An Introduction to Mesoscopic and Nanometer Scale Physics (PDF)

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An Introduction to Mesoscopic and Nanometer Scale Physics (PDF)

An Introduction to Mesoscopic

and Nanometer Scale Physics

Freshman Seminar:

Nanoscience and Nanotechnology

January 26, 2006


A brief review of quantum mechanics

Quantum mechanics tells us that the fundamental building blocks of

nature can be thought of as both particles and waves.

What determines when must we take into account the wave nature of

particles?

λ = h / p

de Broglie wavelength

If the de Broglie wavelength of the particle is less than its physical

dimensions, then classical mechanics provides a good description.

For example, λ of an electron of mass 9.11 x 10 -31 kg traveling at 3

x 10 6 m/s is ~0.24 nm, compared to the electron radius of

~3 femtometers.

A 1 cm 3 droplet of water travelling at the same velocity has a

wavelength of ~2 x 10 -37 m, compared to its physical extent of

~ 1 cm.


Wave Mechanics: Double slit experiment with waves

Consider the classic double slit experiment. We have a wave of some

nature (light, for example, but it could also be water waves). If we have

only slit 1 uncovered, we obtain the intensity pattern I 1 . If we have only

slit 2 uncovered, we obtain I 2 . If we have both uncovered, we obtain the

red curve I 12 , which is of course the interference pattern of the waves

coming from slits 1 and 2.

I 1

I 2

I 1,2

1

y

2


Double slit experiment with electrons

Now consider the same experiment, but with a gun which fires electrons

as a source. Instead of a screen, we use something which counts

individual electrons. Each time an electron hits the screen, we count the

event and where it occurred. By counting the number of hits on average

at a particular value of y, we obtain a probability distribution of the

electrons as they hit the screen.

electron

gun

1

y

2


Interference of electrons

Now electrons, cannot be split up (at least not in this experiment). When we

detect an electron, we detect a whole electron. We then make the very

reasonable assumption that if an electron goes from the electron gun to the

detector screen, it goes through either slit 1 or slit 2. It cannot split itself up and

go through both slits at once. If we measure again the probability distribution

with each slit closed in turn, we obtain the distributions shown below.

electron

gun

1

2

y

Simulation


Interference of electrons

When we let both slits be open, and measure the probability distribution, we

find that it looks like what is shown below, which is very similar to the

probability distribution for a wave. Hence, although the electrons appear to

arrive as particles, in one single “lump” which contains the entire electron, the

final distribution pattern looks very similar to what we found for the intensity

of a wave. Hence the electrons appear to have both particle like and wave like

properties.

P 12 ≠ P 1 + P 2

P 1

P 2

P 1,2

electron

gun

1

y

2


Interference of electrons-effect of observation

One can obtain an interference pattern with only one electron, which can take two

possible paths from the gun to the screen. However, if we put a detector at one

slit to determine which path the electron takes, then we no longer obtain the

interference pattern, but only the two gaussian distributions. Thus, the act of

observation forces the electron to choose one of the two possible paths (or

states), or put another way, the act of measurement affects the observation, a

peculiar property of quantum systems.

P 1

P 2

P 1,2

electron

gun

1

y

2


Wave nature of particles

Like other waves, quantum mechanical waves of particles are described

by functions that satisfy a wave equation, in this case, Schrödinger’s

wave equation

− h2

2m

d 2 ψ

+ V (x)ψ = Eψ

2

dx

where m is the mass of the particle, and E the energy of the particle,

and V(x) the potential. The wave function ψ(x) is in general a complex

function, with an amplitude and a phase

ψ(x) = ψ(x) e iϕ(x )

Once ψ(x) is known, one can determine all the properties of the system.

For example, the probability of finding a particle at a position x is given

by

P(x) = ψ(x) 2


Quantum Tunneling

ψ(x)

V(x)

Classical particle

Quantum description

x

For x


Quantum tunneling through a barrier

ψ(x)

V 0

V(x)

Classical particle cannot penetrate

barrier

x=0 a

Quantum particle: finite probability on opposite side of

barrier- particle tunnels through barrier

x

Potential barrier of height V 0 and width a, where energy E of particle is less than V 0 .

