1. Introduction

Transition metal dichalcogenide (TMD) monolayers host strongly bound excitons which are stable at room temperature and exhibit a wide range of interesting properties [1,2]. Monolayers of TMDs are semiconductors with a direct bandgap in visible and near-infrared ranges making them attractive for optoelectronic applications [3]. Optical properties of various monolayers have been well studied and shown to exhibit exciting phenomena such as circular dichroism [4] and the valley-Hall effect [5]. In order to understand fundamental mechanisms of carrier dynamics, several time-resolved studies have been performed revealing multi-exponential decays indicating various competing relaxation pathways [6,7]. These studies deduced important characteristic times of various recombination mechanisms at room and cryogenic temperatures. The experimentally measured photoluminescence decay times [8] in these TMDs at room temperature [9] were significantly longer than the theoretical predictions (∼270 ps [10]). In contrast, at low temperatures, the observed lifetime matched closely to calculated values [11]. Recently, the mismatch between the calculated and observed values at room temperature was attributed to dark-exciton states [12]. These dark states participate in the bright-exciton dynamics at room temperature leading to longer than calculated effective lifetimes. In monolayer TMDs, three types of dark excitons exist: the spin-mismatched excitons, non-zero orbital angular momentum excitons, and excitons away from the light cone [13]. Spin forbidden dark-exciton states in monolayer TMDs are predicted to lie close to the bright states (within the thermal energy range at room temperature) with the lowest excited state being dark or bright in WSe₂ or MoS₂ respectively [14]. Using an external magnetic field, the spin-forbidden dark-bright exciton splitting for TMDs was found experimentally to agree with the calculated value [15,16]. However, the lowest non-zero orbital angular momentum dark exciton state (P-type) in MoS₂ is predicted to be 0.17 eV above the 1S state for A exciton and 0.21 eV for the B exciton [17]. While the direct population of such dark states was demonstrated [18], their dynamics and subsequent decay studies, which have not been reported yet, can aid in design and development of applications such as quantum memories based on P-type dark excitonic states.

Here, we directly measure the P-type dark-exciton lifetime, by performing two-photon pump and one-photon probe experiments on MoS₂ monolayers to excite non-zero orbital angular momentum dark excitons (P-type) and probe the (1S) bright-exciton population dynamics on femtosecond timescale. Two-photon excitation has been shown experimentally to selectively couple to the P-type dark-exciton states in TMDs [18]. Using a pump pulse, significantly red-detuned relative to the dark-exciton energy, we generate P-type dark-exciton population through the two-photon absorption process. A white-light supercontinuum probe pulse measures bright exciton dynamics at various energies and pump-probe delay times. Since the P-type dark exciton is forbidden to recombine via one-photon processes, the non-radiative recombination rate and, most importantly, the recombination via scattering to lower energy bright state (1S) determine the lifetime of this type of dark exciton. Furthermore, we performed one-photon pump-probe experiments to extract the relevant bright excitonic lifetimes. We carried out a detailed rate analysis of the bright and the dark exciton relaxation and scattering mechanisms to extract the individual decay rates. The extracted bright exciton radiative and Auger recombination rates match the reported values, validating our model. Moreover, we show and contrast dark and bright exciton population dynamics with fs resolution under the aforementioned excitation schemes. Using such a direct measurement we find the 2P-dark → bright exciton scattering time in MoS₂ to be 72.5 ps. In addition to influencing photoluminescence decay time, the interplay between the dark and bright-exciton states also leads to radically different optical properties such as photoluminescence (PL) polarization among the TMD material family members [19]. Detailed understanding of the excitonic lifetimes and dynamics can enable development of practical excitonic devices [20] and novel applications such as quantum memories [21].

