the drude model - uni-osnabrueck.de · the drude model peter hertel overview model dielectric...
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![Page 1: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/1.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
The Drude Model
Peter Hertel
University of Osnabruck, Germany
Lecture presented at APS, Nankai University, China
http://www.home.uni-osnabrueck.de/phertel
October/November 2011
![Page 2: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/2.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Paul Drude, German physicist, 1863-1906
![Page 3: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/3.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Overview
• The Drude model links optical and electric properties of amaterial with the behavior of its electrons or holes
• The model
• Dielectric permittivity
• Permittivity of metals
• Conductivity
• Faraday effect
• Hall effect
![Page 4: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/4.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Overview
• The Drude model links optical and electric properties of amaterial with the behavior of its electrons or holes
• The model
• Dielectric permittivity
• Permittivity of metals
• Conductivity
• Faraday effect
• Hall effect
![Page 5: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/5.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Overview
• The Drude model links optical and electric properties of amaterial with the behavior of its electrons or holes
• The model
• Dielectric permittivity
• Permittivity of metals
• Conductivity
• Faraday effect
• Hall effect
![Page 6: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/6.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Overview
• The Drude model links optical and electric properties of amaterial with the behavior of its electrons or holes
• The model
• Dielectric permittivity
• Permittivity of metals
• Conductivity
• Faraday effect
• Hall effect
![Page 7: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/7.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Overview
• The Drude model links optical and electric properties of amaterial with the behavior of its electrons or holes
• The model
• Dielectric permittivity
• Permittivity of metals
• Conductivity
• Faraday effect
• Hall effect
![Page 8: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/8.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Overview
• The Drude model links optical and electric properties of amaterial with the behavior of its electrons or holes
• The model
• Dielectric permittivity
• Permittivity of metals
• Conductivity
• Faraday effect
• Hall effect
![Page 9: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/9.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Overview
• The Drude model links optical and electric properties of amaterial with the behavior of its electrons or holes
• The model
• Dielectric permittivity
• Permittivity of metals
• Conductivity
• Faraday effect
• Hall effect
![Page 10: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/10.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Overview
• The Drude model links optical and electric properties of amaterial with the behavior of its electrons or holes
• The model
• Dielectric permittivity
• Permittivity of metals
• Conductivity
• Faraday effect
• Hall effect
![Page 11: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/11.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Model
• consider a typical electron
• denote by x = x(t) the deviation from its equilibriumposition
• external electric field strength E = E(t)
• m(x + Γx + Ω2x) = qE
• electron mass m, charge q, friction coefficient mΓ, springconstant mΩ2
• Fourier transform this
• m(−ω2 − iωΓ + Ω2)x = qE
• solution is
x(ω) =q
m
E(ω)
Ω2 − ω2 − iωΓ
![Page 12: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/12.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Model
• consider a typical electron
• denote by x = x(t) the deviation from its equilibriumposition
• external electric field strength E = E(t)
• m(x + Γx + Ω2x) = qE
• electron mass m, charge q, friction coefficient mΓ, springconstant mΩ2
• Fourier transform this
• m(−ω2 − iωΓ + Ω2)x = qE
• solution is
x(ω) =q
m
E(ω)
Ω2 − ω2 − iωΓ
![Page 13: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/13.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Model
• consider a typical electron
• denote by x = x(t) the deviation from its equilibriumposition
• external electric field strength E = E(t)
• m(x + Γx + Ω2x) = qE
• electron mass m, charge q, friction coefficient mΓ, springconstant mΩ2
• Fourier transform this
• m(−ω2 − iωΓ + Ω2)x = qE
• solution is
x(ω) =q
m
E(ω)
Ω2 − ω2 − iωΓ
![Page 14: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/14.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Model
• consider a typical electron
• denote by x = x(t) the deviation from its equilibriumposition
• external electric field strength E = E(t)
• m(x + Γx + Ω2x) = qE
• electron mass m, charge q, friction coefficient mΓ, springconstant mΩ2
• Fourier transform this
• m(−ω2 − iωΓ + Ω2)x = qE
• solution is
x(ω) =q
m
E(ω)
Ω2 − ω2 − iωΓ
![Page 15: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/15.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Model
• consider a typical electron
• denote by x = x(t) the deviation from its equilibriumposition
• external electric field strength E = E(t)
• m(x + Γx + Ω2x) = qE
• electron mass m, charge q, friction coefficient mΓ, springconstant mΩ2
• Fourier transform this
• m(−ω2 − iωΓ + Ω2)x = qE
• solution is
x(ω) =q
m
E(ω)
Ω2 − ω2 − iωΓ
![Page 16: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/16.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Model
• consider a typical electron
• denote by x = x(t) the deviation from its equilibriumposition
• external electric field strength E = E(t)
• m(x + Γx + Ω2x) = qE
• electron mass m, charge q, friction coefficient mΓ, springconstant mΩ2
• Fourier transform this
• m(−ω2 − iωΓ + Ω2)x = qE
• solution is
x(ω) =q
m
E(ω)
Ω2 − ω2 − iωΓ
![Page 17: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/17.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Model
• consider a typical electron
• denote by x = x(t) the deviation from its equilibriumposition
• external electric field strength E = E(t)
• m(x + Γx + Ω2x) = qE
• electron mass m, charge q, friction coefficient mΓ, springconstant mΩ2
• Fourier transform this
• m(−ω2 − iωΓ + Ω2)x = qE
• solution is
x(ω) =q
m
E(ω)
Ω2 − ω2 − iωΓ
