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

The Drude Model Overview Peter Hertel

Overview

Model

The Drude Model Overview Peter Hertel

Overview

Model

The Drude Model Overview Peter Hertel

Overview

Model

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(−ω2 − iωΓ + Ω2)x˜ = qE˜ • solution is q E˜ (ω) x˜(ω) = m Ω2 − ω2 − iωΓ

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(−ω2 − iωΓ + Ω2)x˜ = qE˜ • solution is q E˜ (ω) x˜(ω) = m Ω2 − ω2 − iωΓ

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(−ω2 − iωΓ + Ω2)x˜ = qE˜ • solution is q E˜ (ω) x˜(ω) = m Ω2 − ω2 − iωΓ

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(−ω2 − iωΓ + Ω2)x˜ = qE˜ • solution is q E˜ (ω) x˜(ω) = m Ω2 − ω2 − iωΓ

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(−ω2 − iωΓ + Ω2)x˜ = qE˜ • solution is q E˜ (ω) x˜(ω) = m Ω2 − ω2 − iωΓ

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(−ω2 − iωΓ + Ω2)x˜ = qE˜ • solution is q E˜ (ω) x˜(ω) = m Ω2 − ω2 − iωΓ

The Drude Model Model Peter Hertel

Overview

Model • consider a typical electron Dielectric • medium denote by x = x(t) the deviation from its equilibrium

The Drude Model Model Peter Hertel

Overview

Model • consider a typical electron Dielectric • medium denote by x = x(t) the deviation from its equilibrium

The Drude Model Model Peter Hertel

Overview

Model • consider a typical electron Dielectric • medium denote by x = x(t) the deviation from its equilibrium

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(−ω2 − iωΓ + Ω2)x˜ = qE˜ • solution is q E˜ (ω) x˜(ω) = m Ω2 − ω2 − iωΓ • dipole moment of typical electron is p˜ = qx˜ • recall q E˜ (ω) x˜(ω) = m Ω2 − ω2 − iωΓ • there are N typical electrons per unit volume

• polarization is P˜ = Nqx˜ = 0χE˜ • susceptibility is Nq2 1 χ(ω) = 2 2 0m Ω − ω − iωΓ • 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˜ (ω) x˜(ω) = m Ω2 − ω2 − iωΓ • there are N typical electrons per unit volume

• polarization is P˜ = Nqx˜ = 0χE˜ • susceptibility is Nq2 1 χ(ω) = 2 2 0m Ω − ω − iωΓ • 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(ω) = 2 2 0m Ω − ω − iωΓ • 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˜ (ω) metals x˜(ω) = m Ω2 − ω2 − iωΓ Electrical conductors

Faraday eﬀect

Hall eﬀect • polarization is P˜ = Nqx˜ = 0χE˜ • susceptibility is Nq2 1 χ(ω) = 2 2 0m Ω − ω − iωΓ • 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˜ (ω) metals x˜(ω) = m Ω2 − ω2 − iωΓ Electrical conductors • there are N typical electrons per unit volume Faraday eﬀect

Hall eﬀect • susceptibility is Nq2 1 χ(ω) = 2 2 0m Ω − ω − iωΓ • 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

The Drude Model Polarization Peter Hertel

Overview Model • dipole moment of typical electron is p˜ = qx˜ Dielectric medium • recall

The Drude Model Polarization Peter Hertel

Overview Model • dipole moment of typical electron is p˜ = qx˜ Dielectric medium • recall

Permittivity of q E˜ (ω) metals x˜(ω) = m Ω2 − ω2 − iωΓ Electrical conductors • there are N typical electrons per unit volume Faraday eﬀect • polarization is P˜ = Nqx˜ = 0χE˜ Hall eﬀect • susceptibility is Nq2 1 χ(ω) = 2 2 0m Ω − ω − iωΓ • in particular Nq2 χ(0) = 2 > 0 0mΩ • . . . as it should be • 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 1 − s2 R 0(s) = (1 − s2)2 + γ2s2 • absorption γs D 00(s) = (1 − s2)2 + γ2s2 • limiting cases: s = 0, s = 1, s → ∞, small γ

The Drude Model Discussion I Peter Hertel

Overview

Model

Dielectric medium

Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall eﬀect • Introduce R(ω) = χ(ω)/χ(0), s = ω/Ω and γ = Γ/Ω as normalized quantities. • refraction 1 − s2 R 0(s) = (1 − s2)2 + γ2s2 • absorption γs D 00(s) = (1 − s2)2 + γ2s2 • limiting cases: s = 0, s = 1, s → ∞, small γ

The Drude Model Discussion I Peter Hertel

Overview

Model 0 00 Dielectric • decompose susceptibility χ(ω) = χ (ω) + iχ (ω) into medium refractive part χ 0 and absorptive part χ 00 Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall eﬀect • refraction 1 − s2 R 0(s) = (1 − s2)2 + γ2s2 • absorption γs D 00(s) = (1 − s2)2 + γ2s2 • limiting cases: s = 0, s = 1, s → ∞, small γ

The Drude Model Discussion I Peter Hertel

Overview

Faraday eﬀect

Hall eﬀect • absorption γs D 00(s) = (1 − s2)2 + γ2s2 • limiting cases: s = 0, s = 1, s → ∞, small γ

The Drude Model Discussion I Peter Hertel

Overview

Model 0 00 Dielectric • decompose susceptibility χ(ω) = χ (ω) + iχ (ω) into medium refractive part χ 0 and absorptive part χ 00 Permittivity of metals • Introduce R(ω) = χ(ω)/χ(0), s = ω/Ω and γ = Γ/Ω as Electrical conductors normalized quantities. Faraday eﬀect • refraction Hall eﬀect 1 − s2 R 0(s) = (1 − s2)2 + γ2s2 • limiting cases: s = 0, s = 1, s → ∞, small γ

