John Carroll 1
The Heisenberg uncertainty principle. Some quick particle physics. Beta decay example. The Pauli exclusion principle. The existence of energy bands. Classical conductance. Why it is inaccurate. Quantum conductance. Superconductors Effects of quantum conductance. 2
The uncertainty principle state the position and momentum as well as energy and time cannot be measured with absolute certainty. It further states that as the probability distribution of one begins to narrow the other will in turn broaden. Werner Heisenberg 3
Δx Δp Δp ħ 2 Δt ΔE Δt ħ 2 Where ħ = h/2π and Planck s constant h=6.626 J s 4
What particles make up protons and neutrons? Quarks Up Down Top Bottom Charm Strange The universe is governed by what forces? Strong nuclear Electromagnetic Weak nuclear Gravitation 5
The standard model successfully predicted the existence of several particles before they were observed. The model also predicts the existence of the graviton and the Higg s boson. 6
Proton Neutron Mass = 938MeV/c 2 Mass = 940MeV/c 2 Mass of quarks = 9.6MeV/c 2 Mass of quarks = 12MeV/c 2 7
940MeV/c 2 80,400MeV/c 2 938MeV/c 2 A difference of 79,460MeV/c 2 ħ/2 = 3.29 10 16 ev s ΔE Δt 3.29 10 16 ev s 80.4 10 9 ev/c 2 Δt 3.29 10 16 ev s The W boson exists for 4.10 10 27 s. If the particle traveled at the speed of light it could travel 1.2 10 18 m. That is 1/1000 th the diameter of a proton. 8
Quantum numbers are used to describe the state of a particle on the quantum level. They are most often used to describe the electrons in an atom or molecule. Quantum Number Description The energy shell is described by the value n. The sub shell is denoted by l. Each particle has an angular momentum as denoted by m l. The spin of the particle is described by the value of m s. Wolfgang Pauli 9
The principle states that no two identical fermions in a quantum system can occupy the same quantum state at the exact same time. This means that two atoms within the same system (molecule, crystalline lattice, etc.) cannot have the electrons with identical energies. The electrons of identical atoms in the same system will take on slightly different energies to satisfy this principle. This is how energy bands are formed. 10
ΔE 11
Metals are good electrical conductors and must have free electrons. The free electrons act like a gas and move in all directions though the lattice. Their average velocity is zero but their speed is very high. The electrons collide with ions in the lattice. The motion of the electrons gives rise to Ohm s law. 12
The random motion of the electrons can be effected by an external electric field. The force is proportional to the product of e, the charge of an electron and ε, the electric field. Under this force the electrons accelerate forward in accordance with Newton s second law. Electrons will proceed until they collide with lattice ions transferring kinetic energy to them creating heat. They will accelerate again. This cycle is called the drift velocity. 13
The math developed from the classical assumptions does not match the experimental results. The classical model treats the electrons as particles only, where as they are shown the act as both a particle and wave. This model also predicts that particles at absolute zero will have zero velocity. It also does not take into account the principles of quantum mechanics that have been shown to be experimental true (Heisenberg, Pauli Principle). 14
By definition the conductance is defined as: G = I ΔV Current per potential difference 15
Current is defined as: I = nq Δt Where n is the number of electrons and q is the charge of an electron Potential difference is defined as: ΔV = ΔU q Where n is the number of electrons and U is the electrostatic charge 16
G = nq I 2 ΔV ħ 2 I = nq Δt G ΔV = ΔU q 2nq 2 ħ nq R Δt = 1 G Rnq 2 1027.3 Ω Uncertainty Principle ΔU ΔUΔt R 12.9 kωq when the equation ΔU is ΔE Δt a measure h is used for the uncertainty of energy principle. 17
In the quantum system electrons propagate as waves. The electrons get energy from collisions with ions. The ions have an energy of kt and only the electrons that are within kt of the Fermi level can be promoted to the conduction band. The exclusion principle dictates that there will only be a few electrons at that level. Thermal vibrations cause scattering of the electrons. 18
What about superconductors? 19
If there is a minimum resistance for a conductor than how do superconductors break the rule? BCS theory says that a pair of electrons condensate into a boson like state. The exclusion principle applies to the fermions. Bosons are not subject to the same rules as fermions. 20
Consider a clock speed of 3GHz. This would require as many as 3 10 9 electrons to pass though a single molecule transistor. The current required to support this clock speed would be focused onto a very small area which could exceed the energy of the molecular bonds. 1A = 1C/1s V=IR What we need V=1J/1C R=1027.3Ω 1 amp = 6.242 10 18 e /s e = 1.602 10 19 C 21
3 10 9 e pass per second 3 10 9 e 6.242 10 18 e /A 4.8062 10 10 A V = 1027.3Ω 4.8062 10 10 A 4.9374 10 7 V 1.602 10 19 C/e 3 10 9 e 4.806 10 10 C V = J C J = V C 4.9374 10 7 V 4.806 10 10 C 2.3729 10 16 J 22
Carbon Carbon bond strength = 606.68 kj/mol 606.68 kj/mol 1.0074 10 18 J/bond 6.022 10 23 bonds/mol Recall that the electrons have an energy of 2.3729 10 16 J The electrons have enough energy at a 3 GHz switching speed to break the carbon carbon bond 23
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Metallic Nano Scale Conductors Nebergall (2006) Charge Deficiency, Charge Transport and Comparison of Dimensions Avron, Seiler, Simon (2009) International Technology Roadmap for Semi conductors 2007 Edition Introduction to Quantum Mechanics 2 nd Edition Griffiths (2005) Modern Physics 2 nd Edition Krane (1996) Physical Chemistry 7 th Edition Atkins, de Paula (2002) www.en.wikipedia.org www.images.google.com 25
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