Optical Phase Transition in Semiconductor Quantum Metamaterials

Unexpected light propagation effects, such as negative refraction, have been reported in artificial media. Leveraging on the intersubband resonances in heterostructured semiconductors, we show that all possible optical regimes, ranging from classical dieletric and metal to hyperbolic metamaterial types 1 and 2, can be achieved. As a demonstration, we prove that the negative refraction effect can occur at a designed frequency by controlling the electronic quantum confinement.

Figure 1

(a) Schematic of the toy model structure. A stack along the $z$ direction of two alternating media: layer 1, thickness $l_1$ and a resonant permittivity ${ϵ}_1$; layer 2, thickness $l_2$, dispersionless with a permittivity ${ϵ}_2$. (b) Optical phase diagram showing the sign inversion conditions for both in-plane and $z$ components of the effective permittivity as a function of the normalized frequency $\omega /{\omega }_p$ and of the thickness ratio. 3D inset images: isofrequency surfaces plotted in $k$ space for the different behaviors. The resonance energy is ${\omega }_0={\omega }_p/2$. For simplicity, the curves are plotted imposing ${ϵ}_2={ϵ}_{\infty }=1$ and $\gamma =0$. (c) Comparison between the refraction effect in (right) a conventional and in (left) a negative index material.

Figure 2

Hyperbolic metamaterial condition with a nominal doping of $n\sim 5\times {10}^{19}{\mathrm{cm}}^{-3}$ and $l_{\mathrm{ZnO}}/l_{\mathrm{MgZnO}}=3.5$. (a),(b) Real part of the effective permittivity (a) in-plane and (b) out-of-plane as a function of the frequency and QW thickness. The black solid curves show the sign inversions. (c) Optical phase transition diagram of the system as a function of frequency and QW thickness. (d) $\mathrm{FOM}=\mathrm{Re}(k_{\perp })/\mathrm{Im}(k_{\perp })$, where $k_{\perp }$ is the wave vector in the direction perpendicular to the layers.

Figure 3

(a) MQW measured transmittance under $p$ polarization at a 50° angle of incidence, indicating the absorption at the ISB resonance. (b) Calculated real part of the in-plane and out-of-plane components of the effective permittivity of the sample. The shaded area indicates the region where the structure fulfills the requirements of a type 1 HMM. Ratio of the half-blocked to the unblocked transmittance spectra for (c) $p$ and (d) $s$ polarizations of the incoming light. The spectra at normal incidence have been taken as a baseline and subtracted from all other spectra. The arrows indicate the overall evolution of the spectra with the angle of incidence of the light.