Nonreciprocal and Non-Hermitian Material Response Inspired by Semiconductor Transistors

Here, inspired by the operation of conventional semiconductor transistors, we introduce a novel class of bulk materials with nonreciprocal and non-Hermitian electromagnetic response. Our analysis shows that material nonlinearities combined with a static electric bias may lead to a linearized permittivity tensor that lacks the Hermitian and transpose symmetries. Remarkably, the material can either dissipate or generate energy, depending on the relative phase of the electric field components. We introduce a simple design for an electromagnetic isolator based on an idealized “MOSFET-metamaterial” and show that its performance can in principle surpass conventional Faraday isolators due to the material gain. Furthermore, it is suggested that analogous material responses may be engineered in natural media in nonequilibrium situations. Our solution determines an entirely novel paradigm to break the electromagnetic reciprocity in a bulk nonlinear material using a static electric bias.

Figure 1

(a)(i) Sketch of a MOSFET transistor. (ii) Illustration of a metamaterial formed by a periodic array of MOSFETs. (b) Geometrical relation between the $\mathbf{E}$ field of the plane wave modes in the MOSFET-metamaterial [Eq. (5)] for propagation along $y$.

Figure 2

Normalized Poynting vector as a function of the propagation distance $y$ normalized to the vacuum wavelength ${\lambda }_0=2\pi c/\omega$. The parameters are ${ϵ}_{xx}=2$, ${ϵ}_{zz}=2.1$, ${ϵ}_{xz}=0.2$, $A_e=1$ and $A_o=-A_e[{ϵ}_{xz}/({ϵ}_{zz}-{ϵ}_{xx})]$. The regions shaded in green correspond to active regions, whereas the regions shaded in purple correspond to dissipative regions.

Figure 3

(a) Electromagnetic isolator based on an electrically biased MOSFET-metamaterial placed in between two orthogonal linear polarizers. To ease the visualization the polarizers are represented as wire grids, WGP1 and WGP2, that absorb electric fields parallel to the directions of the wires, i.e., the $x$ and $z$ directions, respectively. The propagation from right to left is forbidden. The direction of the electric bias ${\mathbf{E}}_0$ is also shown. (b) Transmission from left to right $|T_{\mathrm{iso}}^{l\to r}|$ as a function of the isolator thickness $d$ normalized to the free-space wavelength. The parameters are ${ϵ}_{xx}=2$, ${ϵ}_{xz}=0.2$ and (i) ${ϵ}_{zz}=2.1$ and (ii) ${ϵ}_{zz}=2.5$. The red line is the exact solution whereas the blue dashed line is the approximate solution given by Eq. (10). The vertical dotted lines mark the positions of $d_{\max }$.