The Drude Model

The Drude Model Peter Hertel Overview Model The Drude Model Dielectric medium Permittivity of metals Electrical conductors Peter Hertel University of Osnabrück, 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 Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect Overview The Drude Model Overview Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • The Drude model links optical and electric properties of a material with the behavior of its electrons or holes The Drude Model Overview Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • The Drude model links optical and electric properties of a material with the behavior of its electrons or holes • The model The Drude Model Overview Peter Hertel Overview Model Dielectric medium Permittivity of metals • The Drude model links optical and electric properties of a material with the behavior of its electrons or holes Electrical conductors • The model Faraday effect • Dielectric permittivity Hall effect The Drude Model Overview Peter Hertel Overview Model Dielectric medium Permittivity of metals • The Drude model links optical and electric properties of a material with the behavior of its electrons or holes Electrical conductors • The model Faraday effect • Dielectric permittivity Hall effect • Permittivity of metals The Drude Model Overview Peter Hertel Overview Model Dielectric medium Permittivity of metals • The Drude model links optical and electric properties of a material with the behavior of its electrons or holes Electrical conductors • The model Faraday effect • Dielectric permittivity Hall effect • Permittivity of metals • Conductivity The Drude Model Overview Peter Hertel Overview Model Dielectric medium Permittivity of metals • The Drude model links optical and electric properties of a 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 Permittivity of metals • The Drude model links optical and electric properties of a 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 The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect Model The Drude Model Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • consider a typical electron 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 metals Electrical conductors Faraday effect Hall effect position 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 metals Electrical conductors Faraday effect Hall effect position • external electric field strength E = E(t) 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 metals Electrical conductors Faraday effect Hall effect position • external electric field strength E = E(t) • m(ẍ + Γẋ + 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 metals Electrical conductors Faraday effect Hall effect position • external electric field strength E = E(t) • m(ẍ + Γẋ + Ω2 x) = qE • electron mass m, charge q, friction coefficient mΓ, spring constant mΩ2 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 metals Electrical conductors Faraday effect Hall effect position • external electric field strength E = E(t) • m(ẍ + Γẋ + Ω2 x) = qE • electron mass m, charge q, friction coefficient mΓ, spring constant mΩ2 • Fourier transform this 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 metals Electrical conductors Faraday effect Hall effect position • external electric field strength E = E(t) • m(ẍ + Γẋ + Ω2 x) = qE • electron mass m, charge q, friction coefficient mΓ, spring constant mΩ2 • Fourier transform this • m(−ω 2 − iωΓ + Ω2 )x̃ = q Ẽ 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 metals Electrical conductors Faraday effect Hall effect position • external electric field strength E = E(t) • m(ẍ + Γẋ + Ω2 x) = qE • electron mass m, charge q, friction coefficient mΓ, spring constant mΩ2 • Fourier transform this • m(−ω 2 − iωΓ + Ω2 )x̃ = q Ẽ • solution is x̃(ω) = q Ẽ(ω) 2 m Ω − ω 2 − iωΓ The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect Polarization The Drude Model Polarization Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • dipole moment of typical electron is p̃ = q x̃ The Drude Model Polarization Peter Hertel Overview Model • dipole moment of typical electron is p̃ = q x̃ Dielectric medium • recall Permittivity of metals Electrical conductors Faraday effect Hall effect x̃(ω) = q Ẽ(ω) 2 m Ω − ω 2 − iωΓ The Drude Model Polarization Peter Hertel Overview Model • dipole moment of typical electron