---
title: "Dielectric function"
date: 2022-01-24
categories:
- Physics
- Electromagnetism
- Quantum mechanics
layout: "concept"
---

The **dielectric function** or **relative permittivity** $\varepsilon_r$
is a measure of how strongly a given medium counteracts
[electric fields](/know/concept/electric-field/) compared to a vacuum.
Let $\vb{D}$ be the applied external field,
and $\vb{E}$ the effective field inside the material:

$$\begin{aligned}
    \boxed{
        \vb{D} = \varepsilon_0 \varepsilon_r \vb{E}
    }
\end{aligned}$$

If $\varepsilon_r$ is large, then $\vb{D}$ is strongly suppressed,
because the material's electrons and nuclei move to create an opposing field.
In order for $\varepsilon_r$ to be well defined, we only consider linear media,
where the induced polarization $\vb{P}$ is proportional to $\vb{E}$.

We would like to find an alternative definition of $\varepsilon_r$.
Consider that the usual electric fields $\vb{E}$, $\vb{D}$, and $\vb{P}$
can each be written as the gradient of an electrostatic potential like so,
where $\Phi_\mathrm{tot}$, $\Phi_\mathrm{ext}$ and $\Phi_\mathrm{ind}$
are the total, external and induced potentials, respectively:

$$\begin{aligned}
    \vb{E}
    = -\nabla \Phi_\mathrm{tot}
    \qquad \qquad
    \vb{D}
    = - \varepsilon_0 \nabla \Phi_\mathrm{ext}
    \qquad \qquad
    \vb{P}
    = \varepsilon_0 \nabla \Phi_\mathrm{ind}
\end{aligned}$$

Such that $\Phi_\mathrm{tot} = \Phi_\mathrm{ext} + \Phi_\mathrm{ind}$.
Inserting this into $\vb{D} = \varepsilon_0 \varepsilon_r \vb{E}$
then suggests defining:

$$\begin{aligned}
    \boxed{
        \varepsilon_r
        \equiv \frac{\Phi_\mathrm{ext}}{\Phi_\mathrm{tot}}
    }
\end{aligned}$$


## From induced charge density

A common way to calculate $\varepsilon_r$ is from
the induced charge density $\rho_\mathrm{ind}$,
i.e. the offset caused by the material's particles responding to the field.
We start from [Gauss' law](/know/concept/maxwells-equations/) for $\vb{P}$:

$$\begin{aligned}
    \nabla \cdot \vb{P}
    = \varepsilon_0 \nabla^2 \Phi_\mathrm{ind}(\vb{r})
    = - \rho_\mathrm{ind}(\vb{r})
\end{aligned}$$

This is Poisson's equation, which has the following well-known
[Fourier transform](/know/concept/fourier-transform/):

$$\begin{aligned}
    \Phi_\mathrm{ind}(\vb{q})
    = \frac{\rho_\mathrm{ind}(\vb{q})}{\varepsilon_0 |\vb{q}|^2}
    = V(\vb{q}) \: \rho_\mathrm{ind}(\vb{q})
\end{aligned}$$

Where $V(\vb{q})$ represents Coulomb interactions,
and $V(0) = 0$ to ensure overall neutrality:

$$\begin{aligned}
    V(\vb{q})
    = \frac{1}{\varepsilon_0 |\vb{q}|^2}
    \qquad \implies \qquad
    V(\vb{r} - \vb{r}')
    = \frac{1}{4 \pi \varepsilon_0 |\vb{r} - \vb{r}'|}
\end{aligned}$$

The [convolution theorem](/know/concept/convolution-theorem/)
then gives us the solution $\Phi_\mathrm{ind}$ in the $\vb{r}$-domain:

$$\begin{aligned}
    \Phi_\mathrm{ind}(\vb{r})
    = (V * \rho_\mathrm{ind})(\vb{r})
    = \int_{-\infty}^\infty V(\vb{r} - \vb{r}') \: \rho_\mathrm{ind}(\vb{r}') \dd{\vb{r}'}
\end{aligned}$$

To proceed, we need to find an expression for $\rho_\mathrm{ind}$
that is proportional to $\Phi_\mathrm{tot}$ or $\Phi_\mathrm{ext}$,
or some linear combination thereof.
Such an expression must exist for a linear material.

Suppose we can show that $\rho_\mathrm{ind} = C_\mathrm{ext} \Phi_\mathrm{ext}$,
for some $C_\mathrm{ext}$, which may depend on $\vb{q}$. Then:

$$\begin{aligned}
    \Phi_\mathrm{tot}
    = (1 + C_\mathrm{ext} V) \Phi_\mathrm{ext}
    \quad \implies \quad
    \boxed{
        \varepsilon_r(\vb{q})
        = \frac{1}{1 + C_\mathrm{ext}(\vb{q}) V(\vb{q})}
    }
\end{aligned}$$

Similarly, suppose we can show that $\rho_\mathrm{ind} = C_\mathrm{tot} \Phi_\mathrm{tot}$,
for some quantity $C_\mathrm{tot}$, then:

$$\begin{aligned}
    \Phi_\mathrm{ext}
    = (1 - C_\mathrm{tot} V) \Phi_\mathrm{tot}
    \quad \implies \quad
    \boxed{
        \varepsilon_r(\vb{q})
        = 1 - C_\mathrm{tot}(\vb{q}) V(\vb{q})
    }
\end{aligned}$$



## References
1.  H. Bruus, K. Flensberg,
    *Many-body quantum theory in condensed matter physics*,
    2016, Oxford.
2.  M. Fox,
    *Optical properties of solids*, 2nd edition,
    Oxford.