Excitons, trions and Rydberg states in monolayer MoS2 revealed by low temperature photocurrent spectroscopy

We investigate excitonic transitions in a h-BN encapsulated monolayer $\textrm{MoS}_2$ phototransistor by photocurrent spectroscopy at cryogenic temperature (T = 5 K). The spectra presents excitonic peaks with linewidths as low as 8 meV, one order of magnitude lower than in earlier photocurrent spectroscopy measurements. We observe four spectral features corresponding to the ground states of neutral excitons ($\textrm{X}_{\textrm{1s}}^\textrm{A}$ and $\textrm{X}_{\textrm{1s}}^\textrm{B}$) and charged trions ($\textrm{T}^\textrm{A}$ and $\textrm{T}^\textrm{B}$) as well as up to eight additional spectral lines at energies above the $\textrm{X}_{\textrm{1s}}^\textrm{B}$ transition, which we attribute to the Rydberg series of excited states of $\textrm{X}^\textrm{A}$ and $\textrm{X}^\textrm{B}$. The relative intensities of the different spectral features can be tuned by the applied gate and drain-source voltages, with trions and Rydberg excited states becoming more prominent at large gate voltages. Using an effective-mass theory for excitons in two-dimensional transition-metal dichalcogenides we are able to accurately fit the measured spectral lines and unambiguously associate them with their corresponding Rydberg states. The fit also allows us to determine the quasiparticle bandgap and spin-orbit splitting of monolayer $\textrm{MoS}_2$, as well as the exciton binding energies of $\textrm{X}^\textrm{A}$ and $\textrm{X}^\textrm{B}$.


Introduction
Two-dimensional transition metal dichalcogenides (2D-TMDs) are an excellent playground for studying and exploiting exciton physics. This family of materials presents unusually large exciton binding energies and lifetimes, even at room temperature and, in consequence, their optical and optoelectronic properties are largely dominated by excitonic transitions. 1 Until recent years, research of exciton physics in 2D-TMDs have mainly relied on photoluminescence (PL) spectroscopy measurements. This technique and its variants (timeresolved PL, 2 PL excitation, 3 etc.) allowed to obtain detailed information on the properties of light-emitting exciton transitions, including exciton binding energy, 4,5 lifetime, 6,7 spin and valley polarization, 8,9 etc. However, PL spectroscopy relies on spontaneous radiative decay of excitonic states. Thus, if a competing non-radiative exciton relaxation mechanism is present, it can cause a dumping, or even complete disappearance of the corresponding PL peaks, even if they are allowed by optical selection rules. 10 In consequence, other techniques such as absorption spectroscopy, [11][12][13][14][15] or electroluminescence spectroscopy 16 have increasingly gained popularity for investigating excitonic states not accessible by PL. Photocurrent spectroscopy (PCS) 17,18 provides a simple, powerful, and yet largely underused complementary approach for studying excitonic transitions in 2D-TMDs. For this technique, the sample at study is exposed to monochromatic light and the light-induced change in conductivity (photoconductivity) is registered as a function of the illumination wavelength. Since the detection mechanism does not require for excitons to decay radiatively via spontaneous emission, PCS allows detecting exciton transitions regardless of the presence of dominant non-radiative relaxation mechanisms. Further, since this technique is sensitive to the electric charge, it is especially suited to study the transport properties and ionization mechanisms of photogenerated excitons, which are crucial for the development of exciton-based optoelectronics. Up to date, however, only few attempts have been made to use PCS for exciton characterization in 2D materials, 18,19 and measurements at cryogenic temperature and with high-quality samples are still missing in literature.
Here we investigate the excitonic properties of a monolayer (1L) MoS2 phototransistor by PCS at cryogenic temperature (T = 5 K). Our measurements allow us to fully resolve and identify the 1L-MoS2 neutral exciton states X A and X B , originating from the two spin-split band-edge optical transitions at the K point of the reciprocal lattice, as well as their associated charged trion states, T A and T B . Owing to the excellent quality of the fabricated device, in combination with the use of cryogenic temperature, we observe remarkably sharp exciton transitions, with bandwidths as low as 8 meV (full width at half maximum; FWHM), roughly one order of magnitude lower than in earlier PCS measurements, 18,19 and comparable to the bandwidths observed in low-temperature PL experiments for high-quality 1L-MoS2. 20 By applying a gate voltage to increase the charge carrier density in the semiconductor channel we are able to tune the relative intensity of the exciton and trion transitions. This is observed not only for the X A and T A transitionsalready reported in PL measurementsbut also for the X B and T B features, not shown before. Moreover, we find that a drain-source voltage can also be used to modulate the relative intensity of exciton and trion spectral features in a similar way (shown in Supplementary Note 3).
Besides the four mentioned peaks, the measured PC spectra also shows eight additional features at energies above the X B transition, which we attribute to the Rydberg series of excited states of X A and X B . 3,15,21 Using a 2D effective-mass Hamiltonian with a non-hydrogenic Keldysh potential we are able to accurately reproduce the observed spectral features and unambiguously determine their origin. respectively. For optoelectronic measurements, the entire device is exposed to homogeneous monochromatic illumination. (b) Gate transfer characteristic for Vsd = 5 V, showing a clear n-type behaviour. The threshold gate voltage is found to be around Vth = -9 V, as estimated extrapolating the linear region of ISD (solid red line). (c) I-V characteristics of the device measured for different gate voltages. (d) Source-drain current of the device for Vsd = 5 V and g − th = 5V. When the light excitation (hν = 1.94 eV) is turned on, the drain-source current increases by IPC = 4 nA. Figure 1a schematically depicts the geometry of the studied 1L-MoS2 phototransistor. A full description of the device fabrication process and the identification of monolayer flakes is provided in Supplementary Notes 1 and 2, respectively. The 2D channel is fully encapsulated between top and bottom multilayer hexagonal boron nitride (h-BN) flakes in order to better preserve its intrinsic properties. 22 The Ti/Au electrodes follow an edge-contact geometry, as shown at the top-right inset of Figure 1a and described in Supplementary Note 1. We start our measurements by characterizing the electrical response of the device. Unless otherwise stated, all measurements reported below were performed in vacuum and at T = 5 K. Figure 1b shows a transfer curve of the monolayer MoS2 phototransistor, with a clear n-type behaviour. We observe that the semiconductor channel conductivity increases as the back-gate voltage becomes larger than the threshold voltage (Vth = -9 V). Figure 1c shows two-terminal I-V curves of the device at different gate voltages. The curves present a back-to-back diode-like behaviour due to the presence of Schottky barriers at the contacts. 23,24 The different saturation currents for positive and negative voltages are caused by an asymmetry in the Schottky barrier heights. At low temperature ( = 5K), thermionic transport can be neglected as a mechanism for conduction and thus, the source-drain current Isd is mainly generated by tunnelling through the Schottky barriers.