The Drude Model Discussion I Peter Hertel

Overview

Peter Hertel

Overview

Model

Dielectric 5 medium

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 • 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 Discussion II Peter Hertel

Overview

Model

Dielectric medium

Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall eﬀect • 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 Discussion II Peter Hertel

Overview

Model Dielectric • medium For small frequencies (as compared with Ω) the

Permittivity of susceptibility is practically real. metals

Electrical conductors

Faraday eﬀect

Hall eﬀect • ∂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 Discussion II Peter Hertel

Overview

Model Dielectric • medium For small frequencies (as compared with Ω) the

Permittivity of susceptibility is practically real. metals • This is the realm of classical optics Electrical conductors

Faraday eﬀect

Hall eﬀect • 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 Discussion II Peter Hertel

Overview

Model Dielectric • medium For small frequencies (as compared with Ω) the

Permittivity of susceptibility is practically real. metals • This is the realm of classical optics Electrical conductors • ∂χ/∂ω is positive – normal dispersion Faraday eﬀect

Hall eﬀect • 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 Discussion II Peter Hertel

Overview

Model Dielectric • medium For small frequencies (as compared with Ω) the

The Drude Model Discussion II Peter Hertel

Overview

Model Dielectric • medium For small frequencies (as compared with Ω) the

Overview

Model Dielectric • medium For small frequencies (as compared with Ω) the

Permittivity of susceptibility is practically real. metals • This is the realm of classical optics Electrical conductors • ∂χ/∂ω is positive – normal dispersion Faraday effect • In the vicinity of ω = Ω absorption is large. Negative Hall effect 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 . . . • χ(ω) must be the Fourier transform of a causal response function G = G(τ) • as defined in Z P (t) = 0 dτG(τ)E(t − τ) • check this for −iωτ Z dω e G(τ) = a 2π Ω2 − ω2 − iωΓ • poles at iΓ p ω = − ± ω¯ whereω ¯ = + Ω2 − Γ2/4 1,2 2 • Indeed, G(τ) = 0 for τ < 0 • for τ > 0 Nq2 sinωτ ¯ −Γτ/2 G(τ) = e 0m ω¯

The Drude Model Kramers-Kronig relation I Peter Hertel

Overview

Model

Dielectric medium

Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall eﬀect • as deﬁned in Z P (t) = 0 dτG(τ)E(t − τ) • check this for −iωτ Z dω e G(τ) = a 2π Ω2 − ω2 − iωΓ • poles at iΓ p ω = − ± ω¯ whereω ¯ = + Ω2 − Γ2/4 1,2 2 • Indeed, G(τ) = 0 for τ < 0 • for τ > 0 Nq2 sinωτ ¯ −Γτ/2 G(τ) = e 0m ω¯

The Drude Model Kramers-Kronig relation I Peter Hertel

Overview • χ(ω) must be the Fourier transform of a causal response Model function G = G(τ) Dielectric medium

Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall eﬀect • check this for −iωτ Z dω e G(τ) = a 2π Ω2 − ω2 − iωΓ • poles at iΓ p ω = − ± ω¯ whereω ¯ = + Ω2 − Γ2/4 1,2 2 • Indeed, G(τ) = 0 for τ < 0 • for τ > 0 Nq2 sinωτ ¯ −Γτ/2 G(τ) = e 0m ω¯

The Drude Model Kramers-Kronig relation I Peter Hertel

Overview • χ(ω) must be the Fourier transform of a causal response Model function G = G(τ) Dielectric • as deﬁned in medium Z Permittivity of metals P (t) = 0 dτG(τ)E(t − τ)

Electrical conductors

Faraday eﬀect

Hall eﬀect • poles at iΓ p ω = − ± ω¯ whereω ¯ = + Ω2 − Γ2/4 1,2 2 • Indeed, G(τ) = 0 for τ < 0 • for τ > 0 Nq2 sinωτ ¯ −Γτ/2 G(τ) = e 0m ω¯

The Drude Model Kramers-Kronig relation I Peter Hertel

Overview • χ(ω) must be the Fourier transform of a causal response Model function G = G(τ) Dielectric • as deﬁned in medium Z Permittivity of metals P (t) = 0 dτG(τ)E(t − τ)

Electrical conductors • check this for −iωτ Faraday eﬀect Z dω e G(τ) = a Hall eﬀect 2π Ω2 − ω2 − iωΓ • Indeed, G(τ) = 0 for τ < 0 • for τ > 0 Nq2 sinωτ ¯ −Γτ/2 G(τ) = e 0m ω¯

The Drude Model Kramers-Kronig relation I Peter Hertel

Overview • χ(ω) must be the Fourier transform of a causal response Model function G = G(τ) Dielectric • as deﬁned in medium Z Permittivity of metals P (t) = 0 dτG(τ)E(t − τ)

Electrical conductors • check this for −iωτ Faraday eﬀect Z dω e G(τ) = a Hall eﬀect 2π Ω2 − ω2 − iωΓ • poles at iΓ p ω = − ± ω¯ whereω ¯ = + Ω2 − Γ2/4 1,2 2 • for τ > 0 Nq2 sinωτ ¯ −Γτ/2 G(τ) = e 0m ω¯

The Drude Model Kramers-Kronig relation I Peter Hertel

Overview • χ(ω) must be the Fourier transform of a causal response Model function G = G(τ) Dielectric • as deﬁned in medium Z Permittivity of metals P (t) = 0 dτG(τ)E(t − τ)

Electrical conductors • check this for −iωτ Faraday eﬀect Z dω e G(τ) = a Hall eﬀect 2π Ω2 − ω2 − iωΓ • poles at iΓ p ω = − ± ω¯ whereω ¯ = + Ω2 − Γ2/4 1,2 2 • Indeed, G(τ) = 0 for τ < 0 The Drude Model Kramers-Kronig relation I Peter Hertel

Overview • χ(ω) must be the Fourier transform of a causal response Model function G = G(τ) Dielectric • as deﬁned in medium Z Permittivity of metals P (t) = 0 dτG(τ)E(t − τ)

