Statistical analysis of time-resolved emission from ensembles of semiconductor quantum dots: Interpretation of exponential decay models

We present a statistical analysis of time-resolved spontaneous emission decay curves from ensembles of emitters, such as semiconductor quantum dots, with the aim of interpreting ubiquitous non-single-exponential decay. Contrary to what is widely assumed, the density of excited emitters and the intensity in an emission decay curve are not proportional, but the density is a time integral of the intensity. The integral relation is crucial to correctly interpret non-single-exponential decay. We derive the proper normalization for both a discrete and a continuous distribution of rates, where every decay component is multiplied by its radiative decay rate. A central result of our paper is the derivation of the emission decay curve when both radiative and nonradiative decays are independently distributed. In this case, the well-known emission quantum efficiency can no longer be expressed by a single number, but is also distributed. We derive a practical description of non-single-exponential emission decay curves in terms of a single distribution of decay rates; the resulting distribution is identified as the distribution of total decay rates weighted with the radiative rates. We apply our analysis to recent examples of colloidal quantum dot emission in suspensions and in photonic crystals, and we find that this important class of emitters is well described by a log-normal distribution of decay rates with a narrow and a broad distribution, respectively. Finally, we briefly discuss the Kohlrausch stretched-exponential model, and find that its normalization is ill defined for emitters with a realistic quantum efficiency of less than 100%.

Figure 1

Schematic of the relation between decay of an excited state $X^{*}$ to the ground state $X$ and experimental observable parameters. The density of emitters in the excited state is equal to $c\left(t\right)$ and can be probed by transient absorption. The emitted light intensity as a function of time $f\left(t\right)$ is recorded in luminescence decay measurements. In photothermal measurements the released heat $(g-f)\left(t\right)$ after photoexcitation is detected. $g\left(t\right)$ describes the total decay, i.e., the sum of the radiative and the nonradiative decay.

Figure 2

(Color online) Plot of a non-single-exponential decay of the fraction $c\left(t\right)∕c\left(0\right)$ (black solid curve, left axis) and the corresponding total intensity decay curve $g\left(t\right)$ (red dashed curve, right axis). The curves that describe the fraction of excited emitters and the corresponding intensity decay curve are strongly different. In this example, $c\left(t\right)∕c\left(0\right)$ is the Kohlrausch stretched-exponential decay of the fraction [Eq. (19), black solid curve] and $g\left(t\right)$ the corresponding decay curve [Eq. (20), red dashed curve]. We have taken $\beta =0.5$ and ${\Gamma }_{\mathit{str}}=1$.

Figure 3

(Color online) Luminescence decay curve of emission from a dilute suspension of CdSe quantum dots (open dots, left axis). Data were collected at the red side of the emission maximum of the suspension, at $\lambda =650\pm 5\mathrm{nm}$. Single-exponential modeling (red dashed curve, right axis) yields a decay time of $39.0\pm 2.8\mathrm{ns}$ and a ${\chi }_r^2$ of 1.12. The average photon arrival time $⟨t⟩$, calculated with Eq. (7), is $39.1\mathrm{ns}$.

Figure 4

(Color online) Luminescence decay curve of emission from CdSe quantum dots in a titania inverse opal photonic crystal (dots, left axis). The lattice parameter of the titania inverse opal was $340\mathrm{nm}$ and the emission wavelength $\lambda =595\mathrm{nm}$. (a) A log-normal distribution of rates [Eqs. (17, 18), red dashed curve, right axis] models the data extremely well $({\chi }_r^2=1.17)$. The ${\Gamma }_{mf}$ is $91.7\mu {\mathrm{s}}^{-1}$ $(1∕{\Gamma }_{mf}=10.9\mathrm{ns})$ and the width of the distribution $\Delta \Gamma$ is $0.57{\mathrm{ns}}^{-1}$. (b) In contrast, a Kohlrausch stretched-exponential model (red dashed curve, right axis) does not fit the data $({\chi }_r^2=60.7)$. The stretched-exponential curve corresponds to ${\Gamma }_{\mathit{str}}=96.2\mu {\mathrm{s}}^{-1}$ $(1∕{\Gamma }_{\mathit{str}}=10.4\mathrm{ns})$, an average decay time $⟨t⟩$ of $31.1\mathrm{ns}$, and a $\beta$ value of 0.42.

Figure 5

Log-normal distribution of $\Gamma$. This distribution was modeled to the data of Fig. 4 (curve $a$, quantum dots in photonic crystal) and Fig. 3 (curve $b$, quantum dots in a diluted suspension), with ${\Gamma }_{mf}$ and $\Delta \Gamma$ as adjustable parameters. For curve $a$ ${\Gamma }_{mf}$ is $91.7\mu {\mathrm{s}}^{-1}$ $(1∕{\Gamma }_{mf}=10.9\mathrm{ns})$ and the width of the distribution $\Delta \Gamma$ is $0.57{\mathrm{ns}}^{-1}$ and for curve $b$ ${\Gamma }_{mf}$ is $25.8\mu {\mathrm{s}}^{-1}$ $(1∕{\Gamma }_{mf}=38.8\mathrm{ns})$ and the width of the distribution $\Delta \Gamma$ is $0.079{\mathrm{ns}}^{-1}$.