![Page 18: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/18.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Model
• consider a typical electron
• denote by x = x(t) the deviation from its equilibriumposition
• external electric field strength E = E(t)
• m(x + Γx + Ω2x) = qE
• electron mass m, charge q, friction coefficient mΓ, springconstant mΩ2
• Fourier transform this
• m(−ω2 − iωΓ + Ω2)x = qE
• solution is
x(ω) =q
m
E(ω)
Ω2 − ω2 − iωΓ
![Page 19: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/19.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Model
• consider a typical electron
• denote by x = x(t) the deviation from its equilibriumposition
• external electric field strength E = E(t)
• m(x + Γx + Ω2x) = qE
• electron mass m, charge q, friction coefficient mΓ, springconstant mΩ2
• Fourier transform this
• m(−ω2 − iωΓ + Ω2)x = qE
• solution is
x(ω) =q
m
E(ω)
Ω2 − ω2 − iωΓ
![Page 20: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/20.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Polarization
• dipole moment of typical electron is p = qx
• recall
x(ω) =q
m
E(ω)
Ω2 − ω2 − iωΓ• there are N typical electrons per unit volume
• polarization is P = Nqx = ε0χE
• susceptibility is
χ(ω) =Nq2
ε0m
1
Ω2 − ω2 − iωΓ• in particular
χ(0) =Nq2
ε0mΩ2> 0
• . . . as it should be
![Page 21: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/21.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Polarization
• dipole moment of typical electron is p = qx
• recall
x(ω) =q
m
E(ω)
Ω2 − ω2 − iωΓ• there are N typical electrons per unit volume
• polarization is P = Nqx = ε0χE
• susceptibility is
χ(ω) =Nq2
ε0m
1
Ω2 − ω2 − iωΓ• in particular
χ(0) =Nq2
ε0mΩ2> 0
• . . . as it should be
![Page 22: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/22.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Polarization
• dipole moment of typical electron is p = qx
• recall
x(ω) =q
m
E(ω)
Ω2 − ω2 − iωΓ
• there are N typical electrons per unit volume
• polarization is P = Nqx = ε0χE
• susceptibility is
χ(ω) =Nq2
ε0m
1
Ω2 − ω2 − iωΓ• in particular
χ(0) =Nq2
ε0mΩ2> 0
• . . . as it should be
![Page 23: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/23.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Polarization
• dipole moment of typical electron is p = qx
• recall
x(ω) =q
m
E(ω)
Ω2 − ω2 − iωΓ• there are N typical electrons per unit volume
• polarization is P = Nqx = ε0χE
• susceptibility is
χ(ω) =Nq2
ε0m
1
Ω2 − ω2 − iωΓ• in particular
χ(0) =Nq2
ε0mΩ2> 0
• . . . as it should be
![Page 24: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/24.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Polarization
• dipole moment of typical electron is p = qx
• recall
x(ω) =q
m
E(ω)
Ω2 − ω2 − iωΓ• there are N typical electrons per unit volume
• polarization is P = Nqx = ε0χE
• susceptibility is
χ(ω) =Nq2
ε0m
1
Ω2 − ω2 − iωΓ• in particular
χ(0) =Nq2
ε0mΩ2> 0
• . . . as it should be
![Page 25: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/25.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Polarization
• dipole moment of typical electron is p = qx
• recall
x(ω) =q
m
E(ω)
Ω2 − ω2 − iωΓ• there are N typical electrons per unit volume
• polarization is P = Nqx = ε0χE
• susceptibility is
χ(ω) =Nq2
ε0m
1
Ω2 − ω2 − iωΓ
• in particular
χ(0) =Nq2
ε0mΩ2> 0
• . . . as it should be
![Page 26: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/26.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Polarization
• dipole moment of typical electron is p = qx
• recall
x(ω) =q
m
E(ω)
Ω2 − ω2 − iωΓ• there are N typical electrons per unit volume
• polarization is P = Nqx = ε0χE
• susceptibility is
χ(ω) =Nq2
ε0m
1
Ω2 − ω2 − iωΓ• in particular
χ(0) =Nq2
ε0mΩ2> 0
• . . . as it should be
![Page 27: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/27.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Polarization
• dipole moment of typical electron is p = qx
• recall
x(ω) =q
m
E(ω)
Ω2 − ω2 − iωΓ• there are N typical electrons per unit volume
• polarization is P = Nqx = ε0χE
• susceptibility is
χ(ω) =Nq2
ε0m
1
Ω2 − ω2 − iωΓ• in particular
χ(0) =Nq2
ε0mΩ2> 0
• . . . as it should be
![Page 28: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/28.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Discussion I
• decompose susceptibility χ(ω) = χ ′(ω) + iχ ′′(ω) intorefractive part χ ′ and absorptive part χ ′′
• Introduce R(ω) = χ(ω)/χ(0), s = ω/Ω and γ = Γ/Ω asnormalized quantities.
• refraction
R ′(s) =1− s2
(1− s2)2 + γ2s2
• absorption
D ′′(s) =γs
(1− s2)2 + γ2s2
• limiting cases: s = 0, s = 1, s→∞, small γ
![Page 29: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/29.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Discussion I
• decompose susceptibility χ(ω) = χ ′(ω) + iχ ′′(ω) intorefractive part χ ′ and absorptive part χ ′′
• Introduce R(ω) = χ(ω)/χ(0), s = ω/Ω and γ = Γ/Ω asnormalized quantities.
• refraction
R ′(s) =1− s2
(1− s2)2 + γ2s2
• absorption
D ′′(s) =γs
(1− s2)2 + γ2s2
• limiting cases: s = 0, s = 1, s→∞, small γ
![Page 30: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/30.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Discussion I
• decompose susceptibility χ(ω) = χ ′(ω) + iχ ′′(ω) intorefractive part χ ′ and absorptive part χ ′′
• Introduce R(ω) = χ(ω)/χ(0), s = ω/Ω and γ = Γ/Ω asnormalized quantities.
• refraction
R ′(s) =1− s2
(1− s2)2 + γ2s2
• absorption
D ′′(s) =γs
(1− s2)2 + γ2s2
• limiting cases: s = 0, s = 1, s→∞, small γ
![Page 31: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/31.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Discussion I
• decompose susceptibility χ(ω) = χ ′(ω) + iχ ′′(ω) intorefractive part χ ′ and absorptive part χ ′′
• Introduce R(ω) = χ(ω)/χ(0), s = ω/Ω and γ = Γ/Ω asnormalized quantities.
• refraction
R ′(s) =1− s2
(1− s2)2 + γ2s2
• absorption
D ′′(s) =γs
(1− s2)2 + γ2s2
• limiting cases: s = 0, s = 1, s→∞, small γ
![Page 32: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/32.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Discussion I
• decompose susceptibility χ(ω) = χ ′(ω) + iχ ′′(ω) intorefractive part χ ′ and absorptive part χ ′′
• Introduce R(ω) = χ(ω)/χ(0), s = ω/Ω and γ = Γ/Ω asnormalized quantities.
• refraction
R ′(s) =1− s2
(1− s2)2 + γ2s2
• absorption
D ′′(s) =γs
(1− s2)2 + γ2s2
• limiting cases: s = 0, s = 1, s→∞, small γ
![Page 33: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/33.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Discussion I
• decompose susceptibility χ(ω) = χ ′(ω) + iχ ′′(ω) intorefractive part χ ′ and absorptive part χ ′′
• Introduce R(ω) = χ(ω)/χ(0), s = ω/Ω and γ = Γ/Ω asnormalized quantities.