Electrical conductors • check this for −iωτ Faraday eﬀect Z dω e G(τ) = a Hall eﬀect 2π Ω2 − ω2 − iωΓ • poles at iΓ p ω = − ± ω¯ whereω ¯ = + Ω2 − Γ2/4 1,2 2 • Indeed, G(τ) = 0 for τ < 0 • for τ > 0 Nq2 sinωτ ¯ −Γτ/2 G(τ) = e 0m ω¯ • causal response function: G(τ) = θ(τ)G(τ) • apply the convolution theorem Z du χ(ω) = χ(u)θ˜(ω − u) 2π • Fourier transform of Heaviside function is 1 θ˜(ω) = lim 0<η→0 η − iω • dispersion , or Kramers-Kronig relations Z du uχ 00(u) χ 0(ω) = 2Pr π u2 − ω2 Z du ωχ 0(u) χ 00(ω) = 2Pr π ω2 − u2

The Drude Model Kramers-Kronig relation II Peter Hertel

Overview

Model

Dielectric medium

Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall eﬀect Z du ωχ 0(u) χ 00(ω) = 2Pr π ω2 − u2

• apply the convolution theorem Z du χ(ω) = χ(u)θ˜(ω − u) 2π • Fourier transform of Heaviside function is 1 θ˜(ω) = lim 0<η→0 η − iω • dispersion , or Kramers-Kronig relations Z du uχ 00(u) χ 0(ω) = 2Pr π u2 − ω2

The Drude Model Kramers-Kronig relation II Peter Hertel

Overview Model • causal response function: G(τ) = θ(τ)G(τ) Dielectric medium

Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall eﬀect Z du ωχ 0(u) χ 00(ω) = 2Pr π ω2 − u2

• Fourier transform of Heaviside function is 1 θ˜(ω) = lim 0<η→0 η − iω • dispersion , or Kramers-Kronig relations Z du uχ 00(u) χ 0(ω) = 2Pr π u2 − ω2

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 Z du Permittivity of χ(ω) = χ(u)θ˜(ω − u) metals 2π Electrical conductors

Faraday eﬀect

Hall eﬀect Z du ωχ 0(u) χ 00(ω) = 2Pr π ω2 − u2

• dispersion , or Kramers-Kronig relations Z du uχ 00(u) χ 0(ω) = 2Pr π u2 − ω2

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 Z du Permittivity of χ(ω) = χ(u)θ˜(ω − u) metals 2π Electrical conductors • Fourier transform of Heaviside function is 1 Faraday eﬀect θ˜(ω) = lim Hall eﬀect 0<η→0 η − iω Z du ωχ 0(u) χ 00(ω) = 2Pr π ω2 − u2

The Drude Model Kramers-Kronig relation II Peter Hertel

Peter Hertel

Overview

Model

Dielectric medium

Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall eﬀect

Dispersion of white light • consider a typical conduction band electron • it behaves as a free quasi-particle • recall m(x¨ + Γx˙ + Ω2x) = qE • spring constant mΩ2 vanishes • m is eﬀective mass • therefore ω2 (ω) = 1 − p ω2 + iωΓ • plasma frequency ωp 2 2 Nq ωp = 0m • correction for ω ωp ω2 (ω) = − p ∞ ω2 + iωΓ

The Drude Model Free quasi-electrons Peter Hertel

Overview

Model

Dielectric medium

Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall eﬀect • it behaves as a free quasi-particle • recall m(x¨ + Γx˙ + Ω2x) = qE • spring constant mΩ2 vanishes • m is eﬀective mass • therefore ω2 (ω) = 1 − p ω2 + iωΓ • plasma frequency ωp 2 2 Nq ωp = 0m • correction for ω ωp ω2 (ω) = − p ∞ ω2 + iωΓ

The Drude Model Free quasi-electrons Peter Hertel

Overview • consider a typical conduction band electron Model

Dielectric medium

Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall eﬀect • recall m(x¨ + Γx˙ + Ω2x) = qE • spring constant mΩ2 vanishes • m is eﬀective mass • therefore ω2 (ω) = 1 − p ω2 + iωΓ • plasma frequency ωp 2 2 Nq ωp = 0m • correction for ω ωp ω2 (ω) = − p ∞ ω2 + iωΓ

The Drude Model Free quasi-electrons Peter Hertel

Overview • consider a typical conduction band electron Model • it behaves as a free quasi-particle Dielectric medium

Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall eﬀect • spring constant mΩ2 vanishes • m is eﬀective mass • therefore ω2 (ω) = 1 − p ω2 + iωΓ • plasma frequency ωp 2 2 Nq ωp = 0m • correction for ω ωp ω2 (ω) = − p ∞ ω2 + iωΓ

The Drude Model Free quasi-electrons Peter Hertel

Overview • consider a typical conduction band electron Model • it behaves as a free quasi-particle Dielectric 2 medium • recall m(x¨ + Γx˙ + Ω x) = qE Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall eﬀect • m is eﬀective mass • therefore ω2 (ω) = 1 − p ω2 + iωΓ • plasma frequency ωp 2 2 Nq ωp = 0m • correction for ω ωp ω2 (ω) = − p ∞ ω2 + iωΓ

The Drude Model Free quasi-electrons Peter Hertel

Overview • consider a typical conduction band electron Model • it behaves as a free quasi-particle Dielectric 2 medium • recall m(x¨ + Γx˙ + Ω x) = qE Permittivity of 2 metals • spring constant mΩ vanishes

Electrical conductors

Faraday eﬀect

Hall eﬀect • therefore ω2 (ω) = 1 − p ω2 + iωΓ • plasma frequency ωp 2 2 Nq ωp = 0m • correction for ω ωp ω2 (ω) = − p ∞ ω2 + iωΓ

