1. Introduction

Single-photon sources are important for secure communication lines and linear-optics-based quantum computing. For these applications, the single-photon sources need significant improvements in efficiency and speed. Because of the high internal quantum efficiency of quantum dots, the quantum-dot-based single-photon sources are a promising development direction [1,2]. However, the nano-scale engineering of such sources has many challenges, including embedding of the quantum dot into an optical medium or a resonator structure without fluorescence quenching [3], as well as dealing with the short-term instabilities (blinking) [4,5], long-term instabilities (chemical degradation) [6], and homogeneous broadening of the dot’s radiation spectrum due to interaction with phonons. One promising way of increasing the quantum dots’ emission rate (and, hence, the potential speed of quantum communication) is to use the electric field enhancement by metals and metallic resonant structures [7,8]. However, the presence of a material with free electrons (metals) near the dot also induces a strong non-radiative relaxation [9], which limits the benefits of this approach. In microcavity-based single-photon sources (e.g., see review [10]), the emission rate increase originates from both resonant enhancement of the local density of states and from the dielectric properties of the material surrounding the quantum dot. Thus, it is hard to separate and compare the impacts of these two contributions on the total emission rate. While the influence of the nearby metal on the quantum-dot emission has received a lot of attention recently (e.g., see [11,12]), the studies of the role of the surrounding dielectric in quantum-dot fluorescence have been rather limited. Damped-oscillator-like dependence of the fluorescence decay time on the distance between the quantum dot and silicon substrate was demonstrated in [13]. In [14], the reduction of the decay time on a glass surface compared to the decay time in free space was shown. The impact of the interaction within an ensemble of quantum dots on the emission spectra was studied in [15] for several surface types.

The present work experimentally investigates the effect of the surrounding dielectric material on the reduction of the fluorescence decay time of quantum dots. A dot at the surface of a high-refractive-index dielectric is coupled to the denser vacuum field of the dielectric and thus experiences a faster decay. We derive simple theoretical expressions that predict up to one order of magnitude improvement in the decay rate. We analyze experimentally the lifetime of long-wavelength (λ ≈630 nm) CdSe/ZnS core-shell quantum dots either deposited in different ways on a surface of four dielectric media or dissolved in the liquids of reference, toluene and hexane. The experimental data measured for low-index dielectrics (n < 2) match well with the model of a dipole equally shared between two interfacing media, while for high-index dielectrics (n > 2), such as silicon, the correction factors due to reflections and evanescent-field coupling are required.

2. Background

The spontaneous emission rate of an atom in free space is given by [9]:

The atomic transitions are initiated by the interference of vacuum fields coming from different modes. Each mode contains vacuum noise with zero-point energy of one half of a photon. The density of modes in the dielectric is n³ times higher than in free space, but the vacuum field variance is n² times lower. Therefore, the probability of atomic “ignition” is increased n times [factor with the square root of ε_medium in Eq. (2)]. In Eq. (2), the atom is assumed to be contained in a small (less than the average inter-atomic distance) empty virtual spherical cavity formed inside the continuous dielectric medium. The atom sees a higher local field than the average macroscopic field [the former can be expressed as the latter multiplied by a factor (ε_medium + 2) / 3 coming from Lorentz-Lorenz, or Clausius-Mossotti, equation, as illustrated in Fig. 1(a) ], and the resulting radiative decay is faster than that in free space.

 

Fig. 1 (a) Illustration of Lorentz-Lorenz / Clausius-Mossotti relationship between the macroscopic electric field E₁ and the local field E₂ seen by the atom (ε₂ = 1) or quantum dot (ε₂ = ε_dot). (b) Enhancement of the radiative decay rate, compared to the rate in free space, by the refractive index n of the dielectric surrounding the quantum dot. The solid blue line corresponds to the case of complete embedding of the quantum dot into the dielectric host [Eq. (5)]. The solid red line corresponds to the quantum dot positioned at the air-dielectric interface [Eq. (7)]. The dashed purple line shows the asymptotic dependence n⁵/9.