Device fabrication and electrical characterization
Next, in order to characterize the 1L-MoS2 photoresponse, we expose the whole device to a homogeneous monochromatic light source. Figure 1d shows the time-dependent source-drain current sd , measured while the optical excitation is turned on and off at a frequency f = 31.81 Hz, while applying a constant Vsd =10 V and Vg = 20 V. The illumination energy is fixed at h = 1.92 eV, in resonance with the X A exciton (as shown below). When the light is turned on, Ids increases from its value in the dark, ID, by an amount IPC due to photoconductivity. In a 2D-TMD phototransistors, photoconductivity can emerge from two main mechanisms: 18,[25][26][27][28][29] photoconductive effect, where light-induced formation of electron-hole pairs leads to an increased charge carrier density and electrical conductivity; and photovoltaic effect, where the light-induced filling or depletion of localized states causes a shift of the Fermi energy. When the characteristic relaxation times for these localized states are very long, photovoltaic effects are observed as photodoping, rather than as photoconductivity, and the Fermi energy shift remains for a long time, or even permanently, after the optical excitation is removed. 30 As we further discuss in section 4, we believe that in our case the observed photoconductivity is mainly dominated by photovoltaic effects. 25 It is worth remarking that, due to the presence of h-BN layers in contact with the 1L-MoS2 channel, the device is also affected by photodoping, 30 which can result in large and persistent shifts of th after exposure to light. We find that this effect can be prevented to a large extent by carrying out the measurements at a low illumination power density, below 2.5 mW cm -2 . To further avoid inconsistencies in the measurements due to shifts of Vth all the spectral data is presented as a function of Vg-Vth. Figure 2 shows a typical low-temperature PC spectrum of the 1L-MoS2 phototransistor, acquired at = 5K, sd = 10 V and g − th = −20 V. To improve the signal-to-noise ratio, the measurement is performed while switching the illumination on and off at a fixed frequency of 31.81 Hz and the PC is registered using a lock-in amplifier. The PC spectrum is obtained by repeating this measurement while scanning the illumination wavelength in steps of 0.1 nm. A detailed description of the PCS setup can be found in Supplementary Note 4. The resulting spectrum presents two main peaks at 1.918 eV and 2.060 eV, that we associate to the A and B excitonic ground states (X 1s A and X 1s B ) of 1L-MoS2, with an energy splitting of 142 meV. We find that the linewidth of the observed peaks is at its lowest for gate voltages g well below th . In these conditions the X A peak bandwidth is found to be as low as 8 meV (FWHM), comparable with the typical A-exciton bandwidths for low-temperature PL spectroscopy in h-BN encapsulated 1L-MoS2. 20 For the spectral range between 1.85 eV and 2.15 eV the experimental PC spectrum can be very accurately fit by a multi-peak Lorentzian plus an exponential background, which accounts for the tail of the direct interband absorption edge. We find that the spectral profile is best reproduced by a quintuple Lorentzian function, with two main peaks centered at the energies of the X 1s A and X 1s B transitions discussed above plus three smaller peaks at 1.892 eV, 2.035 eV and 2.090 eV. We attribute the first two of these smaller features to the A and B trion states, T A and T B , expected to occur at energies 20-30 meV below X 1s A and X 1s B . While photogenerated trions can be either  positively or negatively charged depending on the nature of the constituent charge carriers, in our sample, the MoS2 channel is strongly n-doped and, thus, we expect that the observed T A and T B features mainly account for negatively charged trions. The feature at 2.090 eV is tentatively assigned to the first excited Rydberg state of the A exciton, X 2s A , recently observed in 1L-MoS2 by low-temperature micro-reflectance and transmission spectroscopy measurements. 15, 21 In fact, we also observe additional features lying at energies above 2.1 eV, indicated by arrows in Figure 2, which we associate to the Rydberg series of excited states of X A and X B , as further discussed in section 5.

