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Observation of carrier transfer in a vertical 0D-CsPbBr3/2D-MoS2 mixed-dimensional van der Waals heterojunction

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Abstract

Two-dimensional transition metal dichalcogenides with outstanding properties open up a new way to develop optoelectronic devices such as phototransistors and light-emitting diodes. Heterostructure with light-harvesting materials can produce many photogenerated carriers via charge and/or energy transfer. In this paper, the ultrafast dynamics of charge transfer in zero-dimensional CsPbBr3 quantum dot/two-dimensional MoS2 van der Waals heterostructures are investigated through femtosecond time-resolved transient absorption spectroscopy. Hole and electron transfers in the ps and fs magnitude at the interfaces between MoS2 and CsPbBr3 are observed by modulating pump wavelengths of the pump-probe configurations. Our study highlights the opportunities for realizing the exciton devices based on quantum dot/two-dimensional semiconductor heterostructures.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

In the past decade, two-dimensional (2D) layered transition metal dichalcogenides (TMDs) have attracted considerable attention due to their excellent physical properties including superior mechanical flexibility, high carrier mobility, and large exciton binding energy [13]. 2D TMDs have been extensively investigated for next-generation optoelectronic device applications such as photodetectors, light-harvesting devices, and light-emitting diodes [4,5]. In particular, the 2D molybdenum disulfide (MoS$_2$) semiconductor has a direct optical energy gap at the K and $\Gamma$ points in the Brillouin zone, as a result of inversion symmetry breaking in its honeycomb lattice structure [68]. The atomically thin monolayer (ML) MoS$_2$ becomes a robust candidate for photonics and optoelectronics due to its unique properties especially strong photoluminescence (PL) resulting from its direct energy-gap transition [9,10].

The hybrid structures or heterostructures (HSs) with low dimensional light-harvesting semiconductors including other 2D materials, nanowires, and quantum dots (QDs) provide an effective way to increase the light-harvesting and photo-conducting performance [1115]. Among them, cesium lead halide (CsPbX$_3$, X=Cl, Br, and I) perovskite QDs possess size-dependent tunable bandgap, wide absorption cross-section from ultraviolet (UV) to near-infrared range, high absorption coefficient, high photoluminescence quantum yield (PLQY), narrow emission spectrum width, significant extinction coefficient, and long carrier diffusion length [1619]. The QDs/2D-TMDs van der Waals (vdW) HSs establish interfacial processes including the efficient transfer of charge and/or energy at the HS interface which multiplies the photoresponsivity and the photocurrent [2023]. In recent years, initial efforts have been devoted to the investigation of the interaction between the CsPbBr$_3$-QDs and 2D-TMDs. For instance, the type-II band alignment obtained for CsPbBr$_3$-QDs/MoS$_2$-ML HS on Si/SiO$_2$ substrate has been reported by J. Ghosh et al. that the conduction band minimum (CBM) is in the MoS$_2$ and the valence band maximum (VBM) locates in the CsPbBr$_3$ [24]. Also, the high responsivity and fast photoresponse for the QD CsPbBr$_3$/MoS$_2$ photodetectors are attributed to the efficient interfacial charge separation and fast electron transfer from CsPbBr$_3$-QDs to ML-MoS$_2$ probed by steady-state and time-resolved PL spectra [21,24]. However, few reports systematically investigate the fundamental characteristics of the charge transfer process, and a deep understanding of the charge transfer mechanism is still highly desired [25,26].

Mixed-dimensional CsPbBr$_3$/MoS$_2$ (0D/2D) HSs are constructed in this work. The PL quenching of CsPbBr$_3$ and MoS$_2$ in the HS is confirmed by the transfer of holes (electrons) from MoS$_2$ (CsPbBr$_3$) to CsPbBr$_3$ (MoS$_2$). The separation and recombination of carriers in the CsPbBr$_3$/MoS$_2$ HS are systemically studied by tuning the pump wavelength and probing in a wide spectra range. By selectively exciting electrons in MoS$_2$ and probing exciton states in MoS$_2$, the hole-transfer process is observed in the HS within 3.94 ps. The electron-transfer is observed in the excited CsPbBr$_3$ side within 0.20 ps. This study provides a comprehensive understanding of the ultrafast exciton dynamics in CsPbBr$_3$/MoS$_2$ HS.