The Drude Model Free quasi-electrons Peter Hertel

Faraday eﬀect

Hall eﬀect • plasma frequency ωp 2 2 Nq ωp = 0m • correction for ω ωp ω2 (ω) = − p ∞ ω2 + iωΓ

The Drude Model Free quasi-electrons Peter Hertel

Faraday eﬀect • therefore 2 Hall eﬀect ω (ω) = 1 − p ω2 + iωΓ • correction for ω ωp ω2 (ω) = − p ∞ ω2 + iωΓ

The Drude Model Free quasi-electrons Peter Hertel

Faraday eﬀect • therefore 2 Hall eﬀect ω (ω) = 1 − p ω2 + iωΓ • plasma frequency ωp 2 2 Nq ωp = 0m The Drude Model Free quasi-electrons Peter Hertel

Faraday eﬀect • therefore 2 Hall eﬀect ω (ω) = 1 − p ω2 + iωΓ • plasma frequency ωp 2 2 Nq ωp = 0m • correction for ω ωp ω2 (ω) = − p ∞ ω2 + iωΓ • 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 Example: gold Peter Hertel

Overview

Model

Dielectric medium

Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall eﬀect • as determined by Johnson and Christy in 1972

The Drude Model Example: gold Peter Hertel

Overview

Model Dielectric • Drude model parameters for gold medium

Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall eﬀect • ∞ = 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 Example: gold Peter Hertel

Overview

Model Dielectric • Drude model parameters for gold medium

Permittivity of • as determined by Johnson and Christy in 1972 metals

Electrical conductors

Faraday eﬀect

Hall eﬀect • ~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 Example: gold Peter Hertel

Overview

Model Dielectric • Drude model parameters for gold medium

Permittivity of • as determined by Johnson and Christy in 1972 metals • ∞ = 9.5 Electrical conductors

Faraday eﬀect

Hall eﬀect • ~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 Example: gold Peter Hertel

Overview

Model Dielectric • Drude model parameters for gold medium

Permittivity of • as determined by Johnson and Christy in 1972 metals • ∞ = 9.5 Electrical conductors • ~ωp = 8.95 eV Faraday eﬀect

Hall eﬀect • 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 Example: gold Peter Hertel

Overview

Model Dielectric • Drude model parameters for gold medium

Permittivity of • as determined by Johnson and Christy in 1972 metals • ∞ = 9.5 Electrical conductors • ~ωp = 8.95 eV Faraday eﬀect • Γ = 0.069 eV Hall eﬀect ~ • 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 Example: gold Peter Hertel

Overview

Model Dielectric • Drude model parameters for gold medium

Permittivity of • as determined by Johnson and Christy in 1972 metals • ∞ = 9.5 Electrical conductors • ~ωp = 8.95 eV Faraday effect • Γ = 0.069 eV Hall effect ~ • with these parameters the Drude model fits optical measurements well for ~ω < 2.25 eV (green) • This allows surface plasmon polaritons (SPP)

The Drude Model Example: gold Peter Hertel

Overview

Model Dielectric • Drude model parameters for gold medium

Overview

Model Dielectric • Drude model parameters for gold medium

Overview

Model 0

Dielectric medium Permittivity of −20 metals

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. • consider a typical charged particle • recall m(x¨ + Γx˙ + Ω2x) = qE • electric current density J = Nqx˙ • Fourier transformed: J˜ = Nq(−iω)x˜ • recall q E˜ (ω) x˜(ω) = m Ω2 − ω2 − iωΓ • Ohm’s law J˜(ω) = σ(ω)E˜ (ω) • conductivity is Nq2 −iω σ(ω) = m Ω2 − ω2 − iωΓ

The Drude Model Electrical conductivity Peter Hertel

Overview

Model

Dielectric medium

Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall eﬀect • recall m(x¨ + Γx˙ + Ω2x) = qE • electric current density J = Nqx˙ • Fourier transformed: J˜ = Nq(−iω)x˜ • recall q E˜ (ω) x˜(ω) = m Ω2 − ω2 − iωΓ • Ohm’s law J˜(ω) = σ(ω)E˜ (ω) • conductivity is Nq2 −iω σ(ω) = m Ω2 − ω2 − iωΓ

The Drude Model Electrical conductivity Peter Hertel

Overview

Model • consider a typical charged particle

Dielectric medium

Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall eﬀect • electric current density J = Nqx˙ • Fourier transformed: J˜ = Nq(−iω)x˜ • recall q E˜ (ω) x˜(ω) = m Ω2 − ω2 − iωΓ • Ohm’s law J˜(ω) = σ(ω)E˜ (ω) • conductivity is Nq2 −iω σ(ω) = m Ω2 − ω2 − iωΓ

The Drude Model Electrical conductivity Peter Hertel

Overview

Model • consider a typical charged particle 2 Dielectric • recall m(x¨ + Γx˙ + Ω x) = qE medium

Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall eﬀect • Fourier transformed: J˜ = Nq(−iω)x˜ • recall q E˜ (ω) x˜(ω) = m Ω2 − ω2 − iωΓ • Ohm’s law J˜(ω) = σ(ω)E˜ (ω) • conductivity is Nq2 −iω σ(ω) = m Ω2 − ω2 − iωΓ

The Drude Model Electrical conductivity Peter Hertel

Overview

Model • consider a typical charged particle 2 Dielectric • recall m(x¨ + Γx˙ + Ω x) = qE medium

Permittivity of • electric current density J = Nqx˙ metals

Electrical conductors

Faraday eﬀect

Hall eﬀect • recall q E˜ (ω) x˜(ω) = m Ω2 − ω2 − iωΓ • Ohm’s law J˜(ω) = σ(ω)E˜ (ω) • conductivity is Nq2 −iω σ(ω) = m Ω2 − ω2 − iωΓ

The Drude Model Electrical conductivity Peter Hertel

Overview

Model • consider a typical charged particle 2 Dielectric • recall m(x¨ + Γx˙ + Ω x) = qE medium