For xa, solution same as before.

For 0>x>a, Schrödinger’s equation is

− h2

2m

d 2 ψ

dx 2 + V 0ψ = Eψ


Solutions are ψ(x) = Ce αx + De −αx

h 2

2m

d 2 ψ

dx + (E −V 0)

2 14 24 3

ψ = 0

< 0

Solutions in barrier are exponentially decaying

Tunneling simulations


More about wave functions

Classically, can specify both position x and momentum p with infinite accuracy

Quantum mechanics?

Consider wave function ψ(x)= ψ 0 e ikx

Corresponds to a particle with definite momentum

p = hk

What about position?

Probability density |ψ(x)| 2 =|ψ 0 | 2

independent of x

cannot tell position of particle at all

If we know the momentum p exactly, the position x of the particle is completely

unknown (and unknowable)

Conversely, if we know the position of a particle exactly, its momentum is completely

unknown


More about wave functions

Suppose we want to know the position of a particle accurately. How do we define

its wavefunction?

ψ(x)

Particle localized within a distance ∆x

ψ(x) is a superposition of waves of different

wavevectors over some range

(Wave packet)

ψ (x) = sin(2πk 1

x) + (1/ 2)sin(2πk 2

x) + (1/ 2)sin(2πk 3

x)

∆x

What about momentum p?

Not well defined..some average of k 1 , k 2 , k 3 .

|ψ(x)| 2

Uncertainty in both position and momentum

Both momentum and position only known to some

uncertainty ∆x, ∆p


Heisenberg uncertainty principle

Impossible to know both momentum and position of a particle precisely

Minimum uncertainty is given by

∆x∆p ≥ h

If position is known precisely (∆x=0), then momentum is completely unknown (∆p=∞)

If momentum is known precisely (∆p=0), then position is completely unknown (∆x=∞)

In general, both position and momentum are not precisely known

True for non-commuting variables


Time evolution of quantum states

Quantum mechanics tells us that solutions of Schrödinger’s equation with

definite energy evolve in time according to the equation ψ(t) =ψ(0)e −iEt / h

Consider a superposition of two states ψ 1 and ψ 2 . The time dependence

is given by

ψ(t) =ψ 1

(t) +ψ 2

(t) =ψ 1

(0)e −iE 1t / h +ψ 2

(0)e −iE 2t / h

As a function of time, the particle will oscillate between the two

eigenstates

QuickTime and a

Microsoft Video 1 decompressor

are needed to see this picture.


Quantum mechanics and measurement

Wave function is frequently superposition of different components with well

defined quantum numbers

Example

Wave function

ψ (x) = sin(2πk 1

x) + (1/ 2)sin(2πk 2

x) + (1/ 2)sin(2πk 3

x)

Measurement of momentum can give three possible values

p 1

= hk 1

p 2

= hk 2

p 3

= hk 3

What is observed in a single measurement? Either p 1

or p 2

or p 3

What is the average momentum recorded after many measurements?

Average determined by weights of different momentum functions

p ave = (2/3) p 1

+ (1/6) p 2

+ (1/6) p 3


Superposition of states: In search of Schrödinger’s Cat

Closed box with Cat and radiation source inside

Source emits radiation intermittently which

kills Cat

=

Question

With the box lid closed, is the Cat dead or alive?

Can only determine by making a measurement!


In search of Schrödinger’s Cat

Is Schrödinger’s Cat dead or alive ?

?

Cat is in a superposition of two quantum states

ψ cat = +

How do we find out? Open the box and look at the Cat

Two possibilities...Cat is either dead or alive (two quantum states with two quantum

numbers)

Act of observation collapses Cat wave function into one of the two superposition

states (dead or alive)!

Quantum mechanically, the act of observation affects the system being observed!


