Understanding Solvent Effects on the Properties of Two-Dimensional

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Understanding Solvent Effects on the Properties of Two-Dimensional Transition Metal Dichalcogenides Jungwook Choi, Hanyu Zhang, Haodong Du, and Jong Hyun Choi ACS Appl. Mater. Interfaces, Just Accepted Manuscript • DOI: 10.1021/acsami.6b01491 • Publication Date (Web): 28 Mar 2016 Downloaded from http://pubs.acs.org on March 29, 2016

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ACS Applied Materials & Interfaces

Understanding Solvent Effects on the Properties of Two-Dimensional Transition Metal Dichalcogenides

Jungwook Choi, Hanyu Zhang, Haodong Du, and Jong Hyun Choi*

School of Mechanical Engineering, Purdue University, West Lafayette, Indiana 47907, United States *

E-mail: [email protected]

Keywords: two-dimensional semiconductor, transition metal dichalcogenide, solvent, photoluminescence, charge transfer, conductive AFM

ABSTRACT Two-dimensional (2D) transition metal dichalcogenides (TMDCs) have emerged as attractive direct bandgap semiconducting materials with remarkable properties. Recently, TMDC-based electronic and optoelectronic systems have been demonstrated with various chemical doping and functionalization approaches for modulating their physical properties and enhancing device performances. However, the dependence of intrinsic properties of TMDCs on diverse solvents, which are used necessarily in fabrication processes and chemical doping, remains largely unaddressed. Here we report a charge transfer mechanism in TMDCs by commonly used solvents such as chloroform, toluene, acetone, and 2-propanol, which significantly changes the physical properties of monolayer MoS2 and WSe2. We find that the relative difference in electronegativity between solvents and TMDCs drives the transfer of electrons from or to the TMDCs, which results in photoluminescence (PL) enhancement or quenching 1

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depending on the change of carrier density in TMDCs. The analysis of exciton and trion spectral weights in MoS2 as a function of solvent electronegativity provides evidence of charge transfer. Finally, conductive atomic force microscopy (C-AFM) on TMDCs before and after immersion in the solvents further supports the charge transfer mechanism and resulting changes in carrier density. Our results highlight the importance of selection of solvents for solution-processed 2D TMDC devices and systems.

Transition metal dichalcogenides (TMDCs) are emerging two-dimensional (2D) semiconducting materials with a wide range of bandgap and band positions.1 Recent advances in materials processing enabled successful isolation and synthesis of monolayer TMDC semiconductors.2 The direct bandgap of TMDCs, unique property found in monolayer flakes, makes them suitable for 2D digital electronics and optoelectronics.3 Monolayer TMDCs such as MoS2 and WSe2 exhibit distinct characteristics including high on/off ratio,4 strong photoluminescence (PL),5 and exceptional photoresponsivity.6 The optoelectronic properties of TMDCs are dominated by interplay between excitons and charge carriers;7 thus, controlling carrier type and density provides an effective means for modulating intrinsic properties of TMDCs and facilitates new designs of future 2D devices and systems. Several mechanisms of modulating electrical and optical properties of TMDCs have been

successfully

demonstrated,

including

external

electrostatic

field,7-9

covalent

functionalization,10,11 photoinduced charge transfer,12 and physical and chemical doping.13-21 Given the scalability and simplicity, solution-based molecular adsorption (i.e., chemical doping) has been widely investigated, which exploits surface charge transfer between adsorbates

and

TMDCs.

Adsorbed

molecules

such

as

2,3,5,6-tetrafluoro-7,7,8,8-

tetracyanoquinodimethane (F4TCNQ),16,17 cesium carbonate,18 benzyl viologen,19 1,22