• refraction
R ′(s) =1− s2
(1− s2)2 + γ2s2
• absorption
D ′′(s) =γs
(1− s2)2 + γ2s2
• limiting cases: s = 0, s = 1, s→∞, small γ
![Page 34: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/34.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
0 0.5 1 1.5 2 2.5 3−5
0
5
10
Refractive part (blue) and absorptive part (red) of thesusceptibility function χ(ω) scaled by the static value χ(0).The abscissa is ω/Ω. Γ/Ω = 0.1
![Page 35: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/35.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Discussion II
• For small frequencies (as compared with Ω) thesusceptibility 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. Negativedispersion ∂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 . . .
![Page 36: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/36.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Discussion II
• For small frequencies (as compared with Ω) thesusceptibility 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. Negativedispersion ∂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 . . .
![Page 37: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/37.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Discussion II
• For small frequencies (as compared with Ω) thesusceptibility 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. Negativedispersion ∂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 . . .
![Page 38: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/38.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Discussion II
• For small frequencies (as compared with Ω) thesusceptibility 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. Negativedispersion ∂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 . . .
![Page 39: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/39.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Discussion II
• For small frequencies (as compared with Ω) thesusceptibility 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. Negativedispersion ∂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 . . .
![Page 40: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/40.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Discussion II
• For small frequencies (as compared with Ω) thesusceptibility 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. Negativedispersion ∂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 . . .
![Page 41: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/41.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Discussion II
• For small frequencies (as compared with Ω) thesusceptibility 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. Negativedispersion ∂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 . . .
![Page 42: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/42.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Kramers-Kronig relation I
• χ(ω) must be the Fourier transform of a causal responsefunction G = G(τ)
• as defined in
P (t) = ε0
∫dτG(τ)E(t− τ)
• check this for
G(τ) = a
∫dω
2π
e−iωτ
Ω2 − ω2 − iωΓ• poles at
ω1,2 = − iΓ
2± ω where ω = +
√Ω2 − Γ2/4
• Indeed, G(τ) = 0 for τ < 0
• for τ > 0
G(τ) =Nq2
ε0m
sin ωτ
ωe−Γτ/2
![Page 43: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/43.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Kramers-Kronig relation I
• χ(ω) must be the Fourier transform of a causal responsefunction G = G(τ)
• as defined in
P (t) = ε0
∫dτG(τ)E(t− τ)
• check this for
G(τ) = a
∫dω
2π
e−iωτ
Ω2 − ω2 − iωΓ• poles at
ω1,2 = − iΓ
2± ω where ω = +
√Ω2 − Γ2/4
• Indeed, G(τ) = 0 for τ < 0
• for τ > 0
G(τ) =Nq2
ε0m
sin ωτ
ωe−Γτ/2
![Page 44: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/44.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Kramers-Kronig relation I
• χ(ω) must be the Fourier transform of a causal responsefunction G = G(τ)
• as defined in
P (t) = ε0
∫dτG(τ)E(t− τ)
• check this for
G(τ) = a
∫dω
2π
e−iωτ
Ω2 − ω2 − iωΓ• poles at
ω1,2 = − iΓ
2± ω where ω = +
√Ω2 − Γ2/4
• Indeed, G(τ) = 0 for τ < 0
• for τ > 0
G(τ) =Nq2
ε0m
sin ωτ
ωe−Γτ/2
![Page 45: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/45.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Kramers-Kronig relation I
• χ(ω) must be the Fourier transform of a causal responsefunction G = G(τ)
• as defined in
P (t) = ε0
∫dτG(τ)E(t− τ)
• check this for
G(τ) = a
∫dω
2π
e−iωτ
Ω2 − ω2 − iωΓ
• poles at
ω1,2 = − iΓ
2± ω where ω = +
√Ω2 − Γ2/4
• Indeed, G(τ) = 0 for τ < 0
• for τ > 0
G(τ) =Nq2
ε0m
sin ωτ
ωe−Γτ/2
![Page 46: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/46.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Kramers-Kronig relation I
• χ(ω) must be the Fourier transform of a causal responsefunction G = G(τ)
• as defined in
P (t) = ε0
∫dτG(τ)E(t− τ)
• check this for
G(τ) = a
∫dω
2π
e−iωτ
Ω2 − ω2 − iωΓ• poles at
ω1,2 = − iΓ
2± ω where ω = +
√Ω2 − Γ2/4
• Indeed, G(τ) = 0 for τ < 0
• for τ > 0
G(τ) =Nq2
ε0m
sin ωτ
ωe−Γτ/2
![Page 47: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/47.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Kramers-Kronig relation I
• χ(ω) must be the Fourier transform of a causal responsefunction G = G(τ)
• as defined in
P (t) = ε0
∫dτG(τ)E(t− τ)
• check this for
G(τ) = a
∫dω
2π
e−iωτ
Ω2 − ω2 − iωΓ• poles at
ω1,2 = − iΓ
2± ω where ω = +
√Ω2 − Γ2/4
• Indeed, G(τ) = 0 for τ < 0
• for τ > 0
G(τ) =Nq2
ε0m
sin ωτ
ωe−Γτ/2
![Page 48: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/48.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Kramers-Kronig relation I
• χ(ω) must be the Fourier transform of a causal responsefunction G = G(τ)
• as defined in
P (t) = ε0
∫dτG(τ)E(t− τ)
• check this for
G(τ) = a
∫dω
2π
e−iωτ
Ω2 − ω2 − iωΓ• poles at
ω1,2 = − iΓ
2± ω where ω = +
√Ω2 − Γ2/4
• Indeed, G(τ) = 0 for τ < 0
• for τ > 0
G(τ) =Nq2
ε0m
sin ωτ
ωe−Γτ/2
![Page 49: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/49.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Kramers-Kronig relation II
• causal response function: G(τ) = θ(τ)G(τ)
• apply the convolution theorem
χ(ω) =
∫du
2πχ(u)θ(ω − u)
• Fourier transform of Heaviside function is
θ(ω) = lim0<η→0
1
η − iω
• dispersion , or Kramers-Kronig relations
χ ′(ω) = 2Pr
∫du
π
uχ ′′(u)
u2 − ω2
χ ′′(ω) = 2Pr
∫du
π
ωχ ′(u)
ω2 − u2
![Page 50: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/50.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Kramers-Kronig relation II