2. Methods

For our experiments, the MoS₂ monolayer samples were mechanically exfoliated onto a SiO₂ substrate from undoped bulk MoS₂ crystals. In our pump-probe setup, a 1 kHz Ti:Sapphire regenerative amplifier delivers 35 fs pulses at 1.55 eV (800 nm). The laser output was split into two linearly polarized beams: the first one is fed to an optical parametric amplifier (OPA), which serves as the wavelength-tunable pump and the second one generates broadband supercontinuum in a single-crystal sapphire plate for the probe. The white-light impinges on the sample at almost normal incidence and is reflected into a spectrometer, which allows us to deduce reflectivity changes induced by the pump as depicted in Fig. 1. Subsequently, a motorized translation stage varies the pump-probe time delay and a pair of quarter-wave plates controls both pulses. The probe signal spot size was set to 30 µm and the pump signal spot size overlapping the probe was about 1 mm. The integration time of the detector was 10 ms and the time resolution observed was about 100 fs. Several flakes were inspected and a flake, identified using Raman and photoluminescence, was used to conduct all the measurements reported.

 

Fig. 1. (a). Illustration of a MoS₂ sample transferred onto a SiO₂ substrate and the pump (red) and the probe (rainbow colors) beams impinging on the sample. (b). Schematic drawing of the experimental setup: OPA - optical parametric amplifier, BS - beam splitter; M - mirror; S - shutter, L - plano-convex lens, ND - neutral density filter, λ/4 – quarter wave plate.

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Reflectivity at normal incidence from the interface between two media with complex refractive indices n₁ and n₂ is given by [22]

3. Results and Discussion

We first performed a one-photon pump-probe experiment, generating carriers in the conduction band and observing subsequent ultrafast exciton dynamics. The monolayer of MoS₂ was excited by 35 fs pump pulses with photon energy of 2.55 eV and fluence of 337µJ/cm². This excitation wavelength was chosen to populate both the A and B exciton series simultaneously. Such as scheme was chosen over resonant excitation to extract population dynamics of both A and B as these one-photon experiments were conducted to serve as a benchmark for the subsequently two-photon measurements. In contrast to the pump, the probe pulse is weak, and therefore non-linear effects such as two-photon absorption and second harmonic generation are negligible and not observed. Differential reflectivity ΔR/R was obtained from the measured spectra of the reflected probe at various pump and one-photon probe time delays and are presented in Fig. 2, where color code represents normalized changes. We observed two prominent spectral features centered near 1.86 eV and 2.01 eV coinciding with the reported resonance values for the A and B excitons in the MoS₂ monolayers, their energies are marked with the black dotted lines in Fig. 2(a).

 

Fig. 2. (a), (c). Differential reflectivity spectra of MoS₂ as a function of energy and pump-probe delay time obtained under one-photon (a) and two-photon (c) excitation schemes, corresponding to pump photon energies of 2.55 eV and 1.01 eV, respectively. Two prominent features centered at 1.86 eV and 2.01 eV correspond to A and B exciton reflectivity changes caused by the pump. Dotted black lines mark A and B exciton energies. (b), (d). Time evolution of ΔR/R at energies shown in (a) and (c) by the two dashed lines, which reflects A and B exciton dynamics. Top inset in (b) shows linear photoluminescence plot showing no trion peaks. Bottom insets in (b) and (d) depict ΔR/R as a function of energy close to the zero pump-probe delay time.

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Since the monolayer was undoped, trion peaks were not [23] in the reflectivity measurements as well as linear photoluminescence measurements (top inset of Fig. 2(b)). Any population to the higher lying 2-S states also quickly scatter down to the 1-S states due to efficient intra-valley scattering processes [24]. The changes in sample absorption after the pump arrival result in alternating differential reflectivity spectra near the mentioned energy values (Fig. 2(b) lower inset).

The minima of ΔR/R (dashed lines in Fig. 2(a)) were used to extract A and B exciton dynamics shown in Fig. 2(b). The absolute value of ΔR/R decreases monotonically with multi-exponential decay. Such behavior in TMDs has been previously observed in one-photon pump-probe measurements and in time-resolved photoluminescence [7,6,9]. This behavior has been attributed to various types of radiative and non-radiative recombination pathways such as exciton-exciton annihilation process, which depend strongly on the excited carrier density, trap or defect assisted non-radiative recombination, as well as scattering into dark-exciton states [13,24] . The scattering to dark excitonic states is difficult to isolate from all the other aforementioned recombination mechanisms. Thus, the direct measurement of the dark state lifetime is virtually impossible with the standard one-photon excitation techniques.

