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NANOPHYSICS
E 304
Dr. MC Ozturk, [email protected]
Early 1900s

Electrons are nice particles
 They

obey laws of classical mechanics
Light behaves just like a wave should
 It
reflects, refracts and diffracts
Then, things began to happen…
Electromagnetic Waves

EM waves include two oscillating components:



EM waves can travel in vacuum


electric field
magnetic field
Mechanical waves (e.g. sound) need a medium (e.g. air)
EM Waves travel at the speed of light, c

c = 299,792,458 m/s
frequency
wavelength
Electromagnetic Spectrum
Thermal Radiation
All hot bodies emit thermal radiation
For human skin, T = 95oF, wavelength is 9.4 micrometer (infrared)
Ultraviolet Catastrophe

Rayleigh-Jeans law described thermal radiation
emitted by a black-body as
This implies that as the wavelength of an EM wave approaches zero (infinite
frequency), its energy will become infinitely large! i.e. will get brighter and
brighter
Experimentally, this was not observed… and this was referred to as the
ultraviolet catastrophe
Rutherford Atom - Challenge
Belief: Attractive force between the positively charged nucleus and an electron orbiting
around is equal to the centrifugal exerted on the electron. This balance determines the
electron’s radius.
Challenge: A force is exerted on the electron, then, the electron should accelerate
continuously according to F = ma. If this is the case, the electron should continuously lose
its energy. According to classical physics, all accelerating bodies must lose energy. Then,
the electrons must collapse with the nucleus.
Photoelectric Effect - Challenge
Light shining on a piece of
metal results in electron
emission from the metal
There is always a threshold frequency of light below which no electron emission
occurs from the metal.
Maximum kinetic energy of the electrons has nothing to do with the intensity of
light. It is determined by the frequency of light.
Photoelectric Effect
The Experiment - 1
Light Source
anode
cathode
Electrons emitted by the cathode are attracted to the positively charged anode.
A photocurrent begins to flow in the loop.
Photoelectric Effect
The Experiment - 2
Light Source
anode
cathode
Electrons emitted by the cathode are repelled by the negatively charged anode.
The photocurrent decreases.
Photoelectric Effect
The Experiment - 3
Current
Increasing
Light
Intensity
Vo
Voltage
Regardless of the light intensity, the photocurrent becomes zero at V = - Vo
At this voltage, every emitted electron is repelled
Therefore, qVo must be the maximum kinetic energy of the electron
This energy is independent of the light intensity
Photoelectric Effect
The Experiment - 4
fo
Frequency
The maximum kinetic energy of electrons is determined by the frequency of light
The slope of this line is Planck’s constant
Increasing the light intensity only increases the number of photons hitting the cathode
Video:
A Brief History of Quantum Mechanics

http://www.youtube.com/watchv=B7pACq_xWyw&list=PLXD9X52rMwwN
J85QDzv3G6cqbQ2ZVlsR8&index=2
Video:
Max Planck & Quantum Physics

https://www.youtube.com/watch?v=2UkO_3NC3F4
NANOPHYSICS
E 304
Dr. MC Ozturk, [email protected]
Hydrogen Atom
Hydrogen Atom
Orbital
three dimensional space around the nucleus of all
the places we are likely to find an electron.
Orbitals & Quantum Numbers
Atoms have infinitely many orbitals
 Each orbital can have at most two electrons
 Each orbital represents a specific

 Energy
level
 Angular momentum
 Magnetic moment
Sub-levels
Quantum Numbers

Principal Quantum Number, n = 1, 2, 3, …
 Determines

Azimuthal Quantum Number, l = 0, 1, 2, …
 Determines

the electron’s angular momentum
Magnetic Quantum Number, m = 0, ± 1, ± 2, …


the electron energy
Determines the electron’s magnetic moment
Spin Quantum Number, s= ± 1/2
 Determines
the electron spin (up or down)
The energy, angular momentum and magnetic moment of an orbital are quantized
i.e. only discrete levels are allowed
Principal Quantum Number