is p̃ = q x̃ Dielectric medium • recall Permittivity of metals Electrical conductors Faraday effect Hall effect q Ẽ(ω) 2 m Ω − ω 2 − iωΓ • there are N typical electrons per unit volume x̃(ω) = The Drude Model Polarization Peter Hertel Overview Model • dipole moment of typical electron is p̃ = q x̃ Dielectric medium • recall Permittivity of metals Electrical conductors Faraday effect Hall effect q Ẽ(ω) 2 m Ω − ω 2 − iωΓ • there are N typical electrons per unit volume x̃(ω) = • polarization is P̃ = N q x̃ = 0 χẼ The Drude Model Polarization Peter Hertel Overview Model • dipole moment of typical electron is p̃ = q x̃ Dielectric medium • recall Permittivity of metals Electrical conductors Faraday effect Hall effect q Ẽ(ω) 2 m Ω − ω 2 − iωΓ • there are N typical electrons per unit volume x̃(ω) = • polarization is P̃ = N q x̃ = 0 χẼ • susceptibility is χ(ω) = 1 N q2 2 0 m Ω − ω 2 − iωΓ The Drude Model Polarization Peter Hertel Overview Model • dipole moment of typical electron is p̃ = q x̃ Dielectric medium • recall Permittivity of metals Electrical conductors Faraday effect Hall effect q Ẽ(ω) 2 m Ω − ω 2 − iωΓ • there are N typical electrons per unit volume x̃(ω) = • polarization is P̃ = N q x̃ = 0 χẼ • susceptibility is 1 N q2 2 0 m Ω − ω 2 − iωΓ • in particular N q2 χ(0) = >0 0 mΩ2 χ(ω) = The Drude Model Polarization Peter Hertel Overview Model • dipole moment of typical electron is p̃ = q x̃ Dielectric medium • recall Permittivity of metals Electrical conductors Faraday effect Hall effect q Ẽ(ω) 2 m Ω − ω 2 − iωΓ • there are N typical electrons per unit volume x̃(ω) = • polarization is P̃ = N q x̃ = 0 χẼ • susceptibility is 1 N q2 2 0 m Ω − ω 2 − iωΓ • in particular N q2 χ(0) = >0 0 mΩ2 • . . . as it should be χ(ω) = The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect Discussion I The Drude Model Discussion I Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • decompose susceptibility χ(ω) = χ 0 (ω) + iχ 00 (ω) into refractive part χ 0 and absorptive part χ 00 The Drude Model Discussion I Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • decompose susceptibility χ(ω) = χ 0 (ω) + iχ 00 (ω) into refractive part χ 0 and absorptive part χ 00 • Introduce R(ω) = χ(ω)/χ(0), s = ω/Ω and γ = Γ/Ω as normalized quantities. The Drude Model Discussion I Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • decompose susceptibility χ(ω) = χ 0 (ω) + iχ 00 (ω) into refractive part χ 0 and absorptive part χ 00 • Introduce R(ω) = χ(ω)/χ(0), s = ω/Ω and γ = Γ/Ω as normalized quantities. • refraction R 0 (s) = 1 − s2 (1 − s2 )2 + γ 2 s2 The Drude Model Discussion I Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors • decompose susceptibility χ(ω) = χ 0 (ω) + iχ 00 (ω) into refractive part χ 0 and absorptive part χ 00 • Introduce R(ω) = χ(ω)/χ(0), s = ω/Ω and γ = Γ/Ω as normalized quantities. Faraday effect • refraction Hall effect 1 − s2 (1 − s2 )2 + γ 2 s2 • absorption γs D 00 (s) = 2 (1 − s )2 + γ 2 s2 R 0 (s) = The Drude Model Discussion I Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors • decompose susceptibility χ(ω) = χ 0 (ω) + iχ 00 (ω) into refractive part χ 0 and absorptive part χ 00 • Introduce R(ω) = χ(ω)/χ(0), s = ω/Ω and γ = Γ/Ω as normalized quantities. Faraday effect • refraction Hall effect 1 − s2 (1 − s2 )2 + γ 2 s2 • absorption γs D 00 (s) = 2 (1 − s )2 + γ 2 s2 • limiting cases: s = 0, s = 1, s → ∞, small γ R 0 (s) = The Drude Model 10 Peter Hertel Overview Model Dielectric medium 5 Permittivity of metals Electrical conductors Faraday effect 0 Hall effect −5 0 0.5 1 1.5 2 2.5 3 Refractive part (blue) and absorptive part (red) of the susceptibility function χ(ω) scaled by the static value χ(0). The abscissa is ω/Ω. Γ/Ω = 0.1 The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect Discussion II The Drude Model Discussion II Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • For small frequencies (as compared with Ω) the susceptibility is practically real. The Drude Model Discussion II Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • For small frequencies (as compared with Ω) the susceptibility is practically real. • This is the realm of classical optics The Drude Model Discussion II Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • For small frequencies (as compared with Ω) the susceptibility is practically real. • This is the realm of classical