Monolayer MoS2 Bandgap Modulation by Dielectric Environments and Tunable Bandgap Transistors

Semiconductors with a moderate bandgap have enabled modern electronic device technology, and the current scaling trends down to nanometer scale have introduced two-dimensional (2D) semiconductors. The bandgap of a semiconductor has been an intrinsic property independent of the environments and determined fundamental semiconductor device characteristics. In contrast to bulk semiconductors, we demonstrate that an atomically thin two-dimensional semiconductor has a bandgap with strong dependence on dielectric environments. Specifically, monolayer MoS2 bandgap is shown to change from 2.8 eV to 1.9 eV by dielectric environment. Utilizing the bandgap modulation property, a tunable bandgap transistor, which can be in general made of a two-dimensional semiconductor, is proposed.


Semiconductors with a moderate bandgap have enabled modern electronic device technology, and the current scaling trends down to nanometer scale have introduced two-dimensional (2D) semiconductors.
The bandgap of a semiconductor has been an intrinsic property independent of the environments and determined fundamental semiconductor device characteristics. In contrast to bulk semiconductors, we demonstrate that an atomically thin two-dimensional semiconductor has a bandgap with strong dependence on dielectric environments. Specifically, monolayer MoS 2 bandgap is shown to change from 2.8 eV to 1.9 eV by dielectric environment. Utilizing the bandgap modulation property, a tunable bandgap transistor, which can be in general made of a two-dimensional semiconductor, is proposed.
Atomically thin two-dimensional (2D) semiconductors have attracted a great deal of attention for their superior properties in electronic devices. Monolayer (ML) molybdenum disulfide (MoS 2 ) has shown high electron mobility of about 217 cm 2 V −1 s −1 and an excessively high current on/off ratio of an order of 10 8 in a field effect transistor (FET) 1,2 . However, the superior properties have been achieved only with a supporting substrate and a gate dielectric in a top gate FET structure, such as the HfO 2 /MoS 2 /SiO 2 stack 1,2 . Without the top gate high-k dielectric, large reduction of the electron mobility has been reported [1][2][3] , and it has been believed to be due to the environmental dielectric screening (EDS) effect suppressing the Coulomb scattering of carriers with charged impurities in the 2D semiconductors 2,4 . The EDS effect has also been reported to change the defect level with the band gap and induce deep-to shallow-level transition of dopants, enhancing the carrier concentrations significantly and the electrical conductivities 5 . Furthermore, the exciton binding energies have also been reported to be affected by the EDS effect strongly in 2D semiconductors 6,7 .
A moderate bandgap size is a determining characteristic property of a semiconductor. Nonetheless, an accurate evaluation of the bandgap in low dimensional semiconductors has not been as simple as in conventional bulk semiconductors. It is well established that the bandgap size of MoS 2 layers has a strong dependence on the number of layers. Furthermore, in a spatially isolated low dimensional system (e.g. freestanding 2D semiconductors), the strong unscreened Coulomb interaction (through the space outside of the 2D materials) makes the quasiparticle (QP) renormalization of electrons huge. Within the GW approximation, the electronic bandgap of a freestanding ML MoS 2 has been predicted to be about 2.8 eV (refs 8-12). Due to the strong exciton binding (~1 eV) [8][9][10][11][12] , the optical bandgap has been obtained to be about 1.8 eV from the photoluminescence (PL) and optical absorption experiments 13 , which agree with the theoretical Bethe-Salpeter-Equation (BSE) calculations [9][10][11][12] . Since the exciton binding energy is large, the measured optical bandgaps are not accurate representation of semiconductor bandgap, determined by the energy difference between valence and conduction band edges. Eliminating the excitonic effect, the measurements of the electronic bandgap have given diverse results. With intercalated potassium (K) in a bulk MoS 2 , a quasi ML MoS 2 has been fabricated from a bulk MoS 2 , and a direct bandgap of 1.86 eV at the K valley has been measured using angle-resolved photo-emission spectroscopy (ARPES) 14 . For a chemical vapor deposition (CVD) grown ML MoS 2 on a Au(111) substrate, the ARPES bandgap of about 1.39 eV has been measured, which is very small 15 . In scanning tunneling spectroscopy (STS) measurements, the bandgap of a ML MoS 2 on graphite substrate has been measured to be 2.15 eV (ref. 16) and that on a bilayer graphene to be 2.16 eV (ref. 17), which are larger than other measured values, but still significantly lower than the predicted GW value of 2.8 eV (refs 8-12). In a ML MoS 2 phototransistor that has the Al 2 O 3 /MoS 2 /SiO 2 stack structure, the electronic transport bandgap of the ML MoS 2 has been measured to be 1.8 eV, in which the optically excited excitons are separated to generate electron and hole carriers by applying the source and drain bias voltages 18 . As such, the measured electronic bandgap sizes have been in a wide range (1.39-2.16 eV), and the bandgap changes have been speculated to be introduced by the EDS 10,15,17 or carrier-induced bandgap renormalization 19 effect. Nevertheless, the measured bandgap sizes are significantly smaller than the accurate GW bandgap of 2.8 eV. These findings indicate that the traditional concept of assigning a well-defined bandgap size for a specific semiconductor (e.g., 1.1 eV for Si) as a fundamental material property may not be applicable to 2D semiconductors, and that the electronic bandgap may have strong dependence on the environments. Considering the fundamental role of the bandgap size in electronic device applications, it is critical to develop a fundamental and quantitative understanding on how the environmental effects change the bandgap sizes of 2D semiconductors.
In this study, as a representative 2D semiconductor, we investigate the bandgaps of a ML MoS 2 with various environments based on GW calculations, and predict a wide range of bandgap size determined by the strong effects on the ML MoS 2 embedded in a device structure. The bandgap of ML MoS 2 is found to change by the surrounding medium according to the dielectric constant (κ E ) of the environment. Specifically, the GW bandgap changes from 2.8 eV of freestanding MoS 2 down to about 1.9 eV for MoS 2 in a sandwich structure between two high-k dielectrics. On the other hand, the bandgap changes down to 2.2 eV for a supported structure on ultrahigh-κ E dielectrics. These GW bandgap changes are continuous functions of dielectric constant of the surrounding medium. Based on this finding, it is suggested that there should be transport barriers to electrons and holes in the ML MoS 2 channel between near the metallic contacts and near the gate dielectric in a device structure, because of the different screening environments (and correspondingly different bandgap sizes) surrounding the ML MoS 2 . When the barriers are controlled by an external source, a tunable bandgap transistor can be made possible utilizing the environment-dependent property of the 2D semiconductors. The electronic structure analysis shows that the orbitals of the valence band maximum (E V ) and the conduction band minimum (E C ) at the K valley are characterized as the Mo 4d atomic orbitals. These atomic orbitals are located in the middle of the three atomic layers of ML MoS 2 layer, and they have negligible hybridization with the orbitals of the nearby surrounding materials that interacts with the ML MoS 2 through the van der Waals gap. The LDA bandgap at the K valley is about 1.8 eV agreeing with previous DFT calculations and found to remain unchanged with different environments. For the metallic Au environments (Fig. 1j,k), the Au related states (6s) are found inside the bandgap of the ML MoS 2 , but they can be clearly distinguished from the ML MoS 2 states (shown as blue dots). Within the LDA, the Au 5d states are found inside the valence bands of the ML MoS 2 , and the Fermi level (E F ) is found to be located at E V + 0.8 eV or E C − 1.0 eV.

