Transcript Part IX
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 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+4pc electric field leads to dr j = qi i dt i P = qiri i e (w ) = 1 + 4p P(w ) E (w ) D = E + 4pP = εE div D = 4pext div E = 4p = 4p ( ext + ind ) E(w , t ) = E(w )e iwt dP = iω P dt P 1 j σ χ= = =i E E iω ω 4π ε(ω) = 1 + i σ(ω) ω j= j(w , t ) ~ j(w )e iwt P(w , t ) ~ P(w )e iwt 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 dp(w , t ) p(w , t ) Newton’s 2nd Law Equation of = eE(w , t ) Motion for the momentum of one dt electron in a time dependent E(w , t ) = E(w )e iwt electric field. Look for a steady p(w , t ) = p(w )e iwt state solution of the form: eE(w ) p(w ) = 1 / iw nep(w ) ( ne 2 / m )E(w ) j(w ) = = m 1 / iw j(w ) = s (w )E(w ) AC conductivity DC conductivity w 1 s (w ) = s0 1 iw e (w ) = 1 + i 4p w s (w ) s0 1 iw ne 2 s (w ) = s0 = s0 Re s (w ) = 1 + w 2 2 m Plasma Frequency 4pne2 e (w ) = 1 mw 2 wp 2 4pne2 = m wp2 e (w ) = 1 2 w A plasma is a medium with positive & negative charges & at least one charge type is mobile. Even more simplified: w 1 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: d 2x m 2 = eE dt eE x= mw 2 ne 2 P = exn = E 2 mw P (w ) 4p ne 2 e (w ) = 1 + 4p = 1 E (w ) mw 2 wp2 4p ne 2 = m wp2 e (w ) = 1 2 w 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 e (w, K ) 2E / t 2 = c 22E in a nonmagnetic isotropic medium. Look for a solution with the dispersion E exp( iwt + iK r ) e (w, K )w 2 = c 2 K 2 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/e1/2 (2) e real & < 0 → for w real, K is imaginary & the wave is damped with a characteristic length 1/|K|: E e Kr (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 wp Dispersion relation for e (w, K )w 2 = c 2 K 2 electromagnetic waves w 2 wp 2 = c2 K 2 (1) For w > wp → K2 > 0, K is real, waves with w > wp propagate in the media with the dispersion relation: w 2 = wp 2 + c2 K 2 The electron gas is transparent. 2 4pne2 = m wp2 e (w ) = 1 2 w e (w ) E e Kr (2) For w < wp → K2 < 0, K is imaginary, waves with w < wp incident on the medium do not propagate, but are totally reflected vgroup = dw dK c v ph = w K c w/wp Metals are shiny due to the reflection of light w/wp (2) w = cK forbidden frequency gap E&M waves are totally reflected from the medium when e is negative cK/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 = 2pc/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 wp 2 4pne2 = m wp2 e (w ) = 1 2 w 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 w2 w2 4p 2 w 4p 2 K = 2 e = 2 1 + i si 2 s c c w c w c r d cl = Kr c (2psw)1 2 The classical skin depth 12 ( 2psw) K= (1 + i ) (2psw)1 2 E exp( iwt + iKr ) exp c E er d = e 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 c c d'= (2ps ' w )1 2 d ' 1 2 2p sw l 13 lc 2 d ' = 2 psw Longitudinal Plasma Oscillations A charge density oscillation, or a longitudinal plasma oscillation, or a plasmon The Nature of Plasma Oscillations: Correspond to a Equation of Motion 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 = 4pnde. 2d Oscillations at the Nm 2 = NeE = Ne( 4pnde) t Plasma Frequency 2 d 2 2 4 p ne 2 + w d = 0 p wp = t 2 m Longitudinal Plasma Oscillations wL= wp w/wp wL 2 e (wL ) = 1 2 = 0 wp Transverse Electromagnetic Waves forbidden frequency gap cK/wp Longitudinal Plasma Oscillations