Permittivity of • electric current density J = Nqx˙ metals • Fourier transformed: J˜ = Nq(−iω)x˜ Electrical conductors

Faraday eﬀect

Hall eﬀect • Ohm’s law J˜(ω) = σ(ω)E˜ (ω) • conductivity is Nq2 −iω σ(ω) = m Ω2 − ω2 − iωΓ

The Drude Model Electrical conductivity Peter Hertel

Overview

Model • consider a typical charged particle 2 Dielectric • recall m(x¨ + Γx˙ + Ω x) = qE medium

The Drude Model Electrical conductivity Peter Hertel

Overview

Model • consider a typical charged particle 2 Dielectric • recall m(x¨ + Γx˙ + Ω x) = qE medium

Overview

Model • consider a typical charged particle 2 Dielectric • recall m(x¨ + Γx˙ + Ω x) = qE medium

Permittivity of • electric current density J = Nqx˙ metals • Fourier transformed: J˜ = Nq(−iω)x˜ Electrical conductors • recall Faraday eﬀect q E˜ (ω) Hall eﬀect x˜(ω) = m Ω2 − ω2 − iωΓ • Ohm’s law J˜(ω) = σ(ω)E˜ (ω) • conductivity is Nq2 −iω σ(ω) = m Ω2 − ω2 − iωΓ • 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. • which means Nq2 1 σ(ω) = m Γ − iω • or σ(ω) 1 = σ(0) 1 − iω/Γ • Note that the DC conductivity is always positive.

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 conductor. • Charged particles must be free, Ω = 0. • which means Nq2 1 σ(ω) = m Γ − iω • or σ(ω) 1 = σ(0) 1 − iω/Γ • Note that the DC conductivity is always positive.

The Drude Model Electrical conductors Peter Hertel

Overview

Model

Dielectric • A material with σ(0) = 0 is an electrical insulator . It medium cannot transport direct currents (DC). Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall eﬀect • Charged particles must be free, Ω = 0. • which means Nq2 1 σ(ω) = m Γ − iω • or σ(ω) 1 = σ(0) 1 − iω/Γ • Note that the DC conductivity is always positive.

The Drude Model Electrical conductors Peter Hertel

Overview

Model

Faraday eﬀect

Hall eﬀect • which means Nq2 1 σ(ω) = m Γ − iω • or σ(ω) 1 = σ(0) 1 − iω/Γ • Note that the DC conductivity is always positive.

The Drude Model Electrical conductors Peter Hertel

Overview

Model

Hall eﬀect • or σ(ω) 1 = σ(0) 1 − iω/Γ • Note that the DC conductivity is always positive.

The Drude Model Electrical conductors Peter Hertel

Overview

Model

Dielectric • A material with σ(0) = 0 is an electrical insulator . It medium cannot transport direct currents (DC). Permittivity of metals • 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 Nq2 1 σ(ω) = m Γ − iω • Note that the DC conductivity is always positive.

The Drude Model Electrical conductors Peter Hertel

Overview

Model

Overview

Model

Peter Hertel

Overview

Model

Dielectric medium

Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall eﬀect

Georg Simon Ohm, German physicist, 1789-1854 • apply a quasi-static external induction B • the typical electron obeys m(x¨ + Γx˙ + Ω2x) = q(E + x˙ × B) • Fourier transform this m(−ω2 − iωΓ + Ω2)x˜ = q(E˜ − iωx˜ × B)

• assume B = Beˆz • assume circularly polarized light √ E˜ = E˜±eˆ± where eˆ± = (eˆx + ieˆy)/ 2

• try x˜ =x ˜±eˆ±

• note eˆ± × eˆz = ∓ieˆ± • therefore 2 2 m(−ω − iωΓ + Ω )˜x± = q(E˜± ∓ ωBx˜±)

The Drude Model External static magnetic ﬁeld Peter Hertel

Overview

Model

Dielectric medium

Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall eﬀect • the typical electron obeys m(x¨ + Γx˙ + Ω2x) = q(E + x˙ × B) • Fourier transform this m(−ω2 − iωΓ + Ω2)x˜ = q(E˜ − iωx˜ × B)

• assume B = Beˆz • assume circularly polarized light √ E˜ = E˜±eˆ± where eˆ± = (eˆx + ieˆy)/ 2

• try x˜ =x ˜±eˆ±

• note eˆ± × eˆz = ∓ieˆ± • therefore 2 2 m(−ω − iωΓ + Ω )˜x± = q(E˜± ∓ ωBx˜±)

The Drude Model External static magnetic ﬁeld Peter Hertel

Overview • apply a quasi-static external induction B Model

Dielectric medium

Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall eﬀect • Fourier transform this m(−ω2 − iωΓ + Ω2)x˜ = q(E˜ − iωx˜ × B)

• assume B = Beˆz • assume circularly polarized light √ E˜ = E˜±eˆ± where eˆ± = (eˆx + ieˆy)/ 2

• try x˜ =x ˜±eˆ±

• note eˆ± × eˆz = ∓ieˆ± • therefore 2 2 m(−ω − iωΓ + Ω )˜x± = q(E˜± ∓ ωBx˜±)

The Drude Model External static magnetic ﬁeld Peter Hertel

Overview • apply a quasi-static external induction B Model • Dielectric the typical electron obeys medium m(x¨ + Γx˙ + Ω2x) = q(E + x˙ × B) Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall eﬀect • assume B = Beˆz • assume circularly polarized light √ E˜ = E˜±eˆ± where eˆ± = (eˆx + ieˆy)/ 2

• try x˜ =x ˜±eˆ±

• note eˆ± × eˆz = ∓ieˆ± • therefore 2 2 m(−ω − iωΓ + Ω )˜x± = q(E˜± ∓ ωBx˜±)

The Drude Model External static magnetic ﬁeld Peter Hertel

Hall eﬀect • assume circularly polarized light √ E˜ = E˜±eˆ± where eˆ± = (eˆx + ieˆy)/ 2

