Lecture 2: Electronics and Mechanics on the Nanometer Scale

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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


,
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

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/48

Lecture
2
: Electronics and Mechanics on the Nanometer Scale

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/48

Lecture
2
: Electronics and Mechanics on the Nanometer Scale

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/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

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/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ihe牥i猠
di獳spain

⡦inie

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