Categories: Electromagnetism, Physics, Quantum mechanics.

Dielectric function

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

D=ε0εrE\begin{aligned} \boxed{ \vb{D} = \varepsilon_0 \varepsilon_r \vb{E} } \end{aligned}

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

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

E=ΦtotD=ε0ΦextP=ε0Φind\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 Φtot=Φext+Φind\Phi_\mathrm{tot} = \Phi_\mathrm{ext} + \Phi_\mathrm{ind}. Inserting this into D=ε0εrE\vb{D} = \varepsilon_0 \varepsilon_r \vb{E} then suggests defining:

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

In practice, a common way to calculate εr\varepsilon_r is from the induced charge density ρind\rho_\mathrm{ind}, i.e. the offset caused by the material’s particles responding to the field. Starting from Gauss’ law for P\vb{P}:

P=ε02Φind(r)=ρind(r)\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:

Φind(q)=ρind(q)ε0q2V(q)ρind(q)\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(q)V(\vb{q}) represents Coulomb interactions, and V(0)0V(0) \equiv 0 to ensure overall neutrality:

V(q)1ε0q2    V(rr)=14πε0rr\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 Φind\Phi_\mathrm{ind} in the r\vb{r}-domain:

Φind(r)=(Vρind)(r)=V(rr)ρind(r)dr\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 εr\varepsilon_r from ρind\rho_\mathrm{ind}, one needs an expression for ρind\rho_\mathrm{ind} that is proportional to Φtot\Phi_\mathrm{tot} or Φext\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 εr\varepsilon_r once such an expression has been found.

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

Φtot=(1+cextV)Φext    εr(q)=11+cext(q)V(q)\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 ρind=ctotΦtot\rho_\mathrm{ind} = c_\mathrm{tot} \Phi_\mathrm{tot} for some quantity ctotc_\mathrm{tot}, then:

Φext=(1ctotV)Φtot    εr(q)=1ctot(q)V(q)\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 ρind=cextΦext ⁣+ ⁣ctotΦtot\rho_\mathrm{ind} = c_\mathrm{ext} \Phi_\mathrm{ext} \!+\! c_\mathrm{tot} \Phi_\mathrm{tot} is found:

(1ctotV)Φtot=(1+cextV)Φext    εr(q)=1ctot(q)V(q)1+cext(q)V(q)\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}


  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.