2. Results and discussion

Figure 1(a) shows a transmission electron microscopy (TEM) image of the CsPbBr$_3$-QDs. The particles are cubic with an average size of 11.2 $\pm$1 nm. The Raman spectra of the MoS$_2$ exhibit two distinct peaks at 378 cm$^{-1}$ and 404 cm$^{-1}$, corresponding to the in-plane mode (E$_{2g}^1$) and extra-surface mode (A$_{1g}$), respectively (Fig. 1(b)) [27,28]. The frequency difference of 26 cm$^{-1}$ between the A$_{1g}$ and E$_{2g}^1$ confirms that MoS$_2$ is an ML. Figure 1(c) shows the steady-state absorption spectra of CsPbBr$_3$-QDs, MoS$_2$-ML, and CsPbBr$_3$/MoS$_2$ HS, respectively. The absorption peak of CsPbBr$_3$-QDs locates at 505 nm. For MoS$_2$-ML, there are three exciton absorption peaks at 660 nm (A-exciton), 613 nm (B-exciton), and 430 nm (C-exciton), respectively, corresponding to the direct exciton transitions from the valence band (VB) to the conduction band (CB) at K-point and $\Gamma$-point in the Brillouin zone [29,30]. The absorption peaks of the CsPbBr$_3$/MoS$_2$ HS are at 430 nm, 505 nm, 613 nm, and 660 nm, as a signal superposition of CsPbBr$_3$-QDs and MoS$_2$-ML. The PL spectra of CsPbBr$_3$, MoS$_2$, and CsPbBr$_3$/MoS$_2$ HS are shown in Fig. 1(d). With the 450 nm excitation, the PL peak of CsPbBr$_3$-QDs is located at 505 nm, and the PL peak from the A-exciton of MoS$_2$ is located at 660 nm. There is an obvious quenching phenomenon of PL intensity at 505 nm and 660 nm for the HS compared with the PL peaks for pure CsPbBr$_3$ and the A-exciton of isolated MoS$_2$. Generally, PL quenching is a typical characterization of the charge and/or energy transfer between two components in the interfaces, which could be verified by transient absorption (TA) spectra and 2D action spectra techniques [31]. The band offset of the type-II band alignment for this HS provides a strong force to separate the electron-hole pairs so that the photogenerated electrons from CsPbBr$_3$ are easily injected to adjacent MoS$_2$ and the holes from MoS$_2$ are transferred to CsPbBr$_3$ side. Thus, the charges transfer and separation at the interface are expected to contribute to the efficient reduction of the electron-hole recombination resulting in the PL quenching in the CsPbBr$_3$/MoS$_2$ HS.

 figure: Fig. 1.

Fig. 1. (a) TEM image of the CsPbBr$_3$ QDs. (b) Raman spectra of MoS$_2$ grown on sapphire. (c) Steady-state absorption spectra of CsPbBr$_3$ QDs, ML MoS$_2$, and CsPbBr$_3$/MoS$_2$ HS on a sapphire substrate. (d) PL spectra of CsPbBr$_3$ QDs, ML MoS$_2$, and CsPbBr$_3$/MoS$_2$ HS on a sapphire substrate.

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The TA spectra of CsPbBr$_3$, MoS$_2$, and CsPbBr$_3$/MoS$_2$ HS are conducted in the range of 400 nm – 680 nm with a pump wavelength of 365 nm and an excitation intensity of 45 $\mu$J/cm$^2$, plotted in the upper panels of Fig. 2(a)–2(c), respectively. For CsPbBr$_3$ QDs, there is a negative signal at 505 nm, named ground-state bleaching (GSB), corresponding to the exciton transition from the VB to the CB. The GSB of ML MoS$_2$ locates at 430 nm, 613 nm, and 660 nm. The GSB peaks of CsPbBr$_3$/MoS$_2$ HS are the sum of the bleached signals for individual CsPbBr$_3$ and MoS$_2$, consistent with the steady-state absorption spectrum in Fig. 1(d). Also, positive signals for the three samples are attributed to excitation state absorption. The lower panels of Fig. 2(a)–2(c) show the temporal evolution of the time-resolved TA spectrum of CsPbBr$_3$, MoS$_2$, and CsPbBr$_3$/MoS$_2$ HS, respectively. For CsPbBr$_3$ QDs, the GSB lifetime at 505 nm is long up to 1 ns. The GSB of CsPbBr$_3$ QDs reaches the maximum of 2.0 ps. However, the GSB arrives at the maximums at around 0.22 ps and 0.43 ps for the isolated MoS$_2$ and CsPbBr$_3$/MoS$_2$ HS, respectively. Generally, the building-up process of the GSB signal is caused by the hot carriers at high energy levels cooling/relaxing to the band edge by emitting phonons to dissipate energies in nanostructured semiconductors [32]. The 2.0 ps building-up of the CsPbBr$_3$ GSB with pump fluence of 45 $\mu$J/cm$^2$ implies that there is a slow relaxation for the photogenerated hot carriers, which can be explained by the strong carrier-phonon coupling due to phonon reabsorption [33,34]. Fig. S1 for TA spectra with 4 times measurements under 45 $\mu$J/cm$^2$ fluence excitation showing the stability of the samples can be found in Supplement 1.

