AC Electrical Conductivity in Metals Brief discussion only

AC Electrical Conductivity in Metals (Brief discussion only, in the free & independent electron approximation) Application to the Propagation of Electromagnetic Radiation in a Metal Consider a time dependent electric field E(t) acting on a metal. Take the case when the wavelength of the field is large compared to the electron mean free path between collisions: l >> l In this limit, the conduction electrons will “see” a homogeneous field when moving between collisions. Write: E(t) = E(ω)e-iωt That is, assume a harmonic dependence on frequency. Next is standard Junior-Senior physics major E&M!

Response to the electric field both in metals and dielectrics electric field leads to mainly described by historically used mainly for electric current j conductivity s = j/E metals polarization P polarizability c = P/E dielectrics dielectric function e =1+4 pc e(w, 0) describes the collective excitations of the electron gas – the plasmons e(0, k) describes the electrostatic screening

AC Electrical Conductivity of a Metal Newton’s 2 nd Law Equation of Motion for the momentum of one electron in a time dependent electric field. Look for a steady state solution of the form: AC conductivity DC conductivity Plasma Frequency A plasma is a medium with positive & negative charges & at least one charge type is mobile.

Even more simplified: No electron collisions (no frictional damping term) Equation of motion of a Free Electron: If x & E have harmonic time dependences e-iωt The polarization P is the dipole moment per unit volume:

Application to the Propagation of Electromagnetic Radiation in a Metal Transverse Electromagnetic Wave T

Application to the Propagation of Electromagnetic Radiation in a Metal The electromagnetic wave equation in a nonmagnetic isotropic medium. Look for a solution with the dispersion relation for electromagnetic waves (1) e real & > 0 → for w real, K is real & the transverse electromagnetic wave propagates with the phase velocity vph= c/e 1/2 (2) e real & < 0 → for w real, K is imaginary & the wave is damped with a characteristic length 1/|K|: (3) e complex → for w real, K is complex & the wave is damped in space (4) e = → → The system has a final response in the absence of an applied force (at E = 0); the poles of e(w, K) define the frequencies of the free oscillations of the medium (5) e = 0 longitudinally polarized waves are possible

Transverse optical modes in a plasma Dispersion relation for electromagnetic waves (1) For w > wp → K 2 > 0, K is real, waves with w > wp propagate in the media with the dispersion relation: The electron gas is transparent. (2) For w < wp → K 2 < 0, K is imaginary, waves with w < wp incident on the medium do not propagate, but are totally reflected Metals are shiny due to the reflection of light w/ wp (2) w = c. K forbidden frequency gap E&M waves are totally reflected from the medium when e is negative c. K/wp (1) E&M waves propagate with no damping when e is positive & real vph > c → vph This does not correspond to the velocity of the propagation of any quantity!!

Ultraviolet Transparency of Metals Plasma Frequency wp & Free Space Wavelength lp = 2 pc/wp Range n, cm-3 wp, Hz lp, cm spectral range Metals 1022 5. 7× 1015 3. 3× 10 -5 UV Semiconductors 1018 5. 7× 1013 3. 3× 10 -3 IF Ionosphere 1010 5. 7× 109 33 The reflection of radio light from a metal The Electron Gas is Transparent when w > wp i. e. l < lp is similar to the reflection of radio waves from the Ionosphere! Plasma Frequency Ionosphere Semiconductors Metals metal ionosphere reflects transparent for visible UV radio visible

Skin Effect When w < wp the electromagnetic wave is reflected. It is damped with a characteristic length d = 1/|K|: The wave penetration – the skin effect The penetration depth d – the skin depth The classical skin depth d >> l The classical skin effect d << l: The anomalous skin effect (pure metals at low temperatures) the usual theory of electrical conductivity is no longer valid; the electric field varies rapidly over l. Further, not all electrons are participating in the wave absorption & reflection. d’ l Only electrons moving inside the skin depth for most of the mean free path l are capable of picking up much energy from the electric field. Only a fraction of the electrons d’/l contribute to the conductivity

Longitudinal Plasma Oscillations A charge density oscillation, or a longitudinal plasma oscillation, or a plasmon The Nature of Plasma Oscillations: Correspond to a displacement of the entire electron gas a distance d with respect to the positive ion background. This creates surface charges s = nde & thus an electric field E = 4 pnde. Oscillations at the Plasma Frequency Equation of Motion Longitudinal Plasma Oscillations w. L = wp w/wp Transverse Electromagnetic Waves forbidden frequency gap c. K/wp Longitudinal Plasma Oscillations