# Size Dependence of the Resonant Third-Order Nonlinear Refraction of Colloidal PbS Quantum Dots

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}S or CsPbBr

_{3}) [3,4,5,6]. The available studies of strongly confined QDs are mostly limited to PbSe and PbS [7,8]. Core–shell materials have also been investigated for nonlinearities [8,9,10]. NLO effects should be more pronounced in QDs, which are in a strong confinement regime. In this regime, both electrons and holes are confined, and optical transitions occur between single-particle eigenstates, thus concentrating the optical transitions in a single energy interval. In strongly confined QDs, Coulomb or exchange interactions can be neglected. Hence, absorption saturation may be considered as the saturation of a two-level system. Absolute changes in the refractive index can be large before the saturation of the transition [11].

## 2. Materials and Methods

_{4}). QDs were synthesized with radii from 1.5 nm to 4.2 nm, size dispersion gradually decreases with size from 9.6% to 4.5%, respectively. QD size and size dispersion have been estimated from the analysis of the absorbance spectra, namely from first excitonic peak position and full width at half maximum [25]. Spectra were captured with a Shimadzu UV3600 spectrophotometer (Shimadzu, Japan). QD solution absorbance spectra are presented in Figure 1.

_{in}—laser beam intensity in the absence of the absorption, λ—laser wavelength (800 nm), L

_{α}—effective length, l—sample thickness, α—linear absorption coefficient S—aperture linear transmittance, r

_{a}and w

_{a}are radii of aperture and beam respectively. The laser intensity within the sample was calculated from Equation (1) using the known value of the quartz NRI (3.28 · 10

^{−16}cm

^{2}/W).

_{0}is the linear refraction coefficient. Both nonlinear susceptibility and NRI are macroscopic parameters, and to correctly assess the nonlinearities of single QDs we have to use microscopic quantities. For third-order nonlinear processes, such a quantity is second-order hyperpolarizability γ. For a two-component solution, we can apply Equation (3) (in electrostatic system of units - esu) [28]:

_{Re}and ε

_{Im}are real and imaginary parts of QD dielectric permittivity at the wavelength of 800 nm; f—local field factor, N

_{QD}—number of QD per unit volume; n

_{eff}—effective refractive index, calculated from Maxwell Garnett model, N

_{a}—the Avogadro number, C

_{QD}—QD molar concentration. The QD concentration was kept sufficiently low to have a negligible impact on the refractive index of the solution. For our solvent, CCl

_{4}(n

_{0}= 1.4607), we obtained the following macroscopic nonlinear parameters: n

_{2}= 5.1 × 10

^{−16}cm

^{2}/W, χ

^{(3)}= 2.76 × 10

^{−14}esu. The real and imaginary parts of QD dielectric permittivity were determined from linear absorbance spectra using the Kramers–Kronig relations. The iterative method for Kramers–Kronig calculations was adopted from Moreels et al. [29].

## 3. Results and Discussion

#### 3.1. Thermal Nonlinearity

^{−5}M) with PRR ranging from 1 Hz to 1000 Hz (see Figure 3A). We revealed that, for our QD solutions, NRI remains almost unchanged for PRR values of 100 Hz and less. When PRR exceeds 100 Hz, NRI increases, which we assume is a sign of “indirect” thermal nonlinearity. To avoid thermal nonlinearities, all further experiments were performed at a PRR of 10 Hz. NRI values obtained are independent of the beam intensity (see Figure 3B).

^{−6}M–1.3 × 10

^{−4}M are shown in Figure 4A. The linear fit for the NRI values can be seen in Figure 4B. The interpolated hyperpolarizability value for 3.3 nm PbS QDs is −(2.5 ± 0.8) × 10

^{−32}esu.

#### 3.2. PbS QD Nonlinear Refraction in the Literature

#### 3.3. Size-Dependent Nonlinear Response

_{x}Se

_{1−x}[31], CuInS and AgInS [32] QDs in the resonant regime, and CdTe QDs in the near-resonant regime [33].

^{2.9}FOM growth is similar to the observed cubic χ

^{(3)}growth for QDs in the weak confinement regime [34].

- Both the linear and nonlinear properties of the QD are governed by its electronic structure. The seemingly featureless absorption spectra of PbS QDs contain rich electronic structure [35]. It has been found that PbS QDs electronic structure induce the natural anomalous size dependence of the excited carrier relaxation, which was explained in terms of phonon-induced transitions from the in-gap states to QD fundamental states [25,36]. When the QD size decreases, the relaxation time increases to the power of ~1.4. That intrinsically modifies FOM size dependence. It has recently been proven that PbS QDs possess two emissive states [37]. The number of optically active states should influence the resonant nonlinear response. The influence of PbS QDs’ electronic system aspects on their nonlinear responses was recently pointed out by Padihla et al. They reported unconventional increase in volume-normalized two-photon cross-section with decreasing QD sizes [38];
- The dependence of nonlinear optical responses on QD size distribution has been theoretically predicted [39]. The broadening of the QD size distribution increases the inhomogeneous linewidth of the QD ensemble, thus weakening their nonlinear optical response. In our case, when the QD radius reduces from 4.2 nm to 1.4 nm, QD size distribution increases from 4.5% to 9.6%, respectively. Temperature-dependent photoluminescence analysis can also be applied to extrapolate inhomogeneous broadening, which increases from 149 meV to 176 meV for 4.5 nm and 3.7 nm QD, respectively [40];
- QD surface has a great impact on QD properties [41,42]. The incomplete passivation of the QD surface forms so-called surface trap states. Photoexcited carriers trapped at the QD surface form a static internal field, reducing the oscillator strength, leading to the saturation of the absorption [43] and reducing the nonlinear response. When the QD radius decreases from 4.3 to 1.5 nm, the QD surface-to-volume ratio increases drastically, which makes the impact of the of surface traps even more pronounced, decreasing the nonlinear response further on.