• causal response function: G(τ) = θ(τ)G(τ)
• apply the convolution theorem
χ(ω) =
∫du
2πχ(u)θ(ω − u)
• Fourier transform of Heaviside function is
θ(ω) = lim0<η→0
1
η − iω
• dispersion , or Kramers-Kronig relations
χ ′(ω) = 2Pr
∫du
π
uχ ′′(u)
u2 − ω2
χ ′′(ω) = 2Pr
∫du
π
ωχ ′(u)
ω2 − u2
![Page 51: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/51.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Kramers-Kronig relation II
• causal response function: G(τ) = θ(τ)G(τ)
• apply the convolution theorem
χ(ω) =
∫du
2πχ(u)θ(ω − u)
• Fourier transform of Heaviside function is
θ(ω) = lim0<η→0
1
η − iω
• dispersion , or Kramers-Kronig relations
χ ′(ω) = 2Pr
∫du
π
uχ ′′(u)
u2 − ω2
χ ′′(ω) = 2Pr
∫du
π
ωχ ′(u)
ω2 − u2
![Page 52: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/52.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Kramers-Kronig relation II
• causal response function: G(τ) = θ(τ)G(τ)
• apply the convolution theorem
χ(ω) =
∫du
2πχ(u)θ(ω − u)
• Fourier transform of Heaviside function is
θ(ω) = lim0<η→0
1
η − iω
• dispersion , or Kramers-Kronig relations
χ ′(ω) = 2Pr
∫du
π
uχ ′′(u)
u2 − ω2
χ ′′(ω) = 2Pr
∫du
π
ωχ ′(u)
ω2 − u2
![Page 53: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/53.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Kramers-Kronig relation II
• causal response function: G(τ) = θ(τ)G(τ)
• apply the convolution theorem
χ(ω) =
∫du
2πχ(u)θ(ω − u)
• Fourier transform of Heaviside function is
θ(ω) = lim0<η→0
1
η − iω
• dispersion , or Kramers-Kronig relations
χ ′(ω) = 2Pr
∫du
π
uχ ′′(u)
u2 − ω2
χ ′′(ω) = 2Pr
∫du
π
ωχ ′(u)
ω2 − u2
![Page 54: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/54.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Kramers-Kronig relation II
• causal response function: G(τ) = θ(τ)G(τ)
• apply the convolution theorem
χ(ω) =
∫du
2πχ(u)θ(ω − u)
• Fourier transform of Heaviside function is
θ(ω) = lim0<η→0
1
η − iω
• dispersion , or Kramers-Kronig relations
χ ′(ω) = 2Pr
∫du
π
uχ ′′(u)
u2 − ω2
χ ′′(ω) = 2Pr
∫du
π
ωχ ′(u)
ω2 − u2
![Page 55: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/55.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Kramers-Kronig relation II
• causal response function: G(τ) = θ(τ)G(τ)
• apply the convolution theorem
χ(ω) =
∫du
2πχ(u)θ(ω − u)
• Fourier transform of Heaviside function is
θ(ω) = lim0<η→0
1
η − iω
• dispersion , or Kramers-Kronig relations
χ ′(ω) = 2Pr
∫du
π
uχ ′′(u)
u2 − ω2
χ ′′(ω) = 2Pr
∫du
π
ωχ ′(u)
ω2 − u2
![Page 56: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/56.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Dispersion of white light
![Page 57: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/57.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Free quasi-electrons
• 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
ε(ω) = 1−ω2p
ω2 + iωΓ• plasma frequency ωp
ω2p =
Nq2
ε0m• correction for ω ωp
ε(ω) = ε∞ −ω2p
ω2 + iωΓ
![Page 58: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/58.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Free quasi-electrons
• 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
ε(ω) = 1−ω2p
ω2 + iωΓ• plasma frequency ωp
ω2p =
Nq2
ε0m• correction for ω ωp
ε(ω) = ε∞ −ω2p
ω2 + iωΓ
![Page 59: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/59.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Free quasi-electrons
• 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
ε(ω) = 1−ω2p
ω2 + iωΓ• plasma frequency ωp
ω2p =
Nq2
ε0m• correction for ω ωp
ε(ω) = ε∞ −ω2p
ω2 + iωΓ
![Page 60: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/60.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Free quasi-electrons
• 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
ε(ω) = 1−ω2p
ω2 + iωΓ• plasma frequency ωp
ω2p =
Nq2
ε0m• correction for ω ωp
ε(ω) = ε∞ −ω2p
ω2 + iωΓ
![Page 61: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/61.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Free quasi-electrons
• 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
ε(ω) = 1−ω2p
ω2 + iωΓ• plasma frequency ωp
ω2p =
Nq2
ε0m• correction for ω ωp
ε(ω) = ε∞ −ω2p
ω2 + iωΓ
![Page 62: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/62.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Free quasi-electrons
• 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
ε(ω) = 1−ω2p
ω2 + iωΓ• plasma frequency ωp
ω2p =
Nq2
ε0m• correction for ω ωp
ε(ω) = ε∞ −ω2p
ω2 + iωΓ
![Page 63: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/63.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Free quasi-electrons
• 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
ε(ω) = 1−ω2p
ω2 + iωΓ
• plasma frequency ωp
ω2p =
Nq2
ε0m• correction for ω ωp
ε(ω) = ε∞ −ω2p
ω2 + iωΓ
![Page 64: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/64.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Free quasi-electrons
• 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
ε(ω) = 1−ω2p
ω2 + iωΓ• plasma frequency ωp
ω2p =
Nq2
ε0m
• correction for ω ωp
ε(ω) = ε∞ −ω2p
ω2 + iωΓ
![Page 65: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/65.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Free quasi-electrons
• 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
ε(ω) = 1−ω2p
ω2 + iωΓ• plasma frequency ωp
ω2p =
Nq2
ε0m• correction for ω ωp
ε(ω) = ε∞ −ω2p
ω2 + iωΓ
![Page 66: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/66.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Example: gold
• Drude model parameters for gold
• as determined by Johnson and Christy in 1972
• ε∞ = 9.5
• ~ωp = 8.95 eV
• ~Γ = 0.069 eV
• with these parameters the Drude model fits opticalmeasurements well for ~ω < 2.25 eV (green)
• The refractive part of the permittivity can be large andnegative while the absorptive part is small.
• This allows surface plasmon polaritons (SPP)
![Page 67: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/67.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Example: gold
• Drude model parameters for gold
• as determined by Johnson and Christy in 1972
• ε∞ = 9.5
• ~ωp = 8.95 eV
• ~Γ = 0.069 eV
• with these parameters the Drude model fits opticalmeasurements well for ~ω < 2.25 eV (green)
• The refractive part of the permittivity can be large andnegative while the absorptive part is small.
