Probing the formation of dark interlayer excitons via ultrafast photocurrent

TL;DR

This paper introduces a time-resolved ultrafast photocurrent technique to directly probe the formation dynamics of optically dark interlayer excitons in 2D heterostructures, revealing femtosecond formation times and fluence dependence.

Contribution

The study develops and demonstrates a novel ultrafast photocurrent method to investigate dark exciton formation and carrier dynamics in layered semiconductors, providing insights inaccessible to traditional spectroscopy.

Findings

Interlayer exciton formation time is approximately 0.4 ps at a specific fluence.

Formation time decreases as excitation fluence increases.

The method enables access to carrier dynamics and interlayer transport in heterostructures.

Abstract

Optically dark excitons determine a wide range of properties of photoexcited semiconductors yet are hard to access via conventional spectroscopies. Here, we develop a time-resolved ultrafast photocurrent technique (trPC) to probe the formation dynamics of optically dark excitons. The nonlinear nature of the trPC makes it particularly sensitive to the formation of excitons occurring at the femtosecond timescale after the excitation. As proof of principle, we extract the interlayer exciton formation time 0.4~ps at 160 $\mu$J/cm$^2$ fluence in a MoS$_2$/MoSe$_2$ heterostructure and show that this time decreases with fluence. In addition, our approach provides access to the dynamics of carriers and their interlayer transport. Overall, our work establishes trPC as a technique to study dark excitons in various systems that are hard to probe by other approaches.

Equations

$$\{\begin{array}{l}\frac{d{N}_e}{dt}=-\frac{N_e}{{\tau }_e}-{\gamma }_{e-h}{N}_e{N}_h+G_e\\ \\ \frac{d{N}_h}{dt}=-\frac{N_h}{{\tau }_h}-{\gamma }_{e-h}{N}_e{N}_h+G_h\end{array}$$

$$N_{ex}(t)=\underset{-\infty }{\overset{t}{\int }}{\gamma }_{e-h}{N}_e{N}_{h}d{t}^{\star }.$$

$$\Delta trPC(\Delta t)\sim {\gamma }_{e-h}{N}_e^0{N}_h^0{\tau }_e{\tau }_h\cdot \exp \left(-\frac{\Delta t}{{\tau }_{e/h}}\right)$$

Full text

††preprint: APS/123-QED

Coulomb-bound electron-hole pairs (excitons) dominate the optical response of 2D semiconductors from the group of transitional metal dichalcogenides (TMDs) [1]. While early studies focused on optically allowed bright excitons, optically forbidden ”dark” excitons are much less studied. The radiative recombination of these latter excitons is suppressed as they involve states with non-zero total momentum, non-integer total spin, or spatially separated electron and hole wavefunctions [1, 2]. Due to the weak interaction with light, these states have a long lifetime. Dark excitons are also the lowest energy excitation in many TMDs [3]. Because of that, dark exciton states likely dominate the long-range transport of excitons [4, 5], determine temperature-dependent optical spectra [6], and are responsible for long-lived spin signals in TMDs [7, 8]. Furthermore, dark excitons are promising for realizing interacting bosonic many-body states including the Bose-Einstein condensate and excitonic Mott insulator [9, 10, 11].

Special approaches are required to investigate the properties of the dark states due to their weak interaction with light. For example, time- and angle-resolved photoemission spectroscopy (trARPES) or spectroscopies in the terahertz and far-infrared frequency ranges have been used to study dark excitons in TMDs and their heterostructures (HS) [12, 13, 14]. These approaches typically require large (hundreds of $\mu$m2 area) homogenous samples or are performed at room temperature. Another approach to probe dark excitons, time-resolved photoluminescence (trPL) [10, 15], has a submicron spatial resolution but features a lower time resolution and does not work for states with vanishingly small oscillator strengths. As a result, many questions related to dark exciton formation, e.g. its timescale or the influence of phonon scattering and electron screening, remain unresolved.