Quantum Two-State Systems: Qubits

Consider an electron, which has a spin angular momentum of 1/2

in units of Planck’s constant

h

According to quantum mechanics, components of angular momentum are

quantized-- can only change in units of

z-component of electron spin can have two values: + 1/2 or - 1/2

h

+ 1/2 - 1/2

(Two orientations like a

computer bit -- quantum

bit or “qubit”)

x, y and z components of angular momentum are non-commuting variables

if z-component is known exactly, not possible to know the x or y

components


Combinations of qubits

Now consider a quantum state involving two electrons

What are the possible spin combinations?

| 1 2 > | 1 2 > |

1 2 > |

1 2

>

(4 states)

Usually organize in terms of total spin

Total spin S=1

| 1 2 >

S z =1

| 1 2 > + |

1 2 >

|

1 2

>

S z =0

S z =-1

Triplet

states

Total spin S=0 | 1 2 > - | 2 >

1 S z =0 Singlet states

Electrons in state are correlated -- “entangled” states


Einstein-Podolski-Rosen (EPR) Paradox

Suppose we prepare two electrons on Earth in a spin-singlet state (S=0)

| 1 2 > - |

1 2 >

Take one electron of the pair to Alice on Venus, take the other to

Bob on Mars. Let Alice measure the spin of her electron

Suppose Alice measures S z =+1/2. What is the spin of Bob’s electron?

QM tells us that a measurement of Bob’s electron must give S z =-1/2

since the electrons are correlated, even though they are separated by

a very long distance. Alice’s measurement of her electron tells us

the spin orientation of Bob’s electron, even without making a

measurement.

In order to remain correlated, the state must retain its coherence


Quantum coherence

Quantum interference in disordered metals: the Aharonov-Bohm effect

Classical charged particle in a magnetic field B

Lorentz force F~ q v B, executes circular cyclotron motion

B

e

No effect in regions where there is no B field


Quantum coherence: The Aharonov-Bohm effect

Annulus enclosing magnetic field B and magnetic flux Φ

but no magnetic field in electron’s path!

B

e

Magnetic field affects electron phase through magnetic

vector potential A

Electron wave function

Absence of magnetic field

r

B = ∇ × A

r

With magnetic field, phase modified

ψ ~ ψ 0

e ir p • r =ψ 0

e iϕ

ϕ → ϕ − 2π e h

b


a

r

A • d r


The Aharonov-Bohm effect

ϕ 1

= ϕ 0

− 2π e h

1

b


a

r

A • dr

e

a

B

b

Total phase difference

ϕ = ϕ 2

−ϕ 1

= 2π e h

2

ϕ 2

= ϕ 0

+ 2π e h


r

A • dr

b


a

r

A • dr

Quantum of magnetic flux

= 2π e h φ = 2π φ φ 0

φ 0

= h /e = 4.14 ×10 −15 T-m 2

Electron current periodic in flux, with fundamental period φ 0


Interference of electrons: the Aharonov-Bohm

effect

1

e

B

2

Like Young’s double slit experiment, except phase difference

controlled by magnetic field instead of path length

Smallness of flux quantum φ 0 means that such interference devices

are very sensitive detectors of magnetic field.

Example: Superconducting Quantum Interference Devices (SQUIDs)

Typical noise levels ~10 µφ 0 /Hz 1/2

For a 1 cm 2 area SQUID, corresponds to a magnetic field noise level of

2 x 10 -16 T/ /Hz 1/2


The Aharonov-Bohm effect in disordered metals

1

e

B

2

Strong elastic impurity scattering, electron motion is diffusive

Elastic scattering length L e

~ 10-100 nm

Electron momentum changes at each scattering even,

is interference pattern observed?

Yes! Relevant length scale is electron

phase coherence length L φ ~1 µm

Samples with dimensions< L φ show

quantum interference effects

But coherence can easily be destroyed by interaction with environment:

thermal vibrations, electromagnetic fields, Coulomb interactions, etc.