• This allows surface plasmon polaritons (SPP)
![Page 68: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/68.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Example: gold
• Drude model parameters for gold
• as determined by Johnson and Christy in 1972
• ε∞ = 9.5
• ~ωp = 8.95 eV
• ~Γ = 0.069 eV
• with these parameters the Drude model fits opticalmeasurements well for ~ω < 2.25 eV (green)
• The refractive part of the permittivity can be large andnegative while the absorptive part is small.
• This allows surface plasmon polaritons (SPP)
![Page 69: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/69.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Example: gold
• Drude model parameters for gold
• as determined by Johnson and Christy in 1972
• ε∞ = 9.5
• ~ωp = 8.95 eV
• ~Γ = 0.069 eV
• with these parameters the Drude model fits opticalmeasurements well for ~ω < 2.25 eV (green)
• The refractive part of the permittivity can be large andnegative while the absorptive part is small.
• This allows surface plasmon polaritons (SPP)
![Page 70: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/70.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Example: gold
• Drude model parameters for gold
• as determined by Johnson and Christy in 1972
• ε∞ = 9.5
• ~ωp = 8.95 eV
• ~Γ = 0.069 eV
• with these parameters the Drude model fits opticalmeasurements well for ~ω < 2.25 eV (green)
• The refractive part of the permittivity can be large andnegative while the absorptive part is small.
• This allows surface plasmon polaritons (SPP)
![Page 71: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/71.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Example: gold
• Drude model parameters for gold
• as determined by Johnson and Christy in 1972
• ε∞ = 9.5
• ~ωp = 8.95 eV
• ~Γ = 0.069 eV
• with these parameters the Drude model fits opticalmeasurements well for ~ω < 2.25 eV (green)
• The refractive part of the permittivity can be large andnegative while the absorptive part is small.
• This allows surface plasmon polaritons (SPP)
![Page 72: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/72.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Example: gold
• Drude model parameters for gold
• as determined by Johnson and Christy in 1972
• ε∞ = 9.5
• ~ωp = 8.95 eV
• ~Γ = 0.069 eV
• with these parameters the Drude model fits opticalmeasurements well for ~ω < 2.25 eV (green)
• The refractive part of the permittivity can be large andnegative while the absorptive part is small.
• This allows surface plasmon polaritons (SPP)
![Page 73: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/73.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Example: gold
• Drude model parameters for gold
• as determined by Johnson and Christy in 1972
• ε∞ = 9.5
• ~ωp = 8.95 eV
• ~Γ = 0.069 eV
• with these parameters the Drude model fits opticalmeasurements well for ~ω < 2.25 eV (green)
• The refractive part of the permittivity can be large andnegative while the absorptive part is small.
• This allows surface plasmon polaritons (SPP)
![Page 74: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/74.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Example: gold
• Drude model parameters for gold
• as determined by Johnson and Christy in 1972
• ε∞ = 9.5
• ~ωp = 8.95 eV
• ~Γ = 0.069 eV
• with these parameters the Drude model fits opticalmeasurements well for ~ω < 2.25 eV (green)
• The refractive part of the permittivity can be large andnegative while the absorptive part is small.
• This allows surface plasmon polaritons (SPP)
![Page 75: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/75.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
1 1.5 2 2.5 3−80
−60
−40
−20
0
20
Refractive (blue) and absorptive part (red) of the permittivityfunction for gold. The abscissa is ~ω in eV.
![Page 76: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/76.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Electrical conductivity
• consider a typical charged particle
• recall m(x + Γx + Ω2x) = qE
• electric current density J = Nqx
• Fourier transformed: J = Nq(−iω)x
• recall
x(ω) =q
m
E(ω)
Ω2 − ω2 − iωΓ
• Ohm’s law
J(ω) = σ(ω)E(ω)
• conductivity is
σ(ω) =Nq2
m
−iω
Ω2 − ω2 − iωΓ
![Page 77: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/77.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Electrical conductivity
• consider a typical charged particle
• recall m(x + Γx + Ω2x) = qE
• electric current density J = Nqx
• Fourier transformed: J = Nq(−iω)x
• recall
x(ω) =q
m
E(ω)
Ω2 − ω2 − iωΓ
• Ohm’s law
J(ω) = σ(ω)E(ω)
• conductivity is
σ(ω) =Nq2
m
−iω
Ω2 − ω2 − iωΓ
![Page 78: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/78.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Electrical conductivity
• consider a typical charged particle
• recall m(x + Γx + Ω2x) = qE
• electric current density J = Nqx
• Fourier transformed: J = Nq(−iω)x
• recall
x(ω) =q
m
E(ω)
Ω2 − ω2 − iωΓ
• Ohm’s law
J(ω) = σ(ω)E(ω)
• conductivity is
σ(ω) =Nq2
m
−iω
Ω2 − ω2 − iωΓ
![Page 79: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/79.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Electrical conductivity
• consider a typical charged particle
• recall m(x + Γx + Ω2x) = qE
• electric current density J = Nqx
• Fourier transformed: J = Nq(−iω)x
• recall
x(ω) =q
m
E(ω)
Ω2 − ω2 − iωΓ
• Ohm’s law
J(ω) = σ(ω)E(ω)
• conductivity is
σ(ω) =Nq2
m
−iω
Ω2 − ω2 − iωΓ
![Page 80: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/80.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Electrical conductivity
• consider a typical charged particle
• recall m(x + Γx + Ω2x) = qE
• electric current density J = Nqx
• Fourier transformed: J = Nq(−iω)x
• recall
x(ω) =q
m
E(ω)
Ω2 − ω2 − iωΓ
• Ohm’s law
J(ω) = σ(ω)E(ω)
• conductivity is
σ(ω) =Nq2
m
−iω
Ω2 − ω2 − iωΓ
![Page 81: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/81.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Electrical conductivity
• consider a typical charged particle
• recall m(x + Γx + Ω2x) = qE
• electric current density J = Nqx
• Fourier transformed: J = Nq(−iω)x
• recall
x(ω) =q
m
E(ω)
Ω2 − ω2 − iωΓ
• Ohm’s law
J(ω) = σ(ω)E(ω)
• conductivity is
σ(ω) =Nq2
m
−iω
Ω2 − ω2 − iωΓ
![Page 82: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/82.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Electrical conductivity
• consider a typical charged particle
• recall m(x + Γx + Ω2x) = qE
• electric current density J = Nqx
• Fourier transformed: J = Nq(−iω)x
• recall
x(ω) =q
m
E(ω)
Ω2 − ω2 − iωΓ
• Ohm’s law
J(ω) = σ(ω)E(ω)
• conductivity is
σ(ω) =Nq2
m
−iω
Ω2 − ω2 − iωΓ
![Page 83: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/83.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Electrical conductivity
• consider a typical charged particle
• recall m(x + Γx + Ω2x) = qE
• electric current density J = Nqx
• Fourier transformed: J = Nq(−iω)x
• recall
x(ω) =q
m
E(ω)
Ω2 − ω2 − iωΓ
• Ohm’s law
J(ω) = σ(ω)E(ω)
• conductivity is
σ(ω) =Nq2
m
−iω
Ω2 − ω2 − iωΓ
![Page 84: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/84.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Electrical conductors
• A material with σ(0) = 0 is an electrical insulator . Itcannot transport direct currents (DC).