Time-resolved photocurrent spectroscopy (trPC) has recently emerged as an approach to study optical processes in (2D) semiconductors [16, 17, 18, 19]. In trPC, a current across the sample is recorded vs. the time delay between two light pulses impinging onto it. Critically, the technique is inherently sensitive to nonlinear processes. The approach applies to devices down to the nm-scale and is compatible with other probes such as magnetic or electric fields, temperature, or strain. Finally, as a transport-based technique, trPC is inherently sensitive to the phenomena at contacts and interfaces. Here, we use the non-linear response of two-color trPC to probe the formation dynamics of dark excitonic species. To test our approach, we interrogate the formation dynamics of the the most studied dark excitons in TMDs: interlayer excitons in MoS2/MoSe2 heterostructures.

Toy model of time-resolved photocurrent. Our first goal is to show that the dynamics of dark excitons, which are not accessible to conventional optical techniques, can be obtained from the time-dependent populations of free carriers. To understand this, we consider a simplified model of an optically excited semiconductor. We track the time-dependent densities of free electrons $N_{e}(t)$ and free holes $N_{h}(t)$. We assume that electron and hole populations can be excited together (direct excitation) or separately (indirect excitation), which we model by generation functions $G_{e}(t)$ and $G_{h}(t)$. We focus on coupled relaxation $\sim N_{e}N_{h}$ which describes the binding of an electron and a hole into an exciton  [20] (other terms are analyzed in the Supplementary Note 1). Overall, the carrier populations are described within our toy model by the following equations:

[TABLE]

Here $\tau_{e/h}$ are linear decay times of electron and hole populations, and $\gamma_{e-h}$ is the nonlinear exciton formation rate. Since the decay of excitons is several orders of magnitude slower compared to the decay/trapping of free electrons and holes  [21, 22], the density of the excitons is given by

[TABLE]

While the excitons described by $N_{ex}(t)$ can be dark (and hence hard to probe), it can be reconstructed if we have experimental access to $N_{e}(t)$ and $N_{h}(t)$. To accomplish this, we numerically solve the above equations for parameters typical for TMD materials (see SI for details). For simplicity, we first assume that holes and electrons can be excited separately. When only holes are excited at $t_{1}=0$ ps and only electrons at, for example, $t_{2}\approx\tau_{h}=4$ ps, the solution yields dynamics shown in Fig. 1a-c. Initially, the excited population of holes decays exponentially. After the second pulse arrives, electrons are generated (Fig. 1b). The decay of holes speeds up due to the formation of excitons if $\gamma_{e-h}$ is non-zero. Interestingly, we see that the population of excitons (Fig. 1c) qualitatively follows the difference between the hole populations with zero and non-zero $\gamma_{e-h}$ (dashed and solid lines in Fig. 1a). For the non-interacting case ($\gamma_{e-h}=0$), the hole density is not affected by the second pulse exciting electrons (as in a single pulse excitation case) and therefore the population of excitons can also be equated to the difference of hole densities between a single pulse excitation ($G_{h}\neq 0$ ; $G_{e}=0$) vs. two pulse excitation ($G_{h}$ $\neq 0$; $G_{e}$ $\neq 0$). We see that, in principle, the dynamics of dark excitons can be obtained from the dynamics of free carriers.

Two obvious challenges arise when applying this toy model to a realistic physical system. First, conventional optical techniques, such as transient reflectivity, detect combined contributions from photoexcited electron ($N_{e}$), hole ($N_{h}$), and exciton ($N_{ex}$) populations. Second, in conventional semiconductors, optical pulses generate electrons and holes simultaneously, so the generation functions $G_{h}$ and $G_{e}$ cannot be separately controlled.