optics • ∂χ/∂ω is positive – normal dispersion The Drude Model Discussion II Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • For small frequencies (as compared with Ω) the susceptibility is practically real. • This is the realm of classical optics • ∂χ/∂ω is positive – normal dispersion • In the vicinity of ω = Ω absorption is large. Negative dispersion ∂χ/∂ω is accompanied by strong absorption. The Drude Model Discussion II Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • For small frequencies (as compared with Ω) the susceptibility is practically real. • This is the realm of classical optics • ∂χ/∂ω is positive – normal dispersion • In the vicinity of ω = Ω absorption is large. Negative dispersion ∂χ/∂ω is accompanied by strong absorption. • For very large frequencies again absorption is negligible, and the susceptibility is negative with normal dispersion. This applies to X rays. The Drude Model Discussion II Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • For small frequencies (as compared with Ω) the susceptibility is practically real. • This is the realm of classical optics • ∂χ/∂ω is positive – normal dispersion • In the vicinity of ω = Ω absorption is large. Negative dispersion ∂χ/∂ω is accompanied by strong absorption. • For very large frequencies again absorption is negligible, and the susceptibility is negative with normal dispersion. This applies to X rays. • χ(∞) = 0 is required by first principles . . . The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect Kramers-Kronig relation I The Drude Model Kramers-Kronig relation I Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • χ(ω) must be the Fourier transform of a causal response function G = G(τ ) The Drude Model Kramers-Kronig relation I Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • χ(ω) must be the Fourier transform of a causal response function G = G(τ ) • as defined Zin P (t) = 0 dτ G(τ )E(t − τ ) The Drude Model Kramers-Kronig relation I Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors • χ(ω) must be the Fourier transform of a causal response function G = G(τ ) • as defined Zin P (t) = 0 • check this for Z Faraday effect Hall effect dτ G(τ )E(t − τ ) G(τ ) = a −iωτ dω e 2π Ω2 − ω 2 − iωΓ The Drude Model Kramers-Kronig relation I Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors • χ(ω) must be the Fourier transform of a causal response function G = G(τ ) • as defined Zin P (t) = 0 • check this for Z Faraday effect Hall effect dτ G(τ )E(t − τ ) G(τ ) = a −iωτ dω e 2π Ω2 − ω 2 − iωΓ • poles at ω1,2 = − p iΓ ± ω̄ where ω̄ = + Ω2 − Γ2 /4 2 The Drude Model Kramers-Kronig relation I Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors • χ(ω) must be the Fourier transform of a causal response function G = G(τ ) • as defined Zin P (t) = 0 • check this for Z Faraday effect Hall effect dτ G(τ )E(t − τ ) G(τ ) = a −iωτ dω e 2π Ω2 − ω 2 − iωΓ • poles at p iΓ ± ω̄ where ω̄ = + Ω2 − Γ2 /4 2 • Indeed, G(τ ) = 0 for τ < 0 ω1,2 = − The Drude Model Kramers-Kronig relation I Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors • χ(ω) must be the Fourier transform of a causal response function G = G(τ ) • as defined Zin P (t) = 0 • check this for Z Faraday effect Hall effect dτ G(τ )E(t − τ ) G(τ ) = a −iωτ dω e 2π Ω2 − ω 2 − iωΓ • poles at p iΓ ± ω̄ where ω̄ = + Ω2 − Γ2 /4 2 • Indeed, G(τ ) = 0 for τ < 0 ω1,2 = − • for τ > 0 G(τ ) = N q 2 sin ω̄τ −Γτ /2 e 0 m ω̄ The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect Kramers-Kronig relation II The Drude Model Kramers-Kronig relation II Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • causal response function: G(τ ) = θ(τ )G(τ ) The Drude Model Kramers-Kronig relation II Peter Hertel Overview Model • causal response function: G(τ ) = θ(τ )G(τ ) Dielectric medium • apply the convolution theorem Permittivity of metals Electrical conductors Faraday effect Hall effect Z χ(ω) = du χ(u)θ̃(ω − u) 2π The Drude Model Kramers-Kronig relation II Peter Hertel Overview Model • causal response function: G(τ ) = θ(τ )G(τ ) Dielectric medium • apply the convolution theorem Permittivity of metals Electrical conductors Faraday effect Hall effect Z du χ(u)θ̃(ω − u) 2π • Fourier transform of Heaviside function is 1 θ̃(ω) = lim 0<η→0 η − iω χ(ω) = The Drude Model Kramers-Kronig relation II Peter Hertel Overview Model • causal response function: G(τ ) = θ(τ )G(τ ) Dielectric medium • apply the convolution theorem Permittivity of metals Electrical conductors Faraday effect Hall effect Z du χ(u)θ̃(ω − u) 2π • Fourier transform of Heaviside function is 1 θ̃(ω) = lim 0<η→0 η − iω χ(ω) = • dispersion , or Kramers-Kronig relations 0 χ (ω) = 2Pr Z du uχ 00 (u) π u2 − ω 2 The Drude Model Kramers-Kronig relation II Peter Hertel Overview Model • causal response function: G(τ ) = θ(τ )G(τ ) Dielectric medium • apply the convolution theorem Permittivity of metals Electrical conductors Faraday effect Hall effect Z du χ(u)θ̃(ω − u) 2π • Fourier transform of Heaviside function is 1 θ̃(ω) = lim 0<η→0 η − iω χ(ω) = • dispersion , or Kramers-Kronig relations 0 χ (ω) = 2Pr Z du uχ 00 (u) π u2 − ω 2 The Drude Model Kramers-Kronig relation II Peter Hertel Overview Model • causal response function: G(τ ) = θ(τ )G(τ ) Dielectric medium • apply the convolution theorem Permittivity of metals Electrical conductors Faraday effect Hall effect Z du χ(u)θ̃(ω − u) 2π • Fourier transform of Heaviside function is 1 θ̃(ω) = lim 0<η→0 η − iω χ(ω) = • dispersion , or Kramers-Kronig relations 0 Z du uχ 00 (u) π u2 − ω 2 Z du ωχ 0 (u) π ω 2 − u2 χ (ω) = 2Pr 00 χ (ω) = 2Pr The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect Dispersion of white light The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect Free quasi-electrons The Drude Model Free quasi-electrons Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • consider a typical conduction band electron The Drude Model Free quasi-electrons Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • consider a typical conduction band electron • it behaves as a free quasi-particle The Drude Model Free quasi-electrons Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • consider a typical conduction band electron • it behaves as a free quasi-particle • recall m(ẍ + Γẋ + Ω2 x) = qE The Drude Model Free quasi-electrons Peter Hertel Overview Model • consider a typical conduction band electron • it behaves as a free quasi-particle Dielectric medium • recall m(ẍ + Γẋ + Ω2 x) = qE Permittivity of metals • spring constant mΩ2 vanishes Electrical conductors Faraday effect Hall effect The Drude Model Free quasi-electrons Peter Hertel Overview Model • consider a typical conduction band electron • it behaves as a free quasi-particle Dielectric medium • recall m(ẍ + Γẋ + Ω2 x) = qE Permittivity of metals • spring constant mΩ2 vanishes Electrical conductors • m is effective mass Faraday effect Hall effect The Drude Model Free quasi-electrons Peter Hertel Overview Model • consider a typical conduction band electron • it behaves as a free quasi-particle Dielectric medium • recall m(ẍ + Γẋ + Ω2 x) = qE Permittivity of metals • spring constant mΩ2 vanishes Electrical conductors • m is effective mass Faraday effect • therefore Hall effect (ω) = 1 − ωp2 ω 2 + iωΓ The Drude Model Free quasi-electrons Peter Hertel Overview Model • consider a typical conduction band electron • it behaves as a free quasi-particle Dielectric medium • recall m(ẍ + Γẋ + Ω2 x) = qE Permittivity of metals • spring constant mΩ2 vanishes Electrical conductors • m is effective mass Faraday effect • therefore Hall effect ωp2 ω 2 + iωΓ • plasma frequency ωp N q2 ωp2 = 0 m (ω) = 1 − The Drude Model Free quasi-electrons Peter Hertel Overview Model • consider a typical conduction band electron • it behaves as a free quasi-particle Dielectric medium • recall m(ẍ + Γẋ + Ω2 x) = qE Permittivity of metals • spring constant mΩ2 vanishes Electrical conductors • m is effective mass Faraday effect • therefore Hall effect ωp2 ω 2 + iωΓ • plasma frequency ωp N q2 ωp2 = 0 m • correction for ω ωp ωp2 (ω) = ∞ − 2 ω + iωΓ (ω) = 1 − The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect Example: gold The Drude Model Example: gold Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • Drude model parameters for gold The Drude Model Example: gold Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • Drude model parameters for gold • as determined by Johnson and Christy in 1972 The Drude Model Example: gold Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • Drude model parameters for gold • as determined by Johnson and Christy in 1972 • ∞ = 9.5 The Drude Model Example: gold Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • Drude model parameters for gold • as determined by