Quantifying carrier density in monolayer MoS₂ by optical spectroscopy

Two-dimensional (2D) layered semiconductors, such as transition metal dichalcogenides (TMDCs), have been studied intensively for over forty years.^1,2 The successful isolation and growth of monolayer TMDCs in the past ten years ushered in the development and studies of TMDC heterostructures for use in optoelectronic, catalytic, and quantum information processing applications,^3–7 due to the emergence of a direct bandgap and other beneficial properties, such as spin–valley locking.^8,9 Despite these advances, several fundamental knowledge gaps remain for monolayer TMDCs and their heterostructures. In general, the optimization of semiconductor heterostructures depends on the knowledge of both ground- and excited-state carrier densities, which dictate the steady-state and transient properties of optoelectronic devices. Quantitative assessment of charge carrier concentrations can be challenging for monolayer TMDCs, since traditional methods rely upon an accurate knowledge of properties, such as dielectric constant,¹⁰ effective mass or mobility,¹¹ and device capacitance,¹² all of which can be challenging to measure and are impacted in uncertain ways by the local environment.¹³ 

Several decades ago, the III–V community developed a powerful, but simple, method based on optical absorption spectroscopy to quantify carrier densities and carrier-density dependent properties in GaAs quantum wells, an early 2D excitonic material. This model, based on the phase space filling effect, related the carrier density in the two-dimensional layer to the resulting quenching of the strong excitonic optical transitions.^14,15 Recent studies have also applied similar models to semiconducting single-walled carbon nanotubes (s-SWCNTs) as a model one-dimensional (1D) excitonic semiconductor.^16,17 A key to this phase-space filling model is the knowledge, or estimate, of the exciton size, since that determines the fraction of exciton oscillator strength that is quenched by a particular charge carrier density. In turn, this exciton size is influenced by the reduced exciton mass and the dielectric constant experienced by the exciton. Applying such a model to monolayer TMDCs is an attractive possibility but is made challenging by the large experimental and theoretical variations in reported dielectric constants (that can depend on sample substrate, thickness, and growth methods)^10,18,19 and excitonic constants.^11,20,21

Applying a phase-space filling model to TMDCs requires a means to systematically vary the monolayer carrier density, a series of experimentally measured absorption or reflection spectra at each carrier density, and a reliable way to quantify the carrier density for each spectrum. Carrier density can be modulated by the gate-voltage in a transistor, via molecular or substitutional ground-state doping, or via dynamic processes, such as photoinduced charge transfer. Some transistor-based studies have reported trends for the dependence of exciton oscillator strength, measured by reflectance, on carrier density in MoS₂,²² WS₂,²³ and MoSe₂²⁴ monolayers, but these studies have not simulated those results with a physically relevant model, such as a phase-space filling dependence. The carrier density in these studies is extracted by considering the gate oxide capacitance and the difference between the applied gate voltage and the gate voltage associated with the charge neutrality point. However, the carrier density is dictated by the total gate capacitance, equal to the sum of the gate oxide capacitance and the channel quantum capacitance. It has been pointed out that the monolayer TMDC quantum capacitance can dominate the total capacitance, leading to errors in extracting carrier density by using the gate oxide capacitance alone.¹² With these considerations in mind, the need remains for a rigorous analysis aimed at quantifying carrier density in monolayer TMDCs by optical spectroscopy.

In this study, we quantify carrier density in monolayer MoS₂ by using an “internal standard,” in this case a one-dimensional (1D) electron donor in a series of mixed-dimensionality (1D/2D) donor/acceptor heterostructures (see Fig. 1),^25,26 and employ a 2D phase-space filling model to simulate the dependence of MoS₂ exciton bleach on carrier density. This study expands upon the concept recently introduced by Sulas-Kern et al.,²⁵ where the optical cross section for charges in a semiconducting single-walled carbon nanotube (s-SWCNT) thin film was used to quantify charge separation quantum yield in a photoexcited s-SWCNT/MoS₂ heterostructure. That proof-of-concept demonstration was limited in that it only utilized a single value of excited-state charge yield and only a single study at the time²⁷ had estimated a charge carrier (hole) cross section in the highly enriched (6,5) s-SWCNTs used as electron donors. Since that study, Eckstein et al. utilized spectro-electrochemistry and a 1D phase-space filling model to quantify hole density in (6,5) s-SWCNTs,¹⁶ and we have performed additional studies on mixed-dimensionality s-SWCNT/MoS₂ heterostructures.^26,28 Furthermore, an experimental diamagnetic shift study²⁰ that came out shortly after Sulas-Kern et al.²⁵ provided a rigorously determined reduced exciton mass for high-quality MoS₂ that allows us to better constrain the 2D phase-space filling model.