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The optical transition of a quantum dot can be approximated as a two-level atom with a large electron orbit and, therefore, with a large dipole moment e<d>. The background vacuum field interacts with this dipole moment and induces the spontaneous emission. In contrast to the atom, the quantum-dot electron levels are spread over thousands of atoms and, therefore, interact with average macroscopic vacuum field. Hence, the conventional Clausius-Mossotti equation does not apply here. Instead, the difference between the dielectric constants of the quantum-dot material ε_dot and the medium ε_medium, where typically ε_dot > ε_medium, reduces the field inside the dot by a factor (ε_medium + 2) / (ε_dot + 2). This factor comes from considering a spherical cavity filled by material of permittivity ε_dot, formed inside the continuous dielectric medium of permittivity ε_medium, i.e. Clausius-Mossotti relation acts in the opposite direction: the dielectric constant of the dot reduces the vacuum field of the outside medium and hence slows down the decay. The decay rate of a quantum dot in vacuum is

After normalization to the quantum-dot’s free-space decay rate of Eq. (3), the factor ε_dot + 2 in Eqs. (3), (4) cancels out, yielding the following expression:

This expression no longer depends on the refractive index of the quantum-dot material. This result can also be obtained from Eq. (2) by noting that γ_free-space in Eq. (3) is a fictitious non-observable quantity, and the inverse Clausius-Mossotti factor in Eq. (3) can be incorporated into γ_free-space by simply redefining the dipole moment of the quantum dot in Eq. (1).

The real core/shell quantum dots have a complex structure (core-shell-ligands) which practically does not affect the coupling with vacuum fields. For the sake of simplicity, the quantum dot can be considered to be a concentric two-sphere structure (core-ligands) with a core diameter much smaller than that of the ligands. In this case, the local-field correction factor stays the same:

This means that Eqs. (4), (5) are also applicable to the core-shell quantum dots. Thus, the dependence of the quantum-dot decay rate on the dielectric constant of the host medium (5) is similar to Eq. (2) for an atom, and is applicable to dots completely immersed into the host dielectric. The typical situation described by this formula is the dot immersed in a liquid.

Another typical situation is the dot attached to a dielectric surface (e.g. glass, sapphire or silicon). If the center of the dot is assumed to be placed exactly at the dielectric interface, this situation can be approximated analytically in a simple way [16]. The resulting decay rate is the sum of the half-rate into the air and the half-rate into the dielectric. Using Eq. (5), we obtain such experimentally observable rate as

Figure 1(b) illustrates the dependence of the decay rates of a dot on the medium’s refractive index $n=\sqrt{{\epsilon }_{medium}}$ for the cases of complete immersion [solid blue line, Eq. (5)] and air-dielectric interface [solid red line, Eq. (7)]. We see that, for high-index dielectrics, the rate enhancement in the bulk host scales as ~n⁵/9 (dashed purple line) and can exceed 100, whereas the enhancement at the interface is about one half of that in the bulk dielectric.

In particular, for a dot embedded in silicon (n ≈3.5), the decay enhancement factor can reach the value of 80. However, the high-index dielectrics and semiconductors often demonstrate optical losses. The near-field longitudinal component of the dot’s radiating field gets absorbed and contributes to the non-radiative losses. The total decay time of the dot in the dielectric host is the inverse of the sum of radiative and non-radiative decay:

For the dot embedded into a high-index dielectric or a semiconductor such as silicon, the radiative decay time can be reduced down to several nanoseconds; however, the non-radiative part of the decay, if significant, may degrade the efficiency.

3. Samples and experimental setup

In the experiments, we use the CdSe/ZnS core/shell quantum dots from Ocean Nanotech with an octadecinine ligand layer, a central wavelength of emission λ₀ = 630 nm, and a spectral width of ~25 nm. These dots are available in powder form, which has longer shelf life than quantum-dot solutions. The typical diameter of the core/shell ligand’s structure is 10...15 nm [17,18]. Our measurements of the fluorescence intensity have confirmed the manufacturer-specified quantum efficiency of the fluorescence in toluene solution of about 50%.

The measurements are performed using an avalanche photon counting module, a fast digital oscilloscope, a digital delay generator and a home-made blue pulsed laser system arranged in the configuration shown in Fig. 2(a) . Data from the digital oscilloscope are transmitted to the computer and analyzed for time-delay probability distribution.

