five-lecture course on the basic physics of nanoelectromechanical devices lecture 1: introduction to...

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 Nano-Electromechanics

• Quantum coherence of electrons• Quantum coherence of mechanical displacements• Quantum nanomechanical operations:

a) shuttling of single electrons;

b) mechanically induced quantum

interferrence of electrons

Outline

Formalization of Heisenberg’s principle:

operators for physical variables

eigenfunctions – quantum states

quantum state with definite momentum

In this state the momentum experiences quantum fluctuations

Quantum Coherence of Electrons

,r p

Classical approach

2p x

Heisenberg principle in quantumapproach

A

( )pr

i

p r Ce

( ) ( )p pp

r a r

Lecture 4: Quantum Nano-Electromechanics 3/29

Stationary Quantum States

Hamiltonian of a single electron:

2ˆˆ ˆ( );2

pH U r p

m i r

Stationary quantum states: H E


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:

Creation and annihilation operators

( )nN r

1

1

ˆ ( ) ( )

ˆ ( ) ( )

N n N n

N n N n

a r r

a r r

ˆ ˆN n a a

ˆ ˆH n

,,a a

fermions

,,a a

bosons


Field Operators

ˆ ˆ ˆˆ( ) ( ); ( ) ( ) ( )

ˆ ˆˆ ˆ( ) ( ) ( )

r a r n r r r

H dr r H r r

1 2 1 2ˆ ˆ( ) , ( ) ( )r r r r


Density Matrix

ˆˆ ˆˆ ˆ{ }; [ , ]A Sp A i Ht

Louville – von Neumann

equation

Zero-Point Oscillations

20

1( )

2U x U kx

2

2

d x Um kx

dt x

k

m

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:

Quantum motion:

Classical vs quantum description: the choice is determined by the parameter

where d is a typical length scale for the problem. “Quantum” when


0x d

0 1x d

Amplitude of zero-point oscillations:

0x m

2 2 2( ) 2 2p x E x mx kx

0( ) min ( )E x E x

Quantum Nanoelectromechanics of Shuttle Systems

If then quantum fluctuations of the

grain significantly affect nanoelectromechanics.

x p 0

22x x

M

( ) ( ) ( )R x x R x R x


Conditions for Quantum Shuttling

021

x

length Tunneling

1. Fullerene based SET

2. Suspended CNT

THz 1

1.00 X Quasiclassical shuttle

vibrations.

μmL - 1 with SWNTfor Hz 1010 98

STM

L

nm 1d

214 Hz 10

L

d

9

0 2x

M

9


Quantum Harmonic Oscillator

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.

22 2ˆ 1ˆ

2 2

pH M x

M ˆ

dp i

dx

1 122 41

| exp2 ! 2 nn

M M x Mn H x

n

1

2nE n

Ledder operators

ˆ ˆ2

ˆ ˆ2

M ia x p

M

M ia x p

M

| | 1

| 1 | 1

a n n n

a n n n

ˆ2

ˆ2

x a aM

p i a aM

1/ 2H a a [ , ] 1a a

Eigenstate of oscillator is a ideal gas of elementary excitations – vibrons, which are a bose particles.


Quantum Shuttle Instability

d

thr

Quantum vibrations, generated by tunneling electrons, remain undamped and accumulate in a coherent “condensate” of phonons, which is classical shuttle oscillations.

k

eEd

2

Shift in oscillator position caused by charging it by a single electron charge.

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)

Phase space trajectory of shuttling. From Ref. (3)

eVe


Hamiltonian ˆ ˆ ˆ ˆ ˆ

leads dot oscillator tunnelingH H H H H

,

2

,

22 2

, ,

ˆ

ˆ ˆ ˆˆ( )

ˆ 1ˆ ˆ2 2

ˆ ˆ( ) . .

leads k k kk

dot q q q ck

oscillator

tunneling kq k qk q

H a a

H c c E N e x N

pH M x

M

H T x a c h c

ˆq q

q

N c c


The Hamiltonian:

Time evolution in Schrödinger picture:

0

1

ˆ ( ) 0ˆ ˆ( )

ˆ0 ( )leads

tt Tr t

t

Theory of Quantum Shuttle

)](ˆ,[)(ˆ tHitt

Reduced density operator

Total density operator

Dot

x

Lead Lead

,

2

,

22 2

, ; ,

ˆ /, 0

ˆ ˆ ˆ ˆ ˆ

ˆ

ˆ ˆ ˆˆ( )

ˆ 1ˆ ˆ2 2

ˆ ˆ( ) . .