• try x˜ =x ˜±eˆ±

• note eˆ± × eˆz = ∓ieˆ± • therefore 2 2 m(−ω − iωΓ + Ω )˜x± = q(E˜± ∓ ωBx˜±)

The Drude Model External static magnetic ﬁeld Peter Hertel

• note eˆ± × eˆz = ∓ieˆ± • therefore 2 2 m(−ω − iωΓ + Ω )˜x± = q(E˜± ∓ ωBx˜±)

The Drude Model External static magnetic ﬁeld Peter Hertel

Overview • apply a quasi-static external induction B Model • Dielectric the typical electron obeys medium m(x¨ + Γx˙ + Ω2x) = q(E + x˙ × B) Permittivity of metals • Fourier transform this Electrical 2 2 conductors m(−ω − iωΓ + Ω )x˜ = q(E˜ − iωx˜ × B) Faraday eﬀect • assume B = Beˆz Hall eﬀect • assume circularly polarized light √ E˜ = E˜±eˆ± where eˆ± = (eˆx + ieˆy)/ 2 • note eˆ± × eˆz = ∓ieˆ± • therefore 2 2 m(−ω − iωΓ + Ω )˜x± = q(E˜± ∓ ωBx˜±)

The Drude Model External static magnetic ﬁeld Peter Hertel

• try x˜ =x ˜±eˆ± • therefore 2 2 m(−ω − iωΓ + Ω )˜x± = q(E˜± ∓ ωBx˜±)

The Drude Model External static magnetic ﬁeld Peter Hertel

• try x˜ =x ˜±eˆ±

• note eˆ± × eˆz = ∓ieˆ± The Drude Model External static magnetic ﬁeld Peter Hertel

• try x˜ =x ˜±eˆ±

• note eˆ± × eˆz = ∓ieˆ± • therefore 2 2 m(−ω − iωΓ + Ω )˜x± = q(E˜± ∓ ωBx˜±) 2 2 • m(−ω − iωΓ + Ω )˜x± = q(E˜± ∓ ωBx˜±) • therefore qE˜ x˜ = ± ± m(Ω2 − iωΓ − ω2) ± qωB

• recall P˜ = Nqx˜ = 0χE˜ • effect of quasi-static induction B is Nq2 1 χ±(ω) = 2 2 0m Ω − iωΓ − ω ± (q/m)ωB • left and right handed polarized light sees different susceptibility • Faraday effect

The Drude Model Faraday eﬀect Peter Hertel

Overview

Model

Dielectric medium

Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall eﬀect • therefore qE˜ x˜ = ± ± m(Ω2 − iωΓ − ω2) ± qωB

The Drude Model Faraday eﬀect Peter Hertel

Overview

Model • m(−ω2 − iωΓ + Ω2)˜x = q(E˜ ∓ ωBx˜ ) Dielectric ± ± ± medium

Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall effect • recall P˜ = Nqx˜ = 0χE˜ • effect of quasi-static induction B is Nq2 1 χ±(ω) = 2 2 0m Ω − iωΓ − ω ± (q/m)ωB • left and right handed polarized light sees different susceptibility • Faraday effect

The Drude Model Faraday eﬀect Peter Hertel

Overview

Model • m(−ω2 − iωΓ + Ω2)˜x = q(E˜ ∓ ωBx˜ ) Dielectric ± ± ± medium • therefore Permittivity of metals qE˜± x˜± = Electrical m(Ω2 − iωΓ − ω2) ± qωB conductors

Faraday eﬀect

Hall effect • effect of quasi-static induction B is Nq2 1 χ±(ω) = 2 2 0m Ω − iωΓ − ω ± (q/m)ωB • left and right handed polarized light sees different susceptibility • Faraday effect

The Drude Model Faraday eﬀect Peter Hertel

Overview

Model • m(−ω2 − iωΓ + Ω2)˜x = q(E˜ ∓ ωBx˜ ) Dielectric ± ± ± medium • therefore Permittivity of metals qE˜± x˜± = Electrical m(Ω2 − iωΓ − ω2) ± qωB conductors ˜ ˜ Faraday effect • recall P = Nqx˜ = 0χE Hall effect • left and right handed polarized light sees different susceptibility • Faraday effect

The Drude Model Faraday eﬀect Peter Hertel

Overview

Model • m(−ω2 − iωΓ + Ω2)˜x = q(E˜ ∓ ωBx˜ ) Dielectric ± ± ± medium • therefore Permittivity of metals qE˜± x˜± = Electrical m(Ω2 − iωΓ − ω2) ± qωB conductors ˜ ˜ Faraday effect • recall P = Nqx˜ = 0χE Hall effect • effect of quasi-static induction B is Nq2 1 χ±(ω) = 2 2 0m Ω − iωΓ − ω ± (q/m)ωB • Faraday effect

The Drude Model Faraday eﬀect Peter Hertel

Overview

Model • m(−ω2 − iωΓ + Ω2)˜x = q(E˜ ∓ ωBx˜ ) Dielectric ± ± ± medium • therefore Permittivity of metals qE˜± x˜± = Electrical m(Ω2 − iωΓ − ω2) ± qωB conductors ˜ ˜ Faraday effect • recall P = Nqx˜ = 0χE Hall effect • effect of quasi-static induction B is Nq2 1 χ±(ω) = 2 2 0m Ω − iωΓ − ω ± (q/m)ωB • left and right handed polarized light sees different susceptibility The Drude Model Faraday effect Peter Hertel

Overview

Model • m(−ω2 − iωΓ + Ω2)˜x = q(E˜ ∓ ωBx˜ ) Dielectric ± ± ± medium • therefore Permittivity of metals qE˜± x˜± = Electrical m(Ω2 − iωΓ − ω2) ± qωB conductors ˜ ˜ Faraday effect • recall P = Nqx˜ = 0χE Hall effect • effect of quasi-static induction B is Nq2 1 χ±(ω) = 2 2 0m Ω − iωΓ − ω ± (q/m)ωB • left and right handed polarized light sees different susceptibility • 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 • B is always small (in natural units)