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dichloroethane,20 and benzimidazoline21 withdraw or donate electrons, resulting in the enhancement or quenching of TMDC PL. Such molecular interactions may also control carrier density or contact resistance in the field-effect transport. To prepare the molecular adsorption, functional molecules are often dissolved in solvents (e.g., toluene) for dropcasting or dip-coating. During the adsorption process, the TMDC layers interact not only with the molecules, but also with the solvents. In parallel, typical micro/nanofabrication processes routinely involve cleaning of TMDC samples with organic solvents such as acetone and isopropyl alcohol (2-propanol). Thus, the interaction between TMDCs and diverse solvents is unavoidable for both chemical doping and fabrication of TMDC-based devices. Given the nature of organic solvents, they may influence the TMDCs significantly, resulting in undesirable and unexpected modulation of optoelectronic responses. Therefore, it is critical to understand the effects of solvents on the properties of TMDCs, which remains largely unaddressed. In this work, we study the solvent-driven tunability of optical and electrical properties of monolayer MoS2 and WSe2 for the first time. The TMDCs are immersed into commonly used solvents such as chloroform, toluene, acetone, and 2-propanol and dried before characterization, simulating typical fabrication and/or chemical doping processes. We find that the charge transfer interaction between the TMDCs and the solvents modulates the PL and carrier density of TMDCs as probed by micro-PL spectroscopy and conductive atomic force microscopy (C-AFM). The PL spectra and C-AFM scans suggest that the charge transfer reaction is dependent on the relative difference in electronegativity between the solvents and the TMDCs; a solvent with a higher electronegativity attracts electrons from TMDCs, whereas a lower electronegative solvent donates electrons to TMDCs. We also show that n-type MoS2 and p-type WSe2 show opposite behaviors in the PL intensity change as the carrier density increases or decreases via interaction with the solvents. Our results indicate 3

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that a solvent, without any dissolved molecules, can modulate the optoelectronic properties of TMDCs, and thus, this solvent effect should be considered in chemical doping and material processing for TMDC-based optoelectronic devices.

Figure 1. Interaction between semiconducting TMDCs and commonly used solvents such as toluene (C7H8), acetone (C3H6O), 2-propanol (C3H8O), and chloroform (CHCl3). (a) Molecular model of MoS2 that consists of Mo atoms sandwiched between two layers of S atoms. TMDCs may gain or lose electrons during their interaction with solvents. (b) PL spectra of MoS2 monolayers after immersion in solvents and blow-drying with air. The spectra are normalized with that of pristine MoS2 before immersion. The emission signatures are all centered at around 660 nm (~1.88 eV), but the intensity strongly depends on the solvent. After monolayer MoS2 is immersed in toluene, acetone, and 2-propanol, the PL intensity decreases, whereas chloroform significantly enhances the PL signature. 4

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Monolayer TMDCs were mechanically exfoliated onto a SiO2/Si substrate and annealed in argon environment at 250 °C for 1 hour to remove any organic residues and contaminants – representative AFM image, height profile, and Raman spectrum are shown in Figure S1. To investigate the solvent effects, TMDCs were immersed in toluene, acetone, 2propanol, or chloroform for 20 min, followed by blow-drying with air. The PL spectra from immersed TMDCs were compared with the intrinsic spectra obtained from the pristine flakes. It should be noted that all PL spectra were obtained under ambient conditions after immersing and drying. Thus, our work is different from solvatochromism or dielectric screening studies which probe the emission signatures within dielectric environments.22,23 To elucidate the solvent effects on the PL emission properties of TMDCs, a number of solvent properties were examined, and we found that the optical behaviors of TMDCs are strongly correlated with the relative electronegativity between solvents and TMDCs. The absolute electronegativity (χ) of molecules is closely related to the electronic chemical potential. The electronegativity is a measure of the tendency of a medium to attract electrons, and indicative of energy balance between two different molecular systems, determining which of the two becomes electron donor or acceptor. Thus, for any two interacting molecules, the difference in electronegativity drives an electron transfer and determines the direction of net electron flow from one to another. It is defined as: ଵ

χ = ଶ (‫ ܲܫ‬+ ‫)ܣܧ‬, where IP is the ionization potential and EA is the electron affinity.24 Each solvent has a unique electronegativity. In this work, we investigated four common solvents: toluene (C7H8), acetone (C3H6O), 2-propanol (C3H8O), and chloroform (CHCl3) whose electronegativity values are 3.86, 4.1, 4.92, and 5.51 eV, respectively. These values are obtained from IP and

EA of each molecular structure.25-28 The first ionization potential of a molecular system is 5

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equal to the negative of the highest occupied molecular orbital (HOMO), while the electron affinity corresponds to the negative of the lowest unoccupied molecular orbital (LUMO).26 The EA of 2-propanol is unknown to the best of our knowledge, so we used that of isopropyl radical for calculating the electronegativity.25 The electronegativity of TMDC (χTMDC) can be expressed as:29,30