In this work, we use a two-photon excitation technique, which directly populates 2P dark-exciton states [18]. The pump energy in the two-photon pump and one-photon probe experiment was set slightly above half of the 2P A-exciton energy. We excited the sample with a photon energy of 1.01 eV with pump fluence of 6.18 mJ/cm², under which two-photon excitation can directly populate 2P state of A-exciton series [18] . The B excitons, lying 20 meV lower than the 2P state are populated [25]. The two-photon absorption selection rule permits selective population of the 2P state and observation of its dynamics to bright states of both A and B excitonic series [18]. Even as the 2S and 2P states are close by in energy, with 2P being slightly lower, the two-photon selection rule selectively populate the 2P state making the population dynamics significantly simpler. The observed dynamics under such excitation is fundamentally different from that obtained under one-photon excitation. Comparing Fig. 2(b) and Fig. 2(d) we stress two important observations, which are related to the central theme of this work: slow and non-monotonic dynamics of the differential reflectivity in the two-photon case. The two-photon differential reflectivity dynamics allow us to directly extract the 2P dark exciton lifetime, determined by 2P → 1S scattering times, in these two-dimensional materials. Based on our experimental data analysis, the 2P dark to bright exciton scattering time at room temperature is estimated to be 72.5 ps in monolayer MoS₂. This value is larger than similar dynamics of nP excitons studied in MoS₂ at 77 K using intra-band infra-red excitations by Cha et. al [17]. This longer lifetime observed at room temperature in our measurements can be explained by population dynamics with other dark excitonic species such as spin flip excitons with much longer lifetime [16].

The exciton energy diagram of MoS₂ monolayer and the one- and two-photon excitation schemes implemented in our experiment are shown in Fig. 3. The bright and dark excitonic states appear in orange and in gray respectively. The electron continuum is colored in blue. Under one-photon excitation (2.55 eV), electrons created in the continuum relax, forming bright and dark excitons of all types as shown in Fig. 3(a). The created excitons can further recombine via either radiative or non-radiative processes such as Auger and trap/defect-assisted recombination. The P-type dark excitons can recombine non radiatively, get scattered to low lying 1S bright states [26]. Moreover, spin-flip mechanisms allow scattering from the bright to the spin forbidden dark states and vice versa. In the two-photon excitation case (1.01 eV), in contrast, the pump mostly creates P-type dark excitons directly via two-photon absorption [18] . Thus, the bright-exciton states, measured by the probe pulse, are populated via scattering from the dark states. The bright state population dynamics measured through time dependent differential reflectivity allows us to extract the scattering time from the dark states to the bright ones. The density dynamics of bright exciton population (X_B) and the dark exciton population (P type $X_D^P$, spin forbidden $X_D^S$, outside the light cone dark $X_D^O$) are modeled as follows:

 

Fig. 3. Monolayer MoS2 exciton band structure and population dynamics under one-photon and two-photon excitations. (a). In the one-photon case (2.55 eV photon energy), continuum states are directly excited and the carriers decay populating both 1S and 2P excitons. (b). In the two-photon case (1.01 eV photon energy), 2P excitons are directly generated and scatter to the 1S states. The reverse process, where the 1s excitons scatters to the dark ones, is less probable due to the energy scales involved. Since the 1S population can be created only from the 2P scattering in this case, the 1S dynamics allows measuring the 2P lifetime directly.

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In the one-photon case, ${G_{xB}}(t )= {\alpha _1}I(t )$ and ${G_{xD}}(t )= 0$, where ${\alpha _1}$ is the one-photon absorption efficiency and $I(t )$ is the pump intensity. In the two-photon case ${G_{xD}} = {\alpha _2}I{(t )^2}$ and ${G_{xB}}(t )= 0$, where ${\alpha _2}\; $ is the two-photon absorption efficiency. The pulse temporal intensity dependence is well described by Gaussian width FWHM of 35 fs. The time-dependent densities of the bright (X_B) and the dark ($X_D^P$) excitons are obtained by relating the differential reflectivity to the absorption change and including photon to exciton conversion efficiency [7].

