The Drude Model Peter Hertel Overview Model The Drude Model Dielectric medium Permittivity of metals Peter Hertel Electrical conductors University of Osnabr¨uck,Germany Faraday effect Hall effect Lecture presented at APS, Nankai University, China http://www.home.uni-osnabrueck.de/phertel October/November 2011 The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect Paul Drude, German physicist, 1863-1906 • The Drude model links optical and electric properties of a material with the behavior of its electrons or holes • The model • Dielectric permittivity • Permittivity of metals • Conductivity • Faraday effect • Hall effect The Drude Model Overview Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • The model • Dielectric permittivity • Permittivity of metals • Conductivity • Faraday effect • Hall effect The Drude Model Overview Peter Hertel Overview Model Dielectric medium • The Drude model links optical and electric properties of a Permittivity of metals material with the behavior of its electrons or holes Electrical conductors Faraday effect Hall effect • Dielectric permittivity • Permittivity of metals • Conductivity • Faraday effect • Hall effect The Drude Model Overview Peter Hertel Overview Model Dielectric medium • The Drude model links optical and electric properties of a Permittivity of metals material with the behavior of its electrons or holes Electrical conductors • The model Faraday effect Hall effect • Permittivity of metals • Conductivity • Faraday effect • Hall effect The Drude Model Overview Peter Hertel Overview Model Dielectric medium • The Drude model links optical and electric properties of a Permittivity of metals material with the behavior of its electrons or holes Electrical conductors • The model Faraday effect • Dielectric permittivity Hall effect • Conductivity • Faraday effect • Hall effect The Drude Model Overview Peter Hertel Overview Model Dielectric medium • The Drude model links optical and electric properties of a Permittivity of metals material with the behavior of its electrons or holes Electrical conductors • The model Faraday effect • Dielectric permittivity Hall effect • Permittivity of metals • Faraday effect • Hall effect The Drude Model Overview Peter Hertel Overview Model Dielectric medium • The Drude model links optical and electric properties of a Permittivity of metals material with the behavior of its electrons or holes Electrical conductors • The model Faraday effect • Dielectric permittivity Hall effect • Permittivity of metals • Conductivity • Hall effect The Drude Model Overview Peter Hertel Overview Model Dielectric medium • The Drude model links optical and electric properties of a Permittivity of metals material with the behavior of its electrons or holes Electrical conductors • The model Faraday effect • Dielectric permittivity Hall effect • Permittivity of metals • Conductivity • Faraday effect The Drude Model Overview Peter Hertel Overview Model Dielectric medium • The Drude model links optical and electric properties of a Permittivity of metals material with the behavior of its electrons or holes Electrical conductors • The model Faraday effect • Dielectric permittivity Hall effect • Permittivity of metals • Conductivity • Faraday effect • Hall effect • consider a typical electron • denote by x = x(t) the deviation from its equilibrium position • external electric field strength E = E(t) • m(x¨ + Γx_ + Ω2x) = qE • electron mass m, charge q, friction coefficient mΓ, spring constant mΩ2 • Fourier transform this • m(−!2 − i!Γ + Ω2)x~ = qE~ • solution is q E~ (!) x~(!) = m Ω2 − !2 − i!Γ The Drude Model Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • denote by x = x(t) the deviation from its equilibrium position • external electric field strength E = E(t) • m(x¨ + Γx_ + Ω2x) = qE • electron mass m, charge q, friction coefficient mΓ, spring constant mΩ2 • Fourier transform this • m(−!2 − i!Γ + Ω2)x~ = qE~ • solution is q E~ (!) x~(!) = m Ω2 − !2 − i!Γ The Drude Model Model Peter Hertel Overview Model • consider a typical electron Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • external electric field strength E = E(t) • m(x¨ + Γx_ + Ω2x) = qE • electron mass m, charge q, friction coefficient mΓ, spring constant mΩ2 • Fourier transform this • m(−!2 − i!Γ + Ω2)x~ = qE~ • solution is q E~ (!) x~(!) = m Ω2 − !2 − i!Γ The Drude Model Model Peter Hertel Overview Model • consider a typical electron Dielectric • medium denote by x = x(t) the deviation from its equilibrium Permittivity of position metals Electrical conductors Faraday effect Hall effect • m(x¨ + Γx_ + Ω2x) = qE • electron mass m, charge q, friction coefficient mΓ, spring constant mΩ2 • Fourier transform this • m(−!2 − i!