references book: andrew n. cleland, foundation of nanomechanics springer,2003 (chapter7,esp.7.1.4,...

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 1Electronics – 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 scaleof a temperature-dependent ”coherence length”. On such scales electronic and mechanical phenomena coexist: Mesoscopic Nanoelectromechanics


Quantum Coherence of Electrons

• Spatial quantization of electronic motion

• Quantum tunneling of electrons

• Resonance transmission phenomenon

• Tunnel charge relaxation and tunnel resistance


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.


Quantum Level Spacing

F

( )n E

E

Estimation of average level spacing, assuming all quantum states are nondegenerate and homogeneously distributed in energy

FE N

N – total number of electrons


Quantum Tunneling

0 1

00 2

2 2( ) exp exp

2 ( )( ) exp

i mE mEx x x x c x

m U Ex x x c x

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:

0

2

1 ( )( ) exp 2 ( ( ) ) | | 1x

x

d xx c dx m U x E if dx

02 ( )

lm U E

Under-the-barrier propagation is called tunneling. Wave function’s decay length is called the tunneling length.


Tunneling through a BarrierDue to quantum tunneling a particle has a finite probability to penetrate through a barrier of arbitrary height.

( ) exp exp

( ) exp

ipx ipxx r

h

ipxx t

2

1

( )

0( )

2 21 2

1exp 2 ( ( ) ) exp

| | | | 1; | | exp ; | | exp

x E

x E

dt dx m U x E

l

t r t t i r r i

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.

2

1

2

1 ( )exp 2 ( ( ) ) , | 1x

x

d xt c dx m U x E dx


Tunneling Width of a Quantum Level

Let N be the number of ”tries” made before the particle finally escapes the dot:

2| | 1N t 21| |

Nt

Escape time

00 2

;| |

hN tunneling level widtht

02

F

LV

L


Resonant Tunneling

2

2 2

20

exp2

exp | | exp2

1 | | exp

n

n

ipdt

ipd i pdT t r

ipdr

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.

42

2

2 2 4 2

| || |

2 21 | | cos | | sin

tD T

pd pdr r

n

np p

d

If p = pn = nh/2d we have D=1 independently of the barrier transparency! (Resonance)

2 2 2 22

2 2

| |( ) ; ; | | ; 1

2n

n nnn

E EnD E E t E

m EE E

Breit-Wigner formula

0

1

0 ( ) exp /

1 ( ) exp 3

....

( ) ( ) exp (2 1) /nn

x d tt ipd

x d t rr t i pd

n x d t rr t i n pd


Tunneling Resistance

p

p 0F No acceleration of electrons

SF p scattering time

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

0b S bF eE F F F


L

Quantization Effects in Electronic Tanspansport

/G I VConductance of a quantum point contact:

2 2

0

; ; 22 2

( )

n x FF d F F

n

p p NE n N N mE

m m d

np quantized transverse momentum

d x

Adiabatic point cointact

Landauer formula

22

0 0

2| | ;d

eG G N t G

h


Charge Relaxation Due to Tunneling

Q -Q

1 1; ; R

dQQ VV I QC Rdt RC RC

If one transfers a charge Q from one conductor to the other, it will first accumulate in surface layerson both sides of the tunnel barrier, and will then relax due to tunneling of electrons .


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= 1-10nmD=0.0001

At low enough temperatures all quantum coherent effects might be experimentaly relevant.


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.


Single Electron Transistor

e

e

Gate

Source Drain

V/2 -V/2

Q ne

GV

, ,S D GC C C - Mutual capacitances

S D GC C C C

2

2 2D S G

n G

C C CQ VE Q V

C C C

1 2 ( , )n n n C GE E E E n N V V ( , )2

D S GG G

C C CN V V V V

e e

2

2C

eE

C


As a result,


I-V curves: Coulomb staircase

How one can calculate the I-V curve?e g c

+-

(Master equation)


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.


