Free electron Fermi gas ( Sommerfeld, 1928 ) M.C. Chang Dept of Phys counting of states Fermi energy, Fermi surface thermal property: specific heat transport property electrical conductivity, Hall effect thermal conductivity In the free electron model, there is neither lattice, nor electron-electron interaction, but it gives good result on electron specific heat, electric and thermal conductivities… etc. Free electron model is most accurate for alkali metals. 1926: Schrodinger eq., FD statistics

… L. Hoddeson et al, Out of the crystal maze, p.104

Plane wave solution “Box” BC k = π/L, 2π/L, 3π/L… Periodic BC (PBC) k = ±2π/L, ±4π/L, ±6π/L… Quantization of k in a 1-dim box

Free electron in a 3-dim box x y z LxLx LyLy LzLz Advantage: allows travelling waves

Quantization of k in a 3-dim box box BCperiodic BC Different BCs give the same Fermi wave vector and the same energy box BC periodic BC Each point can have 2 electrons (because of spin). After filling in N electrons, the result is a spherical sea of electrons called the Fermi sphere. Its radius is called the Fermi wave vector, and the energy of the outermost electron is called the Fermi energy.

Connection between electron density and Fermi energy F For K, the electron density n=1.4×10 28 m -3, therefore ε F is of the order of the atomic energy levels. k F is of the order of a -1.

The Fermi temperature is of the order of 10 4 K Fermi temperature and Fermi velocity

For a 3D Fermi sphere, Density of states D(ε) ( DOS, 態密度 ) D(ε)dε is the number of states within the energy surfaces of ε and ε+dε important

Free electron DOS (per volume) in 1D, 2D, and 3D Multiple bound states in 2D z ⊥

counting of states Fermi energy, Fermi surface thermal property: specific heat transport property electrical conductivity, Hall effect thermal conductivity

tourists Thermal distribution of electrons (fermions) Combine DOS D(E) and thermal dist f(E,T) Hotel rooms money D(  ) important

Electronic specific heat, heuristic argument (see Kittel p.142 for details) The energy absorbed by the electrons is U(T)-U(0) ～ N A (kT) 2 /E F specific heat C e ～ dU/dT = 2R kT/E F = 2R T/T F a factor of T/T F smaller than classical result T/T F ～ 0.01 Therefore usually electron specific is much smaller than phonon specific heat In general C = C e + C p = γT + AT 3 Only the electrons near the Fermi surface are excited by thermal energy kT. The number of excited electrons are roughly of the order of N’ = N（kT /E F ） C e is important only at very low T.

counting of states Fermi energy, Fermi surface thermal property: specific heat transport property electrical conductivity, Hall effect thermal conductivity

Electrical transport Classical view If these two types of scatterings are not related, then scattering rate: Current density (n is electron density) Electric conductivity Electric resistance comes from electron scattering with defects and phonons. Relaxation time

Semi-classical view The center of the Fermi sphere is shifted by Δk = -eEτ. One can show that when Δk<<k F, V 沒重疊 /V 重疊 ～ 3/2(Δk/k F ). Therefore, the number of electrons being perturbed away from equilibrium is only about (Δk/k F )N e, or (v d /v F )N e Semiclassical vs classical: v F vs v d (differ by 10 9 !) (v d /v F ) N e vs N e The results are the same. But the microscopic pictures are very different.

At room temp. The electron density → τ= m/ρne 2 = 2.5× s Calculating the scattering time τ from measured resistivity ρ Fermi velocity of copper ∴ mean free path = v F τ = 40 nm. For a very pure Cu crystal at 4K, the resistivity reduces by a factor of 10 5, which means increases by the same amount ( = 0.4 cm!). This cannot be explained using classical physics. Residual resistance at T=0 dirty clean K For a crystal without any defect, the only resistance comes from phonon. Therefore, at very low T, the electron mean free path theoretically can be infinite.

Hall effect (1879) Classical view: (consider only 2-dim motion) |ρH||ρH| B

藉著測量霍爾係數可以推算自由電子濃度。 Positive Hall coefficient? Can’t be explained by free electron theory. Band theory (next chap) is required. (ρ H )

Quantum Hall effect (von Klitzing, 1979) h/e 2 = ( 95 ) Ω offers one of the most accurate way to determine the Planck constant classical quantum R xy deviates from (h/e 2 )/C 1 by less than 3 ppm on the very first report. This result is independent of the shape/size of sample. optional

An accurate and stable resistance standard ( 1990 ) optional

Thermal conduction in metal Lorentz number: K/σT=2.45×10 -8 watt-ohm/deg 2 Wiedemann-Franz law ( 1853 ): for a metal, thermal conductivity is closely related to electric conductivity. Both electron and phonon can carry thermal energy (Electrons are dominant in metals). Similar to electric conduction, only the electrons near the Fermi energy can contribute thermal current. Heat capacity per unit volume

Thermal conduction in metal Both electron and phonon can carry thermal energy In a metal, electrons are dominant Only electrons near FS contribute to κ Wiedemann-Franz law (1853) Classical: Semi- classical: Lorentz number heat current density (classical theory) Thermal conductivity