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Graduate Lecture Series 29 June – 3 July, 2009 Prof Ngee-Pong Chang Lecture 2 Fermi Gas

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Enrico Fermi 1901 - 1954 Paul Dirac 1902 - 1984

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Ionisation Energy of Sodium 5.14 eV Band Theory of Metals Start with isolated Sodium Atom

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Ionisation Energy of Sodium 5.14 eV Band Theory of Metals

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Splitting of 3s level with 2 Sodium atoms Bring two Sodium Atoms together

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Band Theory of Metals Splitting of 3s level with 6 Sodium atoms Bring six Sodium Atoms together

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Band Theory of Metals Splitting of 3s levels with Sodium atoms in crystalline solid

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Band Theory of Metals

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Fermi-Dirac Distribution electrons holes

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Probability of Occupancy 1.0 0.0 T=0 T > 0

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Probability of Occupancy

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Energy in units Filling up the Fermi Sea In One Dimensional Box 10 20 30 ² F

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1-Dimensional Box

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1-Dimensional Fermi Gas Sum Over Spins

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1-Dimensional Fermi Gas Single Spin Orientation Fermi Surface dependence on number of electrons

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2-Dimensional Fermi Gas Single Spin Orientation Fermi Surface dependence on number of electrons

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3-Dimensional Fermi Gas Single Spin Orientation Fermi Surface dependence on number of electrons

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1-Dimensional Fermi Gas Single Spin Orientation Total Energy

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2-Dimensional Fermi Gas Single Spin Orientation Total Energy

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3-Dimensional Fermi Gas Single Spin Orientation Total Energy

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3-Dimensional Density of States

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2-Dimensional Density of States

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http://upload.wikimedia.org/wikipedia/en/c/c5/DOS_multdim.jpg

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3-Dimensional Fermi Gas Density of States

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2-Dimensional Fermi Gas Density of States

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1-Dimensional Fermi Gas Density of States

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Conduction Band Valence Band

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Woodward Yang harvard lecture notes

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CBO =conduction band offset

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2-Dimensional Density of States

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Quantum Wire

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http://boulder.research.yale.edu/Boulder- 2005/Lectures/Matveev/Boulder%20lecture.pdf

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A graphene nanoribbon field-effect transistor (GNRFET). Here contacts A and B are at two different Fermi levels E F1 and. E F2Fermi levels

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Ballistic Conductor

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i i’ t’ t Landauer formula 1927 - 1999 Conductance

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µ Gas Pressure on the Wall A cos θ v δ t A θ

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Pressure due to Non-Relativistic Degenerate Fermi Gas Equation of state for Fermi Gas since at T = 0 We have for a metal

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White dwarf as seen by Hubble Space Telescope

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What's Inside a White Dwarf? To say that white dwarfs are strange is an understatement. An earth-sized white dwarf has a density ofdensity 1 x 10 9 kg/m 3. In comparison, the earth itself has an average density of only 5.4 x 10 3 kg/m 3. That means a white dwarf is 200,000 times as dense!

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