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### The Applications of Nano Materials

Department of Chemical and Materials Engineering

San Jose State University

Zhen Guo, Ph. D.

- Nano Material Science
- Session II: Atomic Structure/Quantum Mechanics
- Session III: Bonding / Band Structures
- Session IV: Computational Nano Materials Science
- Session V: Surface / Interface Properties

and Atomic Structure

History of Quantum Mechanics

- Physics of 19th Century:
- The framework of physics has been all build up, all left is remodeling work.
- Achievement of classic physics: Finding of all planets in solar system based on classic mechanics; thermodynamics; Maxwell\'s electro magnetic theory
- Lap Lace has claim that if w knew all laws and principles of physics and have all mathematic skills, we can calculate what happened tomorrow without crystal ball.
- Two unsolved problems -- heat capacitance and aether. (Shadows)
- Quantum mechanics is one of the two most important founding stone for modern physics (another is relativity theory)
- Two key features for Quantum mechanics -- Uncertainty principle and duality nature
- Scientifically, it discovered atoms, electrons, nucleus, protons and all other basic particles. It also, for first time, discovered the close relationship between physics and chemistry and understood the principle of periodical table.
- Practically, this is the principle behind transistor, computers and every single electric device and consumer applicants.

Photoelectric Effect

- Critical Frequency u0
- Positive voltage leads to saturation current which is depending upon Intensity
- Stopping Voltage at negative –V0 =>KE=hu-hu0

Electrons – Wave Particle Duality

- Electrons are particles.

-- Charge, Mass, Velocity, Energy

-- Follow Newton’s classic mechanics and Maxwell’s eletro-magnetic theory.

- Electrons also have wave character.

-- Diffraction: Young’s Double split experiment

-- De Broglie Relations: =h/P

Wave Characters of Electrons

- Young’s Double Split

The infringe pattern follows Bragg’s law if electron’s wave length obey De Broglie relations

- Schrödinger Equation

Note: Wave Function itself does not have physical meaning while 2 has (probability of finding electron per unit volume)

Fourier Transformation and Wave Number

- Any wave function can be expressed by Fourier transformation:

where k is the wave number per unit length or k=2p/l

- According to De Broglie Relations:

Schrödinger Equation

- Consider P as an operator:

- Total Energy is sum of kinetic energy and potential energy

1-D Schrödinger Equation

3-D Schrödinger Equation

Example I – 1-D Free Electrons

- 1-D Free Electrons: V=0

Free Electrons: Total Energy is equal to kinetic energy (V=0)

No energy quantizing needed. Electrons can occupy any energy

Example II -- Electrons in 1-D well

Inside Well (0<x<a), V=0

Outside Well (x<0 or x>a),

V=∞=>Y=0

Solutions:

a=5cm, DE=4.53X10-16ev, no Quantum effect;

a=0.5nm, DE=4.53ev, (274nm light) Quantum Well

Implications on Quantum Wells

- Light Absorption and Emission
- Absorption hu= DE=Em-En,

-- Incoming photons absorpted and excited electron from lower quantum state to higher state. Has to be exact wavelength / frequency.

- Emission hu= DE=Em-En,

-- Electron jump back from higher quantum state to lower one. Photons emitted are exact wavelength.

- The light absorbed and emitted is also a function of quantum well size “a”.

Example III -- Hydrogen Atoms

Coulomb Potential Energy

Principle Quantum Number: n=1, 2, 3, 4..... (or n=K, L, M, N...)

Orbital Angular Momentum Quantum Number l=0, 1, 2, ..(n-1), (or l=S, P, D, F...)

Magnetic Quantum Number ml=-l, -(l-1)...0, ...(l-1), l or |ml|<=l

Spin Angular Momentum Quantum Numberms=+1/2, -1/2 or |ms|=1/2

n=1, l=0, ml=0, => 1S state

n=2, l=0, ml=0, => 2S state

l=1, ml=-1, 0, 1 => 2Px, 2Py, 2Pz state.

Pauli Exclusion Principlesand Periodical Table

- Pauli Exclusion principle -- No two electrons within any given system may have all four identical quantum numbers (n, l, ml, ms) Each orbital motion is determined by n, l, ml, so every orbital state can contain a spin paired electron.
- Hund\'s rule: Electron in the same n, l orbital prefer their spins to be parallel.

-- Getting same ms number will allow electron take different ml, and thus different orbital (space) which can increase R12 and decrease Coulomb repulsive energy

Atomic Structure – Bohr’s Model

- Mass and positive charge at nucleus with protons and neutrons.
- Negative charge -- electron occupying several shell orbits, starting from lowest energy (inner shell) to outer shell.
- Full Orbit is stable as inert gas. Electron on un-fulfilled orbit is called valence electron with higher energy
- Useful Link for animation: http://www.colorado.edu/physics/2000/applets/schroedinger.html

Heisenberg Uncertainty Principle

- One of the corner stones and also odd aspect for Quantum Mechanics and direct results from wave-particle duality
- Classically, that is, in our macroscopic world, we can measure the position and momentum of the object to infinite precision (more or less) as they are two independent characters of the particles. There is really no question about a particle\'s position and momentum.

vx or px

wave-particle duality

Easy to define wave length or p but not for x

Classic Particles

x

In Quantum mechanics, k (and p) and x are reciprocally relatedUncertainty Principle (I)

Y(x) in X Space

g(K) in k Space (Reciprocal Space)

y(x)=eik0x, spread in x space

g(k)=k0 is a definite value

y(x)=x0, a delta equation

localized in the space

g(k)= e-ix0k, spread in k space

x

0

-a/2

a/2

g(k)

k

0

4p/a

Uncertainty Principles (II)General property of functions that are Fourier transformation of each other.

- It is impossible to make both Dx and Dk small
- General Features for wave packets

Uncertainty Principles

- In the Quantum Mechanical world, the idea that we can locate objects exactly breaks down. Suppose a particle has momemtum p and position x. In a Quantum Mechanical world, we would not be able to measure p and x precisely. There would be an uncertainty associated with each measurement that we could never get rid of, even in a perfect experiment!The size of the uncertainties are not independent; they are related as

Dp x Dx > h / (2 p) = Planck\'s constant / (2 p)

- The preceding is a statement of the Heisenberg Uncertainty Principle. A consequence of the Uncertainty Principle is that if an object\'s position x is defined precisely then the momentum of the object will be only weakly constrained, and vice versa. One cannot simultaneously find both the position and momentum of an object to arbitrary accuracy.

History and Argument

- Due to uncertainty principle, we can only describe a probability to find a particle at x
- It is then our measurement to determine the experiment results. (Collapse a combined wave function to one state)
- Quantum Mechanics is thus a probilistic science, rather than deterministic world.
- Einstein has a big disagreement on this principles – God is not gambling

Implication to Nano Materials

- For nano materials, the size has been limited to a small dimension.
- So the momentum variation is very big.
- Consequence is the tunneling and Coulombs Blockade which we will discuss them later
- This directly lead to physical limit of transistor scaling => 16nm node with 5nm gate length is the brick wall for Moore’s Law

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