Experimental test: Al-Al203 SET, temperature 30 mK

V=10 μV

Coulomb blockade oscillations


Calculations for different gate potentials

Thermal smearing

Coulomb staircase

Experiment: STM of surface clusters

Coulomb Staircase


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.

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) )

Quantum Fluctuation of Electric Charge

nQ

Charge fluctuations due toquantum tunneling smear the charge quantization . This destroys the Coulomb Blockade.

E

RC

2eE

C Coulomb Blockade is destroyed by quantum fluctuations

of the charge

Coulomb Blockade is restored2

hR

e


Part 2.Mechanics – Classical Dynamics of

Mechanical Deformations


Mechanical Dynamics of Nanostructures

Focus on spatial displacements of bodies and their parts

Examples

m( )F r

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


Classical Mechanics of a Point-Like Mass

Newton’s equation

In most cases we may consider to be of elastic or electric originClassical harmonic oscillator:

2

2( )

d rm F r

dt

( )r t

( )F r

U

x

( )dU

F xdx

2

00

( )x

U x Ux

2

20

d x dxm kx

dt dt

x Cos t Sin t

, exp t


Euler-Bernoulli Equation

P(x)

U(x)

2

2

2 2

2 2

( , )( ) ( )

( ) ( , )

el

el

U x tA P x P x

t

P x EI U x tx x

E – Young’s modulus – represents rigidity of the materialI – 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


Longitudinal and Flexural Vibrations

2 2

2 2

2 4

2 4

( , )

( , )

U x t UA k

t x

U x t UA k

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


Flexural Vibrations of a Strained Beam

2 4 2

2 4 2

u u uA EI T

t x x

APL 78 (2001) 162


Flexural Vibrations of a Doubly Clamped BeamRef: A.N. Cleland, Foundations of Nanoelectromechanics (Springer, 2003), Ch. 7

2 4

2 40

(0) ( ) 0; (́0) (́ ) 0

u uA EI

t xu u L u u L

2

( ) cos cosh sin sinh exp( )

/

; 0, 4.73004, 7.8532,...

/ 1.01781, 0.99923, 1.0000,...

n n n n n n n n

n n

n

n n

u x a x x b x x i t

EI A

L

a b

A: cross-section area (=HW)ρ: mass density of the beamE,I: assumed independent of positionThe solution is:

Silicon: L=1m, H=W=0.1 m, f0=1 GHz

Nanotube:L=100nm, d=1.4 nm, f0=5 GHz


2 4

2 40

(0) (́0) ´́ ( ) ´́ (́ ) 0

u uA EI

t xu u u L u L

A: cross-section area (=HW)ρ: mass density of the beamE,I: assumed independent of positionThe solution is:

...,008.1,9819.0,3622.1/

,...855.7,694.4,875.1;/

)exp(sinhsincoshcos)(

2

nn

nnn

nnnnnnnn

ba

LAEI

tixxbxxaxw

Flexural Vibrations of a CantileverRef: A.N. Cleland, Foundations of Nanoelectromechanics (Springer, 2003)


Damping of the Mechanical MotionSo 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 modesb)Interaction with electronsc)(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.


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


Langevin Equation

)(202

2

tFxmdt

xdm env

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:

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))exp()(;0 00

202

2

tixtxxmdt

xdm


In many dissipative systems the environmental force can be separated into adissipation (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 transferredfrom the harmonic oscillator to the environment.

Thermal equilibrium in a system controlled by the Langevin equation is achievedthrough the second moment of the noise force, which must satisfy:

Dissipation and Noise are Due to the Environment

0)();(202

2

tFtFdt

dxmxm

dt

xdm NN

2 (0) ( )B N Nmk T F F t dt


TkE B 2 2 20

1 1 1

2 2 2 Bm x m x k T

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 Langevinequation will equilibriate to the energy of the environment by losing any initial excess energy to the environment by the velocity-dependent dissipation termand then, gaining and losing energy stochastically through the noise term thenoise force will produce this equilibrium.