χ்ெ஽஼ = ‫ܧ‬௏஻ெ + ‫ܧ‬௘ − 0.5‫ܧ‬௚ , where EVBM is the potential of valence band maximum (VBM) against normal hydrogen electrode (NHE), Ee (~4.5 eV) is the standard electrode potential on the NHE scale, and Eg is the bandgap. Based on the band edge positions of TMDCs,31,32 the electronegativity values of MoS2 and WSe2 are estimated to be 5.06–5.26 and 4.21–4.34 eV, respectively. The electronegativity of TMDCs is related to its intrinsic Fermi level. However, it can be shifted by n- or p-doping, and the degree of doping would be strongly affected by oxygen molecules in air.12,13 Thus, we have adopted the electronegativity for elucidating our mechanism which may not be influenced by unintentional doping effects. The electronegativity differences between TMDCs and solvents induce charge transfer. If two systems with different electronegativity interact each other, the material with a higher electronegativity withdraws electrons from the other, until the associated chemical potential reaches an equilibrium.24 It is also expected that a larger difference in electronegativity would transfer more electrons.26 For examples, MoS2 attracts more electrons from toluene than acetone or 2-propanol, while MoS2 donates electrons to chloroform as illustrated in Figure 1a. The electron transfer results in the PL intensity change of TMDCs. Representative PL spectra of MoS2 are shown in Figure 1b, which are measured from five different monolayer MoS2 and normalized to the PL intensity of pristine MoS2. Toluene, acetone, and 2-propanol increase the electron density in MoS2, leading to the PL quenching, owing to destabilized exciton (X) recombination by an increased formation of charged excitons (negative trions, X-). 6

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In contrast, the decrease of electron density in MoS2 by chloroform results in the PL enhancement. This PL intensity change by solvents is similar to the mechanism of chemical doping of TMDCs16-18 where the carrier density can be controlled by surface charge transfer between TMDCs and adsorbed molecules, ultimately modulating the emission characteristics.

Figure 2. Spectral analysis of pristine and solvent-immersed MoS2. (a–c) PL spectra of pristine and solvent-immersed MoS2 (black) deconvoluted into exciton (X ~ 656 nm; blue) and trion (X- ~ 669 nm; green) peaks with different spectral weight. The convoluted PL spectra are shown in red. The spectral weights of exciton and trion peaks from pristine, toluene-immersed, and chloroform-immersed MoS2 are approximately 0.64 and 0.46 (a), 0.57 and 0.54 (b), and 0.82 and 0.26 (c), respectively. (d,e) Change in exciton (d) and trion (e) weights of MoS2 as a function of solvent electronegativity. Toluene, acetone, and 2-propanol, which have a lower electronegativity than MoS2, donate electrons to MoS2, thereby decreasing the exciton weight and increasing the trion weight. A larger difference in electronegativity leads to a greater change in spectral weights. In contrast, chloroform attracts electrons from MoS2, suppressing recombination of negative trions and increasing the exciton weight. 7

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Figure 2 shows the PL analysis of monolayer MoS2 after immersion in toluene, 2propanol, and chloroform along with pristine MoS2 (also see Figure S2 for details). The pristine MoS2 excited at 633 nm exhibits strong PL signatures at ~660 nm33 via excitonic transitions at the K point of the Brillouin zone.3 The PL spectra can be deconvoluted into two Lorentzian profiles of excitons (X) at ~656 nm (~1.89 eV) and negative trions (X-) at ~669 nm (~1.85 eV),7,12,16 as shown in Figs. 2a–c. The trion emission originates from the transition of two electrons bound to one hole, on account of abundant electrons in pristine n-type MoS2.7,16 The spectral weights of excitons (IX/Itotal) and trions (IX-/Itotal) in pristine MoS2 are measured to be ~0.64 and ~0.46, respectively (Figure 2a). The exciton and trion weights as well as total PL intensity are changed after MoS2 interacts with the solvents. Given the lower electronegativity of toluene than that of MoS2, the electrons transfer from toluene to MoS2. The increased carrier density in MoS2 promotes the recombination of negative trions (IX-/Itotal ~ 0.54), while destabilizing excitonic recombination (IX/Itotal ~ 0.57) (Figure 2b). As a result, the overall PL intensity decreases as shown in Figure 1b, because nonradiative recombination becomes dominant at high carrier density.13 In contrast, chloroform-immersed MoS2 displays the opposite behavior. The spectral weight of excitons increases up to ~0.82, whereas the trion weight decreases significantly to ~0.26 (Figure 2c). It is attributed to the strong electron-withdrawing ability of chloroform, thereby reducing the carrier density, suppressing trion formation, and increasing overall PL intensity. Changes of exciton and trion weights in MoS2, which can be defined as the difference in each spectral weight before and after immersion (e.g., (IX/Itotal)solvent–(IX/Itotal)pristine), are plotted as a function of the solvent electronegativity in Figure 2d and e. The relative difference of electronegativity between MoS2 and solvents determines the change in the spectral weight. Toluene, acetone, and 2-propanol, which have lower electronegativity than 8