ˆ( )

leads dot oscillator tunneling

leads k k kk

dot q q q ck

oscillator

tunneling kq k qL R k q

xL R

H H H H H

H a a

H c c E N e x N

pH M x

M

H T x a c H c

T x T e


Generalized Master Equation:ˆ0 density matrix operator of the uncharged shuttle

:ˆ1 density matrix operator of the charged shuttle

0 0 0 1 0

1 1 1 0 1

ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ[ ( ), ] { ( ), } ( ) ( )

ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ[ , ] { ( ), } ( ) ( )

t osc L R R

t osc R L L

i H e x x x x L

i H eEx x x x L

:ˆˆˆ 10 :ˆˆˆ 10

20 1, '( ) 1,x e x M

ˆ,ˆ,ˆ2

ˆ,ˆ,ˆ2

ˆ xxpxi

L

At large voltages the equations for are local in time:10 ˆ,ˆ


After linearisation in x (using the small parameter xo/l) one finds:

Result: an initial deviation from the equilibrium position grows exponentially if the dissipation is small enough:

Shuttle Instability

2

/

'( 0)

2

x p M

p M x p x q

q q e x

1

ˆ ˆ( ) ( )

ˆ ˆ( ) ( )

ˆ( ) 1 ( )

x t Tr t x

p t Tr t p

q t Tr t

2

'( 0)( ) exp( ); ( ) / 2 ;

2thr thr

e xx t t

M


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.


Try new principles for sensing the quantum displacements!

How to Detect Nanomechanical Vibrations in the Quantum Limit?

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).


Aharonov-Bohm Effect

1 2 1 2 10( ) ( ) ( ) ( ) exp (1 exp ( )

Ab b b b i i

2

2 2 2

00 0

| | | | 1 cos 2 sin 2A

T

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:

The probability for the particle transition through the device is given by:

( )i

i

L

ex dl A

c

0( ) ( ) ( ) exp{ }i iA

i i

b b b i


2 1 00

1 22 ; ; (flux quantum)

2 L S

cd l A ds H

e

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.

Classical and Quantum Vibrations


Quantum Nanomechanical Interferometer

Classical interferometer

Quantum nanomechanicalinterferometer


Electronic Propagation Through Swinging Polaronic States

21

Model

, ,e m

L R L R

H H H H T

, ,H a a

22

22

3 { ( ) ( ) ( , ) ( ) ( )}2e

ieH d r r A r U y u x z r r

m r c

222 2

22

1 ( )ˆ

2 2

L

m L

EI u xH dx

x

( ) ( )q qq

r a r

ˆd

idu


Coupling to the Fundamental Bending Mode

142 2

0 0 00

ˆ ˆˆ( ) ( ) ( ) 2 ; Lu x x u x b b x EI

Only one vibration mode is taken into account

CNT is considered as a complex scatterer for electrons tunneling from one metallic lead to the other.


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

ˆ ˆ( ), , , ,

, ,

ˆ ˆ ( . )2

i b beff R LH a a b b T e a a h c

0 0/ 2gHLY 0 02Y M L *

L Reff

T TT E


Calculation of the Electrical Current

ˆ ˆ( ) 20

0

( ) | | | |

( ) 1 ( ) ( ) 1 ( )

i b b

n l n

l r r l

I G P n n e n l

d f f l f f l

ˆ ˆ( ) 2

0 0

ˆ ˆ( ) 2

0 1

( ) | | | |

2( ) | | | |

exp( ) 1

i b b

n

i b b

n l

G P n n e nG

lkTP n n e n l

lkT


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.