Photocurrent spectra
Although small, the three features assigned to T A , T B and X 2s A are consistently reproduced in multiple spectra acquired at different bias voltages. Furthermore, as we show below, these features become much more prominent when Vg is increased to bring the MoS2 Fermi energy above the edge of the conduction band. Figure 3a shows the gate dependence of IPC for illumination with hν = 1.92 eV, on resonance with the X A transition. We find that the measured photocurrent becomes maximal for gate voltages close to the threshold voltage of the 1L-MoS2 channel, i.e. when the Fermi energy approaches the edge of the conduction band, and largely decreases for sub-threshold voltages. This result is somewhat counterintuitive since for g < th the electronic states at the conduction band should be completely depleted and thus, the probability of interband exciton absorption should be maximal. Indeed, for absorption spectroscopy experiments, the signal maximizes for Fermi energies below the edge of the conduction band, and decreases for larger gate voltages due to Pauli blockade. 31 For PC spectroscopy, however, the situation is more complex, as optical transitions will only be observed if they lead to a change in the device conductivity, either by a photoconductive effect or by a photovoltaic effect. The observed decrease of PC for gate voltages below Vth suggests that the main mechanism for photoresponse in our 1L-MoS2 device is a photovoltaic effect similar to the one described by Furchi et al. 25 For this effect, upon optical excitation, photoexcited carriers decay into localized states within the 1L-MoS2 bandgap, resulting in a shift of the Fermi energy. Differently from the h-BN induced photodoping effects mentioned above, 30 the relaxation time for charge carriers in these impurities is very short, and thus the effect manifests as an increase of conductivity while the device is exposed to light. A characteristic signature of the photovoltaic effect is that the resulting photocurrent is proportional to the transconductance = ds / g of the semiconductor channel. As shown in Figure 3a, the transconductance of the 1L-MoS2 device, measured in the dark at Vsd = 5 V, markedly resembles the gate voltage dependence of IPC, in consistence with the proposed mechanism for photoconductivity.