Quantum interference in disordered metals

Relevant length scale is the electron phase coherence length L φ

Determined by electron-electron, electron-phonon interactions, etc

Increases at low temperatures, but still small! (1 µm)

Need to make devices with small dimensions

Electron-beam lithography

measured at low temperatures!

Cryogenic low-noise measurement techniques

Quantum interference is an example of a mesoscopic effect

microscopic < relevant length scale < macroscopic


The Aharonov-Bohm effect in disordered metals

1

e

B

2

h/e oscillations: Webb et al, PRL (1985) h/2e oscillations: Chandrasekhar et al, PRL (1985)


Quantum devices

The operating principles of almost all modern solid state devices

(transistors, diodes, etc.) are based on quantum mechanical

phenomenon. However, these devices are macroscopic in dimension, and

involve averages over a very large number (10 23 ) number of incoherent

quantum states.

Two factors separate distinctly quantum phenomena

Coherence, or the control or manipulation of the quantum

mechanical phase of the wave function

The ability to probe individual quantum states


Quantum operations

Consider a two-level system like the spin 1/2 electron discussed earlier.

To make the notation applicable to a variety of two-level systems, we

will denote the two states by |0> and |1> (say |0> = and |1>= )

| > | >

What sort of quantum operations can we apply to such a qubit?

There are an infinite number of one-qubit operations that one can apply

Example: Consider the time evolution of quantum states discussed

earlier. Take the energy of the |0> state to be 0, and the

energy of the |1> state to be E. Then the time evolution

in a time interval t is described by the set of operations

|0> -> |0>

|1> -> |1>e iωt

where ω=2πE/h ψ =ψ 0e −iEt / h


Single qubit quantum operations

The time evolution operation can be represented as a matrix that

operates on a general qubit superposition state α|0> + β|1>

where θ=ωt


P(θ) = 1 0 ⎞


⎝ 0 e iθ ⎟



ψ = α ⎞

⎜ ⎟

⎝ β⎠

Other operators:

Identity operator I takes |0> -> |0>, |1> -> |1>

Operator X switches the two states |0> -> |1>, |1> -> |0>


X = 0 1 ⎞


⎝ 1 0⎠

The X operator can be recognized as the analog of the

classical NOT operator


I = 1 0 ⎞

⎜ ⎟

⎝ 0 1⎠


Double qubit quantum operations

Now consider the following double-qubit operation

|a>

|a>

|b>

|a> ⊕ |b>

This gate is defined by the following transformations on the two qubit

states |a b>

|00> -> |00>

|01> -> |01>

|10> -> |11>

|11> -> |10>

This is called a quantum XOR operation (or gate), also known as a

controlled-NOT gate.


Quantum computers

A classical computer can be constructed entirely from logic gates (in

particular, NAND gates)

A quantum computer can be constructed entirely from quantum XOR

gates and single-qubit gates.

How does a quantum computer work?

Prepare a state of n qubits, and let it evolve under a set of quantum

operations.

What is the advantage of a quantum computer?

Logic operations occur on n bits simultaneously -->

massively parallel computation


Advantages of quantum computers

Example: Factorization of large numbers (important in cryptography,

internet security, etc.)

RSA-129 Challenge problem

15 = 5 x 3

4633 = 41 x 113

1143816257578888676692357799761466120102182 9672124236256256184293570693524573389783059 7123563958705058989075147599290026879543541

=34905295108476509491478496199038 98133417764638493387843990820577 X

32769132993266709549961988190834 461413177642967992942539798288533

Took 8 months, 600 people and equivalent of 750 ten-MIPS computers (April, 1994)