 figure: Fig. 2.

Fig. 2. (a-c) TA and evolutionary correlation difference spectra of CsPbBr$_3$, MoS$_2$, and CsPbBr$_3$/MoS$_2$ with 365 nm excitation.

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To further explore the charge-transfer process in the CsPbBr$_3$/MoS$_2$ HS, three excitation wavelengths of 660 nm, 505 nm, and 365 nm are used in the ultrafast TA measurements. (i) 660 nm is used to excite the A-exciton of MoS$_2$ ML (1.88 eV) to probe the hole transfer from MoS$_2$ to CsPbBr$_3$. (ii) 505 nm is selected for CsPbBr$_3$ (2.45 eV) to detect electron-transfer from CsPbBr$_3$ to MoS$_2$. (iii) 365 nm (3.4 eV) is employed for the hole transfer and electron transfer in the HS interface.

2.1 Photoexcitation on MoS$_2$

The excitation wavelength of 660 nm (1.88 eV) with 90 $\mu$J/cm$^2$ is selected to excite the A-exciton of MoS$_2$ in CsPbBr$_3$/MoS$_2$ HS. Figure 3(a) shows the schematic diagram of the hole-transfer process. The holes are generated in MoS$_2$ in the CsPbBr$_3$/MoS$_2$ HS upon photoexcitation of MoS$_2$ and transfer to the CsPbBr$_3$. It should be noted that the pump energy of 1.88 eV can not excite CsPbBr$_3$ with a bandgap energy of 2.03 eV (505 nm). Since the pump blocks the GSB of the MoS$_2$ B-exciton, we probe the time-evolution dynamics of C-exciton of ML MoS$_2$ (blue triangles) and the HS (red squares) at 430 nm, respectively, as shown in Fig. 3(b). To elucidate the charge transfer in the HS, a generalized sequential kinetic model with three-component is utilized to describe the spectral evolution [35]. The TA kinetic curves in Fig. 3(b) are fitted with the triple-exponential equation,

$${\Delta}A(t)=a_1e^{({-}t/\tau_1)}+a_2e^{({-}t/\tau_2)}-ce^{({-}t/\tau_{et})}$$
where a$_1$, a$_2$, and c stand for amplitudes, $\tau _1$, and $\tau _2$ denote decay time constants, and $\tau _{et}$ is the formation time constant [36]. The time constant $\tau _1$ for the fast decay component and $\tau _2$ for the slow decay component are obtained at about 10.1 ps (4.7 ps) and 56.6 ps (23.9 ps) for ML MoS$_2$ (HS), ascribing to the cooling time of the holes and the lifetime of the carriers, respectively. They are in the same magnitude order as the reported values [13,37]. The hole lifetime of 23.9 ps for MoS$_2$ in the HS is nearly half of the hole lifetime of 56.6 ps in isolated MoS$_2$, suggesting that the holes from the MoS$_2$ side transfer to the CsPbBr$_3$ QDs in the HS. Figure 3(c) shows the hole transfer kinetics for the HS which is obtained by using a subtractive procedure. The hole-transfer time is estimated by fitting this kinetic curve with a single exponential decay function [38],
$${\Delta}A(t)=ae^{({-}t/\tau)}$$
where a denotes amplitudes and $\tau$ stands for the time constant of the hole transfer. It should be noted that the holes transfer process is considered inherent in the absence of other processes. The time constant is 3.94$\pm$0.18 ps, larger than the hole transfer time of 90 fs in 3D/2D CsPbBr$_3$/MoS$_2$ HS, due to the poor interfacial contact [22,39].

 figure: Fig. 3.