With two independent estimates now available for (6,5) s-SWCNTs,^16,27 we use this s-SWCNT “internal standard” to rigorously correlate MoS₂ electron yield and exciton quenching in a variety of photoexcited s-SWCNT/MoS₂ heterostructures. This method allows us to greatly constrain the potential range of physical properties used in a 2D phase-space filling model for the MoS₂ component of the heterostructure. The resulting correlation between electron density and MoS₂ carrier-induced exciton quenching can be used by the community to estimate both ground- and excited-state carrier densities in a wide range of MoS₂-based systems, such as photoexcited heterostructures, (photo)transistors, and redox-doped or electrochemically doped monolayers. Our results also provide well-constrained ranges for fundamental MoS₂ monolayer properties, such as exciton size and dielectric constant.

Figure 2(a) displays the representative steady-state absorption spectra of the mixed-dimensionality heterostructures probed in this study. Both types of heterostructures employ highly enriched (6,5) s-SWCNTs as a photoexcited electron donor and MoS₂ as a photoexcited electron acceptor. The first heterostructure is a s-SWCNT/MoS₂ bilayer,^25,26 and the second is a WSe₂/s-SWCNT/MoS₂ trilayer that forms a charge transfer cascade.²⁸ Strong excitonic absorption features related to both the (6,5) s-SWCNTs and TMDC components are labeled in the figure.

Our study employs femtosecond pump–probe transient absorption (TA) spectroscopy to follow photoinduced charge transfer in these heterostructures. Specific excitonic and charge-related features in TMDCs and SWCNTs have revealed that SWCNTs aid in the spatial separation of charges in these TMDC–SWCNT based heterostructures, resulting in exceptionally long (⁠$>μ$s) charge-separated lifetimes.^25,26,28 In this study, we use the magnitude of these transient spectral features following the completion of photoinduced charge transfer to quantify charge carrier quantum yield and ultimately to develop the phase-space filling “calibration curve” for monolayer MoS₂ (Fig. 1). A judicious choice of the pump photon energy is employed to predominantly excite the s-SWCNT or TMDC components of the heterostructures.

Figure 2(b) displays the TA spectra, taken at 5 ps pump–probe delay, for a representative s-SWCNT thin film, a s-SWCNT/MoS₂ bilayer, and a WSe₂/s-SWCNT/MoS₂ trilayer. TA features in the visible range can be assigned to the ground-state bleach (GSB) of the MoS₂ and WSe₂ excitonic transitions and the GSB of the (6,5) S₂₂ excitonic transition. Features in the near-infrared (NIR) range can be assigned to the (6,5) S₁₁ GSB and the trion induced absorption (IA), a feature that is only present when separated charge carriers (electrons or holes) are present on the s-SWCNTs. These distinct spectral features can be used to quantitatively track the temporal evolution of exciton dissociation, charge diffusion, and charge recombination in the mixed-dimensionality heterostructures.^25,26,28

The charge transfer quantum yield (ϕ_CT) is defined as the number of separated charges (i.e., holes and electrons in separate materials) produced per photogenerated exciton. We estimate charge transfer yields for our type II heterostructures using two separate methods that utilize (1) an empirically determined trion absorption cross section that uses a heuristic model of the dependence of exciton bleaching on redox-doping induced carrier density²⁷ and (2) a correlation between electrochemically modulated charge density and exciton bleach that utilizes a 1D phase-space filling model.¹⁶ Each method utilizes a similar overall methodology of tracking the dependence of the ground-state bleach (GSB) and trion IA on carrier density, but they utilize different frameworks for predicting the impact of the SWCNT density of states on this dependence. Until now, the two studies have not been compared to determine how well-matched their values may be for estimating SWCNT (or SWCNT heterostructure) carrier densities.

For each method described above, we first use singular value decomposition, accompanied by the associated kinetic rate equations, to simulate the full two-dimensional array of time-dependent TA spectra.^25,26,28 These simulations produce concentration trajectories for the relevant species (e.g., excitons in photoexcited s-SWCNTs; charges in s-SWCNTs and TMDCs following exciton dissociation) evolving in time following the pump pulse. This method allows us to isolate and quantify charge-related s-SWCNT and MoS₂ TA features.

Both these methods allow us to extract the s-SWCNT carrier density in units of nm⁻¹. Since we are ultimately concerned with deriving an areal carrier density within the MoS₂ layer, we convert this s-SWCNT carrier density to units of cm⁻² by using the absorbance of the s-SWCNT film, the empirical absorbance coefficient for (6,5) s-SWCNTs, and the number of carbon atoms per nm in a (6,5) s-SWCNT (see Sec. 2 of the supplementary material). For s-SWCNT/MoS₂ bilayers, the calculated s-SWCNT hole density equals the MoS₂ electron density, since there is only one exciton dissociation interface and one exciton splits into one electron and one hole. The WSe₂/s-SWCNT/MoS₂ trilayer is more complex because excitons can be dissociated at two separate interfaces to produce electrons in MoS₂ and holes in WSe₂ [see Fig. 2(c) of the supplementary material]. Thus, the concentration trajectories for each species must be used to calculate the fraction of s-SWCNT holes produced by exciton dissociation at the s-SWCNT/MoS₂ interface and the fraction of s-SWCNT electrons produced by exciton dissociation at the s-SWCNT/WSe₂ interface (Figs. S1 and S2). The MoS₂ electron density is then equivalent to the calculated s-SWCNT hole density.

