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ME 381R Lecture 11 & 12: Free Electron Gas, Electronic Specific Heat & Thermal Conductivity

ME 381R Lecture 11 & 12: Free Electron Gas, Electronic Specific Heat & Thermal Conductivity. Dr. Li Shi Department of Mechanical Engineering The University of Texas at Austin Austin, TX 78712 www.me.utexas.edu/~lishi lishi@mail.utexas.edu. Reading: 1-4 in Tien et al

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ME 381R Lecture 11 & 12: Free Electron Gas, Electronic Specific Heat & Thermal Conductivity

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  1. ME 381R Lecture 11 & 12: Free Electron Gas, Electronic Specific Heat & Thermal Conductivity Dr. Li Shi Department of Mechanical Engineering The University of Texas at Austin Austin, TX 78712 www.me.utexas.edu/~lishi lishi@mail.utexas.edu • Reading: 1-4 in Tien et al • References: Ch6 in Kittel

  2. Free electron band Free Electron Theory for Metals • Origin of Energy Bands Na: 1s22s22p63s1 Pauli’s Exclusion Principle

  3. Probability of finding an electron at position x  Wave-Particle Duality of Electrons The double slit experiment Schrodinger eqn. for free electrons Electrons : electron wave function : Planck’s constantm: electron mass E: electron energy Traveling wave solution to Schrodinger’s eqn: k: wave vector = 2p/l

  4. k = 2np/L; n = ±1, ± 2, ± 3, ± 4, ….. (x+L) = (x) 1-D k-space 0 -6p/L -4p/L 4p/L 2p/L -2p/L Electronic States A metal ring with perimeter L Metal x Energy Born-vonKarman cyclic boundary condition L (x+L) = (x)

  5. Fermi Wavevector 2-D projection of 3-D k-space ky dk • Each state can hold 2 electrons • of opposite spin (Pauli’s principle) • To hold N electrons k kx 2p/L kF: Fermi wavevector he: electron number density

  6. Fermi Parameters for Free Electron Metals Vacuum Level F: Work Function Fermi Energy, EF Energy Fermi Energy Fermi Velocity: Fermi Temp. Band Edge

  7. Effect of Temperature Fermi-Dirac equilibrium distribution for the probability of electron occupation of energy level E at temperature T

  8. Number and Energy Densities Summation over k-states Integration over k-states Transformation from k to E variable Integration of E-levels for number and energy densities Number of k-states available between energy E and E+dE Density of States

  9. e In Bulk Solids Increasing Defect Concentration Defect Scattering PhononScattering Temperature, T Electronic Specific Heat and Thermal Conductivity Specific Heat Thermal Conductivity • Electron Scattering Mechanisms • Defect Scattering • Phonon Scattering • Boundary Scattering (Film Thickness, • Grain Boundary)

  10. Thermal Conductivity of Cu and Al Matthiessen Rule:

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