# Physical cause of Negative Permittivity

What is the physical cause behind a material having a negative real part of its dielectric function? Given the complex permittivity, $\epsilon(\omega)=\epsilon(\omega)'+i\epsilon(\omega)''$, the Drude model gives \begin{align} \epsilon'=1-\frac{\omega_{P}^2}{\omega^2+\omega_{\tau}^2} \end{align} where $\omega$ is the frequency of the incoming light, $\omega_{P}=\sqrt{\frac{Ne^2}{m\epsilon_0}}$ is the plasma frequency, $N$ is the electron density, $m$ is the electron's mass, $e$ is the electronic charge, and $\omega_{\tau}$ is the frequency of collisions between conduction electrons and the ion lattice.

If $\omega$ is small enough, then $\epsilon'<0$. But how does this physically happen?

• The wave is reflected. What hapens is that the exponential that describes the solution fo the wave equation becomes a real function of the wave vector, which implies a exponential decay in the field's amplitude, meaning it does not penetrate the material. I'm in my mobile now, tomorrow I'll post a complete answer with some math to explain! Aug 6 '15 at 3:27
• I look forward to reading it! Aug 7 '15 at 14:50
• @Andre although your explanation is definitely right, good and useful, it doesn't answer the question as to what is physically necessary to make this happen. Just mentioning this in the hope you'll incorporate it in your full-fledged answer. May 28 '18 at 11:38

The standard example of an epsilon near zero (ENZ) material is thin metallic film. This supports surface plasmons which have roughly the same dispersion relation as in Drude. In the area of metamaterials, dielectrics can be constructed that have ENZ or negative permittivity. Near resonances things can go crazy. There are even materials, so called double negative index, which have both negative permittivity and negative permeability. These are hard to make but it's been done. For more physical insight, check out a modern book on metamaterials or https://en.wikipedia.org/wiki/Metamaterial

TL;DR: depending on the frequency, the free electrons react with a large lag with respect to the driving field, thus making the real part of $\epsilon$ negative (damped oscillator model).

The permittivity links the displacement field $\textbf{D}$ to the electric field $\textbf{E}$. Roughly speaking it just tells you how much the material reacts to an electric field. Now, for various reasons Maxwell's equations are usually written in frequency domain where, for linear, isotropic and homogenous material the relation between the two fields at one frequency is $\textbf{D}=\epsilon(\omega)\textbf{E}$.

The Drude model is a particular case of the Lorentz model. The latter assumes that electrons are linked to atoms with "springs" and thus experience a linear restoring force. The system is totally analogous to a damped harmonic oscillator model (e.g. mass on a spring). In the case of metals, one can assume that the electrons are free and thus are not attached to the atoms. The spring constant in the Lorentz model is set to 0 and one obtains the Drude model.

Now this model simply describes massive particles reacting to a periodic oscillating force. Depending on their masses (and the material conductivity) and the frequency considered, they can either be in phase with the force (i.e. the real part of the response is positive), or lagging behind (real part negative). The negative part of the real component of the permittivity simply tells you that the $\textbf{D}$ field is lagging behind the $\textbf{E}$ due to the inertia of the electrons.

It is surprising only if one forgets that it occurs only for the permittivity expressed in frequency domain. In time domain the relation between $\textbf{D}$ and $\textbf{E}$ would be a convolution with a causal "response function" that is linked to the Fourier transform of the complex frequency dependent permittivity.

To be a little more precise, the electrons will always lag behind, it is just when this lag is larger than a certain value that the real part is negative (it is $\pi/2$ for a harmonic oscillator, but it must be slightly different in the Drude model due to the +1 constant). For more information also check the Kramers-Kronig relations, its link to causality and the damped harmonic oscillator model.