To address the first problem, we use time-resolved photocurrent spectroscopy as our measurement technique. Generally, photocurrent spectroscopies have the advantage of being directly sensitive to photogenerated electrons/holes while being insensitive to (neutral) excitons  [17, 18]. In trPC, the system is illuminated by two optical pulses separated by the time interval $\Delta t$: $G_{h}\left(t_{0}\right)$ and $G_{e}\left(t_{0}+\Delta t\right)$, which generate populations of $N_{h}^{0}$ and $N_{e}^{0}$, respectively. The DC current across the material is recorded. The trPC signal is defined as the difference in current with both pulses being present versus only a single pulse. In general, photocurrent is proportional to the total amount of free carriers generated in a system over time. For the systems under study, TMDs, the direct contribution of electrons to the photocurrent can be neglected as their lifetime is much lower and contact resistance is higher than holes [17, 23] (see Supplementary Note 1). In that case, the photocurrent produced by only the first pulse is given by the area under the dashed orange curve in Fig. 1a. The decay of the hole population, in this case, depends only on $\tau_{h}$ as $G_{e}=0$ for single pulse excitation. The same holds for $\gamma_{e-h}=0$ since the nonlinear term in Eq. 1 vanishes for both cases. The photocurrent produced by two pulses is given by the area under the solid curve in Fig. 1a and is smaller than that of a single pulse because of holes recombining with photoexcited electrons. It can be shown analytically (Supplementary Note 4) that in the limit of small $\gamma_{e-h}$ and $\tau_{h}\gg\tau_{e}$, the trPC scales linearly with the exciton formation rate,

[TABLE]

Here the decay time (denominator in the exponent) is $\tau_{h}$ if holes are excited first (positive delay time $\Delta t>0$) and $\tau_{e}$ if electrons are excited first (negative delay time $\Delta t<0$). To summarize, trPC can be used to measure both free carrier parameters ($\tau_{e},\tau_{h})$ as well as the exciton formation rate ($\gamma_{e-h})$.

To address the second problem, we use a 2D heterostructure MoS2/MoSe2 as a system where the generation rates for electrons and holes can be controlled separately. Indeed, the conduction band minimum (CBM) and valence band maximum (VBM) of the heterostructure reside in different materials, MoS2 and MoSe2 respectively (Fig. 1d). Because of that, an optical pulse in resonance with e.g., MoSe2 bandgap (${P\textsubscript{MoSe2}}$) excites holes in the VBM of the structure (MoSe2), while the excited electrons can relax to the CBM (MoS2) through tunneling. Crucially, only around $m\approx 60\%$ of these electrons reach the CBM of the structure in MoS2 [24] (Fig. 1d). The remaining electrons are trapped and do not contribute to photocurrent [25, 26]. These electrons are not affected by the second pulse [27]. Similarly, a pulse resonant with MoS2 bandgap (${P\textsubscript{MoS2}}$) excites electrons in CBM while $m\approx 60\%$ of holes reach VBM of the structure. We see that the excitation of MoSe2 produces predominantly free holes, while the excitation of MoS2 produces predominantly free electrons.

We now apply Eq. 1 to model the excitation dynamics of the MoS2/MoSe2 heterostructure. The parameter $N_{e}(t)$ describes the electron density in the CBM of the structure (MoS2) and $N_{h}(t)$ – the hole density in the VBM (MoSe2). The free electron/hole decay time (the term linear with $N_{e/h}$) describes the combined contributions of defect capture [28], intervalley scattering [29], and radiative decay processes [25, 26]. The rate $\gamma_{e-h}$ describes the formation of (dark) interlayer excitons. Of course, intralayer excitons are also formed by optical pulses. However, optically excited intralayer excitons decay much faster (within $<100$ fs[12]) compared to intralayer exciton recombination (> ps) and electron/hole population cooling rate [30, 31, 32] via charge separation across the heterostructure and are therefore subsume in the generation functions $G_{e}$ and $G_{h}$. The latter contain contributions from both pulses (i.e., $G_{e}(t)=P_{\text{MoS2}}\left(t,t_{0}\right)+m\cdot P_{\text{MoSe2}}(t,t_{0}+\Delta t)$, see effect of $m$ on dynamics of charge carriers in Fig. S5). Finally, the excitonic ground state of MoS2/MoSe2 is an interlayer exciton comprised of an electron in MoS2 bound to a hole in MoSe2 (Fig. 1f). When the twist angle between the heterostructure layers ($\theta$) is non-zero, the interlayer exciton has large in-plane momentum: $k\sim\frac{\theta}{a}$, where $a$ is the averaged lattice constant of the heterostructure[33, 34]. In this case, the radiative recombination must involve a phonon and the state is dark[34]. Thus, the interlayer exciton decay (> 100 ps) can be neglected on the time scales of the population build-up [35]. Our next goal is to obtain the dynamics of $N_{ex}(t)$ via trPC.