Gate modulation of photoconductivity and spectral features
Next we address the effect of the gate voltage on the PC spectral features. For simplicity we restrict our discussion here to the five spectral lines discussed in section 3 (higher-energy spectral features will be discussed in section 5). Figure 3b shows a color map of the photocurrent as a function of the excitation energy hν and the gate voltage Vg -Vth. For each value of Vg -Vth the photocurrent data has been normalized to the value measured at hν = 1.92 eV, on resonance with the X A transition. The five excitonic features discussed in section 3 are also apparent here, and their relative intensities are largely modulated by the gate voltage, as more clearly observed in the individual spectra shown in Figure 3c and discussed below. A similar modulation of spectral features is also observed when tuning the drain-source voltage, as described in Supplementary Note 3. (c) Individual photocurrent spectra acquired at different gate voltages, from 20 to 31.6 V above Vth (gray lines). The dashed lines are fits of the measured spectra to a multi-peak Lorentzian plus an exponential background. For clarity, the spectra have been shifted vertically in steps of 0.6 nA.
When Vg is increased, the trion transition T A becomes progressively larger, even becoming more prominent than X A for Vg -Vth = 24.2 V. A similar gate voltage modulation of the X 1s A and T A spectral features in 1L-TMDs has been reported in literature for photoluminescence, 32,33 electroluminescence, 16 and absorption spectroscopy 31 measurements. Typically, the T A spectral feature becomes more prominent when the Fermi energy is set above the conduction band edge, since in this situation, excess electrons in the semiconductor channel can efficiently bind with photoexcited electron-hole pairs to form trions. 31 Similar to T A , we find that the T B transition also becomes more prominent as the electron density in the conduction band is increased by the gate voltage.
As Vg increases, we also observe a strengthening of the peak associated to the X 2s A excited state, which even becomes larger than X 1s B for a certain gate voltage range. As further discussed in next section and in Supplementary Note 6, we observe a similar gate modulation for excited Rydberg states of X A and X B laying at higher energies.

Rydberg series
We now turn to the study of the different spectral features observed for energies above X 1s B . Figure   4a shows the photocurrent spectrum of the 1L-MoS2 device at T = 5 K, Vsd = 10 V and g − th = 17.2 V. In addition to the peaks associated to X 1s A , X 1s B , T A and T B , we observe eight additional peaks, which we tentatively assign to the Rydberg series of excited states of X A and X B , as labelled in the figure and enlisted in Table 1. Although some of these peaks are relatively weak compared to the X 1s A and X 1s B features, they consistently appear in spectra acquired for different gate and drain-source voltages (see Supplementary Note 6). It is worth noting that, while s-type transitions are predicted to be dipole-allowed, and therefore visible in the linear photoconductivity spectra, 34 p and d excited states do not directly couple to light. Thus, in the following we restrict our discussion to s states only. Figure 4b shows the spectral position of the observed peaks and their tentatively associated quantum number n. In order to confirm unambiguously the spectral assignments of the peaks we fit their spectral positions using an effective-mass theory for excitons in 2D-TMDs. 35 We solve numerically the effective-mass Schrödinger equation for the radially symmetric exciton states ns, with energies En: Here g is the quasiparticle bandgap and H is the Hamiltonian for the relative coordinate = |r − r |, namely For the A and B exciton effective masses we take µA = 0.27 m0 21 and µB = 0.28 m0, 35 respectively, being m0 the free electron mass. In thin semiconductor layers, the electron-hole interaction ( ) cannot be simply modelled as a Coulomb potential because it is largely affected by nonlocal screening from the embedding medium. Instead, the screened electron-hole interaction is accurately described by the Keldysh potential 36 Here 0 and 0 are zero-order Struve and Bessel functions and κ is the dielectric constant of the embedding medium (h-BN in our case). The parameter 0 is related to the screening length due to the 2D polarizability of the 1L-MoS2. For the discussion below, it is worth mentioning that the Keldysh potential approaches the Coulomb potential ( ) ≈ − 2 ⁄ at large distance ( ≫ 0 / ) but diverges logarithmically at short distance ( ≪ 0 / ).
Thus, we are left with three fitting parameters for each excitonic Rydberg series, namely E g , and 0 . A first estimation of the quasiparticle bandgap Eg can be deduced from the spectral position of the highly excited states ( > 2). For these states, the radius of the exciton lies in the region where ( ) ≈ − 2 ⁄ and the corresponding energy levels can be nicely fitted by 2D hydrogenic Rydberg series From the fitting we get g = 2.203 eV and g = 2.387 eV. We then use these values as an initial guess and perform a more accurate fit using the Keldysh potential and numerically solving the corresponding Schrödinger equation (discussed in Supplementary Note 5).
The fit reproduces with great accuracy the energies of the experimentally observed spectral features as shown in Figure 4b, and allows us to estimate the optoelectronic parameters of 1L-MoS2. The results of the fit are summarized in yield an even larger SO ≈ 290 meV.