Johnson and Christy in 1972 • ∞ = 9.5 • ~ωp = 8.95 eV The Drude Model Example: gold Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • Drude model parameters for gold • as determined by Johnson and Christy in 1972 • ∞ = 9.5 • ~ωp = 8.95 eV • ~Γ = 0.069 eV The Drude Model Example: gold Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • Drude model parameters for gold • as determined by Johnson and Christy in 1972 • ∞ = 9.5 • ~ωp = 8.95 eV • ~Γ = 0.069 eV • with these parameters the Drude model fits optical measurements well for ~ω < 2.25 eV (green) The Drude Model Example: gold Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • Drude model parameters for gold • as determined by Johnson and Christy in 1972 • ∞ = 9.5 • ~ωp = 8.95 eV • ~Γ = 0.069 eV • with these parameters the Drude model fits optical measurements well for ~ω < 2.25 eV (green) • The refractive part of the permittivity can be large and negative while the absorptive part is small. The Drude Model Example: gold Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • Drude model parameters for gold • as determined by Johnson and Christy in 1972 • ∞ = 9.5 • ~ωp = 8.95 eV • ~Γ = 0.069 eV • with these parameters the Drude model fits optical measurements well for ~ω < 2.25 eV (green) • The refractive part of the permittivity can be large and negative while the absorptive part is small. • This allows surface plasmon polaritons (SPP) The Drude Model 20 Peter Hertel Overview Model 0 Dielectric medium Permittivity of metals −20 Electrical conductors Faraday effect −40 Hall effect −60 −80 1 1.5 2 2.5 3 Refractive (blue) and absorptive part (red) of the permittivity function for gold. The abscissa is ~ω in eV. The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect Electrical conductivity The Drude Model Electrical conductivity Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • consider a typical charged particle The Drude Model Electrical conductivity Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • consider a typical charged particle • recall m(ẍ + Γẋ + Ω2 x) = qE The Drude Model Electrical conductivity Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • consider a typical charged particle • recall m(ẍ + Γẋ + Ω2 x) = qE • electric current density J = N q ẋ The Drude Model Electrical conductivity Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • consider a typical charged particle • recall m(ẍ + Γẋ + Ω2 x) = qE • electric current density J = N q ẋ • Fourier transformed: J̃ = N q(−iω)x̃ The Drude Model Electrical conductivity Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors • consider a typical charged particle • recall m(ẍ + Γẋ + Ω2 x) = qE • electric current density J = N q ẋ • Fourier transformed: J̃ = N q(−iω)x̃ • recall Faraday effect Hall effect x̃(ω) = q Ẽ(ω) m Ω2 − ω 2 − iωΓ The Drude Model Electrical conductivity Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • consider a typical charged particle • recall m(ẍ + Γẋ + Ω2 x) = qE • electric current density J = N q ẋ • Fourier transformed: J̃ = N q(−iω)x̃ • recall q Ẽ(ω) m Ω2 − ω 2 − iωΓ • Ohm’s law x̃(ω) = J̃ (ω) = σ(ω)Ẽ(ω) The Drude Model Electrical conductivity Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • consider a typical charged particle • recall m(ẍ + Γẋ + Ω2 x) = qE • electric current density J = N q ẋ • Fourier transformed: J̃ = N q(−iω)x̃ • recall q Ẽ(ω) m Ω2 − ω 2 − iωΓ • Ohm’s law x̃(ω) = J̃ (ω) = σ(ω)Ẽ(ω) • conductivity is σ(ω) = N q2 −iω 2 m Ω − ω 2 − iωΓ The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect Electrical conductors The Drude Model Electrical conductors Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • A material with σ(0) = 0 is an electrical insulator . It cannot transport direct currents (DC). The Drude Model Electrical conductors Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • A material with σ(0) = 0 is an electrical insulator . It cannot transport direct currents (DC). • A material with σ(0) > 0 is an electrical conductor . The Drude Model Electrical conductors Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • A material with σ(0) = 0 is an electrical insulator . It cannot transport direct currents (DC). • A material with σ(0) > 0 is an electrical conductor . • Charged particles must be free, Ω = 0. The Drude Model Electrical conductors Peter Hertel Overview Model Dielectric medium