Time-resolved photocurrent. For trPC measurements, we fabricate samples, MoS2/MoSe2 on hBN, (Fig. 2a; see Supplementary Note 2 for details). We observe the characteristic intralayer A and B excitons for both materials in static photocurrent spectroscopy in the heterostructure region (green dots in Fig. 2b and Fig. S1). In addition, weak photoluminescence (PL) due to interlayer excitons (IX) is observed at 1.3 eV [36] (blue line in Fig. 2b). For time-resolved photocurrent (trPC) measurements, the sample is illuminated with two time-relayed (with $\sim$10 fs precision) optical pulses, one in resonance with the MoS2 bandgap and another with the MoSe2 bandgap. The photocurrent is measured with lock-in amplifier synchronized to an optical chopper in one of the beam paths with no bias voltage applied. This measurement effectively allows us to evaluate the difference between single- and two-pulse responses, which corresponds to the area between dashed and solid curves in Fig. 1a.

Figure 3a shows experimental trPC data (green dots) of the MoS2/MoSe2 sample vs. delay $\Delta t$ between the excitation pulses. Positive delay corresponds to the pulses resonant with the MoS2 bandgap arriving first. The most prominent features of the data are a strong dip at zero time delay and a pronounced asymmetry between positive and negative delays. We now show that these features can be understood within our toy model. First, a large drop in trPC suggests non-linear interaction between the carrier populations produced by both pulses, described by $\gamma_{e-h}$ in our model (Eq. 2). Second, the asymmetry can be understood from Eq. 2. It suggests that the lifetime of electrons photoexcited in MoS2 is much smaller than the lifetime of holes excited in MoSe2 (see insets in Fig. 3a for the illustration of trPC at negative, zero, and positive time delays). Third, we note a slower decaying component at $\Delta t>3$ ps. This minor effect is missing in Fig. 1a and occurs for $m\neq 0$ (Supplementary Note 5). Its origin is the transfer of holes to the VBM (MoSe2). The ratio between fast and slow decaying components is proportional to $m^{2}$ (Eq. S12).

To obtain the precise values of the model parameters, we match the numerical solution of Eq. 1 (the solid line in Fig. 3a) with the experimental data. We obtain decay times $\tau_{h}=6.0\pm 0.5$ ps, $\tau_{e}=1.0\pm 0.2$  ps, interaction strength $\gamma_{e-h}=0.13\pm 0.04$ cm2/s, and transfer efficiency $m=55\pm 5\%$. The effect of every parameter is shown in Fig. S7. Using the extracted parameters, we plot the generation dynamics of interlayer excitons (black dashed line in Fig. 3b). Since the exciton formation is a nonlinear process, its acceleration is expected with a higher density of electrons/holes which matches simulations at higher fluence (solid lines Fig. 3b). We also extract the dynamics of electrons and holes transferred after photoexcitation within our model (Fig. S2a). These transferred carriers can be measured with two-color time-resolved reflectivity (trRef) measurements (Supplementary Note 3). Overall, the simulations suggest two key behaviors. First, we see a much faster decay rate for electrons compared to holes. Second, we obtain a formation time of the exciton, $g_{ex}=0.4$ ps, at our experimental incident fluence P_{\text{MoS2}}=160$$~{}\muJ/cm2. This time is expected to be strongly fluence dependent, dropping by a factor of three for tripled fluence of ${P\textsubscript{MoS2}}$ (inset in Fig. 3b).