_{2}~10

^{−16}–10

^{−15}cm

^{2}/W [45], and all-optical switching devices can work with n

_{2}~10

^{−22}cm

^{2}/W [46]. To date, such values for QD solutions have already been achieved. Further increases in the magnitude of QD nonlinear response allow us to achieve the same macroscopic nonlinear parameters (n

_{2}or χ

^{(3)}) with lower QD concentrations, thus lowering the amount of required material. The improvement of the nonlinear response can be achieved by creating QDs with less defects, which will prevent the charge trapping and oscillator strength reduction. The photoluminescence quantum yield of the used QDs was ~25%, which leaves enough room for improving the surface passivation and reducing the defect-state density. Additional increases in the nonlinear response may be achieved by narrowing the QD size distribution [39]. Assuming that inhomogeneous linewidth broadening changes the nonlinear response in the same way as homogeneous broadening, the QDs’ nonlinear response can be increased by three to four times. In summary, the nonlinear response can be enhanced by an order of magnitude that is suitable for the use of non-concentrated media with PbS QDs in most applications. Lastly, the QD ligand shell greatly affects the QD properties, including the nonlinear response; it has been recently shown that changing the shell can even revert the sign of the NRI [21]. However, further studies are required to truly understand the impact of the ligand shell on QDs’ nonlinear properties.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Quantum dot (QD) solution absorbance spectra; inset—QD carrier relaxation time, obtained from time-resolved photoluminescence measurement.

**Figure 3.**(

**A**)—QD nonlinear refraction index measured in range of peak repetition rate of 1 Hz–1000 Hz, red line is given as a guide to the eye; (

**B**)—QD nonlinear refractive index (NRI) is independent of incident power density.

**Figure 4.**(

**A**)—Z-scan traces of the QD solutions with QD concentrations 5 × 10

^{−6}M–1.3 × 10

^{−4}M. (

**B**)—QD NRI is linearly scaled with QD concentration, the point with zero concentration corresponds to the NRI of the pure solvent, blue line—linear fit of the experimental data.

**Figure 5.**(

**A**)—Z-scan traces for QD solutions with different sizes (black squares—pure CCl

_{4}); (

**B**)—QD second-order hyperpolarizability vs. QD size; (

**C**)—nonlinearity figure of merit (FOM) of PbS QDs vs. QD size.

QD Diameter, nm | n_{2}, cm^{2}/W | n_{2}/C, cm^{2}/(W·M) | τ_{p}|PRR | Host | Ref. |
---|---|---|---|---|---|

3÷8.4 | −10^{−16} | −(0.4÷6.8) × 10^{−10} | 35 fs|10 Hz | solution | This work |

3.8÷6.4 | −(5÷35) × 10^{−12} | −(1÷7) × 10^{−6} | 150 fs|76 MHz | solution | [20] |

4.6÷11 | −(0.5÷4.2) × 10^{−3} | −(0.1÷0.8) | 300 fs|1 kHz | solution | [22] |

10 | −3 × 10^{−14} | n/a | ~ns|n/a | solution | [21] |

10 | −(0.9÷3.4) × 10^{−15} | n/a | 130 fs|76 MHz | solution | [30] |

1.6 | −1.05 × 10^{−}^{9} | n/a | cw | solution | [8] |

3.8 | −3.16 × 10^{−12} | n/a | 4 ns|10 Hz | solution | [23] |

2.4÷5 | −(3.5÷8.4) × 10^{−11} | n/a | cw | solution | [24] |

3.5 | −2 × 10^{−23} | n/a | ~fs|n/a | glass | [14] |

<15 | −10^{−16} | n/a | 50 ps|1 Hz | PVA sol | [15] |

<1.3 | −(2.8÷7.0) × 10^{−10} | n/a | 50 ps|10 Hz | zeolite | [16] |

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Skurlov, I.D.; Ponomareva, E.A.; Ismagilov, A.O.; Putilin, S.E.; Vovk, I.A.; Sokolova, A.V.; Tcypkin, A.N.; Litvin, A.P. Size Dependence of the Resonant Third-Order Nonlinear Refraction of Colloidal PbS Quantum Dots. *Photonics* **2020**, *7*, 39.
https://doi.org/10.3390/photonics7020039

**AMA Style**

Skurlov ID, Ponomareva EA, Ismagilov AO, Putilin SE, Vovk IA, Sokolova AV, Tcypkin AN, Litvin AP. Size Dependence of the Resonant Third-Order Nonlinear Refraction of Colloidal PbS Quantum Dots. *Photonics*. 2020; 7(2):39.
https://doi.org/10.3390/photonics7020039

**Chicago/Turabian Style**

Skurlov, Ivan D., Evgeniia A. Ponomareva, Azat O. Ismagilov, Sergey E. Putilin, Ilia A. Vovk, Anastasiia V. Sokolova, Anton N. Tcypkin, and Aleksandr P. Litvin. 2020. "Size Dependence of the Resonant Third-Order Nonlinear Refraction of Colloidal PbS Quantum Dots" *Photonics* 7, no. 2: 39.
https://doi.org/10.3390/photonics7020039