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MoS2, decrease the exciton weight of MoS2. In contrast, chloroform depletes electrons in MoS2 due to its high electronegativity, inducing an increase of the exciton weight (Figure 2d). The trion spectral weight in MoS2 shows the opposite behavior as shown in Figure 2e; the trion weight increases with decreasing electronegativity of solvents. In both exciton and trion weights, a larger difference in electronegativity between MoS2 and solvent induces a greater change. Our spectral analysis shows that the electron density in MoS2 can be modulated by interaction with other medium with different electronegativity, affecting overall PL responses. It is noted that the PL spectra can vary even on a TMDC flake due to the substrate roughness, charges on substrate, and distribution of defects in the TMDCs. Given the flake size of mechanically exfoliated TMDCs (~1–2 µm2) and the spot diameter of our laser beam (~1 µm2), the area of the flake exposed to laser irradiation could also vary in every measurement, resulting in different PL intensity. To obtain PL responses before and after immersion in solvents in Figures 1 and 2, we measured at least three PL spectra from a TMDC flake for each solvents (Figure S3). In addition, we have examined spatially resolved PL intensity of a MoS2 monolayer before and after immersion in chloroform (Figure S4). The variation of PL intensity across the MoS2 flake is much smaller than the intensity change after chloroform immersion, which clearly demonstrates that solvent effects dominate the entire PL responses over spatial variation within the flake. We also have performed micro-Raman measurements to confirm that solvent residues do not remain on the surface of TMDCs after immersion and blow-drying. As presented in Figure S5, the Raman spectrum of chloroform was recorded and compared with those of MoS2 before and after immersion in chloroform. A prominent Raman signature of chloroform at ~3020 cm-1 is completely absent in the Raman spectra of MoS2, indicating that the solvent residues are not present or detectable. Thus, we conclude that there are no significant solvent residues on the surface of TMDCs, and it is not expected that solvatochromism or dielectric 9

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screening modulate the PL responses.

Figure 3. Change of WSe2 PL intensity as a function of the electronegativity of solvents. Holes, the major carrier in p-type WSe2, can be depleted by interaction with the solvents with lower electronegativity than WSe2. Toluene and acetone increase the PL intensity by suppressing the recombination of positive trions, whereas 2-propanol and chloroform can decrease the PL intensity. Note that MoS2 and WSe2 shows opposite behaviors due to the difference in their major carriers.

The PL modulation due to the charge transfer interaction between TMDCs and solvents is observed not only in n-type MoS2, but also in p-type WSe2. Figure 3 presents a change in PL intensity of monolayer WSe2 as a function of the electronegativity of solvents. Given the electronegativity of WSe2 (~4.21–4.34 eV), toluene and acetone donate electrons to WSe2. Thus, the holes, the major carrier in WSe2, are depleted, thereby suppressing the recombination of positive trions and enhancing total PL intensity (Figure S6). In contrast, 2propanol and chloroform can accumulate the holes in WSe2 by attracting electrons, leading to the decreased PL intensity. All the collected PL spectra of WSe2 are also presented in Figure S7. The opposite behaviors in n-type and p-type TMDCs are consistent with the PL 10

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enhancement and quenching observed in MoS2 and WSe2 by physisorption of oxygen and water molecules.13

