1. Introduction

The discovery of graphene opened up the research on other two-dimensional (2D) materials, like monolayer (1L) semiconducting transition metal dichalcogenides (TMDs), e.g. MoS$_2$, MoSe$_2$, WS$_2$ and WSe$_2$, which are 2D direct bandgap semiconductors at the forefront of both scientific research and technological innovation [1–4]. Their atomic thickness makes TMDs ideal candidates for next-generation ultrafast and ultrathin (opto)-electronic devices like sensors [2], transistors and photodetectors [5–7], also integrable with silicon platforms [8]. With charge carriers confined to two dimensions, the Coulomb screening in 1L-TMDs is strongly reduced, and their equilibrium optical response is dominated by tightly bound excitons, with large binding energies up to hundreds of meV [9–11]. The lowest energy excitonic transitions, i.e., the A and B excitons, involve electronic states at the $K/K'$ edges of the Brillouin zone [12]. At higher energies, the C exciton originates from band nesting effects caused by the parallel conduction and valence bands [13,14].

Despite their atomic thickness, TMDs exhibit strong light–matter interaction [15,16]. Unveiling the ultrafast processes occurring after photoexcitation of carriers in these quantum confined materials is key both for fundamental science and for technological applications. For instance, the carrier lifetime determines the responsivity of a photodetector or the performance of a saturable absorber within a mode-locked laser. For these reasons, a large body of studies has addressed the dynamical relaxation pathways of TMDs after optical excitation, predominantly using transient absorption/reflectivity (TA/TR) [17]. From the rise and decay times of the transient signals one can extract information on the time-scales of the formation of electronic correlations [18–20], exciton dissociation, charge transfer [21] and cooling [22] processes. The spectral shape of the resulting signal, on the other hand, indicates which photoinduced effects determine the dynamics at a certain time. At the optical bandgap of TMDs, the most commonly observed pump-induced changes to the excitonic peaks are: (1) a red/blue shift of the resonance [23,24] commonly caused by a change in screening; (2) a reduction of the oscillator strength due to the Pauli-blocking effect at low excitation densities and increased screening at high excitation densities [25]; and (3) a broadening of the excitonic linewidth [24,26] due to enhanced scattering.

TA/TR experiments on 1L-TMDs have been performed on a variety of samples fabricated by different techniques and in different experimental configurations. The TMD monolayers are typically produced either by mechanical exfoliation [27] or by chemical vapor deposition (CVD) [28]. CVD consistently provides large area (cm size) polycrystalline monolayers with comparatively high defect densities, resulting in broad exciton linewidths (tens of meV) dominated by inhomogeneous broadening [28,29]. By contrast, mechanical exfoliation by standard scotch-tape technique produces — although with low throughput — very high quality ${\mathrm{\mu} {\rm m}}$-sized single-crystal lattices with narrow exciton linewidths (down to few meV) [30].

Depending on their final application, TMD monolayers are transferred onto different substrates. For fundamental spectroscopic studies they are usually deposited on thin transparent substrates like quartz, sapphire or fused silica (SiO$_2$). In this case the out-of-equilibrium optical properties of the TMD are measured via the transmittivity $T(\omega )$ [Fig. 1, top schematic]. When TMDs are instead exploited as active elements of optoelectronic devices they are typically exfoliated on reflective substrates, i.e., wafers of silica on silicon, which consist of a thin (a few hundreds nanometers) SiO$_2$ layer on top of a thick silicon (Si) substrate. In this case, the transient optical response of the device is monitored by the reflectivity $R(\omega )$ [Fig. 1, bottom schematic].

 

Fig. 1. Substrate-dependent pump-probe spectra of 1L-MoS$_2$. Schematic of the experimental geometry to measure the transient reflection ($\Delta R/R$) and transmission ($\Delta T/T$) of 1L-MoS$_2$ on a thin transparent substrate (top) and on wafers of SiO$_2$/Si with SiO$_2$ thicknesses of 90 nm and 285 nm (bottom left and right). Center: transient spectra recorded for the SiO$_2$ substrate (a-c) and the SiO$_2$/Si substrate (d-f). We tune the pump photon energy at 1.88 eV (a,d), 2.48 eV (b,e) and 3.10 eV (c,f).

