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Categories: Electromagnetism, Physics, Quantum mechanics.

Dielectric function

The dielectric function or relative permittivity $\varepsilon_r$ is a measure of how strongly a given medium counteracts electric fields compared to a vacuum. Let $\vb{D}$ be the applied external field, and $\vb{E}$ the effective field inside the material, then $\varepsilon_r$ is defined such that:

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

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:

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

In practice, 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. Starting from Gauss’ law 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 a well-known solution via Fourier transformation:

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

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

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

Note that the 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 to calculate $\varepsilon_r$ from $\rho_\mathrm{ind}$, one needs 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 medium, but the details depend on the physics being considered and are thus beyond our current scope; we will just show the general form of $\varepsilon_r$ once such an expression has been found.

Suppose we know that $\rho_\mathrm{ind} = c_\mathrm{ext} \Phi_\mathrm{ext}$ for some factor $c_\mathrm{ext}$, which may depend on $\vb{q}$. Then, since $\Phi_\mathrm{tot} = \Phi_\mathrm{ext} \!+\! \Phi_\mathrm{ind}$, we find in the $\vb{q}$-domain:

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

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

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

And in the unlikely event that an expression of the form $\rho_\mathrm{ind} = c_\mathrm{ext} \Phi_\mathrm{ext} \!+\! c_\mathrm{tot} \Phi_\mathrm{tot}$ is found:

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

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.

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