Vibrational system is in equilibrium

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 magneto-current of 0.1-0.3 pA.

Linear Conductance

2

0 0

exp , 1G

G kT

2

0 0

11 , 1

6

G

G kT kT

0 04 ,gLx H h e


Quantum Nanomechanical Magnetoresistor


Temperature

Magnetic field

R.I. Shekhter et al., PRL 97, 156801 (2006)

Magnetic Field Dependent Offset Current

0 ( ) ;I I V I

2

0

eV

I

V

I

2

00

eV

)(0 VI

0V

2

00

eVI

I


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); 31

Lecture 4: Quantum Nano-electromechanics

Speaker: Professor Robert Shekhter, Gothenburg University 2009 31/41

CNT with an Enclosed FullereneShallow harmonic potential for the

fullerene position vibrations along CNT

Strong quantum fluctuations of thefullerene position, comparable toFermi wave vector of electrons.

0 2x

M

3232



Electronic Transport through 1-D Channel

yLeft

;0QR

VI

33

Right

h

eGRQ

2

01 4

Right

• Weak interaction with the impurity

• Only a small fraction of electrons scatter back.

• Elastic and inelastic backscattering channels

I. No backscatteringLeft and right side reservoirs inject electrons with energy distribution according to Hibbs

with different chemical potentials

Left

py

33



II. Backflow Current

backback

back

dt

dNeI

III

0 , ,

2

( ) 1 ( ) ( )Right Leftp q n lback

back

dN rf p f q P n

dt h

n V n l

..

..ˆ

..)()()(ˆ

ˆ2

ˆ2

,

†

††

chreV

cheaarH

chXyyyrH

Xikback

Xikq

qppback

LRback

F

F

q

iqyqL

p

ipypR

eay

eay

††

††

)(

)(

34

Backscattering potential

34



The probability of backscattering sums up all backscattering channels. The result yields classical formula for non-movable target.

However the sum rule does not apply as Pauli principle puts restrictions on allowed transitions .

n

n

movablenonbacknback WWW

Backscattering of Electrons due to the Presence of Fullerene

35



×

Bias potential across tube selects allowed transitions through oscillator as fermionic nature of electrons has to be considered.

Pauli Restrictions on Allowed Transitions through Oscillator

36



Excess Current

0 0lim (1 )excess

eVI I G V r G r

e

αħω=4εFm/M, excess current independent of confining potential, voltage and temperature. Scales as ratio of masses in the system.

3737



Model

, ,e m

L R L R

H H H H T

, ,H a a

38

22

3 { ( ) ( ) ( , ) ( ) ( )}2e

ieH d r r A r U y u x z r r

m r c

22 2

22

1 ( )( )

2

L

m L

u xH dx x EI

x

n

n

movablenonbacknback WWW

39

Backscattering of Electrons on Vibrating Nanowire.

The probability of backscattering sums up all backscattering channels.The result yields classical formula for non-movable target.

However the sum rule does not apply as Pauli principle puts restrictions on allowed transitions .

×

The applied bias voltage selects the allowed inelastic transitions through vibrating nanowire as fermionic nature of electrons has to be considered.

Pauli Restrictions on Allowed Transitions Through Vibrating Nanowire

40

4-5

41

4-6

42

43



44

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.

44



Magnetic Field Dependent TunnelingIn order to proceed it is convenient to make the unitary transformation

ˆ ˆˆiS iSe He

3

0

ˆ ( )ˆ ˆ ˆ ˆˆ ˆ( ) ( ) ( ) ( ) ( )xr eH

S i d r u x r i dx u x r ry c

2 2

2

222

20

ˆ ˆ ˆ( ) ( )2

ˆ1 ( )ˆ ˆ ˆˆ[ ( ) ( ) ( )]2 2

el

x

mech

H dx x xm x

eH EI u xH dx x dx x x

c x

2

0

ˆ ˆˆ( ) exp ( ) (0) . .

L

l lr r

eHT H T i dxu x T h c

c


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