• A material with σ(0) > 0 is an electrical conductor .
• Charged particles must be free, Ω = 0.
• which means
σ(ω) =Nq2
m
1
Γ− iω• orσ(ω)
σ(0)=
1
1− iω/Γ
• Note that the DC conductivity is always positive .
![Page 85: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/85.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Electrical conductors
• A material with σ(0) = 0 is an electrical insulator . Itcannot transport direct currents (DC).
• A material with σ(0) > 0 is an electrical conductor .
• Charged particles must be free, Ω = 0.
• which means
σ(ω) =Nq2
m
1
Γ− iω• orσ(ω)
σ(0)=
1
1− iω/Γ
• Note that the DC conductivity is always positive .
![Page 86: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/86.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Electrical conductors
• A material with σ(0) = 0 is an electrical insulator . Itcannot transport direct currents (DC).
• A material with σ(0) > 0 is an electrical conductor .
• Charged particles must be free, Ω = 0.
• which means
σ(ω) =Nq2
m
1
Γ− iω• orσ(ω)
σ(0)=
1
1− iω/Γ
• Note that the DC conductivity is always positive .
![Page 87: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/87.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Electrical conductors
• A material with σ(0) = 0 is an electrical insulator . Itcannot transport direct currents (DC).
• A material with σ(0) > 0 is an electrical conductor .
• Charged particles must be free, Ω = 0.
• which means
σ(ω) =Nq2
m
1
Γ− iω• orσ(ω)
σ(0)=
1
1− iω/Γ
• Note that the DC conductivity is always positive .
![Page 88: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/88.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Electrical conductors
• A material with σ(0) = 0 is an electrical insulator . Itcannot transport direct currents (DC).
• A material with σ(0) > 0 is an electrical conductor .
• Charged particles must be free, Ω = 0.
• which means
σ(ω) =Nq2
m
1
Γ− iω
• orσ(ω)
σ(0)=
1
1− iω/Γ
• Note that the DC conductivity is always positive .
![Page 89: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/89.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Electrical conductors
• A material with σ(0) = 0 is an electrical insulator . Itcannot transport direct currents (DC).
• A material with σ(0) > 0 is an electrical conductor .
• Charged particles must be free, Ω = 0.
• which means
σ(ω) =Nq2
m
1
Γ− iω• orσ(ω)
σ(0)=
1
1− iω/Γ
• Note that the DC conductivity is always positive .
![Page 90: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/90.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Electrical conductors
• A material with σ(0) = 0 is an electrical insulator . Itcannot transport direct currents (DC).
• A material with σ(0) > 0 is an electrical conductor .
• Charged particles must be free, Ω = 0.
• which means
σ(ω) =Nq2
m
1
Γ− iω• orσ(ω)
σ(0)=
1
1− iω/Γ
• Note that the DC conductivity is always positive .
![Page 91: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/91.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Georg Simon Ohm, German physicist, 1789-1854
![Page 92: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/92.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
External static magnetic field
• 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 = Bez• assume circularly polarized light
E = E±e± where e± = (ex + iey)/√
2
• try x = x±e±
• note e± × ez = ∓ie±
• therefore
m(−ω2 − iωΓ + Ω2)x± = q(E± ∓ ωBx±)
![Page 93: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/93.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
External static magnetic field
• 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 = Bez• assume circularly polarized light
E = E±e± where e± = (ex + iey)/√
2
• try x = x±e±
• note e± × ez = ∓ie±
• therefore
m(−ω2 − iωΓ + Ω2)x± = q(E± ∓ ωBx±)
![Page 94: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/94.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
External static magnetic field
• 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 = Bez• assume circularly polarized light
E = E±e± where e± = (ex + iey)/√
2
• try x = x±e±
• note e± × ez = ∓ie±
• therefore
m(−ω2 − iωΓ + Ω2)x± = q(E± ∓ ωBx±)
![Page 95: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/95.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
External static magnetic field
• 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 = Bez• assume circularly polarized light
E = E±e± where e± = (ex + iey)/√
2
• try x = x±e±
• note e± × ez = ∓ie±
• therefore
m(−ω2 − iωΓ + Ω2)x± = q(E± ∓ ωBx±)
![Page 96: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/96.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
External static magnetic field
• 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 = Bez
• assume circularly polarized light
E = E±e± where e± = (ex + iey)/√
2
• try x = x±e±
• note e± × ez = ∓ie±
• therefore
m(−ω2 − iωΓ + Ω2)x± = q(E± ∓ ωBx±)
![Page 97: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/97.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
External static magnetic field
• 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 = Bez• assume circularly polarized light
E = E±e± where e± = (ex + iey)/√
2
• try x = x±e±
• note e± × ez = ∓ie±
• therefore
m(−ω2 − iωΓ + Ω2)x± = q(E± ∓ ωBx±)
![Page 98: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/98.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
External static magnetic field
• 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 = Bez• assume circularly polarized light
E = E±e± where e± = (ex + iey)/√
2
• try x = x±e±
• note e± × ez = ∓ie±
• therefore
m(−ω2 − iωΓ + Ω2)x± = q(E± ∓ ωBx±)
![Page 99: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/99.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
External static magnetic field
• 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 = Bez• assume circularly polarized light
E = E±e± where e± = (ex + iey)/√
2
• try x = x±e±
• note e± × ez = ∓ie±
• therefore
m(−ω2 − iωΓ + Ω2)x± = q(E± ∓ ωBx±)
![Page 100: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/100.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
External static magnetic field
• 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 = Bez• assume circularly polarized light
E = E±e± where e± = (ex + iey)/√
2
• try x = x±e±
• note e± × ez = ∓ie±
• therefore
m(−ω2 − iωΓ + Ω2)x± = q(E± ∓ ωBx±)
![Page 101: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/101.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Faraday effect
• m(−ω2 − iωΓ + Ω2)x± = q(E± ∓ ωBx±)
• therefore
x± =qE±
m(Ω2 − iωΓ− ω2)± qωB• recall P = Nqx = ε0χE
• effect of quasi-static induction B is
χ±(ω) =Nq2
ε0m
1
Ω2 − iωΓ− ω2 ± (q/m)ωB• left and right handed polarized light sees different
susceptibility
• Faraday effect
![Page 102: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/102.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Faraday effect