Fig. 3. (a) Schematic diagram of hole transfer from MoS$_2$ to CsPbBr$_3$. (b) TA kinetic curves probed at 430 nm for ML MoS$_2$ and CsPbBr$_3$/MoS$_2$. (c) Hole transfer kinetics for HS.

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2.2 Photoexcitation on CsPbBr$_3$

Electron transfer is expected from CsPbBr$_3$ to MoS$_2$ in HS. Figure 4(a) shows the schematic diagram of electrons transferring to the MoS$_2$ layer while the holes are localized in the CsPbBr$_3$ QDs. A 505 nm (2.45 eV) wavelength pump pulse with 75 $\mu$J/cm$^2$ fluence was selected to excite CsPbBr$_3$ and a 660 nm (1.88 eV) probe pulse was tuned to the exciton resonance of MoS$_2$. Figure 4(b) shows the normalized TA dynamic curves of ML MoS$_2$ and CsPbBr$_3$/MoS$_2$ detected at 660 nm. Compared with isolated MoS$_2$, there is a longer decay time for the HS. The rising and decay of the negative signal in the HS correspond to the electron transfer process and the interfacial exciton recombination process, respectively [40]. Figure 4(c) shows the instrument response function (IRF) and the time-evolution dynamics of the normalized GSB signal for the HS with a timescale of 1.0 ps. The negative rising signal at 660 nm reaches the maximum at 320 fs, almost 3 times larger than the instrument response time, indicating that the injected electrons in CsPbBr$_3$ are quickly transferred to MoS$_2$ via the vdW interface. By fitting the TA curves with the tri-exponential equation, the fast decay constant $\tau _1$ and the slow decay constant $\tau _2$ for ML MoS$_2$ are obtained to be 0.46 ps and 5.63 ps, respectively. For CsPbBr$_3$/MoS$_2$ HS, the $\tau _1$, $\tau _2$ and $\tau _{et}$ are 0.54 ps, 14.80 ps, and 0.20 ps, respectively. The rising time of 0.20 ps for the HS confirms the electron-transfer process in CsPbBr$_3$/MoS$_2$ HS which promotes the formation of interfacial excitons with a lifetime of 14.80 ps. The carrier life in CsPbBr$_3$/MoS$_2$ HS is approximately 1.5 times the exciton life in ML MoS$_2$ owing to charge separation. Due to the interfacial coupling between CsPbBr$_3$ and MoS$_2$, the life of the interfacial excitons in HS is longer than that of the intralayer excitons in ML MoS$_2$ [13,39].

 figure: Fig. 4.

Fig. 4. (a) Schematic diagram of electron transfer from CsPbBr$_3$ to MoS$_2$ when exciting CsPbBr$_3$ with 505 nm excitation wavelength. (b) TA kinetics curves detected at 660 nm. (c) Instrument response function and GSB dynamic curve of the HS within 1.0 ps.

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2.3 Photoexcitation on both MoS$_2$ and CsPbBr$_3$