Without proof we state that:

2 2 20

1 1

2 2E m x m x


1( ) 2 ( )

2i B

B

m k TS m k T e d

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):

( ) ( )́ 2 ( )́N N BF t F t m k T t t

Noise Dissipation Temperature

detFtFS iNN

)()(2

1)(


Dissipation and Quality Factor (Q)In the absence of the noise term the solution to the Langevin equation

is x(t)=x0exp(-iωt+φ), where the complex-valued frequency is given by

The frequency ω has both real and imaginary parts, ω = ωR + i ωI:

The quality factor Q is defined as:

)(202

2

tFdt

dxmxm

dt

xdm N

020

2 i

2/,4/220 IR

)(/4/

2/1 00220

ifQR

I


Now, since

the oscillation amplitude damps as

and the energy damps as

Damping of Mechanical Oscillations

)exp()exp()exp()( 00 ttixtixtx IR

)2/exp()2/exp()( 0 Qtttx

)/exp()( 02 Qttx


Recall the Euler-Bernoulli equation:

A: cross-section area (=HW)ρ: mass density of the beamAnd its solution

The imaginary part of ’n indicates that the n:th eigenmode will decay inamplitude as exp(-n/2Q), similar to the damped harmonic oscillator

2

( ) cos cosh sin sinh exp( )

' 1 / 2 / 1 / 2 ; 0, 4.73004,...

/ 1.01781, 0.99923, 1.0000,...

n n n n n n n n

n n n n

n n

u x a x x b x x i t

i Q EI A i Q L

a b

Dissipation in Nanoscale Mechanical Resonators

2 4

2 4( ) 0

(0) ( ) 0; (́0) (́ ) 0

u uA E I

t xu u L u u L

Different with dissipation!


We add a harmonic driving force F(x,t)=f(x)exp(-ict), where f(x) is a position-dependent force per unit length and c is the drive – or carrier – frequency. Theequation of motion is now:

Solve this for times longer than the damping time for the beam by expansionin terms of eigenfunctions:

The equation for the expansion coefficients an is

Driven Damped Beams

2 4

2 4( ) ( ) exp( )

u uA E I f x i t

t x

1

( , ) ( ) exp( )n nn

u x t a u x i t

42

41 1

( )( ) ( ) ( )n

n n nn n

u xA a u x E I a f x

x

3,

0

( ) ( )L

n m m nu x u x dx L


Using the definitions of the eigenfunctions and their properties, and thedefinition of the complex-valued eigenfrequencies ’n this can be written as:

For close to 1, only the n=1 term has a significant amplitude, given by:

For a uniform force distribution, f(x)=f0 , the integral is evaluated to L2, 1=0.8309 and we have, since ’n=(1-i/Q)n:

2 230

1( ) ( )

L

n n na u x f x dxAL

1 13 2 2 21 1 0

1 1( ) ( )

/

L

a u x f x dxAL i Q

011 2 2 2

1 1 /

fa

i Q M


The displacement of a forced damped beam driven near its fundamentalfrequence is – as we have seen – given by

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):

As we discussed already dissipation drives the beam to equilibrium with itsenvironment at temperature T and the stochastic noise force maintains the equilibrium.

0112 2 2

1 1

( , ) ( ) exp( )/

fu x t u x i t

i Q M

,1

1( ) ( ) ( )N N n n

n

f t f t u xL

Dissipation-Induced Amplitude NoiseLecture 2: Electronics and Mechanics on the Nanometer Scale 48/48

Without driving force the mean total energy for each mode is kBT. This requiresthe spectral density of the noise force fN,n(t) to be:

2

2)(

, QL

TMkS nB

f nN

Force per length, hence theterm L2, which is not therefor a simple harmonic osc.

Using this result we can calculate the spectral density for the thermally drivenamplitude as

222222

2

/)(

QL

TMk

QS nB

nn

nan



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.)


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


54

An estimation of the frequency of the nanovibrations.

What is the meaning of the note ”not harmonic”?


55

Would be nice to get comments to ”W” and ”G” which appear on the slide

references book: andrew n. cleland, foundation of nanomechanics springer,2003 (chapter7,esp.7.1.4,...

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