Permittivity of metals • A material with σ(0) = 0 is an electrical insulator . It cannot transport direct currents (DC). • A material with σ(0) > 0 is an electrical conductor . Electrical conductors • Charged particles must be free, Ω = 0. Faraday effect • which means Hall effect σ(ω) = N q2 1 m Γ − iω The Drude Model Electrical conductors Peter Hertel Overview Model Dielectric medium Permittivity of metals • A material with σ(0) = 0 is an electrical insulator . It cannot transport direct currents (DC). • A material with σ(0) > 0 is an electrical conductor . Electrical conductors • Charged particles must be free, Ω = 0. Faraday effect • which means Hall effect σ(ω) = • or N q2 1 m Γ − iω σ(ω) 1 = σ(0) 1 − iω/Γ The Drude Model Electrical conductors Peter Hertel Overview Model Dielectric medium Permittivity of metals • A material with σ(0) = 0 is an electrical insulator . It cannot transport direct currents (DC). • A material with σ(0) > 0 is an electrical conductor . Electrical conductors • Charged particles must be free, Ω = 0. Faraday effect • which means Hall effect σ(ω) = • or N q2 1 m Γ − iω σ(ω) 1 = σ(0) 1 − iω/Γ • Note that the DC conductivity is always positive . The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect Georg Simon Ohm, German physicist, 1789-1854 The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect External static magnetic field The Drude Model External static magnetic field Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • apply a quasi-static external induction B The Drude Model External static magnetic field Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • apply a quasi-static external induction B • the typical electron obeys m(ẍ + Γẋ + Ω2 x) = q(E + ẋ × B) The Drude Model External static magnetic field Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • apply a quasi-static external induction B • the typical electron obeys m(ẍ + Γẋ + Ω2 x) = q(E + ẋ × B) • Fourier transform this m(−ω 2 − iωΓ + Ω2 )x̃ = q(Ẽ − iω x̃ × B) The Drude Model External static magnetic field Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • apply a quasi-static external induction B • the typical electron obeys m(ẍ + Γẋ + Ω2 x) = q(E + ẋ × B) • Fourier transform this m(−ω 2 − iωΓ + Ω2 )x̃ = q(Ẽ − iω x̃ × B) • assume B = Bêz The Drude Model External static magnetic field Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • apply a quasi-static external induction B • the typical electron obeys m(ẍ + Γẋ + Ω2 x) = q(E + ẋ × B) • Fourier transform this m(−ω 2 − iωΓ + Ω2 )x̃ = q(Ẽ − iω x̃ × B) • assume B = Bêz • assume circularly polarized light √ Ẽ = Ẽ± ê± where ê± = (êx + iêy )/ 2 The Drude Model External static magnetic field Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • apply a quasi-static external induction B • the typical electron obeys m(ẍ + Γẋ + Ω2 x) = q(E + ẋ × B) • Fourier transform this m(−ω 2 − iωΓ + Ω2 )x̃ = q(Ẽ − iω x̃ × B) • assume B = Bêz • assume circularly polarized light √ Ẽ = Ẽ± ê± where ê± = (êx + iêy )/ 2 • try x̃ = x̃± ê± The Drude Model External static magnetic field Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • apply a quasi-static external induction B • the typical electron obeys m(ẍ + Γẋ + Ω2 x) = q(E + ẋ × B) • Fourier transform this m(−ω 2 − iωΓ + Ω2 )x̃ = q(Ẽ − iω x̃ × B) • assume B = Bêz • assume circularly polarized light √ Ẽ = Ẽ± ê± where ê± = (êx + iêy )/ 2 • try x̃ = x̃± ê± • note ê± × êz = ∓iê± The Drude Model External static magnetic field Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • apply a quasi-static external induction B • the typical electron obeys m(ẍ + Γẋ + Ω2 x) = q(E + ẋ × B) • Fourier transform this m(−ω 2 − iωΓ + Ω2 )x̃ = q(Ẽ − iω x̃ × B) • assume B = Bêz • assume circularly polarized light √ Ẽ = Ẽ± ê± where ê± = (êx + iêy )/ 2 • try x̃ = x̃± ê± • note ê± × êz = ∓iê± • therefore m(−ω 2 − iωΓ + Ω2 )x̃± = q(Ẽ± ∓ ωBx̃± ) The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect Faraday effect The Drude Model Faraday effect Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • m(−ω 2 − iωΓ + Ω2 )x̃± = q(Ẽ± ∓ ωBx̃± ) The Drude Model Faraday effect Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • m(−ω 2 − iωΓ + Ω2 )x̃± = q(Ẽ± ∓ ωBx̃± ) • therefore x̃± = m(Ω2 q Ẽ± − iωΓ − ω 2 ) ± qωB The Drude Model Faraday effect Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • m(−ω 2 − iωΓ + Ω2 )x̃± = q(Ẽ± ∓ ωBx̃± ) • therefore x̃± = m(Ω2 q Ẽ± − iωΓ − ω 2 ) ± qωB • recall P̃ = N q x̃ = 0 χẼ The Drude Model Faraday effect Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors • m(−ω 2 − iωΓ + Ω2 )x̃± = q(Ẽ± ∓ ωBx̃± ) • therefore x̃± = m(Ω2 q Ẽ± − iωΓ − ω 2 ) ± qωB Faraday effect • recall P̃ = N q x̃ = 0 χẼ Hall effect • effect of quasi-static induction B is χ± (ω) = N q2 1 2 0 m Ω − iωΓ − ω 2 ± (q/m)ωB The Drude Model Faraday effect Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors • m(−ω 2 − iωΓ + Ω2 )x̃± = q(Ẽ± ∓ ωBx̃± ) • therefore x̃± = m(Ω2 q Ẽ± − iωΓ − ω 2 ) ± qωB Faraday effect • recall P̃ = N q x̃ = 0 χẼ Hall effect • effect of quasi-static induction B is N q2 1 2 0 m Ω − iωΓ − ω 2 ± (q/m)ωB • left and right handed polarized light sees different susceptibility χ± (ω) = The Drude Model Faraday effect Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors • m(−ω 2 − iωΓ + Ω2 )x̃± = q(Ẽ± ∓ ωBx̃± ) • therefore x̃± = m(Ω2 q Ẽ± − iωΓ − ω 2 ) ± qωB Faraday effect • recall P̃ = N q x̃ = 0 χẼ Hall effect • effect of quasi-static induction B is N q2 1 2 0 m Ω − iωΓ − ω 2 ± (q/m)ωB • left and right handed polarized light sees different susceptibility χ± (ω) = • Faraday effect The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect Michael Faraday, English physicist, 1791-1867 The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect Remarks The Drude Model Remarks Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • B is always small (in natural units) The Drude Model Remarks Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • B is always small (in natural units) • ij (ω; B) = ij (ω; 0) + iK(ω)ijk Bk The Drude Model Remarks Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • B is always small (in natural units) • ij (ω; B) = ij (ω; 0) + iK(ω)ijk Bk • linear magneto-optic effect The Drude Model Remarks Peter Hertel Overview Model Dielectric medium • B is always small (in natural units) • ij (ω; B) = ij (ω; 0) + iK(ω)ijk Bk Permittivity of metals • linear magneto-optic effect Electrical conductors • Faraday constant is Faraday effect Hall effect K(ω) = N q3 ω 2 2 0 m (Ω − iωΓ − ω 2 )2 The Drude Model Remarks Peter Hertel Overview Model Dielectric medium • B is always small (in natural units) • ij (ω; B) = ij (ω; 0) + iK(ω)ijk Bk Permittivity of metals • linear magneto-optic effect Electrical conductors • Faraday constant is Faraday effect Hall effect N q3 ω 2 2 0 m (Ω − iωΓ − ω 2 )2 • K(ω) is real in transparency window K(ω) = The Drude Model Remarks Peter Hertel Overview Model Dielectric medium • B is always small (in natural units) • ij (ω; B) = ij (ω; 0) + iK(ω)ijk Bk Permittivity of metals • linear magneto-optic effect Electrical conductors • Faraday constant is Faraday effect Hall effect N q3 ω 2 2 0 m (Ω − iωΓ − ω 2 )2 • K(ω) is real in transparency window K(ω) = • i. e. if ω is far away form Ω The Drude Model Remarks Peter Hertel Overview Model Dielectric medium • B is always small (in natural units) • ij (ω; B) = ij (ω; 0) + iK(ω)ijk Bk Permittivity of metals • linear magneto-optic effect Electrical conductors • Faraday constant is Faraday effect Hall effect N q3 ω 2 2 0 m (Ω − iωΓ − ω 2 )2 • K(ω) is real in transparency window K(ω) = • i. e. if ω is far away form Ω • Faraday effect distinguishes between forward and backward propagation The Drude Model Remarks Peter Hertel Overview Model Dielectric medium • B is always small (in natural units) • ij (ω; B) = ij (ω; 0) + iK(ω)ijk Bk Permittivity of metals • linear magneto-optic effect Electrical conductors • Faraday constant is Faraday effect Hall effect N q3 ω 2 2 0 m (Ω − iωΓ − ω 2 )2 • K(ω) is real in transparency window K(ω) = • i. e. if ω is far away form Ω • Faraday effect distinguishes between forward and backward propagation • optical isolator The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect Conduction in a magnetic field The Drude Model Conduction in a magnetic field Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • set the spring constant mΩ2 = 0 The Drude Model Conduction in a magnetic field Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • set the spring constant mΩ2 = 0 • study AC electric field Ẽ The Drude Model Conduction in a magnetic field Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • set the spring constant mΩ2 = 0 • study AC electric field Ẽ • and static magnetic induction B