NANOPHYSICS Dr. MC Ozturk, mco@ncsu.eduE 304 3.1

NANOPHYSICS

Dr. MC Ozturk, mco@ncsu.eduE 304

3.1

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: electric field magnetic field

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

Thermal Radiation

For human skin, T = 95oF, wavelength is 9.4 micrometer (infrared)

All hot bodies emit thermal radiation

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 EffectThe Experiment - 1

Light Sourceanode

cathode

Electrons emitted by the cathode are attracted to the positively charged anode.

A photocurrent begins to flow in the loop.

Photoelectric EffectThe Experiment - 2

Light Sourceanode

cathode

Electrons emitted by the cathode are repelled by the negatively charged anode.

The photocurrent decreases.

Photoelectric EffectThe Experiment - 3

Voltage

Current

IncreasingLight

Intensity

Regardless of the light intensity, the photocurrent becomes zero at V = - Vo

At this voltage, every emitted electron is repelledTherefore, qVo must be the maximum kinetic energy of the electron

This energy is independent of the light intensity

Vo

Photoelectric EffectThe Experiment - 4

fo

The maximum kinetic energy of electrons is determined by the frequency of light

The slope of this line is Planck’s constantIncreasing the light intensity only increases the number of photons

hitting the cathode

Frequency

Video: A Brief History of Quantum Mechanics http://www.youtube.com/watchv

=B7pACq_xWyw&list=PLXD9X52rMwwNJ85QDzv3G6cqbQ2ZVlsR8&index=2

Video:Max Planck & Quantum Physics https://www.youtube.com/watch?

v=2UkO_3NC3F4

NANOPHYSICS

Dr. MC Ozturk, mco@ncsu.eduE 304

3.2

Hydrogen Atom

Hydrogen Atom

Orbitalthree 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 the electron energy

Azimuthal Quantum Number, l = 0, 1, 2, … Determines the electron’s angular

momentum Magnetic Quantum Number, m = 0, ± 1, ± 2,

… 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

Each n yields n – 1 sub levels

l = 0 l = 1 l = 2 l = 3

n = 1 a

n = 2 a a

n = 3 a a a

n = 4 a a a a

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

1 orbital 3 orbitals 5 orbitals 7 orbitals

Spin 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

1 orbital2 states

3 orbitals6 states

5 orbitals10 states

7 orbitals14 states

(only two possibilities)

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

1 s

2 s

2 p

3 s

3 p

4 s

3 d

Silicon has 14 electrons

1s22s22p63s23p2

Empty States

Occupied States

Example 2: Titanium Atom

1 s

2 s

2 p

3 s

3 p

4 s

3 d

Titanium has 22 electrons

1s22s22p63s23p63d24s2

Empty States

Occupied States

Atomic Orbitals

Hydrogen

Larger Atoms

s (l=0) p (l = 1) d (l = 2) f (l = 3)

Video - Orbitals

http://www.youtube.com/watchv=drCg4ruJCfA&list=PLREtcqhPesTcTAI6di_ysff0ckrjKe83I&index=9

NANOPHYSICS

Dr. MC Ozturk, mco@ncsu.eduE 304

3.3

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. Electric field around the electron accelerates

2. The field nearest to the electron reacts first

3. Outer field lags behind4. Electric field is distorted – bend in

the field5. The bend moves away from the

electron6. 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 = nhfOrbital’s energy level Principa

l Quantu

m Numbe

r

Planck’s Constant6.626 X 10-34 m2/ kg-s

Frequency at which the atom vibrates

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

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=Q1Yqg

PAtzho&list=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_ysff0ckrjKe83I&index=3

NANOPHYSICS

Dr. MC Ozturk, mco@ncsu.eduE 304

3.4

Erwin Schrodinger

1887-1961 Austrian Physicist Formulated the wave

equation in quantum physics

1933 Nobel Prize 1937 Max Planck

Medal

Schrodinger’s Equation

Schrodinger’s Equation is one of the most important equations in modern physics.

E = Energy

Wave Function – Physical Meaning A wave function is a complex quantity of

the form

The probability of finding an electron at a given location is given by

This is the ONLY physical meaning attached to the wave function

where

Complex Numbers – A Brief Review

a

ib

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…

Free Particle – Continued

What does this mean?

A free particle has kinetic energy only…

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…

Which implies…

∞∞

- L/2 + L/20

Infinite Potential Well – Continued The solutions are of the form

Verify:

Inside the well, the electron’s energy is purely kinetic (the potential is zero)

Infinite Potential Well – Solutions The solution was

Applying the boundary conditions

Adding and subtracting the equations:

Infinite Potential Well – Solutions We must satisfy

We have two options

Allowed Momenta

Only discrete momentum values are allowed!

Momentum is quantized…

Allowed Wavelengths

Only certain wavelengths are allowed!

Allowed Energy Levels

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

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

- L/2 + L/20

Vo Vo

Infinite vs. Finite Potential WellWave 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.

Electron tunneling is an important topic in nanoelectronics

The frequency of the wave is related to the momentum and the kinetic energy of the particle.