Discussion
As we showed in section 3, low-temperature PCS allows us to observe very sharp excitonic spectral features, with linewidths as low as 8 meV (FWHM). While similar bandwidths for exciton features can also be achieved by PL or optical spectroscopy techniques, this typically requires the use of a microscope objective to concentrate the beam on a small area of the sample (in the order of 1 μm 2 ) to prevent peak broadening due to sample inhomogeneities. In our case however, the area of the sample that contributes to the observed spectrum is delimited by the spacing between the drain and source contacts. This allows exposing the whole sample to light without losing spectral resolution, largely simplifying the experimental setup, as well as the procedure for optical alignment.
Using PCS we were able to address spectral features typically difficult to observe in PL due to their relatively low PL emission intensity. Owing to this fact, we could observe the electric field modulation of not only the X A and T A transitions, already described in literature for PL, but also of the X B and T B transitions, which to our knowledge was not reported in literature so far. Finally, we were also able to clearly observe the excited Rydberg states of X A and X B , up to n=5. Similar spectral features have been also observed in previous experimental works by PL spectroscopy, PL emission and micro reflectance, 3,15,38 as well as predicted in theoretical studies. 34,35,39,40 However, earlier experiments only revealed excited states corresponding to the Rydberg series of either X A or X B , but never both combined, making difficult to unequivocally label the observed spectral features. The PC spectra presented here, on the other hand, cannot be explained by considering only one series of Rydberg states, but require accounting for excited states of both X A and X B . This strongly constrains the possible spectral assignments. The peak fittings also allow us to extract optoelectronic material parameters for encapsulated 1L-MoS2 that agree well with theoretical and experimental literature, further supporting the proposed peak assignments. Further, thanks to the simultaneous measurement of X A or X B , we could estimate the valence band spinorbit splitting of 1L-MoS2, not calculated before by this method, obtaining SO ≈ g B − g A = 180 meV. Earlier theoretical works give slightly lower values for SO , within 140-170 meV. 41,42 A recent ARPES measurement 37 for epitaxial 1L-MoS2 on gold also gave a lower value, SO = 145 ± 4 meV. However, this lower value could be explained by the effect of enhanced screening by the gold substrate. 40 In all, this work demonstrates low-temperature photocurrent spectroscopy as a simple and powerful technique with great potential for the study of optoelectronics and exciton physics in two-dimensional materials. We thus expect that, in the near future, this characterization technique will progressively become more popular among the 2D optoelectronics community.

5.
Theoretical modelling and exciton energy 6. Additional spectral measurements Figure S1 summarizes the main steps for the device fabrication. The process starts with the stacking of the heterostructure of single layer (1L) MoS2 completely encapsulated in hexagonal boron nitride (h-BN), using a dry-transfer method similar to the polypropylene carbonate (PPC) method for van der Waals heterostructures 1 . We first exfoliate MoS2 and h-BN flakes by the standard scotch-tape method and transfer the flakes onto SiO2 substrates. Then, we inspect the substrates through optical microscope ( Figure S1a), identify the 1L-MoS2 flakes by their optical contrast and confirm their thickness by micro-Raman spectroscopy as further detailed in section S2. We also use optical miscroscopy to identify and select two h-BN flakes with a thickness of 15-20 nm for the top layer h-BN and 25-30 nm for the bottom layer one.

