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The Quantized Free Electron Theory Energy E Spatial coordinate x Nucleus with localized core electrons Jellium model: electrons shield potential to a large extent Electron “sees” effective smeared potential

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Electron in a box In one dimension: In three dimensions: where and

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Fixed boundary conditions: x 0 L Periodic boundary conditions: and kxkx “free electron parabola” density of states Remember the concept of # of states in

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1. approach use the technique already applied for phonon density of states where Density of states per unit volume

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1/ Volume occupied by a state in k-space kxkx kyky kzkz Volume( )

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Free electron gas: Independent from and Independent from and 2 Each k-state can be occupied with 2 electrons of spin up/down k2k2 dk

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2. approach calculate the volume in k-space enclosed by the spheres and kxkx kyky # of states between spheres with k and k+dk : with 2 2 spin states

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E D(E) E’E’+dE D(E)dE =# of states in dE / Volume

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Statistics of the electrons (fermions) Fermions are indistinguishable particles which obey the Pauli exclusion principle T=0 E n=1 E n=2 E n=3 E n=4 E n=5 Let us distribute 4 electrons spin E n=6 Occupation number 0 for state Occupation number 1 for states of a given spin E f(E,T=0) Probability that a qm state is occupied EFEF 1 x

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Fermi Dirac distribution function at T>0 With accuracy sufficient for many estimations: f(E,T) linearized at E F here chemical potential Fermi energy

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More detailed approach to Fermi statistics The grand canonical ensemble Heat Reservoir R T=const. Particle reservoir System Average energy Average particle # Normalized probabilities Now we consider independent particles Total energy of N fermion system occupation # n i =0,1 of single particle state i with energy i where

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average occupation of state j is given by Chemical potential For details see& additional info see where the summation means Repeat this step

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The Fermi gas at T=0 E f(E,T=0) EFEF 1 E D(E) EF0EF0 Electron density #of states in [E,E+dE]/volume Fermi energy depends on T Probability that state is occupied T=0

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Energy of the electron T=0: there is an average energy of per electron without thermal stimulation with electron densitywe obtain Click for a table of Fermi energies, Fermi temperatures and Fermi velocities

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Specific Heat of a Degenerate Electron Gas here: strong deviation from classical value only a few electrons in the vicinity of E F can be scattered by thermal energy into free states Specific heat much smaller than expected from classical consideration D(E) Density of occupied states E EFEF energy of electron state #states in [E,E+dE] probability of occupation, average occupation # 2k B T Before we calculate U let us estimate: These # of electrons increase energy from to

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π2π2 3 subsequent more precise calculation Calculation of C el from Trick: Significant contributions only in the vicinity of E F

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with and E D(E) EFEF decreases rapidly to zero for

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withand in comparison with for relevant temperatures W.H. Lien and N.E. Phillips, Phys. Rev. 133, A1370 (1964) Heat capacity of a metal: electronic contribution lattice T<<Ө D

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Selected phenomena which don’t require detailed knowledge of the band structure Temperature dependence of the electrical resistance T Scattering of electrons: deviations from a perfect periodic potential Impurities: temperature independent imperfection scattering phonon scattering Matthiessen’s rule:

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Simple approach to understand for T>>Ө D Remember Drude expression: scattering rate #of scattering centers/volume scattering cross section Fermi velocity of electrons: compare lecture notes: Thermal Properties of Crystal Lattices

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with and Let us consider the high temperature limit: T>>Ө D Note: T 5 –low temperature dependence not described by this simple approach Lindemann melting temperature T M : where emperical value average atomic spacing

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Thermionic Emission Finite barrier height of the potential E x EFEF work function E vac Current density for homogeneous velocity generalized Current density for k-dependent velocity

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Again: Occupied and Spin degeneracy Since Fermi distribution approximated by Maxwell Boltzmann distribution approximated

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Let us investigate the integral Remember integrals of the type: Richardson-Dushman

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Richardson Constant Typical value of Tungsten: Nobel prizeNobel prize in 1928 "for his work on the thermionic phenomenon and especially for the discovery of the law named after him". Owen Willans Richardson Universal constant: A=1.2 X 10 6 A/m 2 K Reflection at the potential step A (1-r)=0.72 X 10 6 A/m 2 K Vacuum tube

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Field-Aided Emission EFEF E vac E x Image potential Electric field in very high electric fields of tunneling through thin barrier cold emission

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