• m(−ω2 − iωΓ + Ω2)x± = q(E± ∓ ωBx±)
• therefore
x± =qE±
m(Ω2 − iωΓ− ω2)± qωB• recall P = Nqx = ε0χE
• effect of quasi-static induction B is
χ±(ω) =Nq2
ε0m
1
Ω2 − iωΓ− ω2 ± (q/m)ωB• left and right handed polarized light sees different
susceptibility
• Faraday effect
![Page 103: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/103.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Faraday effect
• m(−ω2 − iωΓ + Ω2)x± = q(E± ∓ ωBx±)
• therefore
x± =qE±
m(Ω2 − iωΓ− ω2)± qωB
• recall P = Nqx = ε0χE
• effect of quasi-static induction B is
χ±(ω) =Nq2
ε0m
1
Ω2 − iωΓ− ω2 ± (q/m)ωB• left and right handed polarized light sees different
susceptibility
• Faraday effect
![Page 104: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/104.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Faraday effect
• m(−ω2 − iωΓ + Ω2)x± = q(E± ∓ ωBx±)
• therefore
x± =qE±
m(Ω2 − iωΓ− ω2)± qωB• recall P = Nqx = ε0χE
• effect of quasi-static induction B is
χ±(ω) =Nq2
ε0m
1
Ω2 − iωΓ− ω2 ± (q/m)ωB• left and right handed polarized light sees different
susceptibility
• Faraday effect
![Page 105: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/105.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Faraday effect
• m(−ω2 − iωΓ + Ω2)x± = q(E± ∓ ωBx±)
• therefore
x± =qE±
m(Ω2 − iωΓ− ω2)± qωB• recall P = Nqx = ε0χE
• effect of quasi-static induction B is
χ±(ω) =Nq2
ε0m
1
Ω2 − iωΓ− ω2 ± (q/m)ωB
• left and right handed polarized light sees differentsusceptibility
• Faraday effect
![Page 106: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/106.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Faraday effect
• m(−ω2 − iωΓ + Ω2)x± = q(E± ∓ ωBx±)
• therefore
x± =qE±
m(Ω2 − iωΓ− ω2)± qωB• recall P = Nqx = ε0χE
• effect of quasi-static induction B is
χ±(ω) =Nq2
ε0m
1
Ω2 − iωΓ− ω2 ± (q/m)ωB• left and right handed polarized light sees different
susceptibility
• Faraday effect
![Page 107: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/107.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Faraday effect
• m(−ω2 − iωΓ + Ω2)x± = q(E± ∓ ωBx±)
• therefore
x± =qE±
m(Ω2 − iωΓ− ω2)± qωB• recall P = Nqx = ε0χE
• effect of quasi-static induction B is
χ±(ω) =Nq2
ε0m
1
Ω2 − iωΓ− ω2 ± (q/m)ωB• left and right handed polarized light sees different
susceptibility
• Faraday effect
![Page 108: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/108.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Michael Faraday, English physicist, 1791-1867
![Page 109: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/109.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Remarks
• B is always small (in natural units)
• εij(ω;B) = εij(ω; 0) + iK(ω)εijkBk• linear magneto-optic effect
• Faraday constant is
K(ω) =Nq3
ε0m2
ω
(Ω2 − iωΓ− ω2)2
• K(ω) is real in transparency window
• i. e. if ω is far away form Ω
• Faraday effect distinguishes between forward and backwardpropagation
• optical isolator
![Page 110: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/110.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Remarks
• B is always small (in natural units)
• εij(ω;B) = εij(ω; 0) + iK(ω)εijkBk• linear magneto-optic effect
• Faraday constant is
K(ω) =Nq3
ε0m2
ω
(Ω2 − iωΓ− ω2)2
• K(ω) is real in transparency window
• i. e. if ω is far away form Ω
• Faraday effect distinguishes between forward and backwardpropagation
• optical isolator
![Page 111: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/111.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Remarks
• B is always small (in natural units)
• εij(ω;B) = εij(ω; 0) + iK(ω)εijkBk
• linear magneto-optic effect
• Faraday constant is
K(ω) =Nq3
ε0m2
ω
(Ω2 − iωΓ− ω2)2
• K(ω) is real in transparency window
• i. e. if ω is far away form Ω
• Faraday effect distinguishes between forward and backwardpropagation
• optical isolator
![Page 112: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/112.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Remarks
• B is always small (in natural units)
• εij(ω;B) = εij(ω; 0) + iK(ω)εijkBk• linear magneto-optic effect
• Faraday constant is
K(ω) =Nq3
ε0m2
ω
(Ω2 − iωΓ− ω2)2
• K(ω) is real in transparency window
• i. e. if ω is far away form Ω
• Faraday effect distinguishes between forward and backwardpropagation
• optical isolator
![Page 113: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/113.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Remarks
• B is always small (in natural units)
• εij(ω;B) = εij(ω; 0) + iK(ω)εijkBk• linear magneto-optic effect
• Faraday constant is
K(ω) =Nq3
ε0m2
ω
(Ω2 − iωΓ− ω2)2
• K(ω) is real in transparency window
• i. e. if ω is far away form Ω
• Faraday effect distinguishes between forward and backwardpropagation
• optical isolator
![Page 114: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/114.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Remarks
• B is always small (in natural units)
• εij(ω;B) = εij(ω; 0) + iK(ω)εijkBk• linear magneto-optic effect
• Faraday constant is
K(ω) =Nq3
ε0m2
ω
(Ω2 − iωΓ− ω2)2
• K(ω) is real in transparency window
• i. e. if ω is far away form Ω
• Faraday effect distinguishes between forward and backwardpropagation
• optical isolator
![Page 115: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/115.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Remarks
• B is always small (in natural units)
• εij(ω;B) = εij(ω; 0) + iK(ω)εijkBk• linear magneto-optic effect
• Faraday constant is
K(ω) =Nq3
ε0m2
ω
(Ω2 − iωΓ− ω2)2
• K(ω) is real in transparency window
• i. e. if ω is far away form Ω
• Faraday effect distinguishes between forward and backwardpropagation
• optical isolator
![Page 116: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/116.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Remarks
• B is always small (in natural units)
• εij(ω;B) = εij(ω; 0) + iK(ω)εijkBk• linear magneto-optic effect
• Faraday constant is
K(ω) =Nq3
ε0m2
ω
(Ω2 − iωΓ− ω2)2
• K(ω) is real in transparency window
• i. e. if ω is far away form Ω
• Faraday effect distinguishes between forward and backwardpropagation
• optical isolator
![Page 117: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/117.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Remarks
• B is always small (in natural units)
• εij(ω;B) = εij(ω; 0) + iK(ω)εijkBk• linear magneto-optic effect
• Faraday constant is
K(ω) =Nq3
ε0m2
ω
(Ω2 − iωΓ− ω2)2
• K(ω) is real in transparency window
• i. e. if ω is far away form Ω
• Faraday effect distinguishes between forward and backwardpropagation
• optical isolator
![Page 118: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/118.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Conduction in a magnetic field
• 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
x =q
m
1
−iω
1
Γ− iωE − iωx×B
• by iteration
x = . . . E +q
m
1
Γ− iωE ×B