Supplementary Note 1: Device fabrication and contact geometry
Next, we start the stacking process by transferring the top h-BN layer onto the MoS2 flake ( Figure  S1b). Subsequently we clean the substrate containing the top h-BN/MoS2 heterostructure with anisole, acetone and isopropanol (IPA) for few minutes. Both flakes are then picked up together using a PPC film, as described in ref. 1 , and transferred onto the bottom layer h-BN, previously cleaned with acetone and IPA and annealed at 380 ºC for 15 minutes in an Argon atmosphere.
Once the entire heterostructure is assembled ( Figure S1c) we perform a final cleaning step using anisole, acetone and IPA, followed by a second annealing in argon, with the same parameters described above. This final annealing step is crucial to remove contaminant and residual PC, as well as eventual blisters from the heterostructure 2 . Once the h-BN/MoS2/h-BN heterostructure is assembled, the next step is the geometrical definition of the device by electron beam lithhography (EBL) with a Raith Elphy Plus EBL system. We use a homemade PMMA (4% in chlorobenzene) as resist, spin coated at 4000 rpm for 1 minute and baked at 160 ºC for 10 minutes. The use of chlorobenzene instead of commercial PMMA in anisole permits an easier and very homogeneous resist coating without the need of previous treatment, as well as a thicker coating, useful for the etching mask and the final lift off-process.
After the EBL exposure (electron dose: 250 C/cm 2 at 15 kV), we develop the resist with a mixture of 1 part MIBK to 3 parts of isopropanol, which represents a good compromise between very good contrast and enough sensitivity 3 . The resulting structure is shown in Figure S1d. Next, ( Figure S1e) we etch away the EBL-exposed areas by dry plasma etching with an ICP-RIE Plasma Pro Cobra 100 in SF6 atmosphere (40 sccm, P=75W, process pressure 6 mTorr and T= 10 ºC) 4 . The etching rate is fixed to 2 nm/s and controlled by a DC bias. As further discussed below, the sides of the etched structure have a pyramidal profile, fundamental for the consequent achievement of the edge contacts ( Figure S1e). After this etching process, we clean the sample in acetone and IPA and carry a new annealing process, similar to the previous one, to remove residual contaminants of the PMMA resist.
After defining the stack geometry, a second EBL process (electron dose: 270 C/cm 2 at 15 kV) is used to define the contact geometry ( Figure S1f). For this step, special care was taken while designing the electrodes to avoid contact with eventual multilayer MoS2 flakes.
Finally, we deposit titanium and gold (5/45 nm) by e-beam evaporation. The evaporation takes place at very low pressure (10 -8 mbar) in a main chamber with a base pressure of 10 -10 mbar. The final device, after a lift-off process in acetone, is shown in Figure S1g. We carry all the fabrication steps for the device in one day. In particular it is very important to avoid any possible oxidation of the edge contacts. For this reason, it is crucial to spend the minimum time between the etching process and the loading into the pre-chamber of the e-beam evaporator.

Etching process for the edge contacts
The etching process has a fundamental role for the edge contacts fabrication in high-quality encapsulated devices, where the encapsulation guarantees an isolation from the external environment. In the case of edge contacts, the sides of the etched structure should have a pyramidal profile, 5 as this improves the contact between the atomic layer of MoS2 and the adhesion layer metal (in our case Ti).
In Figure S2 we compare two SEM tilted images for two different etching processes of h-BN. In Figure S2(a) we used the same recipe (in SF6+Ar atmosphere) of Jain et al. 6 , in which the authors recently obtained low resistance edge contacts in encapsulated MoS2 devices. In Figure S2(b) we show the etched interface using our etching recipe (only SF6 atmosphere but at very low pressure).
In both cases the interface presents a clean pyramidal profile and the substrate shows negligible degradation during the etching process. However, our recipe results in a more homogeneous etching.