Figure 5(a) shows the schematic diagram that a 365 nm (3.40 eV) pump pulse with 45 $\mu$J/cm$^2$ fluence is used to excite both the MoS$_2$ and CsPbBr$_3$ in the HS and 505 nm (2.45 eV) and 660 nm (1.78 eV) light are adopted to probe CsPbBr$_3$ excitons and MoS$_2$ A-excitons, respectively. In this configuration, the holes of MoS$_2$ and the electrons of CsPbBr$_3$ are anticipated to move to CsPbBr$_3$ and MoS$_2$, respectively, to form interfacial excitons. Figure 5(b) shows the TA dynamic signals of CsPbBr$_3$ and MoS$_2$/CsPbBr$_3$ HS detected at 505 nm in 9 ps. The formation time is 1.50 ps and 8.80 ps for CsPbBr$_3$ QDs and MoS$_2$/CsPbBr$_3$ HS, respectively, confirming the presence of significant hole-transfer in MoS$_2$/CsPbBr$_3$. The fast decay $\tau _1$ and slow decay $\tau _2$ are 75.42 ps and 662.5 ps for MoS$_2$/CsPbBr$_3$ HS, larger than the values of 40.55 ps and 576.80 ps for pure CsPbBr$_3$ QDs, respectively. In pure CsPbBr$_3$, the fast decay $\tau _1$ often occurred in the exciton-exciton annihilation and Auger recombination process at high excitation intensity, while the slow decay $\tau _2$ is attributed to exciton recombination. The dynamic curves probed at 660 nm for ML MoS$_2$ and MoS$_2$/CsPbBr$_3$ HS are shown in Fig. 5(c), where the signals rise immediately after being excited and follow a similar slow decay trend for both samples. The fast decay $\tau _1$ for the carrier cooling time and the slow decay component $\tau _2$ for the recombination time of the interfacial excitons in the HS are 0.78 ps and 13.40 ps, larger than the respective constants of 0.59 ps and 3.56 ps in ML MoS$_2$, respectively. The exciton lifetime in HS is longer than that in isolated MoS$_2$ ML or CsPbBr$_3$ QDs, due to the slow recombination of the electrons from the CB of MoS$_2$ and the holes in the VB of CsPbBr$_3$ [41]. The faster-rising component probed at 660 nm in Fig. 5(c) indicates that the electron-transfer time is shorter than the hole-transfer time from the rising component detected at 505 nm in Fig. 5(b) for MoS$_2$/CsPbBr$_3$ HS.

 figure: Fig. 5.

Fig. 5. (a) Schematic diagram of photoexcitation for CsPbBr$_3$/MoS$_2$ with 365 nm excitation. (b) TA kinetic curves detected at 505 nm for MoS$_2$ and CsPbBr$_3$/MoS$_2$. (c) TA kinetic curves detected at 660 nm for MoS$_2$ and CsPbBr$_3$/MoS$_2$.

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3. Conclusion

The interfacial charge transfer in CsPbBr$_3$/MoS$_2$ HS is investigated by absorption, PL, and TA spectra. By selectively injecting excitons in the MoS$_2$ layer, the holes injected into MoS$_2$ are transferred to CsPbBr$_3$ in 3.94 ps. Simultaneously, the electrons in CsPbBr$_3$ rapidly move to MoS$_2$ at 0.20 ps. Besides, the recombination time of the interfacial excitons is longer than that of the intralayer excitons. These results guide the understanding of the interaction between QD CsPbBr$_3$ and 2D MoS$_2$ and push the development of optoelectronic devices based on CsPbBr$_3$/MoS$_2$ HS to practical application.

4. Experimental section

Cesium carbonate (Cs$_2$CO$_3$, 99%, 0.8 g), oleic acid (OA, 99%, 2.5 ml), and octadecene (ODE, 99%, 30 ml) was mixed into a 50 ml three-necked flask and degassed for 1 h at 130 $^{\circ }$C. Then the reaction solution was kept at 150 $^{\circ }$C for 0.5 h until Cs$_2$CO$_3$ was fully dissolved. After it was cooled down to 100 $^{\circ }$C naturally, the Cs-precursor was kept in an Ar$_2$ atmosphere. Lead bromide (PbBr$_2$, 99.9%, 0.1321 g) was dissolved in a mixture with ODE (10 ml), OA (1 ml), and oleylamine (OAm, 99%, 1 ml) in another three-necked flask and dried in a vacuum for 1 h at 130 $^{\circ }$C. The temperature was increased to 160 $^{\circ }$C and kept for 10 minutes when the PbBr$_2$ completely dissolves. After a rapid injection of 1 ml of Cs-precursor into the PbBr$_2$ solution, cooled it quickly in an ice bath to form QDs. The QDs were purified by centrifugation with 8000 rounds per minute (RPM) for 10 minutes to remove unreacted salts and other ligands. Discarded the supernatant, kept the precipitate, and dispersed the precipitate in 4 ml N-octane. And centrifuged again with 4000 RPM for 5 minutes.

ML MoS$_2$ films purchased from Six Carbon Inc. (Shenzhen, China) were deposited on sapphire substrates via the chemical vapor deposition technique. QD CsPbBr$_3$/2D MoS$_2$ vertical vdW HSs were fabricated by using the spin-coating method (4000 RPM, 50 s) and the annealing procedure (80 $^{\circ }$C, 10 minutes).