Always a positive integer, n = 1, 2, 3, …
Determines the energy of the electron in each
orbital.
Sub-levels with the the same principal quantum
number have the same energy
Only certain (discrete) energy levels are allowed!
Azimuthal Quantum Number
l=0
l=1
l=2
n=1
a
n=2
a
a
n=3
a
a
a
n=4
a
a
a
l=3
a
Each n yields n – 1 sub levels
Magnetic Quantum Number
l=0
l=1
l=2
l=3
n=1
m=0
n=2
0
-1, 0,+1
n=3
0
-1, 0,+1
-2, -1, 0,+1,+2
n=4
0
-1, 0,+1
-2, -1, 0,+1,+2 -3,-2, -1, 0,+1,+2,+3
s
p
d
f
3 orbitals
5 orbitals
7 orbitals
1 orbital
Spin Quantum Number
spin = up or down
(only two possibilities)
l=0
l=1
l=2
l=3
n=1
m=0
n=2
0
-1, 0,+1
n=3
0
-1, 0,+1
-2, -1, 0,+1,+2
n=4
0
-1, 0,+1
-2, -1, 0,+1,+2 -3,-2, -1, 0,+1,+2,+3
s
p
d
f
3 orbitals
6 states
5 orbitals
10 states
7 orbitals
14 states
1 orbital
2 states
Levels, Sublevels of Atomic Orbitals

http://en.wikipedia.org/wiki/Atomic_orbital
Electrons fill the lowest energy states first
Atomic Number
Example 1: Silicon Atom
Silicon has 14 electrons
3d
Occupied States
4s
Empty States
3p
3s
2p
2s
1s
1s22s22p63s23p2
Example 2: Titanium Atom
Titanium has 22 electrons
3d
Occupied States
4s
Empty States
3p
3s
2p
2s
1s
1s22s22p63s23p63d24s2
Atomic Orbitals
Larger Atoms
Hydrogen
s (l=0)
p (l = 1)
d (l = 2)
f (l = 3)
Video - Orbitals

http://www.youtube.com/watchv=drCg4ruJCfA&list=PLREtcqh
PesTcTAI6di_ysff0ckrjKe83I&index=9
NANOPHYSICS
E 304
Dr. MC Ozturk, [email protected]
Electromagnetic Waves

EM waves can travel in vacuum
 Mechanical

waves (e.g. sound) need a medium (e.g. air)
EM Waves travel at the speed of light, c
c
= 299,792,458 m/s
frequency
wavelength
Electromagnetic Spectrum
How EM Waves are made?
1.
2.
3.
4.
5.
6.
Electric field around the electron accelerates
The field nearest to the electron reacts first
Outer field lags behind
Electric field is distorted – bend in the field
The bend moves away from the electron
The bend carries energy
Charges often accelerate and decelerate in an oscillatory manner – sinusoidal waves
Energy is Quantized

Always a positive integer, n = 1, 2, 3, …
E = nhf
Orbital’s
energy
level
Frequency at which
the atom vibrates
Principal
Quantum
Number
Planck’s Constant
6.626 X 10-34 m2/ kgs
Only certain (discrete) energy levels are allowed!
Photons & Electrons


Atoms gain and lose energy as electrons make
transitions between different quantum states
A photon is either absorbed or emitted during these
transitions
n=1
n=2
n=3
A photon is absorbed
for this transition
Bohr’s radius correspond to distance from the nucleus where the probability of finding the electron
is highest in a given orbital.
Hydrogen Atom
En =
mq 4
32p 2e02
( h 2p )
2
1
n2
n = 1, 2, 3, …
As n approaches infinity, energy approaches zero.
E1 = 13.6 eV - Ground energy of the electron in the hydrogen atom
If you provide this much energy to the electron, it can leave the hydrogen atom
Photon & Electrons

The momentum of a photon (or an electron) is given by

This relationship is true for all particles
 Even
large particles…
This equation was postulated for electrons by de Broglie in 1924
Video - Double Slit Experiment


This is the experiment that confirmed the wave
nature of electrons
http://www.youtube.com/watchv=Q1YqgPAtzho&li
st=PLREtcqhPesTcTAI6di_ysff0ckrjKe83I&index=1
Bullets Thru Double-Slit
P12 = P1 + P2
Waves Thru Double-Slit
P12 ≠ P1 + P2
I12 ≠ I1 + I2 + 2sqrt(I1I2) cos(Phi)
Electrons Thru Double-Slit
P12 ≠ P1 + P2
Electrons Thru Double-Slit
Electrons Observed Thru Double-Slit
No device can determine which slit the e- passes thru, w/o changing the interference.
Photon has momentum – after the collision between the photon and the electron, the electron’s
momentum is no longer the same and we do not know what it is althought we know electron’s
location rather precisely.
Heisenberg Uncertainty Principle
“Accepting quantum mechanics means feeling certain that you are uncertain”
…a great statement from your textbook
Video –
Heisenberg’s Uncertainty Principle
http://www.youtube.com/watch?v=Fw6dI7cguCg&list=PLREtcqhPesTcTAI6di_ysf
f0ckrjKe83I&index=3
NANOPHYSICS
E 304
Dr. MC Ozturk, [email protected]
Erwin Schrodinger





