Free electron Fermi gas (Sommerfeld, 1928) M.C. Chang Dept of Phys Free electron Fermi gas (Sommerfeld, 1928) 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 are 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.

Energy levels in an empty 1-dim box Plane wave solution “Box” BC Periodic BC k = ±2π/L, ±4π/L, ±6π/L… k = π/L, 2π/L, 3π/L…

Energy levels in an empty 3-dim box 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 For K, the electron density N/V=1.4×1028 m-3, therefore F is roughly the order of the atomic binding energy. kFa is of the order of 1. The Fermi temperature defined by EF=kTF is of the order of 10000 K Fermi energy and fermi wave vector for typical metals

Density of states (for electrons) The number of states within the shell bounded by the constant energy surfaces with energy E and E+dE is D(E)dE What’s the DOS for free electron gas in 3 dim? Free electron DOS and dimension E D(E) 3D 2D 1D

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

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

Electronic specific heat (heuristic argument, see Kittel p Electronic specific heat (heuristic argument, see Kittel p.152 for details) 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/EF） The energy absorbed by the electrons is U(T)-U(0)  NA (kT)2/EF specific heat Ce  dU/dT = 2R kT/EF = 2R 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 + Cp = γT + AT3 Ce becomes more important at very low T

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

Electrical transport Classical view Current density conductivity The source of resistance comes from electron scattering with defects and phonons. If these two types of scatterings are not related, then Current density conductivity

Semi-classical view On the average, the center of the Fermi sphere is shifted by k=eE. One can show that when k<<kF, V沒重疊/V重疊3/2(k/kF). Therefore, the number of electrons being perturbed away from equilibrium is only about (vd/vF) Ne Semiclassical vs classical The results are the same. But the microscopic pictures are very different! vF vs vd (vd/vF) N vs N

Calculating the scattering time from measured resistivity  At room temp The electron density τ= m/ne2 = 2.5×10-14 s Fermi velocity of copper mean free path  = F  = 40 nm For a very pure Cu crystal at 4K, the resistivity reduces by a factor of 100000, which means  increases by the same amount ( = 0.4 cm). This cannot be explained using the classical theory. 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 dirty clean K Residual resistance

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

Positive Hall coefficient? 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 is required.

Quantum Hall effect (von Klitzing, 1979) h/e2=25812.807572(95)  offers one of the most accurate way to determine the Planck constant Classical prediction 1985 Also, an accurate and stable resistance standard (1990)

Thermal conduction in metal 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. Wiedemann-Franz law, 1853 Lorentz number: K/σT=2.45×10-8 watt-ohm/deg2