For a 300-digit number, a 1 THz classical computer will take

150,000 years

with a 1 THz quantum computer, it will take less than 1 second


Experimental realizations of qubits

Atomic and molecular systems

NMR on ensembles of spins

Atomic energy levels

Trapped ions

Photons

Solid state qubits

Electron spins in quantum dots

Superconducting devices

Other systems

Electrons on liquid helium

Spins in semiconductors

Nano-electromechanical systems


Fabrication of nanogap devices

High resolution electron beam lithography

e-

Au film

PMMA

Co-PMMA

SiO 2

Si

25-50 nm

best is ~5 nm


Gallery of electron-beam fabricated samples

GaAs 2DEG double-quantum dot

“F-SQUID” Ferromagnetic/superconductor structure

Ferromagnet/superconductor/normal metal device

DNA/Au nanoparticle device


Sample measurement

Dilution or 3 He refrigerator

temperatures down to 20 mK

..can also apply magnetic fields

Cooling electrons is a problem-loss of

contact with phonons

Need to take special care to filter

noise, especially radio-frequency

noise

Use home-made resistance bridges and

electronics

Low measurement currents (20 nA)


Solid state qubits

Double quantum dot in semicondutor 2DEG (Loss and DiVincenzo)

Each quantum dot is a qubit

with a spin 1/2 electron

(simplest case)

α|0> + β|1>

Exchange interaction creates

coupling, resulting in singlet

and triplet states

Need to control interaction

prepare states

read out states

Marcus group, Harvard


Superconducting qubits- the Cooper pair box

Superconducting island isolated by tunnel junctions (charge qubit)

Can control the number of Cooper pairs by varying gate voltage

Slide 38

Two states:

|0> = no Cooper pair on island

|1> = single Cooper pair on

island

Near degeneracy point, superposition

of the two states


Superconducting qubits- flux qubits

(H. Takayanagi, NTT Research)

NTT Atsugi

States controlled by applying magnetic field

Transitions between states can be induced

by radiating with microwave photons

E 0 (1)

∆E

Level

splitting

Φ

M

/I p

Qubit

dc-SQUID

Φ/Φ 0

Classical states

Quantum ground state |0>

Quantum first excited state |1>


Nano Electro-Mechanical Systems (NEMS)

Doubly Clamped Flexural Resonator

t

L

Cleland Group, UCSB

Fundamental

Ω0 E t

= 1. 03


ρ 2

L

Amplitude

1st harmonic

Ω1 = 2.

75Ω0

2nd harmonic

Ω2 = 5.

40Ω0

DC response: ∆z = F / k eff

Fundamental: ∆z = Q F / k eff

Q ~ 10 4 for nm-resonators

0 2 4 6

Frequency (GHz)


Quantum Behavior in Mechanical Systems

Simple Harmonic Oscillator

Cleland Group, UCSB

E

E = (1/2) kx 2

E

E = (n +1/2) hω

n=2

ω

1

0

Classical

x

Quantum

x

To measure transition:

• k B T < hω : T min » 50 mK ω/2π 1 GHz

• (nearly) quantum limited detection: ε ~ (1-10) hω


Detection of Nanometer Scale Motion

Cleland Group, UCSB

DC Bias

+

RF Drive

C

Beam Motion

Beam motion ∆x changes C:

∆C

=

∂C

∆x

∂x

Voltage V changes Q:

∆Q

= ∆C

⋅V

∆I

I

SET

I ds

Charge ∆Q induces change ∆I:

∆I

∂I

= ∆Q

∂Q

∂I

= ∆x

∂x

Sensitivities of 10 -16 m/Hz 1/2

at 1 GHz are possible

∆Q

Q

Blencowe and Wybourne (2000)


Quantum limited detection using a SET

Cleland Group, UCSB

Knobel, Cleland, Nature (2003)

High Q resonator:

• ω o /2π ~ 115 MHz

•Q ~ 10 3

• T ~ 0.03 K

rf-coupled SET:

• operates to ω ~ 1 GHz

• bandwidth ∆ω ~ 20 kHz


Quantum limit in mechanical oscillations

Mohanty group, BU (PRL, 2005)

Classical case, displacement continuous

Quantum case

Displacement

quantized


Summary

Nanometer scale solid-state devices allow investigation of fundamental

quantum mechanical phenomena

Challenges lie in fabrication of devices, sophisticated new measurement

techniques

Need to understand and reduce sources of decoherence (noise,

fundamental limits)

Potentially large payoff (e.g., quantum computers)

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