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As we show here experimentally and theoretically, the transient optical response of 1L-TMDs is strongly affected by the experimental geometry, in particular the choice of substrate. However, in the current literature on transient optical properties one often finds the implicit assumption that the spectral shape is dominated by the optical properties of the monolayer and essentially independent of the substrate used in the experiment [17,31]. This assumption must be scrutinized when the thickness of the SiO$_2$ layer on top of the reflective substrate is of the order of the wavelength of the incident light, so that it gives rise to interference effects. Indeed, the underlying substrate has been shown to play a key role in several optical properties: for example, the visibility of graphene [32–34] and TMDs [35] strongly depends on the thickness of the SiO$_2$ layer, which may also affect the measurement of quantitative Raman spectra [36]. A detailed study on the influence of the photonic structure created by the substrate on the TA/TR spectra of 1L-TMDs is, however, still missing.

In this article, we perform ultrafast TA/TR spectroscopy on exfoliated 1L-MoS$_2$ on transparent (SiO$_2$) and reflective (SiO$_2$/Si) substrates, with SiO$_2$ of various thicknesses. To explain the differences observed in the TA/TR spectra from different substrates with the same type of MoS$_2$ monolayer, we model the photoinduced variation of the complex material susceptibility using a Lorentzian ansatz. We then use a transmission transfer matrix (TTM) approach to describe photonic effects and predict the transient properties. Comparison of the calculated and measured spectra shows that photonic effects explain the large differences in the transient spectra of the same type of monolayer placed on different substrates, modifying the intensities, linewidths and shifts of the excitonic peaks. Indeed, when accounting for the substrate we can fit all measured spectra assuming the same photoinduced modification of the monolayer susceptibility. Our results set the stage for a rational comparison of the plethora of ultrafast optical experiments on 1L-TMDs to extract the underlying photophysics.

2. Results and discussion

2.1 Ultrafast pump-probe spectroscopy on monolayer MoS$_2$

In order to determine how the substrate photonic effects affect the transient optical response of monolayer semiconductors, we perform femtosecond pump-probe spectroscopy (for the description of the experimental setup see Supplement 1) at normal incidence of 1L-MoS$_2$ placed on a 200-${\mathrm{\mu} {\rm m}}$-thick SiO$_2$ substrate and on reflective wafers of SiO$_2$/Si, with SiO$_2$ thicknesses of $d_{\mathrm {SiO}_2}=90$ nm and $d_{\mathrm {SiO}_2}=285$ nm. 1L-MoS$_2$ is produced by gold-assisted mechanical exfoliation [37] (see Supplement 1). We measure transient spectra of 1L-MoS$_2$ at three different pump photon energies [Fig. 1]: (1) a low-energy pump tuned at the optical bandgap [${1.88}\,{\textrm{eV}}$, Fig. 1(a), (d)], (2) a mid-energy pump tuned above the quasi-particle bandgap [${2.48}\,{\textrm{eV}}$, Fig. 1(b), (e)] and (3) a high-energy pump tuned above the band nesting region [${3.10}\,{\textrm{eV}}$, Fig. 1(c), (f)]. For each pump photon energy we adjust the pump fluence in the range $1-{10}{\mathrm{\mu} {\rm J}/{\textrm{cm}}^2}$ such that the differential transmission ($\Delta T/T$) through the transparent substrate is $\sim 0.5\%$. We subsequently keep the pump fluence constant for all the substrates. We first compare the transient spectra of 1L-MoS$_2$ placed on one substrate and excited with different photon energies $\hbar \omega _{\mathrm {pump}}$. All spectra shown are recorded at the pump-probe delay at which the signal reaches its maximum, i.e., where they are least influenced by potentially different decay dynamics. For both the transparent SiO$_2$ substrate [Fig. 1(a)–(c)] and the reflective SiO$_2$/Si substrates [Fig. 1(d)–(f)] the amplitude and the shape of the transient signals change considerably with the pump photon energy. The measured differences originate from (1) the different absorption of the monolayer at each pump photon energy, and thus the different transient reduction of the exciton oscillator strength due to the Pauli blocking effect; (2) the different photoinduced spectral broadening due to the increase of the electronic temperature in the material; (3) many-body effects, which induce an energy shift of the excitonic resonances as a function of $\omega _{\mathrm {pump}}$. The complex interplay between many-body correlations and excitonic interactions determines the final shape and amplitude of the non-equilibrium optical spectra of the monolayer [23,38]. Disentangling all of these effects from the transient optical response of 1L-MoS$_2$ is, however, beyond the scope of the present work. Here we focus on the role of substrate-induced photonic effects and how they determine the differences in the transient spectra of the same monolayer under identical $\omega _{\mathrm {pump}}$ and fluence.