The microstructures were characterized by using transmission electron microscopy (TEM, JEOL JEM 2010 FET). The Raman spectra were measured by a scanning near-field optical microscopy (alpha300, WITec). The steady-state absorption spectrum was tested with an ultraviolet-visible–near-infrared (UV–VIS-NIR) spectrometer (Cary-5000, Agilent). PL spectra were performed using an optical fiber spectrometer (USB-4000, Ocean Optics).

The TA spectra were measured by using an fs pump-probe system consisting of the laser of a Ti:sapphire laser (Coherent, 800 nm, 35 fs, 7 mJ/pulse, and 1 kHz) and a spectrometer (Helios, Ultrafast Systems). The output laser light was split into two beams by using a separator. One output beam enters an optical parametric amplifier (OPA, 800 fs), which was mainly used to output an fs laser light. Here, 365 nm, 515 nm, and 670 nm pulses were used as the pump light to excite the samples. Another output light passes through a CaF$_2$ crystal to form probe light from 320 nm to 680 nm. An optical delay-line kit was used to change the delay time between the pump and probe light. The pump light is cut off when it goes into the chopper before arriving at the sample. The sample was in excited states and not excited states alternately. A detector was used to record the relative intensity of [I($\lambda$)$_{pro}$/I($\lambda$)$_{ref}$]$_{pump}$ and [I($\lambda$)$_{pro}$/I($\lambda$)$_{ref}$]$_{unpump}$ of the probe pulse for the ’excited’ and ’unexcited’ sample by turns where I($\lambda$)$_{ref}$ is the optical intensity of the reference beam. All measurements were performed at room temperature and the data were analyzed by Surface Xplore software (version 4.2.0).

Funding

National Natural Science Foundation of China (12074104, 12174090); Natural Science Foundation of Henan Province (222300420057); National Key Research and Development Program of China (2022YFA1604302); Young Backbone Teacher Training Program in Higher Education of Henan Province (2019GGJS065).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented herein are not publicly available currently but can be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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Supplementary Material (1)

NameDescription
Supplement 1       Supplement 1

Data availability

Data underlying the results presented herein are not publicly available currently but can be obtained from the authors upon reasonable request.

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Figures (5)

Fig. 1.
Fig. 1. (a) TEM image of the CsPbBr$_3$ QDs. (b) Raman spectra of MoS$_2$ grown on sapphire. (c) Steady-state absorption spectra of CsPbBr$_3$ QDs, ML MoS$_2$, and CsPbBr$_3$/MoS$_2$ HS on a sapphire substrate. (d) PL spectra of CsPbBr$_3$ QDs, ML MoS$_2$, and CsPbBr$_3$/MoS$_2$ HS on a sapphire substrate.
Fig. 2.
Fig. 2. (a-c) TA and evolutionary correlation difference spectra of CsPbBr$_3$, MoS$_2$, and CsPbBr$_3$/MoS$_2$ with 365 nm excitation.
Fig. 3.
Fig. 3. (a) Schematic diagram of hole transfer from MoS$_2$ to CsPbBr$_3$. (b) TA kinetic curves probed at 430 nm for ML MoS$_2$ and CsPbBr$_3$/MoS$_2$. (c) Hole transfer kinetics for HS.
Fig. 4.
Fig. 4. (a) Schematic diagram of electron transfer from CsPbBr$_3$ to MoS$_2$ when exciting CsPbBr$_3$ with 505 nm excitation wavelength. (b) TA kinetics curves detected at 660 nm. (c) Instrument response function and GSB dynamic curve of the HS within 1.0 ps.
Fig. 5.
Fig. 5. (a) Schematic diagram of photoexcitation for CsPbBr$_3$/MoS$_2$ with 365 nm excitation. (b) TA kinetic curves detected at 505 nm for MoS$_2$ and CsPbBr$_3$/MoS$_2$. (c) TA kinetic curves detected at 660 nm for MoS$_2$ and CsPbBr$_3$/MoS$_2$.

Equations (2)

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Δ A ( t ) = a 1 e ( t / τ 1 ) + a 2 e ( t / τ 2 ) c e ( t / τ e t )
Δ A ( t ) = a e ( t / τ )
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