider a system of inertial mass
m
that interacts with its
environment
through a conservative
potential
U(x)=kx
2
/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
x
0
and
φ
are
determined by the initial conditions
:
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 environ

ment. It can be included by adding a time

dependent environmental
force term to Newton’s equation
Paul Langevin
(1872

1946)
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
39
/48
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
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:
Dissipation and Noise are Due to the
Environment
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
40
/48
Dissipation and Environmental Noise Drives the
System to Equilibrium and Maintains Equilibrium
The mean energy of a harmonic oscillator is
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:
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
41
/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
We can define a spectral density for the (noise) force

force correlation function
as:
For white noise the
spectral density
is constant (independent of frequency):
Noise
Dissipation
Temperature
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
42
/48
Dissipation and Quality Factor (Q)
In the absence of the noise term the solution to the
Langevin equation
is
x(t)=x
0
exp(

i
ω
t+
φ
)
,
where the complex

valued frequency is given by
The frequency
ω
has both real and imaginary parts,
ω
=
ω
R
+ i
ω
I
:
The quality factor
Q
is defined as:
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
43
/48
Now, since
the oscillation
amplitude
damps as
and the
energy
damps as
Damping of Mechanical Oscillations
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
44
/48
Recall the Euler

Bernoulli equation:
A
: cross

section area
(=HW)
ρ
: mass density of the beam
And its
solution
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
Dissipation in Nanoscale Mechanical
Resonators
Different with dissipation!
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
45
/48
We add a harmonic
driving force
F(x,t)=f(x)exp(

i
c
t)
,
where
f(x)
is a position

dependent force per unit length and
c
is the drive
–
or carrier
–
frequency. The
equation of motion is now:
Solve this for times longer than the damping time for the beam by expansion
in terms of eigenfunctions:
The equation for the
expansion coefficients
a
n
is
Driven Damped Beams
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
46
/48
Using the definitions of the eigenfunctions and their properties, and the
definition of the complex

valued
eigenfrequencies
’
n
this can be written as:
For
捬獥
1
, only the
n=1
term has a
significant amplitude
,
given by:
For a uniform force distribution
,
f(x)=f
0
,
the integral is evaluated to
1
L
2
,
1
=0.8309
and we have, since
’
n
=(1

i/Q)
n
:
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
47
/48
The displacement of a forced damped beam driven near its fundamental
frequence is
–
as we have seen
–
given by
In the absence of noise the motion is purely harmonic at the carrier frequency
.䉵ihe牥i猠
di獳spain
⡦inie
Q
), there is also necessarily
noise
and a
noise force
f
N
(t)
that can be expanded in terms of the eigenfunctions
u
n
(x)
:
As we discussed already dissipation drives the beam to equilibrium with its
environment at temperature
T
and the stochastic noise force maintains the
equilibrium.
Dissipation

Induced Amplitude Noise
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
48
/48
Without driving force the mean total energy for each mode is
k
B
T
.
This requires
the spectral density of the noise force
f
N,n
(t)
to be:
Force per length, hence the
term
L
2
, which is not there
for a simple harmonic osc.
Using this result we can calculate the spectral density for the thermally driven
amplitude as
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
49
/48
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
Speaker: Professor
Robert
Shekhter
, Gothenburg University
2009
51
/52
Comments to the next slide
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.
)
Lecture
2
: Electronics and Mechanics on the Nanometer Scale
Speaker: Professor
Robert
Shekhter
, Gothenburg University
2009
52
/52
Coments to the next slide
53
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
Coments to the next slide
54
An estimation of the frequency of the nanovibrations.
What is the meaning of the note ”not harmonic”?
Coments to the next slide
55
Would be nice to get comments to ”W” and ”G” which appear on the slide
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