Optical conductivity

This article was registered in the quality assurance of the physics editorial team . If you are familiar with the topic, you are welcome to participate in the review and possible improvement of the article. The exchange of views about this is currently not taking place on the article discussion page , but on the quality assurance side of physics.

The optical conductivity generalizes the electrical conductivity (which is often understood as a dc conductivity) to attend their frequency dependence.

Well-known models that describe the optical conductivity are the Drude theory / Drude-Sommerfeld theory for free charge carriers in metals and the model of the Lorentz oscillator for bound charge carriers.

Maxwell's equations in matter

If one assumes that there is no external current, only the bound and unbound electrons that are influenced by an electric field can contribute to the current. With Ohm's law, the contribution of the unbound electrons is expressed in the following way:

${\displaystyle J_{cond}=\sigma _{1}E,}$

${\displaystyle \sigma _{1}}$denotes the conductivity, which does not depend on the electrical field . Furthermore, a homogeneously polarized distribution can be assumed for the charge distribution of the electrons, in which the positive and negative charges are balanced and which is also described. If you combine these two terms and combine them with the electrical flux density , you arrive at Gaussian law for matter ${\displaystyle \mathbf {E} }$ ${\displaystyle \rho }$ ${\displaystyle \rho =-\nabla \cdot P}$ ${\displaystyle \mathbf {D} }$

${\displaystyle D=\epsilon _{1}\mathbf {E} =(1+4\pi \chi _{\text{e}})\mathbf {E} =\mathbf {E} +4\pi \mathbf {P} }$,

where denotes the dielectric susceptibility and denotes the real part of the permittivity. In addition, with these equations and the extended flow law, the complex-valued permittivity can be set up as ${\displaystyle \chi _{\text{e}}}$ ${\displaystyle \epsilon _{1}}$

${\displaystyle {\hat {\epsilon }}=\epsilon _{1}+i{\frac {4\pi \sigma _{1}}{\omega }}=\epsilon _{1}+\epsilon _{2}}$.

Theory according to Drude

In the Drude model, the movement of electrons is described in a similar way to the kinetic gas theory. The relaxation time and the plasma frequency are of decisive importance here . With the Drude-Sommerfeld model, the direct current conductivity can be expressed as follows: ${\displaystyle \tau }$ ${\displaystyle \omega _{\text{p}}}$ ${\displaystyle \sigma _{\text{dc}}}$

${\displaystyle \sigma _{\text{dc}}={\frac {Ne^{2}\tau }{m}}={\frac {1}{4\pi }}\omega _{\text{p}}^{2}\tau }$

and derives from this the frequency-dependent conductivity

${\displaystyle {\hat {\sigma }}(\omega )={\frac {\sigma _{\text{dc}}}{1-{\text{i}}\omega \tau }}}$

If you expand the complex part below the fraction line, you also get the real and imaginary part of the optical conductivity

${\displaystyle \sigma _{1}(\omega )={\frac {\omega _{\text{p}}^{2}\tau }{4\pi }}{\frac {1}{1+\omega ^{2}\tau ^{2}}}}$

and

${\displaystyle \sigma _{2}(\omega )={\frac {\omega _{\text{p}}^{2}\tau }{4\pi }}{\frac {\omega \tau }{1+\omega ^{2}\tau ^{2}}}}$,

Schematic representation of the optical conductivity according to Drude

in which:

• ${\displaystyle \omega _{p}^{2}}$: Plasma resonance of the electron
• N = carrier density
• e = elementary charge
• m = mass

Lorentz oscillator model

The model of the Lorentz oscillator can be seen as an extension of the Drude model. It is assumed here that electrons are stimulated to produce harmonic movements around the core of the atom by electrical fields. The complex conductivity is then described as follows:

${\displaystyle \sigma (\omega )={\frac {Ne^{2}}{m}}{\frac {\omega }{{\text{i}}(\omega _{0}-\omega ^{2})+\omega /\tau }}}$

If you set the center frequency , you get the Drude model again. With this theory, strip transitions in metals are now also taken into account compared to the Drude model. ${\displaystyle \omega _{0}=0}$

literature

• Martin Dressel and George Grüner : Electrodynamics of Solids . 2nd Edition. Cambridge University Press, 2003, ISBN 0-521-59253-4 .