Lect20 - Lecture 20: Consequences of Quantum Mechanics:...

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Lecture 20, p 1 Lecture 20: Consequences of Quantum Mechanics: Effects on our everyday lives
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Lecture 20, p 2 Today Electron energy bands in Solids States in atoms with many electrons – filled according to the Pauli exclusion principle Why do some solids conduct – others do not – others are intermediate Metals, Insulators and Semiconductors Understood in terms of Energy Bands and the Exclusion Principle Solid-state semiconductor devices The electronic states in semiconductors Transistors, . . . Superconductivity Electrical conduction with zero resistance! All the electrons in a metal cooperate to form a single quantum state
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Lecture 20, p 3 Electron states in a crystal (1) Again start with a simple atomic state: +e r n = 1 ψ A Bring N atoms together to form a 1-d crystal (a periodic lattice). N atomic states N crystal states. What do these crystal states look like? Like molecular bonding, the total wave function is (approximately) just a superposition of 1-atom orbitals.
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Lecture 20, p 4
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Lecture 20, p 5 Electron states in a crystal (2) The lowest energy combination is just the sum of the atomic states. This is a generalization of the 2-atom bonding state. The highest energy state is the one where every adjacent pair of atoms has a minus sign: There are N states, with energies lying between these extremes. No nodes! Lowest energy. N-1 nodes! Highest energy.
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Lecture 20, p 6 FYI: These states are called “Bloch states” after Felix Bloch who derived the mathematical form in 1929. They can be written as: where u is an atomic-like function and the exponential is a convenient way to represent both sin and cos functions Energy Band Wave Functions ψ 3 ψ 1 ψ 2 ψ 4 ψ 5 ψ 6 Energy Lowest energy wave function Highest energy wavefunction Closely spaced energy levels form a “band” – a continuum of energies between the max and min energies Example with six atoms six crystal wave functions in each band. u(x)e (x) x ik n n = ψ
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Lecture 20, p 7 Energy Bands and Band Gaps In a crystal the number of atoms is very large and the states approach a continuum of energies between the lowest and highest b a “band” of energies. A band has exactly enough states to hold 2 electrons per atom (spin up and spin down). Each 1-atom state leads to an energy band. A crystal has multiple energy bands. Band gaps (regions of disallowed energies) lie between the bands. 1s band 2s band Band gap
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Lecture 20, p 8
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Electron in a crystal – a periodic array of atoms Bands and Band Gaps Occur for all Types of Waves in Periodic Systems Energy Bands Band Gap: no states Light propagating through a periodic set of layers with different index of refraction – an interference filter Photon Transmitted? Reflected?
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This note was uploaded on 01/19/2012 for the course PHYS 214 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.

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Lect20 - Lecture 20: Consequences of Quantum Mechanics:...

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