The set of equations (2–6) were solved self-consistently and the characteristic times (including the previously unknown 2P dark→ bright scattering time) of various scattering events were obtained from theoretical fits to the experimentally observed differential reflectivity (Table 1).

Table 1. Parameters for rate Eq. 2–6.

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Figure 4(a), and 4(b) present A and B exciton dynamics, respectively, monitored through differential reflectivity temporal changes in the case of 2.55 eV one-photon excitation energy. The pump induced carrier density also leads to renormalization of the exciton energy levels as previously observed [28,29] and the experimentally observed renormalized energy levels and their time dynamics are given as an inset in each case. They match closely with previous reports [29] with excited carrier densities of about 3.5 × 10¹² cm⁻² A excitons and 7.6 × 10¹¹cm⁻² B excitons for one-photon excitation and 4.2 × 10¹¹ cm⁻² and 5 × 10¹⁰ cm⁻² respectively for two photon excitation. The calculated bright and the dark exciton densities are shown in red and gray solid lines, respectively, and match well the measured dynamics. In the one-photon excitation case, the decay is monotonic as the generated bright excitons decay via Auger or radiative pathways as well as via scattering to the underlying bright and dark states. The extracted radiative lifetime, using the previously reported times for exciton-exciton (2 × 10⁻² cm²/s) and trap-assisted Auger scattering (5 × 10⁻²⁰ cm⁶/s), match well with the previously reported values [7,6].

 

Fig. 4. (a),(b). Temporal A and B exciton density dynamics, respectively, in the case of one-photon excitation with photon energy of 2.55 eV. (c),(d). Temporal A and B exciton density dynamics, respectively, in the case of two-photon excitation with photon energy of 1.01 eV. The blue circles are the measured values, the red and the gray lines are the calculated bright and dark exciton densities, respectively. Carrier density dependent renormalization of exciton peak position is given as an inset in each case.

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The 2P → 1S bright exciton scattering time was observed to be 72.5 ps. Since the non-radiative recombination is much slower (lifetime to be about 1 ns) the effective lifetime of 2P dark exciton state is dominated by the 2P dark → bright scattering rates. Figure 4c and d present A and B exciton dynamics, respectively, in the case of 1.01 eV two-photon excitation energy. Here, it must be noted that in our experiment two-photon excitation can populate only the 2P states of A series, whereas 2P of B exciton is about 170 meV higher than the excitation energy. B-exciton population dynamics observed here is due to indirect population of the 1S B exciton states [26,28]. However, the main difference when comparing with the one-photon case is the non-monotonic bright exciton population dynamics exhibiting a rise in ΔR/R during the first 15 ps for the A exciton and 40 ps for the B exciton and subsequent monotonic decay, significantly longer than the one-photon case decay. Since, the P-type dark excitons have complex population dynamics including several pathways, a detailed rate equation model such as the one described before, in contrast to a much simpler one used in other reports [30], was necessary to extract the exact lifetime. The P-type dark excitons, which are exclusively populated by the two-photon excitation process, are forbidden to decay radiatively via one-photon processes. This makes the 2P dark scattering to the bright states, followed by the bright exciton decay, and recombination via non-radiative processes the most probable pathways for recombination as shown in Fig. 3(b). The bright excitons undergo fast recombination, in comparison to the scattering rate, and thus the dark to bright scattering time, as measured here, strongly determines the dark excitonic lifetime.

4. Conclusions

We demonstrated bright and dark exciton formation and decay using one- and two-photon pump and one-photon probe spectroscopy, and identified the characteristic times of various decay processes. We showed that in MoS₂, the dark excitons are directly generated by the two-photon excitation and undergo 2P dark → bright scattering with a characteristic time of ∼72.5 ps. Such direct measurements of radiative and scattering times can aid in better understanding of proposed optoelectronic devices using these novel materials. We believe that the knowledge of the dark exciton formation and scattering times to the bright counterparts could be largely beneficial for devices based on quantum memory and other valleytronic/valley-optoelectronic applications. With the help of the constructed model, one can predict the lifetimes of various processes such as defect/trap mediated Auger recombination and radiative recombination to engineer highly efficient light emitters, photodetectors, and solar cells.