Table I compares the s-SWCNT and MoS₂ carrier densities extracted from each of the methods described above. The two methods agree reasonably well, with percent difference ranging from ∼3% to 37% (excluding the bilayer excited at 532 nm). The good agreement between the methods proposed by Dowgiallo et al.²⁷ and Eckstein et al.¹⁶ suggest that each method is viable for estimating s-SWCNT ground- or excited-state carrier density and provides additional confidence in the extracted MoS₂ carrier densities used in the following calibration curve.

Turning to the MoS₂ components of the heterostructures, we can now correlate the extracted MoS₂ carrier densities to the associated MoS₂ A exciton bleach values. To produce the phase-space filling correlation we ultimately desire,^14,15 we must calculate the extent of MoS₂ exciton bleaching for a given photoinduced carrier density. This calculation is achieved by performing a spectral deconvolution of the steady-state and transient absorbance spectra in the region of the MoS₂ A exciton, as shown in Fig. 2(c) (see also Figs. 6 and 7 of the supplementary material). All steady-state absorbance features have a positive sign, so once the peak for the A exciton is extracted via spectral deconvolution, the area of this peak is used for the original A exciton absorbance (A₀). The TA spectrum can be more complex, since spectral shifts and transfer of oscillator strength to trion absorption can induce positive features in addition to the negative GSB.²⁹ As such, the spectrum is fit to a combination of both negative and positive peaks and the total area of the ground-state bleach (GSB) is taken as the summation of the areas (absolute values) of each of these peaks.²⁹ 

The experimental data fall within a dielectric constant range of 5.5–12.5, with a best fit of ∼8.0 for the apparent local dielectric constant. To rationalize this range, we consider the average dielectric constant, ɛ_avg = 1/2 (ɛ_top + ɛ_bottom), of the materials surrounding the MoS₂ layer. While this approach is commonly applied to understand Coulomb screening, phase space filling, and bandgap renormalization in monolayer TMDCs, some studies use the static dielectric constant of the adjacent materials,³⁰ while other studies utilize the high-frequency dielectric constants.^20,31 If we first consider the static dielectric constants of sapphire [ɛ₀ = 9–9.4 (in-plane), ɛ₀ = 11.6–11.8 (out-of-plane), and ɛ_avg = 10.5]^32,33 and SWCNTs (ɛ₀ = 4.0),^34–36 we arrive at ɛ_avg = 7.3. Turning to the high-frequency dielectric constants, using ɛ₀ = 4.0 for the s-SWCNT film and ɛ_∞ = 3.05 for sapphire³³ leads to ɛ_avg = 3.5. While there are insufficient reports of the high-frequency dielectric constant of highly enriched (near-monochiral) s-SWCNTs in the literature, a recent report suggests that the dielectric function is relatively flat in the far-infrared regime.³⁷ 

At first glance, this analysis suggests that the use of static dielectric constants might be most appropriate for describing phase-space filling in these photoexcited heterojunctions. To test this, we studied s-SWCNT/MoS₂ bilayers prepared on MgF₂ (ɛ₀ = 5.2, ɛ_∞ = 1.7, square data points in Fig. 3)^32,38 instead of sapphire. The average dielectric environment between MgF₂ and s-SWCNTs would be ∼ɛ_avg = 4.1 and ɛ_avg = 2.4, using static and high-frequency dielectric constants, respectively. The MgF₂ results all fall near the ɛ = 5.5 fit line, in reasonable agreement with the average static dielectric constants, but incommensurate with the high-frequency dielectric constants. As such, we tentatively conclude that phase-space filling in these photoexcited monolayer MoS₂ heterostructures is best described by considering the static dielectric constants of the surrounding materials.

To put our results into context with the existing literature, we have also plotted the results of several studies that correlated exciton quenching with the carrier density of monolayer TMDC transistor channels. As shown in Fig. 3(b), the bounds of the phase-space filling model employed in our study do a good job of describing existing monolayer MoS₂ transistor-based measurements.²² This comparison provides additional confidence in our results and may also suggest that using the gate oxide capacitance to extract carrier density is a reasonable approximation for the transistors studied by Kravets et al.²² Similarly, separately derived phase-space filling simulations do a good job of describing the transistor-based trends found for WS₂²³ and MoSe₂²⁴ monolayers (Fig. S9).

