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 electronseffect 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 noncommuting 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 TwoState 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
zcomponent 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 noncommuting variables
if zcomponent 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
EinsteinPodolskiRosen (EPR) Paradox
Suppose we prepare two electrons on Earth in a spinsinglet 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 AharonovBohm 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 AharonovBohm 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 AharonovBohm 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 Tm 2
Electron current periodic in flux, with fundamental period φ 0
Interference of electrons: the AharonovBohm
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 AharonovBohm effect in disordered metals
1
e
B
2
Strong elastic impurity scattering, electron motion is diffusive
Elastic scattering length L e
~ 10100 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 electronelectron, electronphonon interactions, etc
Increases at low temperatures, but still small! (1 µm)
Need to make devices with small dimensions
Electronbeam lithography
measured at low temperatures!
Cryogenic lownoise measurement techniques
Quantum interference is an example of a mesoscopic effect
microscopic < relevant length scale < macroscopic
The AharonovBohm 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 twolevel system like the spin 1/2 electron discussed earlier.
To make the notation applicable to a variety of twolevel 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 onequbit 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 α0> + β1>
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 doublequbit 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
controlledNOT 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 singlequbit 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.)
RSA129 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 tenMIPS computers (April, 1994)
For a 300digit 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
Nanoelectromechanical systems
Fabrication of nanogap devices
High resolution electron beam lithography
e
Au film
PMMA
CoPMMA
SiO 2
Si
2550 nm
best is ~5 nm
Gallery of electronbeam fabricated samples
GaAs 2DEG doublequantum dot
“FSQUID” 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 problemloss of
contact with phonons
Need to take special care to filter
noise, especially radiofrequency
noise
Use homemade 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)
α0> + β1>
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
dcSQUID
Φ/Φ 0
Classical states
Quantum ground state 0>
Quantum first excited state 1>
Nano ElectroMechanical Systems (NEMS)
Doubly Clamped Flexural Resonator
t
L
Cleland Group, UCSB
Fundamental
Ω0 E t
= 1. 03
2π
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 nmresonators
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 ~ (110) 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
rfcoupled 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 solidstate 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)