 

Fig. 2 (a) Experimental setup. Quantum dots deposited on the dielectric are positioned on a 3D piezo-electric translation stage with 50 nm resolution. (b) Spontaneous decay time as a function of refractive index of the dielectric host or interface. The theoretical curves are Eq. (7) and Eq. (5) for the air-dielectric interface (blue) and liquid (green), respectively. Data for low-index dielectrics fit acceptably with the simple dipole-on-the-surface model of Eq. (7). However, the silicon sample shows considerably slower decay compared to the predicted values and requires a more complicated model of Eq. (9) (red), accounting for the distance between the quantum dot and the surface.

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The pump excitation pulses at wavelength 405 nm come at a rate of ~30 kHz. For each measurement of the time decay, between 40,000 and 300,000 time-delay samples are taken. The fluorescence is collected by a single-mode fiber from the dot’s image plane formed by a 1:40 telescope with a numerical aperture NA = 0.95. The single-dot measurements have a strongly pronounced blinking behavior [4]. To avoid the distortion of lifetime data by the parasitic surface-state fluorescence, we discard all photon counting events occurring during the dark state of the quantum dot (“off” state of blinking). The resulting fluorescence decay times, shown in Fig. 2(b), are obtained by averaging several individual measurements, examples of which are presented in Fig. 3(a) . In addition to the dependence on the refractive index of the host medium, the decay time is also a function of the dot concentration, of the state of the dielectric surface, and of the deposition method. Two methods are used in our experiments: spin-on of the quantum dots dissolved in toluene and spin-on from the solution in toluene with PMMA [19]. The first method brings high repeatability in the measurements of decay time. The second results in a better long-term chemical stability of fluorescence efficiency but provides less reliable results for the decay time. Both deposition procedures are performed as follows. The quantum dots are stored in a powder form. Prior to the sample preparation, a small grain of quantum-dot powder (typical size ~100–150 μm) is diluted in toluene to reach a concentration of 3 × 10¹¹ dots/cm³. Then the solution is quickly spun-on to the desired surface. In the case of the second method, the weight percentage of PMMA in toluene is chosen to be around 1% to form a 10–20 nm film over the surface. A typical scanning electron microscope image of the quantum dots deposited on a silicon substrate is shown in Fig. 3(b). The experimental setup shown in Fig. 2(a) has a confocal scanning capability for visualizing and measuring the spatial distribution of the fluorescence. Figure 3(c) shows a typical placement of dots at the surface of a glass as observed by the confocal fluorescence setup prior to the time-decay measurements. To eliminate the impact of closely spaced quantum dots (clusters) on the lifetime data, we select for the measurements only the isolated dots with the average intensity and small size in the observed confocal fluorescence image. Similar pictures are obtained for the three other dielectrics. The fluorescence count rates decrease as the refractive index of the dielectric increases: count rates from the quantum dots on the surface of silicon are almost an order of magnitude smaller than those on the surface of CaF₂. This reduction agrees with the reduction in the collection efficiency by a factor ${\gamma }_{dot}^{free-space}/(2{\gamma }_{dot}^{interface})$, as most of the light emitted by the dot goes into the high-index dielectric and is not collected by the objective on the free-space side of the interface. Hence, there is no indication of change in the intrinsic 50-% fluorescence efficiency of the quantum dot with the refractive index of the interface.

 

Fig. 3 (a) Examples of the raw individual time-decay measurements used in generating the data points in Fig. 2(b). The straight lines are the exponential fits yielding τ = 13.6, 4.6, and 25.6 ns for toluene, silicon, and CaF₂, respectively. (b) Scanning electron microscope image of quantum dots deposited on a silicon substrate. (c) The confocal fluorescence image of quantum dots deposited on a glass substrate. Red circles show the single dots selected for measurements after image analysis. These three dots satisfy the three important criteria: the brightness and the dot size correspond to a single dot particle, and these dots have largest mean square average distance from the neighbors (the typical distance is larger than λ₀).