From the comparison of Fig. 1(a) and (d), Fig. 1(b), and (e) and Fig. 1(c) and (f) it is clear that $\omega _{\mathrm {pump}}$ is not the only factor determining the transient spectra. In fact, for the same excitation conditions, the amplitudes of the transient signals strongly vary depending on the substrate: the sign of $\Delta R/R$ measured on SiO$_2$ is opposite to the sign of $\Delta T/T$ and to the sign of $\Delta R/R$ measured on the reflective SiO$_2$/Si substrate. The absolute magnitude of the signal originating from the SiO$_2$/Si substrate with 90 nm Si thickness by far exceeds the signals from the other substrates. Intriguingly, also the shapes and peak positions observed on the various substrates differ: for the smallest $\hbar \omega _{\mathrm {pump}} = 1.88$ eV [Fig. 1(a) and (d)] the maximum and minimum of $\Delta R/R$ from the $d_{\mathrm {SiO}_2}=285$ nm substrate are almost the same in absolute value, while we find much more pronounced positive than negative peaks for the other three settings; the peaks resulting from the medium [$\hbar \omega _{\mathrm {pump}} = 2.48$ eV, Fig. 1(b) and (e)] and high energy pump [$\hbar \omega _{\mathrm {pump}} = 3.10$ eV, Fig. 1(c) and (f)] occur at different energies depending on the substrate. Furthermore, the negative part of the signal below the first peak is clearly missing entirely from the measurements on the $d_{\mathrm {SiO}_2}=285$ nm substrate. The pronounced variation in the transient signals for different substrates further corroborates the importance of considering photonic substrate effects, not only for a quantitative analysis but also for correctly differentiating between concurrent physical processes, i.e., distinguishing between resonance broadening, Pauli blocking or energy-shifts due to bandgap renormalization.

2.2 Origin of the substrate-dependent signals

To understand the difference between signals measured on the different substrates we build a theoretical model encompassing the effect of the substrate on the pump-pulse-absorption and on the transient change of the monolayer absorption/reflectivity measured by the probe pulse. We describe the substrate-dependence of the incoming (pump) and outgoing (probe) optical signal via the TTM method, which includes solving Fresnel’s equations. Here we consider normal incidence (as in the experiment) and thus do not see polarization effects. We model the optical properties of the monolayer by a sum of Lorentz oscillators, each of the form

Effect of the substrate on pump pulse absorption. The layered structure of the Si/SiO$_2$ substrate leads to constructive or destructive interference of the waves at the surface determining the absorption of the pump pulse by the monolayer. To account for this, we fit the static absorbance measured on the SiO$_2$ substrate $A^\mathrm {ML}_\mathrm {SiO_2}$ and extract the equilibrium dielectric function $\varepsilon ^\mathrm {eq}(\omega )$ of 1L-MoS$_2$ [Fig. 2(a)]. Combined with the photonic properties of the substrate we can calculate the absorbance of 1L-MoS$_2$ on the different substrates $A^\mathrm {ML}_\mathrm {substrate}$ [Fig. 2(b)]. It is clear that the substrate strongly modulates the absorption of the pump pulse at all energies. We extract the absorbance inside the monolayer on the different substrates for our pump photon energies (${1.88}\,{\textrm{eV}}$, ${2.48}\,{\textrm{eV}}$ and ${3.10}\,{\textrm{eV}}$) and normalize the values to the absorbance when the monolayer is placed on transparent SiO$_2$ (listed in Table 1).

