6. Free Electron Fermi Gas Energy Levels in One Dimension Effect of Temperature on the Fermi-Dirac Distribution Free Electron Gas in Three Dimensions Heat Capacity of the Electron Gas Electrical Conductivity and Ohm’s Law Motion in Magnetic Fields Thermal Conductivity of Metals Nanostructures

Introduction Free electron model: Works best for alkali metals (Group I: Li, Na, K, Cs, Rb) Na: ionic radius ~.98A, n.n. dist ~ 1.83A. Successes of classical model: Ohm’s law. σ / κ Failures of classical model: Heat capacity. Magnetic susceptibility. Mean free path. Quantum model ~ Drude model

Energy Levels in One Dimension Orbital: solution of a 1-e Schrodinger equation Boundary conditions:Particle in a box

Pauli-exclusion principle: No two electrons can occupy the same quantum state. Quantum numbers for free electrons: (n, m s ) Degeneracy: number of orbitals having the same energy. Fermi energy ε F = energy of topmost filled orbital when system is in ground state. N free electrons:

Effect of Temperature on the Fermi-Dirac Distribution Fermi-Dirac distribution : Chemical potential μ = μ(T) is determined by g = density of states At T = 0: → For all T : For ε >> μ : (Boltzmann distribution) 3D e-gas

Free Electron Gas in Three Dimensions Particle in a box (fixed) boundary conditions: Periodic boundary conditions: Standing waves → → Traveling waves

→ ψ k is a momentum eigenstate with eigenvalue  k. N free electrons:

Density of states: →

Heat Capacity of the Electron Gas (Classical) partition theorem: kinetic energy per particle = (3/2) k B T. N free electrons: ( 2 orders of magnitude too large at room temp) Pauli exclusion principle → T F ~ 10 4 K for metal free electrons Using the Sommerfeld expansion formula

→ 3-D e-gas for 3-D e-gas

for 1-D e-gas

Experimental Heat Capacity of Metals For T <<  and T << T F : el + ph Deviation from e-gas value is described by m th :

Possible causes: e-ph interaction e-e interaction Heavy fermion: m th ~ 1000 m UBe 3, CeAl 3, CeCu 2 Si 2.

Electrical Conductivity and Ohm’s Law Lorentz force on free electron: No collision: Collision time  : Ohm’s law Heisenberg picture: Free particle in constant E field

Experimental Electrical Resistivity of Metals Dominant mechanisms high T: e-ph collision. low T:e-impurity collision. phononimpurity Matthiessen’s rule: Sample dependent Sample independent Residual resistivity: Resistivity ratio:  imp ~ 1  ohm-cm per atomic percent of impurity K  imp indep of T ( collision freq additive )

Consider Cu with resistivity ratio of 1000: Impurity concentration:= 17 ppm Very pure Cu sample:  For T >  : See App.J From Table 3, we have   imp ~ 1  ohm-cm per atomic percent of impurity

Umklapp Scattering Normal: Umklapp: Large scattering angle ( ~  ) possible Number of phonon available for U-process  exp(  U /T ) For Fermi sphere completely inside BZ, U-processes are possible only for q > q 0 q 0 = k F for 1e /atom Fermi sphere inside a bcc BZ. For K,  U = 23K,  = 91K  U-process negligible for T < 2K

Motion in Magnetic Fields Equation of motion with relaxation time  : be a right-handed orthogonal basis Let Steady state: = cyclotron frequency q = –e for electrons

Hall Effect →  Hall coefficient: electrons

Thermal Conductivity of Metals From Chap 5: Fermi gas: In pure metal, K el >> K ph for all T. Wiedemann-Franz Law: Lorenz number: for free electrons