Γ + Ω2)x~ = qE~ • solution is q E~ (!) x~(!) = m Ω2 − !2 − i!Γ The Drude Model Model Peter Hertel Overview Model • consider a typical electron Dielectric • medium denote by x = x(t) the deviation from its equilibrium Permittivity of position metals • external electric field strength E = E(t) Electrical conductors Faraday effect Hall effect • electron mass m, charge q, friction coefficient mΓ, spring constant mΩ2 • Fourier transform this • m(−!2 − i!Γ + Ω2)x~ = qE~ • solution is q E~ (!) x~(!) = m Ω2 − !2 − i!Γ The Drude Model Model Peter Hertel Overview Model • consider a typical electron Dielectric • medium denote by x = x(t) the deviation from its equilibrium Permittivity of position metals • external electric field strength E = E(t) Electrical conductors • m(x¨ + Γx_ + Ω2x) = qE Faraday effect Hall effect • Fourier transform this • m(−!2 − i!Γ + Ω2)x~ = qE~ • solution is q E~ (!) x~(!) = m Ω2 − !2 − i!Γ The Drude Model Model Peter Hertel Overview Model • consider a typical electron Dielectric • medium denote by x = x(t) the deviation from its equilibrium Permittivity of position metals • external electric field strength E = E(t) Electrical conductors • m(x¨ + Γx_ + Ω2x) = qE Faraday effect • electron mass m, charge q, friction coefficient mΓ, spring Hall effect constant mΩ2 • m(−!2 − i!Γ + Ω2)x~ = qE~ • solution is q E~ (!) x~(!) = m Ω2 − !2 − i!Γ The Drude Model Model Peter Hertel Overview Model • consider a typical electron Dielectric • medium denote by x = x(t) the deviation from its equilibrium Permittivity of position metals • external electric field strength E = E(t) Electrical conductors • m(x¨ + Γx_ + Ω2x) = qE Faraday effect • electron mass m, charge q, friction coefficient mΓ, spring Hall effect constant mΩ2 • Fourier transform this • solution is q E~ (!) x~(!) = m Ω2 − !2 − i!Γ The Drude Model Model Peter Hertel Overview Model • consider a typical electron Dielectric • medium denote by x = x(t) the deviation from its equilibrium Permittivity of position metals • external electric field strength E = E(t) Electrical conductors • m(x¨ + Γx_ + Ω2x) = qE Faraday effect • electron mass m, charge q, friction coefficient mΓ, spring Hall effect constant mΩ2 • Fourier transform this • m(−!2 − i!Γ + Ω2)x~ = qE~ The Drude Model Model Peter Hertel Overview Model • consider a typical electron Dielectric • medium denote by x = x(t) the deviation from its equilibrium Permittivity of position metals • external electric field strength E = E(t) Electrical conductors • m(x¨ + Γx_ + Ω2x) = qE Faraday effect • electron mass m, charge q, friction coefficient mΓ, spring Hall effect constant mΩ2 • Fourier transform this • m(−!2 − i!Γ + Ω2)x~ = qE~ • solution is q E~ (!) x~(!) = m Ω2 − !2 − i!Γ • dipole moment of typical electron is p~ = qx~ • recall q E~ (!) x~(!) = m Ω2 − !2 − i!Γ • there are N typical electrons per unit volume • polarization is P~ = Nqx~ = 0χE~ • susceptibility is Nq2 1 χ(!) = 2 2 0m Ω − ! − i!Γ • in particular Nq2 χ(0) = 2 > 0 0mΩ • . as it should be The Drude Model Polarization Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • recall q E~ (!) x~(!) = m Ω2 − !2 − i!Γ • there are N typical electrons per unit volume • polarization is P~ = Nqx~ = 0χE~ • susceptibility is Nq2 1 χ(!) = 2 2 0m Ω − ! − i!Γ • in particular Nq2 χ(0) = 2 > 0 0mΩ • . as it should be The Drude Model Polarization Peter Hertel Overview Model • dipole moment of typical electron is p~ = qx~ Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • there are N typical electrons per unit volume • polarization is P~ = Nqx~ = 0χE~ • susceptibility is Nq2 1 χ(!) = 2 2 0m Ω − ! − i!Γ • in particular Nq2 χ(0) = 2 > 0 0mΩ • . as it should be The Drude Model Polarization Peter Hertel Overview Model • dipole moment of typical electron is p~ = qx~ Dielectric medium • recall Permittivity of q E~ (!) metals x~(!) = m Ω2 − !2 − i!Γ Electrical conductors Faraday effect Hall effect • polarization is P~ = Nqx~ = 0χE~ • susceptibility is Nq2 1 χ(!) = 2 2 0m Ω − ! − i!Γ • in particular Nq2 χ(0) = 2 > 0 0mΩ • . as it should be The Drude Model Polarization Peter Hertel Overview Model • dipole moment of typical electron is p~ = qx~ Dielectric medium • recall Permittivity of q E~ (!) metals x~(!) = m Ω2 − !2 − i!Γ Electrical conductors • there are N typical electrons per unit volume Faraday effect Hall effect • susceptibility is Nq2 1 χ(!) = 2 2 0m Ω − ! − i!Γ • in particular Nq2 χ(0) = 2 > 0 0mΩ • . as it should be The Drude Model Polarization Peter Hertel Overview Model • dipole moment of typical electron is p~ = qx~ Dielectric medium • recall Permittivity of q