• Ohmic current ∝ E and Hall current ∝ E ×B
![Page 119: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/119.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Conduction in a magnetic field
• 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
x =q
m
1
−iω
1
Γ− iωE − iωx×B
• by iteration
x = . . . E +q
m
1
Γ− iωE ×B
• Ohmic current ∝ E and Hall current ∝ E ×B
![Page 120: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/120.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Conduction in a magnetic field
• 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
x =q
m
1
−iω
1
Γ− iωE − iωx×B
• by iteration
x = . . . E +q
m
1
Γ− iωE ×B
• Ohmic current ∝ E and Hall current ∝ E ×B
![Page 121: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/121.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Conduction in a magnetic field
• 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
x =q
m
1
−iω
1
Γ− iωE − iωx×B
• by iteration
x = . . . E +q
m
1
Γ− iωE ×B
• Ohmic current ∝ E and Hall current ∝ E ×B
![Page 122: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/122.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Conduction in a magnetic field
• 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
x =q
m
1
−iω
1
Γ− iωE − iωx×B
• by iteration
x = . . . E +q
m
1
Γ− iωE ×B
• Ohmic current ∝ E and Hall current ∝ E ×B
![Page 123: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/123.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Conduction in a magnetic field
• 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
x =q
m
1
−iω
1
Γ− iωE − iωx×B
• by iteration
x = . . . E +q
m
1
Γ− iωE ×B
• Ohmic current ∝ E and Hall current ∝ E ×B
![Page 124: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/124.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Conduction in a magnetic field
• 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
x =q
m
1
−iω
1
Γ− iωE − iωx×B
• by iteration
x = . . . E +q
m
1
Γ− iωE ×B
• Ohmic current ∝ E and Hall current ∝ E ×B
![Page 125: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/125.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Conduction in a magnetic field
• 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
x =q
m
1
−iω
1
Γ− iωE − iωx×B
• by iteration
x = . . . E +q
m
1
Γ− iωE ×B
• Ohmic current ∝ E and Hall current ∝ E ×B
![Page 126: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/126.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Hall effect, schematilly
![Page 127: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/127.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Hall effect
• Hall current usually forbidden by boundary conditions
• Hall field
EH = − q
m
1
Γ− iωE ×B
• replace E by E + EH
• EH ×B can be neglected
• current J(ω) = σ(ω)E(ω) as usual
• additional Hall field EH(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
![Page 128: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/128.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Hall effect
• Hall current usually forbidden by boundary conditions
• Hall field
EH = − q
m
1
Γ− iωE ×B
• replace E by E + EH
• EH ×B can be neglected
• current J(ω) = σ(ω)E(ω) as usual
• additional Hall field EH(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
![Page 129: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/129.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Hall effect
• Hall current usually forbidden by boundary conditions
• Hall field
EH = − q
m
1
Γ− iωE ×B
• replace E by E + EH
• EH ×B can be neglected
• current J(ω) = σ(ω)E(ω) as usual
• additional Hall field EH(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
![Page 130: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/130.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Hall effect
• Hall current usually forbidden by boundary conditions
• Hall field
EH = − q
m
1
Γ− iωE ×B
• replace E by E + EH
• EH ×B can be neglected
• current J(ω) = σ(ω)E(ω) as usual
• additional Hall field EH(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
![Page 131: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/131.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Hall effect
• Hall current usually forbidden by boundary conditions
• Hall field
EH = − q
m
1
Γ− iωE ×B
• replace E by E + EH
• EH ×B can be neglected
• current J(ω) = σ(ω)E(ω) as usual
• additional Hall field EH(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
![Page 132: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/132.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Hall effect
• Hall current usually forbidden by boundary conditions
• Hall field
EH = − q
m
1
Γ− iωE ×B
• replace E by E + EH
• EH ×B can be neglected
• current J(ω) = σ(ω)E(ω) as usual
• additional Hall field EH(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
![Page 133: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/133.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Hall effect
• Hall current usually forbidden by boundary conditions
• Hall field
EH = − q
m
1
Γ− iωE ×B
• replace E by E + EH
• EH ×B can be neglected
• current J(ω) = σ(ω)E(ω) as usual
• additional Hall field EH(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
![Page 134: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/134.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Hall effect
• Hall current usually forbidden by boundary conditions
• Hall field
EH = − q
m
1
Γ− iωE ×B
• replace E by E + EH
• EH ×B can be neglected
• current J(ω) = σ(ω)E(ω) as usual
• additional Hall field EH(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
![Page 135: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/135.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Hall effect
• Hall current usually forbidden by boundary conditions
• Hall field
EH = − q
m
1
Γ− iωE ×B
• replace E by E + EH
• EH ×B can be neglected
• current J(ω) = σ(ω)E(ω) as usual
• additional Hall field EH(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
![Page 136: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/136.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Hall effect
• Hall current usually forbidden by boundary conditions
• Hall field
EH = − q
m
1
Γ− iωE ×B
• replace E by E + EH
• EH ×B can be neglected
• current J(ω) = σ(ω)E(ω) as usual
• additional Hall field EH(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
![Page 137: The Drude Model - uni-osnabrueck.de · The Drude Model Peter Hertel Overview Model Dielectric medium Permittivity of metals Electrical conductors Faraday e ect Hall e ect Paul Drude,](https://reader033.vdocuments.mx/reader033/viewer/2022052320/6054141e4ad57f09e46ccab3/html5/thumbnails/137.jpg)
The DrudeModel
Peter Hertel
Overview
Model
Dielectricmedium
Permittivity ofmetals
Electricalconductors
Faraday effect
Hall effect
Edwin Hall, US-American physicist, 1855-1938