• ij(ω; B) = ij(ω; 0) + iK(ω)ijkBk • linear magneto-optic eﬀect • Faraday constant is Nq3 ω K(ω) = 2 2 2 2 0m (Ω − iωΓ − ω ) • K(ω) is real in transparency window • i. e. if ω is far away form Ω • Faraday eﬀect distinguishes between forward and backward propagation • optical isolator

The Drude Model Remarks Peter Hertel

Overview

Model

Dielectric medium

Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall effect • ij(ω; B) = ij(ω; 0) + iK(ω)ijkBk • linear magneto-optic effect • Faraday constant is Nq3 ω K(ω) = 2 2 2 2 0m (Ω − iωΓ − ω ) • K(ω) is real in transparency window • i. e. if ω is far away form Ω • Faraday effect distinguishes between forward and backward propagation • optical isolator

The Drude Model Remarks Peter Hertel

Overview Model • B is always small (in natural units) Dielectric medium

Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall effect • linear magneto-optic effect • Faraday constant is Nq3 ω K(ω) = 2 2 2 2 0m (Ω − iωΓ − ω ) • K(ω) is real in transparency window • i. e. if ω is far away form Ω • Faraday effect distinguishes between forward and backward propagation • optical isolator

The Drude Model Remarks Peter Hertel

Overview Model • B is always small (in natural units) Dielectric medium • ij(ω; B) = ij(ω; 0) + iK(ω)ijkBk Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall eﬀect • Faraday constant is Nq3 ω K(ω) = 2 2 2 2 0m (Ω − iωΓ − ω ) • K(ω) is real in transparency window • i. e. if ω is far away form Ω • Faraday eﬀect distinguishes between forward and backward propagation • optical isolator

The Drude Model Remarks Peter Hertel

Overview Model • B is always small (in natural units) Dielectric medium • ij(ω; B) = ij(ω; 0) + iK(ω)ijkBk Permittivity of metals • linear magneto-optic eﬀect Electrical conductors

Faraday eﬀect

Hall eﬀect • K(ω) is real in transparency window • i. e. if ω is far away form Ω • Faraday eﬀect distinguishes between forward and backward propagation • optical isolator

The Drude Model Remarks Peter Hertel

Overview Model • B is always small (in natural units) Dielectric medium • ij(ω; B) = ij(ω; 0) + iK(ω)ijkBk Permittivity of metals • linear magneto-optic effect Electrical conductors • Faraday constant is 3 Faraday effect Nq ω K(ω) = 2 2 2 2 Hall effect 0m (Ω − iωΓ − ω ) • K(ω) is real in transparency window • i. e. if ω is far away form Ω • Faraday effect distinguishes between forward and backward propagation • optical isolator • set the spring constant mΩ2 = 0 • study AC electric field E˜ • and static magnetic induction B • solve m(−ω2 − iΓω)x˜ = q(E˜ − iωx˜ × B) • or q 1 1 x˜ = {E˜ − iωx˜ × B} m −iω Γ − iω • by iteration q 1 x˜ = ... {E˜ + E˜ × B} m Γ − iω • Ohmic current ∝ E˜ and Hall current ∝ E˜ × B

The Drude Model Conduction in a magnetic ﬁeld Peter Hertel

Overview

Model

Dielectric medium

Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall eﬀect • study AC electric ﬁeld E˜ • and static magnetic induction B • solve m(−ω2 − iΓω)x˜ = q(E˜ − iωx˜ × B) • or q 1 1 x˜ = {E˜ − iωx˜ × B} m −iω Γ − iω • by iteration q 1 x˜ = ... {E˜ + E˜ × B} m Γ − iω • Ohmic current ∝ E˜ and Hall current ∝ E˜ × B

The Drude Model Conduction in a magnetic ﬁeld Peter Hertel

Overview Model • set the spring constant mΩ2 = 0 Dielectric medium

Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall eﬀect • and static magnetic induction B • solve m(−ω2 − iΓω)x˜ = q(E˜ − iωx˜ × B) • or q 1 1 x˜ = {E˜ − iωx˜ × B} m −iω Γ − iω • by iteration q 1 x˜ = ... {E˜ + E˜ × B} m Γ − iω • Ohmic current ∝ E˜ and Hall current ∝ E˜ × B

The Drude Model Conduction in a magnetic ﬁeld Peter Hertel

Overview Model • set the spring constant mΩ2 = 0 Dielectric medium • study AC electric ﬁeld E˜ Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall eﬀect • solve m(−ω2 − iΓω)x˜ = q(E˜ − iωx˜ × B) • or q 1 1 x˜ = {E˜ − iωx˜ × B} m −iω Γ − iω • by iteration q 1 x˜ = ... {E˜ + E˜ × B} m Γ − iω • Ohmic current ∝ E˜ and Hall current ∝ E˜ × B

The Drude Model Conduction in a magnetic ﬁeld Peter Hertel

Overview Model • set the spring constant mΩ2 = 0 Dielectric medium • study AC electric ﬁeld E˜ Permittivity of metals • and static magnetic induction B Electrical conductors

Faraday eﬀect

Hall eﬀect • or q 1 1 x˜ = {E˜ − iωx˜ × B} m −iω Γ − iω • by iteration q 1 x˜ = ... {E˜ + E˜ × B} m Γ − iω • Ohmic current ∝ E˜ and Hall current ∝ E˜ × B

The Drude Model Conduction in a magnetic ﬁeld Peter Hertel

Overview Model • set the spring constant mΩ2 = 0 Dielectric medium • study AC electric field E˜ Permittivity of metals • and static magnetic induction B Electrical • conductors solve 2 Faraday effect m(−ω − iΓω)x˜ = q(E˜ − iωx˜ × B) Hall effect • by iteration q 1 x˜ = ... {E˜ + E˜ × B} m Γ − iω • Ohmic current ∝ E˜ and Hall current ∝ E˜ × B

