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: >> In this limit, the conduction electrons will see a homogeneous field when moving between collisions. Write: E(t) = E()e-it 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 for electric current j electric field leads to polarization P dr j qi i dt i P qiri i ( ) 1 4 P( ) E ( ) D E 4P E divD 4 ext divE 4 4 ( ext ind ) mainly described by conductivity j/E
metals polarizability P/E dielectrics dielectric function dP i P dt P 1 j i i E E i 4 ()i() 1 i ()i() j historically used E(, t ) E( )e it j(, t ) ~ j( )e it P(, t ) ~ P( )e it (,0) describes the collective excitations of the electron gas the plasmons (0,k)) describes the electrostatic screening AC Electrical Conductivity of a Metal dp(, t ) p( , t ) Newtons 2nd Law Equation of eE( , t ) dt Motion for the momentum of one electron in a time dependent E( , t ) E( )e it electric field. Look for a steady p(, t ) p( )e it state solution of the form: eE( )
AC conductivity DC conductivity 1 0 ( ) 1 i ( ) 1 i 4 ( ) j( ) ( )E( ) 0 ( ) 1 i ne 2 0 m p( ) 1 / i nep( ) (ne 2 / m )E( ) j( ) m 1 / i 0 Re ( ) 1 2 2 Plasma Frequency 4ne 2 ( ) 1 m 2 4ne 2 p m 2 p ( ) 1 2 2
A plasma is a medium with positive & negative charges & at least one charge type is mobile. Even more simplified: 1 No electron collisions (no frictional damping term) Equation of motion of a Free Electron: If x & E have harmonic time dependences e-it The polarization P is the dipole moment per unit volume: d 2x m 2 eE dt eE x 2 m ne 2 P exn E 2 m P ( ) 4 ne 2 ( ) 1 4 1 E ( ) m 2 2 4 ne p2 m p2 ( ) 1 2 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 ( , K )2E / t 2 c 22E in a nonmagnetic isotropic medium. E exp( it iK r ) Look for a solution with the dispersion ( , K ) 2 c 2 K 2 relation for electromagnetic waves real & > 0 for real, K is real & the transverse electromagnetic wave propagates with the phase velocity vph= c/ real & < 0 for real, K is imaginary & the wave is damped with a characteristic length 1/|K|: Ee Kr (3) complex for real, K is complex & the wave is damped in space (4) = The system has a final response in the absence of an applied force (at E = 0); the poles of (,K) define the frequencies of the free oscillations of the medium (5) = 0 longitudinally polarized waves are possible 4ne 2 p m 2 p ( ) 1 2 Transverse optical modes in a plasma 2 Dispersion relation for (, K ) 2 c 2 K 2 electromagnetic waves 2 2 p c 2 K 2 (1) For > p K2 > 0, K is real, waves with > p propagate in the media2 with 2 2 2
c K p the dispersion relation: The electron gas is transparent. Ee ( ) Kr (2) For < p K2 < 0, K is imaginary, waves with < p incident on the medium do not propagate, but are totally reflected v group d dK c v ph K c / p Metals are shiny due to the reflection of light /p (2) = cKcK forbidden frequency gap E&M waves are totally reflected from the medium when is negative cK/p (1) E&M waves propagate with no damping when 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 p & Free Space Wavelength p = 2c/p
Range n, cm-3 p, cKHz p, cm spectral range Metals 1022 5.71015 3.310-5 UV Semiconductors 1018 5.71013 3.310-3 IF 4ne 2 p m 2 p ( ) 1 2 2 Ionosphere 1010 5.7109 The reflection of 33 light from a metal radio The Electron Gas is Transparent when > p i.e. < p 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 Sk)in Effect When < p the electromagnetic wave is reflected. It is damped with a characteristic length = 1/|K|: The wave penetration the sk)in effect The penetration depth the sk)in depth 2 2 4 2 4 K 2 2 1 i i 2 c c c 2 c r cl Kr c 2 1 2 The classical skin depth 12 2 K (1 i ) 2 1 2
E exp( it iKr ) exp c E e r e >> The classical sk)in effect << : The anomalous sk)in effect (pure metals at low temperatures) the usual theory of electrical conductivity is no longer valid; the electric field varies rapidly over cK. cK Further, not all electrons are participating in the wave absorption & reflection. l Only electrons moving inside the skin depth for most of the mean free path are capable of picking up much energy from the electric field. Only a fraction of the electrons / contribute to the conductivity c c ' 2 ' 1 2 ' 1 2 2 l lc 2 ' 2 13 Longitudinal Plasma A charge density oscillation, or a longitudinal plasma Oscillations 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 = nde & thus an electric field E = 4nde. 2d Oscillations at the Nm 2 NeE Ne(4nde) t Plasma Frequency 2 d 2 2 4 ne 2 d 0 p p t 2 m Longitudinal Plasma Oscillations L= p /p 2 L 1 L 2 0 p Transverse Electromagnetic Waves forbidden frequency gap cK/p Longitudinal Plasma Oscillations
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