1887-1961
Austrian Physicist
Formulated the wave
equation in quantum physics
1933 Nobel Prize
1937 Max Planck Medal
Schrodinger’s Equation
y ( x ) = Wave Function
E = Energy
Schrodinger’s Equation is one of the most important equations in modern physics.
Wave Function – Physical Meaning

A wave function is a complex quantity of the form
y ( x ) = a + ib

where
i = -1
The probability of finding an electron at a given
location is given by
P ( x) = y * y = y

2
This is the ONLY physical meaning attached to the
wave function
Complex Numbers – A Brief Review
y = a + ib
y * = a - ib
ib
a
y * y = ( a + ib) ( a - ib)
= a 2 - iab + iab - i 2 b 2
where i 2 = -1
= a2 + b2
=y
2
= square of the magnitude
Free Particle

A free particle is not bound to anything
 It
can freely move and go anywhere…
 Its energy must be purely kinetic energy

Schrodinger’s Equation

The solution of this equation is…
y ( x ) = Aeikx
A, k = constants
Free Particle – Continued
y ( x ) = Aeikx
dy ( x )
dx
d 2y ( x )
dx 2
= ikAeikx
= i 2 k 2 Aeikx = -k 2 Aeikx
What does this mean?

A free particle has kinetic energy only…
1 2 p2
E = mv =
2
2m

But we found…

This mean, the electron momentum is given by
Infinite Potential Well
∞
∞

A single electron is placed
in an infinite potential well
 The
walls are infinitely high
 The electron is trapped

The probability of finding
the electron outside is…
2
y =0

- L/2
0
+ L/2
Which implies…
L
L
y ( x ) = 0 at - ³ x ³
2
2
Infinite Potential Well – Continued

The solutions are of the form
y ( x ) = Asin kx + Bcoskx A, B = constants

Verify:
dy ( x )
= kA coskx - kBsin kx
dx
d 2y ( x )
dx 2
= -k 2 Asin kx - k 2 B coskx
= -k 2 ( Asin kx + B coskx )
= -k 2y ( x )
Inside the well, the electron’s energy is purely kinetic (the potential is zero)
Infinite Potential Well – Solutions

The solution was
y ( x ) = Asin kx + Bcoskx A, B = constants

Applying the boundary conditions
L
L
y (- L 2) = 0 = -Asin k + B cosk
2
2
L
L
y (+ L 2) = 0 = +Asin k + B cosk
2
2

Adding and subtracting the equations:
L
B cosk = 0 and
2
L
Asin k = 0
2
Infinite Potential Well – Solutions

We must satisfy
Asin

kL
= 0 and
2
We have two options
A = 0 & cos
kL
= 0 or
2
kL
p
= odd multiples of
2
2
i.e. all multiples of
p
2
B cos
kL
=0
2
B = 0 & sin
kL
=0
2
and even multiples of
np
Þ k=
L
p
2
n = 1, 2, 3, 4...
Allowed Momenta
Only discrete momentum values are allowed!
Momentum is quantized…
Allowed Wavelengths
nh
pn =
2L
n = 1, 2, 3, 4,...
h
ln =
pn
2L
ln =
n
Only certain wavelengths are allowed!
Allowed Energy Levels
nh
pn =
2L
n = 1, 2, 3, 4,...
p2 æ h2 ö 2
En =
= çç
÷÷ n
2
2m è 8mL ø
n = 1, 2, 3, 4,...
Only discrete energy levels are allowed!
Energy is quantized…
Particle in a
Well
The result is strikingly similar to
atomic orbitals in atoms
Recall:
For a hydrogen atom,
En = - 13.6 / n2 eV
Finite Potential Well

Vo
Vo


- L/2
0
+ L/2
A particle has a finite
number of allowed
energy levels in the
potential well
A particle with E > Vo
is not bound to the
potential well
A particle with E < Vo
has a finite probability
of escaping the well
Infinite vs. Finite Potential Well
Wave Functions
Vo
The wavefunctions are
decaying exponentially
outside the potential well
There is a finite probability of finding the electron outside the potential well
Particle (e.g. electron) Tunneling

An electron can tunnel
through a potential
barrier even though its
initial kinetic energy is
smaller than the
potential barrier.
The frequency of the wave is related
to the momentum and the kinetic
energy of the particle.
Electron tunneling is an important topic in nanoelectronics