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?
Best Answer
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