We can also connect our results to the prevailing literature exploring the carrier density-dependent renormalization of electronic (quasiparticle) bandgap and exciton binding energy in monolayer MoS₂. Bandgap renormalization is known to be substantial in doped TMDCs and has been studied extensively for MoS₂.^39–41 Since the phase-space filling model depends on the exciton binding energy, we turn to the study of Yao et al. that discriminated between the impacts of injected charge carriers on the electronic bandgap and exciton binding energy.⁴² In Fig. S10, we plot the carrier density-dependent change in exciton binding energy, normalized to the original binding energy, for the best-fit and upper/lower bounds of the phase space filling simulation in Fig. 3, along with the same trend found by Yao et al. (green data points). The data from Yao et al.⁴² fall nicely within the bounds predicted by our phase-space simulation, providing additional confidence in our results.

Returning to Fig. 3(b), while the data at low carrier densities are most consistent with an apparent local dielectric constant in the range of ∼5.5, the data move progressively toward fit lines corresponding to higher apparent dielectric constants (up to ∼12.5) as carrier density increases. This trend suggests that, although it appears to capture the expected excitonic transition bleaching by charge carriers quite well, the 2D phase-space filling model may not capture all of the underlying physics at play in monolayer MoS₂. Since our spectroscopic data and data from transistor-based measurements both seem to fall off slightly faster than might be predicted by the phase-space filling model, we consider it most likely that the behavior is intrinsic to TMDCs and does not reflect a shortcoming of either type of measurement methods.

The phase-space filling curve elucidated here (with upper/lower bounds) can be used by the community to determine monolayer MoS₂ carrier density in a number of different experiments, as long as the samples/devices can be interrogated by transmission or reflection-based optical spectroscopy. These include experiments and devices where steady-state carrier densities are injected into the monolayer via electrostatic [e.g., (photo)transistors]^43,44 or electrochemical gating (e.g., ion-gated transistors or spectro-electrochemical cells)⁴⁰ or by adsorbed molecular redox dopants,⁴⁵ as well as time-resolved experiments where excited-state carrier densities are modulated by dynamic processes, such as photoinduced charge transfer.³ To recreate the full phase-space filling model shown in Fig. 3, one can use Eqs. (3) and (4) and the most appropriate estimate for the dielectric environment [ɛ_avg = 1/2 (ɛ_top + ɛ_bottom)] of the monolayer sample, to generate the relevant dependence. We suggest the use of the recently derived reduced exciton mass μ = 0.275m_e²⁰ and, based on the empirical data shown here, the static dielectric constants of the surrounding materials.

In this study, we develop a scaling curve that correlates the loss of A exciton oscillator strength in monolayer MoS₂ to the 2D carrier density by means of a s-SWCNT internal standard in a series of photoinduced mixed-dimensionality type II heterostructures. The trends in Fig. 3 highlight that determining an appropriate value for the dielectric constant experienced by an exciton in a monolayer MoS₂/s-SWCNT system is not trivial. The scaling curve utilizes a phase-space filling model originally developed for 2D quantum wells and incorporates empirically valid values for the physical properties of 2D MoS₂ excitons. The consistency between various s-SWCNT/MoS₂ samples within the phase-space filling model supports the use of similar carrier density scaling curves proposed recently by Eckstein et al. and Dowgiallo et al., expanding the relatively underdeveloped assessment of charge carrier concentration in monolayer TMDCs. This study also demonstrates that the well-defined charge-associated transient absorption spectral features s-SWCNTs provide a useful guidepost for understanding how to reliably quantify charge transfer quantum yield and the role of dielectric environment on charge separation and recombination when paired with a variety of TMDCs.

The supplementary material contains materials and methods and calculations for Nexc, charge yield, and the phase-space filling model.

This work was authored by the National Renewable Energy Laboratory, operated by the Alliance for Sustainable Energy, LLC, for the U.S. Department of Energy (DOE) under Contract No. DE-AC36-08GO28308. This study was supported by the Solar Photochemistry Program, Division of Chemical Sciences, Geosciences, and Biosciences, Office of Basic Energy Sciences, U.S. DOE. The views expressed in the article do not necessarily represent views of the DOE or the U.S. Government.

The authors have no conflicts to disclose.

Alexis R. Myers: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Dana B. Sulas-Kern: Data curation (equal); Investigation (equal); Writing – review & editing (equal). Rao Fei: Data curation (equal); Investigation (equal); Writing – review & editing (equal). Debjit Ghoshal: Data curation (equal); Formal analysis (equal); Investigation (equal); Writing – review & editing (equal). M. Alejandra Hermosilla-Palacios: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Jeffrey L. Blackburn: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

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