Optical Properties

dielectric function in micro-scale

$$\begin{array}{r}𝛿𝜌=𝜒𝛿{𝜙}_{ext}=𝑃𝛿𝜙\\ 𝜖𝛿𝜙=𝛿{𝜙}_{\mathrm{ext}}\\ 𝛿𝜙={𝜖}^{-1}𝛿{𝜙}_{\mathrm{ext}}\end{array}$$

where:

• $𝛿𝜌$: Change in charge density
• $𝜒$: Electric susceptibility
• $𝛿{𝜙}_{ext}$: External potential change
• $𝑃$: Polarization function
• $𝜖$: dielectric function
• $𝛿𝜙=𝛿{𝜙}_{\mathrm{ext}}+𝛿{𝜙}_{\mathrm{ind}}$: total potential change, which is sum of changes in the external potential and induced one

INFO

the short formula is used above equations

$$(𝜒𝛿{𝜙}_{ext})(𝒓,{𝒓}^{\prime },𝜔)=\int d{𝒓}^{″}𝜒(𝒓,{𝒓}^{″},𝜔)𝛿{𝜙}_{ext}({𝒓}^{″},{𝒓}^{\prime },𝜔)$$

From these equations, we obtain the following equations straightforwardly

$$\begin{array}{rl}𝜖& =1-𝑣𝑃\\ {𝜖}^{-1}& =1+𝑣𝜒\\ 𝜒& =𝑃+𝑃𝑣𝜒\end{array}$$

where:

• $𝑣$: Coulomb potential

RPA Approxiamation:

$$𝑃=𝛱$$

where

• $𝛱$: Kohn-Sham density response function

$$ϵ(\mathbf{r},{\mathbf{r}}^{\prime },\omega )=\delta (\mathbf{r}-{\mathbf{r}}^{\prime })-\int d{\mathbf{r}}^{″}v(\mathbf{r}-{\mathbf{r}}^{″})𝛱({\mathbf{r}}^{″},{\mathbf{r}}^{\prime },\omega )$$

macroscale dielectric function

$$\widehat{𝒒}\cdot {𝜖}_{\mathrm{mac}}(𝒒,𝜔)\cdot \widehat{𝒒}=1/[{𝜖}^{-1}(𝒒,𝜔){]}_{0,0}$$

without local field correction

$${𝜖}_{\mathrm{mac}}(𝒒,𝜔)\simeq {𝜖}_{0,0}(𝒒,𝜔)$$

in this case, the macroscoptic dielectric function is obtained by following way

$${𝜖}_{0,0}(𝒒,𝜔)=1-{𝑣}_0(𝒒){𝑃}_{0,0}(𝒒,𝜔)$$

only, $0,0$ component of $𝑃$ is needed, being the low computational cost.

local field correction

$$\widehat{𝒒}\cdot {𝜖}_{\mathrm{mac}}(𝒒,𝜔)\cdot \widehat{𝒒}=1/[{𝜖}^{-1}(𝒒,𝜔){]}_{0,0}$$

in this case, all matrix element of ${𝜖}_{\mu \nu }(𝒒,𝜔)$ is needed to calculate the inverse of matrix. See the Adler for the details.

relation with optical functions

• $𝑛$: refractive index

$$𝑛=\sqrt{\frac{{𝜖}_{\mathrm{mac},1}+\sqrt{{𝜖}_{\mathrm{mac},1}^2+{𝜖}_{\mathrm{mac},2}^2}}{2}}$$

• $𝜅$: extinction coeffcient

$$𝜅=\sqrt{\frac{-{𝜖}_{\mathrm{mac},1}+\sqrt{{𝜖}_{\mathrm{mac},1}^2+{𝜖}_{\mathrm{mac},2}^2}}{2}}$$

• $𝑅$: reflectance

$$𝑅=\frac{(𝑛-1{)}^2+{𝜅}^2}{(𝑛+1{)}^2+{𝜅}^2}$$