Figure 4. Change in the current flow in monolayer MoS2 before and after immersion in solvents measured by C-AFM. (a,b) AFM height image of MoS2 exfoliated onto an ITO substrate (a) and corresponding current difference map (b). The current difference map (∆i = ichloroform – ipristine) is reconstructed by subtracting the current map of pristine MoS2 (ipristine) from the current map of chloroform-immersed MoS2 (ichloroform). The ∆i in the monolayer MoS2 region (denoted as 1L) is approximately -6.8 nA, indicating a decrease of electron density after immersion in chloroform. (c,d) AFM height image of MoS2 (c) and corresponding ∆i map in case of toluene immersion (d). In contrast to the chloroformimmersed MoS2, the current increases in the monolayer region (∆i of ~4.5 nA) due to the 11

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increased electron density. (e) The change in the currents (∆i/ipristine) after immersion in solvents (red filled objects with error bars) are fitted with the estimated electron density (blue solid line) as a function of measured trion weight of MoS2 (blue filled objects with error bars). The ∆i/ipristine for each solvent is measured by C-AFM in the same manner as (b) and (d), and the electron density is calculated based on the law of mass action. The standard deviations associated with the averaged current values are used for plotting the error bars.

We further analyzed the change of the electron density in MoS2 before and after immersion in solvents by measuring currents flowing through the MoS2 samples using CAFM. The C-AFM is an ideal tool for investigating the effect of solvents on intrinsic MoS2, since it does not require lithography and lift-off processes for fabrication of electrodes which necessarily use solvents. The MoS2 flakes were exfoliated onto a conductive ITO substrate, and the local current flow through MoS2 was measured using a Pt-Ir coated probe with a bias voltage applied between the probe tip and the ITO. We measured the spatially resolved current map of MoS2 before and after immersion in the solvents. The current difference (∆i = isolvent – ipristine) was then obtained by subtracting the current map of pristine MoS2 (ipristine) from the current map of the solvent-immersed MoS2 (isolvent) with an identical bias voltage. It is noted that the currents obtained at each position have slightly different values even on the same flake (Figure S9). This spatial inhomogeneity could originate from the high roughness of the ITO substrate which would lead to different localized contact areas between the TMDC and the ITO surfaces. Thus, we averaged the current flow through monolayer MoS2 over 3,000 data points before and after immersion in solvents. Figure 4a shows the AFM height image of monolayer (1L) MoS2 with a measured thickness of ~1 nm, and the corresponding current difference before and after immersion in chloroform is visualized in Figure 4b. The dark area in 1L MoS2 confirms that the current flow through MoS2 is decreased (∆i ~ -6.8 nA) after immersion in chloroform. Because the 12

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current density is proportional to the electron density, it directly evidences the decrease of the electron density in MoS2, which is consistent with the results from our PL study. In contrast, the currents flowing through the toluene-immersed MoS2 increase by approximately 4.5 nA in the monolayer region as shown in the AFM height image and the corresponding current difference map of Figure 4c and d. Both AFM images in Figure 4a and c were measured after immersion in solvents, and there are no observable height or topography changes of MoS2 after immersion and drying as presented in Figure S8. Figure 4e shows the change in currents (∆i/ipristine) and the electron density as a function of the trion weight in solvent-immersed MoS2. The relative change of currents in MoS2 was approximately -0.5 ± 0.28, 0.29 ± 0.71, 0.56 ± 0.51, and 0.88 ± 0.28 for chloroform, 2-propanol, acetone, and toluene-immersed flakes, respectively. All current maps before and after immersion in the solvent are presented in Figure S9. The horizontal streaks are usually observed on thicker MoS2 layers where the probe tip experiences sudden height changes. It could be also due to the interaction between the tip and the sample, such as the difference of adhesion associated with the substrate roughness and inhomogeneity. It should be noted that we have used the same measurement conditions before and after immersion to isolate the solvent effects on the current values of monolayer MoS2. The relationship between the electron density (ne) in MoS2 and the concentrations of excitons (NX) and trions (NX-) can be estimated by the mass action law,16,17 which is expressed as:

ܰ௑ ݊௘ 4݉௑ ݉௘ ‫ܧ‬௕ =൬ ଶ ൰ ݇஻ ܶexp ൬− ൰ ܰ௑ ష ߨℏ ݉௑ ష ݇஻ ܶ where mX and mX- are the exciton and trion effective masses, which can be defined as mX = me + mh and mX- = 2me + mh (the effective mass of electron me ~ 0.35m0 and that of hole mh ~ 0.45m0, where m0 is the mass of a free electron).34 ћ is the reduced Planck’s constant, kB is the 13

