Controlling the Spontaneous Emission Rate of Quantum Wells in Rolled-Up Hyperbolic Metamaterials

We experimentally demonstrate the enhancement of the spontaneous emission rate of GaAs quantum wells embedded in rolled-up metamaterials. We fabricate microtubes whose walls consist of alternating Ag and (In)(Al)GaAs layers with incorporated active GaAs quantum-well structures. By variation of the layer thickness ratio of the Ag and (In)(Al)GaAs layers we control the effective permittivity tensor of the metamaterial according to an effective medium approach. Thereby, we can design samples with elliptic or hyperbolic dispersion. Time-resolved low temperature photoluminescence spectroscopy supported by finite-difference time-domain simulations reveal a decrease of the quantum well’s spontaneous emission lifetime in our metamaterials as a signature of the crossover from elliptic to hyperbolic dispersion.

Figure 1

(a) Sketch of the sample geometry showing a microtube (diameter $d\sim 5\mu \mathrm{m}$) that was rolled up from the metal coated substrate. Four different samples were prepared each of which with a different Ag layer thickness ranging between 6 and 39 nm. (b) Detailed multilayer structure with integrated QW structures from which microtubes were rolled up by self-rolling of strained layers. Layer composition: GaAs substrate layer, AlAs sacrificial layer (40 nm), InGaAs strained layer (9 nm), $\mathrm{AlGaAs}/\mathrm{GaAs}/\mathrm{AlGaAs}$ QW ($7/4/7\mathrm{nm}$), GaAs coating layer (3 nm), and Ag layer (6–39 nm). (c) Magnified wall of the microtube representing a three-dimensional radial metamaterial consisting of alternating Ag and GaAs QW layers. QWs in the tube wall emit radiation after excitation with the pump laser. Emission of lower lying QW layers is attenuated due to absorption in the tube wall. (d) Model of the microtube wall showing a QW layer emitting radiation on top of an effective medium.

Figure 2

(a) QW lifetimes ${\tau }_{\mathrm{UC}}$ (gray dots), ${\tau }_C$ (gray triangles), and ${\tau }_{\mathrm{RMT}}$ (black squares) on the samples $S_6$, $S_{17}$, $S_{22}$, and $S_{39}$. The top of the plot shows the corresponding sample label; the bottom shows the Ag layer thickness $d_{\mathrm{Ag}}$ on the respective samples. Red highlighted areas indicate Ag layer thicknesses $d_{\mathrm{Ag}}$ for which the RMTs exhibit a hyperbolic dispersion. (b) Normalized lifetime $({\tau }_{\mathrm{RMT}}/{\tau }_C)$ plotted against the Ag layer thickness $d_{\mathrm{Ag}}$ on the respective samples. The blue framed figures (c)–(f) correspond to ${\mathrm{RMT}}_6$ ($d_{\mathrm{Ag}}=6\mathrm{nm}$) and the red framed figures (g)–(j) to ${\mathrm{RMT}}_{22}$ ($d_{\mathrm{Ag}}=22\mathrm{nm}$). The QW lifetimes of ${\mathrm{RMT}}_6$ and ${\mathrm{RMT}}_{22}$ are highlighted in (a) with a blue and red circle, respectively. [(c) and (g)] Real parts of the effective permittivities ${ϵ}_{x,y}$ and ${ϵ}_z$ for ${\mathrm{RMT}}_6$ and ${\mathrm{RMT}}_{22}$ based on an effective medium model. The red highlighted areas mark photon energies $E_{\mathrm{ph}}$ for which $\mathrm{Re}({ϵ}_{x,y})<0$ and $\mathrm{Re}({ϵ}_z)>0$; the black dashed line indicates the QW’s emission energy $E_{\mathrm{ph},\mathrm{QW}}=1.64\mathrm{eV}$. [(d) and (h)] Isofrequency plots of the dispersion relation for (d) ${\mathrm{RMT}}_6$ and (h) ${\mathrm{RMT}}_{22}$ for a photon energy of $E_{\mathrm{ph},\mathrm{QW}}=1.64\mathrm{eV}$. [(e) and (i)] Absolute values of the magnetic field amplitude in linear scale obtained from finite-difference time-domain (FDTD) simulations for a dipole antenna of length 6 nm (blue dot) emitting radiation at an energy $E=1.64\mathrm{eV}$ in the top layer of (e) ${\mathrm{RMT}}_6$ and (i) ${\mathrm{RMT}}_{22}$. [(f) and (j)] Absolute values of the magnetic field amplitude for an idealized homogenous structure (i.e., no individual layers) with anisotropic material parameters ${ϵ}_{x,y}$ and ${ϵ}_z$ as determined from the effective medium model for (f) ${\mathrm{RMT}}_6$ and (j) ${\mathrm{RMT}}_{22}$.