Strongly nonexponential time-resolved fluorescence of quantum-dot ensembles in three-dimensional photonic crystals

We observe experimentally that ensembles of quantum dots in three-dimensional (3D) photonic crystals reveal strongly nonexponential time-resolved emission. These complex emission decay curves are analyzed with a continuous distribution of decay rates. The log-normal distribution describes the decays well for all studied lattice parameters. The distribution width is identified with variations of the radiative emission rates of quantum dots with various positions and dipole orientations in the unit cell. We find a striking sixfold change of the width of the distribution by varying the lattice parameter. This interpretation qualitatively agrees with the calculations of the 3D projected local density of states. We therefore conclude that fluorescence decay of ensembles of quantum dots is highly nonexponential to an extent that is controlled by photonic crystals.

Figure 1

(a) Electron-microscope image of the (111) surface of an inverse opal consisting of air spheres in $\mathrm{Ti}{\mathrm{O}}_2$ with lattice parameter $a=425\mathrm{nm}$. (b) Schematic of quantum dots (black spots) on the internal titania-air surfaces at symmetry-inequivalent positions ${\mathsf{r}}_{\mathsf{i}}$ in the crystal unit cell.

Figure 2

(Color online) Fluorescence decay curves recorded at $\lambda =615\mathrm{nm}$ and $T=295\mathrm{K}$ from QDs in inverse opals with lattice parameters of $a=425\mathrm{nm}$ (1), $a=255\mathrm{nm}$ (2), and $a=540\mathrm{nm}$ (3), and from QDs in a different chemical environment, a chloroform suspension (S). The solid lines are fits of the log-normal distribution of decay rates to the data. The goodness of fit ${\chi }_{\mathit{red}}^2$ varies from 1.1 to 1.4, close to the ideal value of 1. The dashed line in (3) shows a heuristic biexponential model, which does not agree with the data $({\chi }_{\mathit{red}}^2>1.9)$.

Figure 3

(Color online) Decay-rate distributions $\varphi \left(\gamma \right)$ for the inverse opals with lattice parameters $a=425\mathrm{nm}$ (1), $a=255\mathrm{nm}$ (2), and $a=540\mathrm{nm}$ (3), corresponding to the data shown in Fig. 2. The inset shows $\varphi \left(\gamma \right)$ for $a=255\mathrm{nm}$ in the semilogarithmic scale. Clear modifications of $\Delta \gamma$ and ${\gamma }_{\mathrm{MF}}$ with varying lattice parameter of the inverse opals are seen.

Figure 4

(a) Width of the decay-rate distribution $\Delta \gamma$ vs lattice parameter at a fixed emission wavelength (triangles) (the dashed curve is a guide to the eye). (b) Difference between ${\gamma }_{\mathrm{MF}}$ measured on a photonic sample and ${\gamma }_{\mathrm{MF}}^{\mathit{ref}}=0.05{\mathrm{ns}}^{-1}$ of the low-frequency reference (squares). The error bars are estimated from the largest difference between data on the samples with similar $a$. The dashed curve represents the relative DOS in the inverse opals. The relative (L)DOS in the inverse opals is the (L)DOS divided by DOS in a homogeneous medium with the same average refractive index $(n=1.27)$. (c) Relative LDOS at two positions on the internal $\mathrm{Ti}{\mathrm{O}}_2$-air surfaces projected on dipole orientations parallel $(\Vert )$ or normal $(\perp )$ to the internal surface: at ${\mathsf{r}}_{\mathsf{a}}=0.144(1,1,2)$ and ${\mathsf{r}}_{\mathsf{b}}=0.2(1,1,1)$.