Next, we test the predictions of these simulations. To independently check the dynamics of free carriers, we carry out two-color time-resolved reflectivity [22] (Supplementary Note 3). The resulting time constants of electrons and holes, $\tau_{e}=1.4$ ps and $\tau_{h}=6.3$ ps, are close to what is obtained from trPC. Moreover, our model for trRef suggests that the formation of dark excitons is the reason behind the bi-exponential decay observed by us and in other works [22, 23]. To check the dependence of exciton formation on fluence given by our model (Fig. 3b), we carried out fluence-dependent trPC measurements. We keep the incident fluence of ${P\textsubscript{MoSe2}}$ fixed, while ${P\textsubscript{MoS2}}$ is varied in the range used in simulation: $75$ $\mu$J/cm2 – $450$ $\mu$J/cm2 (dots in Fig. 5). We see that the magnitude of the drop of the trPC at zero time delay increases with fluence, from around $30\%$ at $75$ $\mu$J/cm2 up to $50\%$ at $450$ $\mu$J/cm2. The experimental data closely follow the independent predictions of the model (lines in Fig. 3c) where we used the same parameters as Fig. 3a and only changed the fluence of the beams. We extract the formation time of interlayer excitons $g_{ex}$ (inset in Fig. 3b) and find that it is more than halved in the given fluence range from 0.5 ps to 0.2 ps.

Conclusion and outlook. To summarize the discussion above, the proposed model matches all observed features of the trPC behavior. We extract the parameters of the model: e-h coupling ($\gamma_{e-h}=0.13\pm 0.04$ cm2/s), decay times of electrons and holes ($\tau_{e}=1.0\pm 0.2$ ps and $\tau_{h}=6.0\pm 0.5$ ps), and efficiency of interlayer transport ($m=55\pm 5\%$), for all fluence regimes. The interlayer exciton formation time varies from 0.2 ps to 0.5 ps in the range of fluences 450 $\mu$J/cm2 to 75 $\mu$J/cm2. It is useful to compare these values with those obtained by other approaches. Electron/hole lifetimes are consistent with those broadly reported from optical measurements [7, 25, 32, 37, 38]. The shorter lifetime of electrons is likely related to defect states being closer to the conduction band [39, 40]. The observed exciton formation time $g_{ex}=0.4$ ps at 160 $\mu$J/cm2 fluence matches the time scales reported in trARPES ($\sim$230 fs) [12], trTHz reflectivity ($\sim$350 fs) [41], and trFIR ($\sim$800 fs) [42] experiments. The interlayer transfer efficiency $m$ has been estimated from THz measurements to be 50–70% [24], also close to the values here. Overall, our approach provides simple access to the dynamics of (dark) interlayer excitons. Moreover, the proposed model describes trRef dynamics of heterostructures and explains the biexponential decay reported before [22, 23].

To conclude, we demonstrate an approach for studying dark exciton formation dynamics. In future, this approach can be used to study other dark excitons in TMDs. Unlike other approaches, trPC is fully compatible with other optical techniques (trRef and PL shown here, Kerr and ellipticity spectroscopies, second harmonic generation), has hundreds of nanometers spatial resolution, and works at cryogenic temperatures. It will be particularly interesting to use trPC to uncover the effects of many-body interaction (exciton Mott transition, localization at low temperature), electric field, and twist angle on the exciton formation time.

Acknowledgment

The authors thank Nele Stetzuhn for her comments on the paper. The authors acknowledge the German Research Foundation (DFG) for financial support through the Collaborative Research Center TRR 227 Ultrafast Spin Dynamics (project B08).

Conflict of Interest

The authors declare no conflict of interest.

Author Contribution

D.Y., K.I.B., and C.G. conceived and designed the experiments, D.Y., E.A., A.K., and J.R. prepared the samples, D.Y., A.K. and E.A. performed the optical measurements, D.Y. analyzed the data, E.A. wrote software for simulations, D.Y., E.A., performed the calculations and help to rationalize the experimental data, D.Y. and K.I.B. wrote the manuscript with input from all co-authors.

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.