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Drude Dielectric Function (Permittivity)

Complex conductivity for different frequencies assuming that τ = 10⁻⁵ and that σ₀ = 1.

The Drude model is often combined with other oscillator models to reproduce the observed permittivity of metals.

Stories to Tell

• Relationship between optical conductivity, permittivity and refractive index.
  • Where does the formula come from?
• Decomposition of optical properties of metals into interband and intraband components.

Physical Interpretation

The relaxation time $\tau$ and it's inverse, the scattering rate $\gamma$ are dominated by electron-phonon scattering in good metals like gold, silver, aluminum, lithium. For such metals, the scattering rate is well approximated from the temperature dependent DC resistivity:

$1/\tau (T)=\gamma (T)=\epsilon _{0}\omega _{p}^{2}\rho _{DC}(T)$

also predict the current as a response to a time-dependent electric field with an angular frequency ω. The complex conductivity is

$\sigma (\omega )={\frac {\sigma _{0}}{1-i\omega \tau }}={\frac {\sigma _{0}}{1+\omega ^{2}\tau ^{2}}}+i\omega \tau {\frac {\sigma _{0}}{1+\omega ^{2}\tau ^{2}}}.$

Errors in Time

The shape parameter is $\beta$ and the scale parameter is $\alpha$. $T$ is the time at which we want to calculate the reliability, $R=1-{\textrm {CFR}}$:

${\textrm {CFR}}=1-\exp \left(-\left({\frac {T}{\alpha }}\right)^{\beta }\right)$

$\ln(R)=-\left({\frac {T}{\alpha }}\right)^{\beta }$

$\ln(-\ln(R))=\beta \ln \left({\frac {T}{\alpha }}\right)$

$\ln(-\ln(R))=\beta \ln \left({\frac {T}{\alpha }}\right)$

$\ln(-\ln(R))=\beta \left(\ln(T)-\ln(\alpha )\right)$

Let $u=\ln(T)$

$u={\frac {1}{\beta }}\ln(-\ln(R))+\ln(\alpha )$

In the following, ${\hat {u}}$ is the expected value of $u$ from a fit and $K_{\alpha _{\mathrm {CI} }}=z$ is the number of std deviations away from the norm that we will calculate the CI for. In Python, z = -scipy.stats.norm.ppf((1 - CI) / 2). The Confidence bounds are estimated by:

$u_{\mathrm {U} }={\hat {u}}+K_{\alpha _{\mathrm {CI} }}{\sqrt {\mathrm {Var} \left({\hat {u}}\right)}}$

$u_{\mathrm {L} }={\hat {u}}-K_{\alpha _{\mathrm {CI} }}{\sqrt {\mathrm {Var} \left({\hat {u}}\right)}}$

$\mathrm {Var} \left({\hat {u}}\right)={\frac {\mathrm {Var} ({\hat {\alpha }})}{\hat {\alpha ^{2}}}}+{\frac {\mathrm {Var} ({\hat {\beta }})}{\hat {\beta ^{4}}}}\left(\ln(-\ln(R))\right)^{2}-2\left({\frac {\mathrm {Cov} ({\hat {\beta }},{\hat {\alpha }})}{{\hat {\alpha }}{\hat {\beta ^{2}}}}}\right)\ln(-\ln(R))$