Supplementary Note 2: Raman and photoluminescence characterization
We determine the thickness of the MoS2 flakes used for device fabrication by a combination of optical microscopy, Raman mapping and Photoluminescence. Figure S3a shows an optical microscope image of the MoS2 flake used to fabricate the device described in the main text, and Figure S3b shows a false color map of the ratio between the summed intensities of the A1g + E 1 2g Raman peaks of MoS2 and the intensity of the Si peak, in logarithmic scale. The different thicknesses can be clearly distinguished in the figure. Figure S3c shows individual spectra acquired at the different regions labelled in Figure S3a. The number of layers can be here confirmed by the difference between the spectral positions of the E 1 2g and A1g peaks 7,8 . For the thinnest region we obtain Δ = 19.4 −1 , compatible with the values given in literature for 1L-MoS2.
We further confirm the thickness of the MoS2 flakes by measuring the position of the A exciton peak in their photoluminescence spectrum. Figure S4 shows room-temperature photoluminescence spectra acquired at two separate monolayer MoS2 flakes. The A exciton peak can be clearly observed at around 1.8 eV, in good agreement with the values found in literature [8][9][10] .
For different monolayers we observe small fluctuations (~30 meV) of the A peak position, which we attribute to differences in the relative strength of their exciton and trion transitions. Figure S5a shows two I-V characteristics measured for the same gate voltage while keeping the 1L-MoS2 device in the dark (orange curve) and under illumination on resonance with the X A transition. We observe that the effect of exposure to light is similar to increasing the gate voltage, as expected when photoconductivity is dominated by the photovoltaic effect. In particular, for Vsd above the saturation current ( sat ≈ 1.7 V), the photocurrent remains constant, while for the photoconductive effect it should increase as the drain-source electric field becomes larger.

Supplementary Note 3: Drain-source voltage dependence of the photocurrent spectra
Supplementary Figure 4. Room-temperature photoluminescence spectra of two different monolayer MoS2 crystals on SiO2 under 530 nm excitation. The two observed peaks correspond to A and B exciton transitions. Figure S5b shows PC spectra acquired for at g − th = 9 V for different drain-source voltages.
We observe that the relative weight of the X B and T B features modulate with Vsd, with T B becoming larger at higher voltages. A similar trend is observed as well for the X A and T A features. Changing the drain-source voltage also allows to modulate the intensity of the X 2s A peak, which becomes larger for low drain-source voltages.

Supplementary Note 4: Experimental setup for low-temperature Photocurrent Spectroscopy
The experimental setup for PCS is schematically depicted in Supplementary Figure 6. The sample is placed inside a pulse-tube cryostat with an optical access at 5K and exposed to laser illumination. The light source is a supercontinuum (white) laser, and the excitation wavelength will be selected using a monochromator. This allows to scan the visible and NIR spectral range, roughly from 450 nm to 840 nm. The setup also includes a halogen lamp and a CCD camera, aligned with the laser excitation via two beamsplitters, which allows for an easy sample alignment with micrometric resolution. In order to improve the signal-to-noise ratio of the optoelectronic measurements, the excitation signal is modulated by an optical chopper and the electrical response of the device is registered using a lock-in amplifier with the same modulation frequency.
Supplementary Figure 5. I-V characteristics of the 1L-MoS2 device measured in the dark and under optical excitation on resonance with the X A transition.

Supplementary Note 5: Theoretical modelling and exciton energy
As discussed in the text, nonlocal screening effects alter the energy levels of 2D hydrogenic excitons, especially the lower ones since the radius is of the order or smaller than the screening length. The electron-hole interaction is found to diverge logarithmically on approaching each other instead of the usual Coulomb behavior of the form 1/ , where = |r − r | is the electron-hole separation. Nonlocal screening effects in thin semiconductor are usually described within the Keldysh approach 37 . The potential that interpolates between the Coulomb potential for large separation and logarithm divergence at small electron-hole distance is given, in CGS units, as follows where is the dielectric constant of the embedding medium, 0 is related to the screening length due to the 2D polarizability of the 1L-MoS2 and ( ) = 0 ( ) − 0 ( ).
Here 0 and 0 stands for the zeroth order Struve function and Bessel function of the second kind, respectively. The effective-mass equation for the ns exciton states with energy En is expressed as in terms of the quasiparticle bandgap Eg and the Hamiltonian H for the relative coordinate Let max be the maximum value of the dimensionless relative coordinate considered in the numerical solution of equation (S5). We now set a grid of + 1 equally spaced points, = ℎ, with ℎ = max ⁄ and = 0,1, ⋯ , . For brevity we define = ( ℎ) and ( ) = s ( ℎ).
Therefore, the discretized effective-mass equation can be cast in the form with boundary conditions 0 ( ) = ( ) = 0. Hence, the problem is reduced to finding the eigenvalues of a real, symmetric and tridiagonal matrix for a given set of parameters , and 0 .
Once the lowest eigenvalues are found, the exciton levels are given by