Dept of Phys Free electron Fermi gas Sommerfeld

Dept of Phys Free electron Fermi gas (Sommerfeld, 1928) • counting of states 1926: Schrodinger eq. , FD statistics M. C. Chang • 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.

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

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

Free electron in a 3 -dim box z Lz Lx x Ly y Advantage: allows travelling waves

Quantization of k in a 3 -dim 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. • Different BCs give the same Fermi wave vector and the same energy box BC periodic BC

Connection between electron density and Fermi energy F • For K, the electron density n=1. 4× 1028 m-3, therefore • εF is of the order of the atomic energy levels. • k. F is of the order of a-1.

Fermi temperature and Fermi velocity • The Fermi temperature is of the order of 104 K

important 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 • For a 3 D Fermi sphere,

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

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

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

Electronic specific heat, heuristic argument (see Kittel p. 142 for details) • Only the electrons near the Fermi surface are excited by thermal energy k. T. The number of excited electrons are roughly of the order of N’ = N（k. T /EF） • The energy absorbed by the electrons is U(T)-U(0) ～ NA (k. T)2/EF • specific heat Ce ～ d. U/d. T = 2 R k. T/EF = 2 R T/TF a factor of T/TF smaller than classical result • T/TF ～ 0. 01 Therefore usually electron specific is much smaller than phonon specific heat • In general C = Ce + C p = γT + AT 3 Ce 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 Relaxation time • Electric resistance comes from electron scattering with defects and phonons. • If these two types of scatterings are not related, then scattering rate: • Current density (n is electron density) Electric conductivity

Semi-classical view • The center of the Fermi sphere is shifted by Δk = -e. Eτ. • 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)Ne, or (vd/v. F)Ne • Semiclassical vs classical: The results are the same. But the microscopic pictures are very different. • v. F vs vd (differ by 109 !) • (vd /v. F) Ne vs Ne

Calculating the scattering time τ from measured resistivity ρ • At room temp . The electron density →τ= m/ρne 2 = 2. 5× 10 -14 s • Fermi velocity of copper ∴ mean free path = v. Fτ= 40 nm. • For a very pure Cu crystal at 4 K, the resistivity reduces by a factor of 105, which means increases by the same amount ( = 0. 4 cm!). This cannot be explained using classical physics. • 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. K dirty clean Residual resistance at T=0

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

(ρH) 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.

optional Quantum Hall effect (von Klitzing, 1979) quantum classical 1985 • Rxy 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. • h/e 2=25812. 807572(95) Ω offers one of the most accurate way to determine the Planck constant.

optional An accurate and stable resistance standard (1990)

Thermal conduction in metal • Both electron and phonon 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 • Wiedemann-Franz law (1853): for a metal, thermal conductivity is closely related to electric conductivity. Lorentz number: K/σT=2. 45× 10 -8 watt-ohm/deg 2