 

Fig. 2. Theoretical model. (a) Experimental absorbance and fit for the extraction of the equilibrium dielectric function. (b) Calculated absorbance spectra of 1L-MoS$_2$ on different substrates, for the retrieval of the substrate-dependent absorbed pump fluence. (c) Inferred transient spectrum, following the extraction of the refined equilibrium dielectric function, adapted to the local experimental conditions, and the non-equilibrium dielectric function from $\Delta R/R$ measured on transparent SiO$_2$.

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Table 1. Ratios of the absorbance inside the monolayer on different layered substrates to the absorbance inside the monolayer on the transparent SiO$_2$ sample $A^\mathrm {ML}=A^\mathrm {ML}_\mathrm {substrate}/A^\mathrm {ML}_{\mathrm {SiO_2}}$ at the different pump photon energies.

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Measured transient properties. Since we consider the transient signals for the same type of monolayer, the remaining two factors that determine the difference between the signals from the different substrates should be the substrate-dependent pump-pulse-absorption $A^\mathrm {ML}$ and the substrate-induced interference effects which shape the TA/TR signal. To verify this assumption, we first extract the transient dielectric properties of the monolayer $\varepsilon ^\mathrm {neq}(\omega,\tau )$ from $\Delta R/R$ from SiO$_2$ at the maximum of the signal by fitting the model for each pump photon energy [Fig. 2(c)]. We then infer the other signals ($\Delta T/T$ through SiO$_2$, $\Delta R/R$ from SiO$_2$/Si) by assuming a linear dependence of the transient energy shift $\delta \omega _0$, the oscillator strength reduction $\delta a$ and the broadening of the model resonances $\delta \gamma$ on $A^\mathrm {ML}$. Note that these quantities otherwise do not depend on the specific substrate, but are fitted only once.

Despite the relative simplicity of a Lorentzian model for the susceptibility of the monolayer, we obtain very accurate fits to $\Delta R/R$ from the transparent substrate (see Fig. 3) for all pump photon energies. Likewise, the calculated $\Delta T/T$ on the same substrate follows the experimental curve but slightly overestimates the transient transmission. We assign this difference to averaging and finite angle effects in the experiment. Also the predicted $\Delta R/R$ spectra for 1L-MoS$_2$ on the SiO$_2$/Si substrates, assuming the same transient dielectric function, match well with the experimental signals. Notably, we accurately reproduce both the amplitudes and the spectral shapes of the transient signals. If we omit the variation in the pump-pulse absorption, we are unable to reproduce the experimental results (not shown).

 

Fig. 3. Comparison between theory and experiment. Measured (circle) and calculated (line) transient spectra of 1L-MoS$_2$ on (a, b, c) SiO$_2$ and (d, e, f) SiO$_2$/Si. The pump photon energies are 1.88 eV (a, d), 2.48 eV (b, e) and 3.10 eV (c, f), respectively.

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There are also small differences between experiment and the model. Within the model, we do not account for the slight variations in the intrinsic monolayer properties on the different substrates induced by the fabrication process (e.g., defect densities or strain). Since the transient signals are sensitive to even small variations in the monolayer properties, this leads to deviations between the measurement and the modeled signal. Nevertheless, considering the simplicity of our model for the monolayer, the observed correspondence between experiment and theory is remarkable and clearly shows the importance of substrate effects in determining both pump absorption and $\Delta T/T$ and $\Delta R/R$ signals.