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The time decay of individual dots strongly fluctuates from one dot to another. This is most likely due to the following factors: the dot size diversity, as well as the variations in the orientation of the dots at the boundary and in the distance between the dots and the interface. Accurate measurement of the decay requires averaging the data from several dots. Points for dielectric interfaces in Fig. 2(b) are averaged over 5–9 single-dot measurements. Points for liquids are averaged over 3–4 measurements with various concentrations. The quantum dots without PMMA show fast degradation of the fluorescence efficiency. For example, in toluene or hexane solution the typical long-term decay time of fluorescence count rate is about 3–6 hours, while the long-term decay time for the dots on the surface of dielectric without PMMA varies between 1 and 10 hours. For a given dot, we have observed that the long-term decline of the fluorescence count rate does not affect the fluorescence decay time.

4. Results and discussion

Measurements of the decay time for quantum dots on the surface of a dielectric and for dots in toluene and hexane are presented in Fig. 2(b). The data are in a good agreement with theory. With the increase of the refractive index, the time decay of the dot becomes consistently shorter. The results for liquids fit Eq. (5) well, with only one fitting parameter — decay time in free space $\tau =1/{\gamma }_{dot}^{free-space}$ = 35 ns. The dots on the dielectric interfaces, on the other hand, do not follow the simple equally-shared-dipole model: although the data for n < 2 are in a reasonable agreement with the theory, the measured decay time for high-index material (silicon) is significantly longer than that predicted by Eq. (7).

The reason for this discrepancy is that the simple model (7) of a dipole equally shared between the two interfacing media is not well suited for high-index dielectric. Position of the dot is inherently asymmetric with respect to the surface boundary: the dot is not centered on the interface, but instead is positioned slightly above it. In the case of a high-index dielectric, the characteristic penetration length of the evanescent vacuum field from the dielectric becomes comparable with the real distance between the dot and the surface, and hence the ratio of the dot’s coupling to the two media’s continua of modes changes significantly. For the high-index case, the solid angle of propagating modes from the dielectric half-space (light cone) coupled to the dot becomes much less than 2π, and the main contribution on the dielectric side comes from the evanescent field. On the air side, the density of states is modified by a strong reflected field. Without attempting an exact analytical solution, we have phenomenologically modified Eq. (7) to account for the contributions of the evanescent and reflected fields at small distances from the interface by introducing an exponential coupling factor [20,21]:

Equation (9) fits well the entirety of the obtained data, using the distance parameter a = 40 nm and $\tau =1/{\gamma }_{\text{dot}}^{\text{free-space}}$ = 35 ns. A posteriori measurements of the surface of silicon revealed the presence of 25-nm-thick thermal oxide, which, along with 10...15 nm size estimate of a quantum dot, justifies the non-zero value of the parameter a used in the fitting.

In our model, we have not considered the presence of the non-zero imaginary part of the refractive index. This imaginary part exists only in the silicon case (where the energy of the emitted photons exceeds the bandgap) and may lead to an increase in the non-radiative recombination rate. Even though our data do not indicate any decrease in the quantum dot emission efficiency for silicon, we are planning to conduct a separate study with a different type of quantum dot, whose emission energy is below the silicon bandgap, to verify this.

The obtained results confirm the model’s prediction of the significant decay rate increase at the surface of a dielectric: decay time decreases from 35 ns in free space to ~5 ns on a silicon surface. Complete embedding of the quantum dot into a high-index material is expected to reduce the decay time to a sub-nanosecond scale. By designing the high-index dielectric in the shape of a semi-sphere, for example, one can efficiently collect the fluorescent photons on the dielectric side. Additionally, this purely dielectric broadband mechanism of the decay time reduction can be combined with dielectric resonant cavity design. Such a structure can modify the emission spectra of the quantum dot and allow the generation of photons-on-demand in a single-mode regime.

5. Conclusion

We have analyzed the theoretical dependence of the decay rate on the refractive index for a quantum dot in a dielectric medium or at the boundary of two media. Theory predicts up to an 80-times increase in the spontaneous emission rate for the dots embedded into a high-index dielectric. The experiments conducted with CdSe/ZnS core-shell quantum dots show a reasonable agreement with the theory. This indicates that one can expect to design and fabricate the colloidal-quantum-dot-based single-photon emitters with near-nanosecond or even sub-nanosecond decay times and high fluorescence efficiency.