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Boltzmann constant, T is the temperature, and Eb is the binding energy of trion. Assuming that the total PL intensity is the sum of exciton and trion recombination, the trion weight (IX/Itotal) can be expressed as:

‫ܫ‬௑ ష

‫ܫ‬௧௢௧௔௟

ߛ௧௥ ܰ௑ ష ߛ௧௥ ܰ௑ ష =൬ ൰൘൬1 + ൰ ߛ௘௫ ܰ௑ ߛ௘௫ ܰ௑

where γtr and γex are the radiative decay rates of trions and excitons.16 Finally, the electron density in MoS2 as a function of the trion weight can be estimated from the above equations and is plotted in blue solid line of Figure 4e. The change in the currents driven by solvents is well fitted to the estimated electron density in MoS2. This result confirms that the charge transfer between MoS2 and solvents changes the electron density of MoS2 and ultimately modulates its overall PL responses. Given the trion weight of pristine MoS2 (~0.46), the intrinsic electron density in MoS2 can be estimated to be approximately 2 × 1013 cm-2, which reasonably agrees with previous reports on MoS2 (~6 × 1013 cm-2).16,17 The slightly lower electron density of our pristine MoS2 may be attributed to oxygen adsorption after thermal annealing that can deplete abundant electrons and suppress trion formation.12,13 From the curve-fit in Figure 4e, the decrease and increase of the electron density before and after immersion in chloroform and toluene can be estimated to be ~1.5 × 1013 and ~2 × 1013 cm-2, respectively. The degree of the electron density change in our experiment is less than that in the chemical doping by F4TCNQ (~5.8–7.4 × 1013 cm-2) which is a strong electronwithdrawing molecule with high electronegativity of ~6.79 eV.16,17 Our study indicates that even without any dissolved dopants, the effect of solvents should not be ignored as it can change the electrical and optical properties of TMDCs via charge transfer by the electronegativity differences. To further verify our mechanisms, we have performed C-AFM measurements of WSe2 flakes before and after immersion in chloroform and toluene (Figures S10 and S11). As 14

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demonstrated in the PL studies, it can be expected that p-type WSe2 behaves differently from n-type MoS2. For example, highly electronegative chloroform withdraws electrons from WSe2 during the immersion, which promotes the hole transport and ultimately increases the currents (Figure S10). For toluene, a decrease of currents is expected from WSe2, whereas MoS2 displays an increase (Figure S11). Indeed, we observed the opposite behaviors with WSe2 flakes after immersing in chloroform and toluene. Overall, when the major carrier density of TMDCs increases (decreases) due to the interaction with solvents, C-AFM measures a greater (smaller) current flow. These results verify the proposed mechanism of charge transfer induced modulation of carrier densities in TMDCs by solvent immersion. Further, the opposite trends between p-type WSe2 and n-type MoS2 evidence that our mechanism may not be related to solvent residues or dielectric screening. In summary, we have demonstrated that PL intensity of monolayer TMDCs can be modulated through the interaction with solvents routinely used for fabrication and doping processes. The PL enhancement/quenching behaviors are attributed to the change in the carrier density in TMDCs, as a result of charge transfer driven by the electronegativity difference between the TMDCs and the solvents. The increase or decrease of the carrier density, which leads to the change in spectral weight of excitons and trions, is also verified by measuring the current flow through TMDC samples before and after immersion in the solvents using C-AFM. Given the layer-number dependent bandgap and band positions of TMDCs,1,3 thicker TMDCs may exhibit different electronegativity and demonstrate different behaviors. Our results should be practically useful in designing a fabrication process flow for TMDC-based devices and functionalizing TMDCs via wet chemistry. Finally, the distinguishable responses of PL and currents in monolayer TMDCs depending on the solvent electronegativity could be advantageous for highly sensitive and selective chemical sensing applications. 15

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Conflict of Interest: The authors declare no competing financial interest.

Supporting Information Available: Materials and methods; AFM image, height profile, and Raman spectrum of monolayer MoS2; PL and Raman spectra of MoS2 and WSe2 before and after immersion in various solvents; AFM images and C-AFM current maps of MoS2 and WSe2 before and after immersion in solvents. This material is available free of charge via the Internet at http://pubs.acs.org.

ACKNOWLEDGEMENTS This work was supported by the National Science Foundation.

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