The Drude Model Conduction in a magnetic ﬁeld Peter Hertel

Overview Model • set the spring constant mΩ2 = 0 Dielectric medium • study AC electric field E˜ Permittivity of metals • and static magnetic induction B Electrical • conductors solve 2 Faraday effect m(−ω − iΓω)x˜ = q(E˜ − iωx˜ × B) Hall effect • or q 1 1 x˜ = {E˜ − iωx˜ × B} m −iω Γ − iω • Ohmic current ∝ E˜ and Hall current ∝ E˜ × B

The Drude Model Conduction in a magnetic ﬁeld Peter Hertel

Overview Model • set the spring constant mΩ2 = 0 Dielectric medium • study AC electric field E˜ Permittivity of metals • and static magnetic induction B Electrical • conductors solve 2 Faraday effect m(−ω − iΓω)x˜ = q(E˜ − iωx˜ × B) Hall effect • or q 1 1 x˜ = {E˜ − iωx˜ × B} m −iω Γ − iω • by iteration q 1 x˜ = ... {E˜ + E˜ × B} m Γ − iω The Drude Model Conduction in a magnetic field Peter Hertel

Overview Model • set the spring constant mΩ2 = 0 Dielectric medium • study AC electric field E˜ Permittivity of metals • and static magnetic induction B Electrical • conductors solve 2 Faraday effect m(−ω − iΓω)x˜ = q(E˜ − iωx˜ × B) Hall effect • or q 1 1 x˜ = {E˜ − iωx˜ × B} m −iω Γ − iω • by iteration q 1 x˜ = ... {E˜ + E˜ × B} m Γ − iω • Ohmic current ∝ E˜ and Hall current ∝ E˜ × B The Drude Model

Peter Hertel

Overview

Model

Dielectric medium

Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall eﬀect

Hall eﬀect, schematilly • Hall current usually forbidden by boundary conditions • Hall ﬁeld q 1 E˜ = − E˜ × B H m Γ − iω

• replace E˜ by E˜ + E˜ H

• E˜ H × B can be neglected • current J˜(ω) = σ(ω)E˜(ω) as usual

• additional Hall ﬁeld E˜ H(ω) = R(ω)J˜(ω) × B • Hall constant R = −1/Nq does not depend on ω • . . . if there is a dominant charge carrier. • R has diﬀerent sign for electrons and holes

The Drude Model Hall eﬀect Peter Hertel

Overview

Model

Dielectric medium

Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall eﬀect • Hall ﬁeld q 1 E˜ = − E˜ × B H m Γ − iω

• replace E˜ by E˜ + E˜ H

• E˜ H × B can be neglected • current J˜(ω) = σ(ω)E˜(ω) as usual

The Drude Model Hall eﬀect Peter Hertel

Overview

Model • Hall current usually forbidden by boundary conditions Dielectric medium

Permittivity of metals

Electrical conductors

Faraday eﬀect

Hall eﬀect • replace E˜ by E˜ + E˜ H

• E˜ H × B can be neglected • current J˜(ω) = σ(ω)E˜(ω) as usual

The Drude Model Hall eﬀect Peter Hertel

Overview

Model • Hall current usually forbidden by boundary conditions Dielectric medium • Hall ﬁeld Permittivity of metals q 1 E˜ H = − E˜ × B Electrical m Γ − iω conductors

Faraday eﬀect

Hall eﬀect • E˜ H × B can be neglected • current J˜(ω) = σ(ω)E˜(ω) as usual

The Drude Model Hall eﬀect Peter Hertel

Overview

Hall eﬀect • current J˜(ω) = σ(ω)E˜(ω) as usual

The Drude Model Hall eﬀect Peter Hertel

Overview

Model • Hall current usually forbidden by boundary conditions Dielectric medium • Hall field Permittivity of metals q 1 E˜ H = − E˜ × B Electrical m Γ − iω conductors • replace E˜ by E˜ + E˜ H Faraday effect ˜ Hall effect • E H × B can be neglected • additional Hall field E˜ H(ω) = R(ω)J˜(ω) × B • Hall constant R = −1/Nq does not depend on ω • . . . if there is a dominant charge carrier. • R has different sign for electrons and holes

The Drude Model Hall eﬀect Peter Hertel

Overview

Model • Hall current usually forbidden by boundary conditions Dielectric medium • Hall field Permittivity of metals q 1 E˜ H = − E˜ × B Electrical m Γ − iω conductors • replace E˜ by E˜ + E˜ H Faraday effect ˜ Hall effect • E H × B can be neglected • current J˜(ω) = σ(ω)E˜(ω) as usual • Hall constant R = −1/Nq does not depend on ω • . . . if there is a dominant charge carrier. • R has different sign for electrons and holes

The Drude Model Hall eﬀect Peter Hertel

Overview

• additional Hall ﬁeld E˜ H(ω) = R(ω)J˜(ω) × B • . . . if there is a dominant charge carrier. • R has diﬀerent sign for electrons and holes

The Drude Model Hall eﬀect Peter Hertel

Overview

• additional Hall ﬁeld E˜ H(ω) = R(ω)J˜(ω) × B • Hall constant R = −1/Nq does not depend on ω • R has diﬀerent sign for electrons and holes

The Drude Model Hall eﬀect Peter Hertel

Overview

• additional Hall ﬁeld E˜ H(ω) = R(ω)J˜(ω) × B • Hall constant R = −1/Nq does not depend on ω • . . . if there is a dominant charge carrier. The Drude Model Hall eﬀect Peter Hertel

Overview

• additional Hall ﬁeld E˜ H(ω) = R(ω)J˜(ω) × B • Hall constant R = −1/Nq 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