# NANOPHYSICS Dr. MC Ozturk, 304 3.1.

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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 = 95 o F, 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 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. 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. Electrons emitted by the cathode are repelled by the negatively charged anode. The photocurrent decreases.

Photoelectric Effect The Experiment - 3 Voltage Current Increasing Light Intensity Regardless of the light intensity, the photocurrent becomes zero at V = - V o At this voltage, every emitted electron is repelled Therefore, qV o must be the maximum kinetic energy of the electron This energy is independent of the light intensity Regardless of the light intensity, the photocurrent becomes zero at V = - V o At this voltage, every emitted electron is repelled Therefore, qV o must be the maximum kinetic energy of the electron This energy is independent of the light intensity VoVo

Photoelectric Effect The Experiment - 4 fofo 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 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 Frequency

Video: A Brief History of Quantum Mechanics  http://www.youtube.com/watchv=B7pACq_xWyw&list=PLXD9X52rMwwN J85QDzv3G6cqbQ2ZVlsR8&index=2 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 Dr. MC Ozturk, mco@ncsu.eduE 304 3.2

Hydrogen Atom

Orbital three dimensional space around the nucleus of all the places we are likely to find an electron. 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 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 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 = 0l = 1l = 2l = 3 n = 1  n = 2  n = 3  n = 4 

Magnetic Quantum Number l = 0l = 1l = 2l = 3 n = 1m=0 n = 20-1, 0,+1 n = 30-1, 0,+1-2, -1, 0,+1,+2 n = 40-1, 0,+1-2, -1, 0,+1,+2-3,-2, -1, 0,+1,+2,+3 spdf 1 orbital3 orbitals5 orbitals7 orbitals

Spin Quantum Number l = 0l = 1l = 2l = 3 n = 1m=0 n = 20-1, 0,+1 n = 30-1, 0,+1-2, -1, 0,+1,+2 n = 40-1, 0,+1-2, -1, 0,+1,+2-3,-2, -1, 0,+1,+2,+3 spdf 1 orbital 2 states 3 orbitals 6 states 5 orbitals 10 states 7 orbitals 14 states (only two possibilities)

Levels, Sublevels of Atomic Orbitals  http://en.wikipedia.org/wiki/Atomic_orbital 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 1s 2 2s 2 2p 6 3s 2 3p 2 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 1s 2 2s 2 2p 6 3s 2 3p 6 3d 2 4s 2 Empty States Occupied States

Atomic Orbitals Hydrogen Larger Atoms

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

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 behind 4.Electric field is distorted – bend in the field 5.The bend moves away from the electron 6.The bend carries energy 1.Electric field around the electron accelerates 2.The field nearest to the electron reacts first 3.Outer field lags behind 4.Electric field is distorted – bend in the field 5.The bend moves away from the electron 6.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 Principal Quantum Number Planck’s Constant 6.626 X 10 -34 m 2 / kg- s Planck’s Constant 6.626 X 10 -34 m 2 / 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. E 1 = 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 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 I 12 ≠ I 1 + I 2 + 2sqrt(I 1 I 2 ) 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 “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 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 ibib

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, E n = - 13.6 / n 2 eV

Finite Potential Well  A particle has a finite number of allowed energy levels in the potential well  A particle with E > V o is not bound to the potential well  A particle with E < V o has a finite probability of escaping the well - L/2+ L/20 VoVo VoVo

Infinite vs. Finite Potential Well Wave Functions VoVo 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.