The Drude Model Conduction in a magnetic field Peter Hertel Overview Model Dielectric medium • set the spring constant mΩ2 = 0 • study AC electric field Ẽ Permittivity of metals • and static magnetic induction B Electrical conductors • solve Faraday effect Hall effect m(−ω 2 − iΓω)x̃ = q(Ẽ − iω x̃ × B) The Drude Model Conduction in a magnetic field Peter Hertel Overview Model Dielectric medium • set the spring constant mΩ2 = 0 • study AC electric field Ẽ Permittivity of metals • and static magnetic induction B Electrical conductors • solve Faraday effect Hall effect m(−ω 2 − iΓω)x̃ = q(Ẽ − iω x̃ × B) • or x̃ = q 1 1 {Ẽ − iω x̃ × B} m −iω Γ − iω The Drude Model Conduction in a magnetic field Peter Hertel Overview Model Dielectric medium • set the spring constant mΩ2 = 0 • study AC electric field Ẽ Permittivity of metals • and static magnetic induction B Electrical conductors • solve Faraday effect Hall effect m(−ω 2 − iΓω)x̃ = q(Ẽ − iω x̃ × B) • or q 1 1 {Ẽ − iω x̃ × B} m −iω Γ − iω • by iteration q 1 x̃ = . . . {Ẽ + Ẽ × B} m Γ − iω x̃ = The Drude Model Conduction in a magnetic field Peter Hertel Overview Model Dielectric medium • set the spring constant mΩ2 = 0 • study AC electric field Ẽ Permittivity of metals • and static magnetic induction B Electrical conductors • solve Faraday effect Hall effect m(−ω 2 − iΓω)x̃ = q(Ẽ − iω x̃ × B) • or q 1 1 {Ẽ − iω x̃ × B} m −iω Γ − iω • by iteration q 1 x̃ = . . . {Ẽ + Ẽ × B} m Γ − iω • Ohmic current ∝ Ẽ and Hall current ∝ Ẽ × B x̃ = The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect Hall effect, schematilly The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect Hall effect The Drude Model Hall effect Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • Hall current usually forbidden by boundary conditions The Drude Model Hall effect Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • Hall current usually forbidden by boundary conditions • Hall field Ẽ H = − q 1 Ẽ × B m Γ − iω The Drude Model Hall effect Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • Hall current usually forbidden by boundary conditions • Hall field q 1 Ẽ × B m Γ − iω • replace Ẽ by Ẽ + Ẽ H Ẽ H = − The Drude Model Hall effect Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • Hall current usually forbidden by boundary conditions • Hall field q 1 Ẽ × B m Γ − iω • replace Ẽ by Ẽ + Ẽ H Ẽ H = − • Ẽ H × B can be neglected The Drude Model Hall effect Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • Hall current usually forbidden by boundary conditions • Hall field q 1 Ẽ × B m Γ − iω • replace Ẽ by Ẽ + Ẽ H Ẽ H = − • Ẽ H × B can be neglected • current J̃ (ω) = σ(ω)Ẽ(ω) as usual The Drude Model Hall effect Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • Hall current usually forbidden by boundary conditions • Hall field q 1 Ẽ × B m Γ − iω • replace Ẽ by Ẽ + Ẽ H Ẽ H = − • Ẽ H × B can be neglected • current J̃ (ω) = σ(ω)Ẽ(ω) as usual • additional Hall field Ẽ H (ω) = R(ω)J̃ (ω) × B The Drude Model Hall effect Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • Hall current usually forbidden by boundary conditions • Hall field q 1 Ẽ × B m Γ − iω • replace Ẽ by Ẽ + Ẽ H Ẽ H = − • Ẽ H × B can be neglected • current J̃ (ω) = σ(ω)Ẽ(ω) as usual • additional Hall field Ẽ H (ω) = R(ω)J̃ (ω) × B • Hall constant R = −1/N q does not depend on ω The Drude Model Hall effect Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • Hall current usually forbidden by boundary conditions • Hall field q 1 Ẽ × B m Γ − iω • replace Ẽ by Ẽ + Ẽ H Ẽ H = − • Ẽ H × B can be neglected • current J̃ (ω) = σ(ω)Ẽ(ω) as usual • additional Hall field Ẽ H (ω) = R(ω)J̃ (ω) × B • Hall constant R = −1/N q does not depend on ω • . . . if there is a dominant charge carrier. The Drude Model Hall effect Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect • Hall current usually forbidden by boundary conditions • Hall field q 1 Ẽ × B m Γ − iω • replace Ẽ by Ẽ + Ẽ H Ẽ H = − • Ẽ H × B can be neglected • current J̃ (ω) = σ(ω)Ẽ(ω) as usual • additional Hall field Ẽ H (ω) = R(ω)J̃ (ω) × B • Hall constant R = −1/N q does not depend on ω • . . . if there is a dominant charge carrier. • R has different sign for electrons and holes The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday effect Hall effect Edwin Hall, US-American physicist, 1855-1938

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