2.3 Simple model for the substrate-dependence of the transient signals

To round off the discussion, we further simplify our model to qualitatively and intuitively understand how the photonic structure created by the layered substrate alters the transient signal. The basis of the following discussion are the expressions for the absorbance $A(\omega )$, the transient transmission $\Delta T(\omega,\tau )/T$ and the transient reflection $\Delta R(\omega,\tau )/R$ that can be derived from a simple thin film model of the monolayer on a substrate based on Ref. [39] (see Supplement 1 for details).

Here, $\delta _0(\omega )=2\pi d_\mathrm {ML}/\lambda =d_\mathrm {ML}\omega /c$ is the vacuum phase shift over the thickness of the monolayer $d_\mathrm {ML}=0.7$ nm, $\chi (\omega )$ is the complex dielectric susceptibility of the monolayer with $\Delta \chi (\omega,\tau )$ being its change after the system has been excited. $\eta _s(\omega )$ is the complex refractive index of the substrate.

We start with some general observations. The absorbance is dominated by the imaginary part of $\chi$ and modulated by the substrate properties, i.e., if $\eta _s$ is almost constant and $|\eta _s|\gg |\chi \delta _0|$ then $A$ is proportional to the imaginary, i.e., absorptive, part of the monolayer susceptibility. If $\eta _s(\omega )$ is real, as is the case with the transparent SiO$_2$ substrate, $\Delta R/R$ is again purely determined by the change of the imaginary part of the susceptibility $\Delta \chi (\omega,\tau )$ of the monolayer. In contrast, a complex $\eta _s(\omega )$ mixes the absorptive (imaginary) and dispersive (real) part of $\chi (\omega,\tau )$.

To understand the effect of the layered substrate, we determine the reflection coefficient $r_s(\omega )$ of the substrate in air (without monolayer) using the TTM. Then we invert Fresnel’s reflection coefficient $r_s(\omega )=(1-\eta ^\mathrm {eff}_s(\omega ))/(1+\eta ^\mathrm {eff}_s(\omega ))$ to define an effective refractive index $\eta ^\mathrm {eff}_s(\omega )=(1-r_s(\omega ))/(1+r_s(\omega ))$. The effective refractive index for the SiO$_2$/Si substrates [Fig. 4] is not only complex but also shows strong resonances at certain frequencies determined by the SiO$_2$ layer thickness. This alters the pump excitation by suppressing absorption into the monolayer on the $d_{\mathrm {SiO}_2}=285$ nm SiO$_2$/Si substrate [compare Fig. 2(b)]. It also determines the strength of the contributions of the real and the imaginary part of the monolayer susceptibility measured in the $\Delta R/R$ spectra. These expressions also clearly show that a direct connection between the transient reflection spectra from a layered substrate and the absorption properties of the monolayer is not straightforward.

 

Fig. 4. Effective refractive index. (a) Real and (b) imaginary part of the substrate effective refractive index $\eta ^\mathrm {eff}_s$ of pure SiO$_2$ (orange), 90 nm SiO$_2$ on Si (green) and 285 nm SiO$_2$ on Si (blue).

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3. Conclusions

In this work we have experimentally compared the transient optical spectra of monolayer MoS$_2$ placed on a transparent SiO$_2$ substrate and on reflective SiO$_2$/Si substrates, with the dielectric SiO$_2$ slab of various thicknesses. We interpret the experimental observations using a simple model that combines a Lorentzian ansatz for the complex material susceptibility with a transmission transfer matrix approach to describe photonic effects. Our results show that pure interference effects explain not only the large differences in the amplitudes of the transient spectra of the same monolayer placed on different substrates, but also changes in the shape of the spectra. Our findings caution against oversimplified interpretation of the photo-physics of optoelectronic devices based on TMDs: we show that — regardless of the material under study — the substrate strongly affects the transient reflectivity/transmission signal at the optical bandgap, i.e., red/blue shift, broadening and reduction of the oscillator strength. Moreover, optical resonances due to interference effects make these changes strongly energy dependent. The right choice of substrate and excitation photon energy is thus key to achieve the desired transient optical response of the monolayer TMD.