Results and Discussion
The calculated GW band structures of the ML MoS 2 with the same environments are plotted in Fig. 1m-q. They also show that the ML MoS 2 has a direct bandgap at the K valley irrespective of the environments, as in the case of the LDA. The direct bandgap at the K valley of the freestanding ML MoS 2 ( Fig. 1m) is 2.8 eV in GW, close to the previous calculations [8][9][10][11][12] . The GW bandgap of the ML MoS 2 on HfO 2 ( Fig. 1n) is calculated to be 2.6 eV, and that of the ML MoS 2 sandwiched by HfO 2 (Fig. 1o) is 2.4 eV, which are smaller than that of the freestanding ML MoS 2 (2.8 eV). Since the electronic orbital hybridization between the MoS 2 and the nearby HfO 2 dielectric is negligible, as shown in the LDA results (Fig. 1h,i), the main cause of the bandgap reduction in the GW calculations is expected to be the EDS effect on the QP bandgap renormalization. The size of the bandgap reduction is significant, up to by 0.4 eV, in the presence of HfO 2 layers. The ML MoS 2 on the Au metallic substrate (Fig. 1p) is found to have a direct bandgap of 2.3 eV at the K valley in the GW calculations, and the ML MoS 2 sandwiched by the Au layers (Fig. 1q) has a GW bandgap of 2.1 eV. The bandgap reduction is even more significant, up to by 0.7 eV, by the metallic Au environments. GW bandgaps of ML MoS 2 with effective medium. In order to investigate the primary effect of EDS on the bandgap of ML MoS 2 , we incorporate the effective environmental dielectric constant (κ E ) into the dielectric matrix of the screened Coulomb (W) interaction in GW calculations. Details of the procedure are described in the Supplementary Materials. With this approach that includes the EDS effectively in GW, there are several advantages besides making it possible to study separately the EDS effect: reducing the computational costs of GW calculations with environments, and making it possible to include additional polarizability into the screened Coulomb (W) interaction. Note that the dielectric effect of liquid medium on ML MoS 2 bandgap can be modeled by the effective dielectric medium. The quasiparticle renormalization in the GW approximation ( Fig. 1m-r) includes only the electronic contribution of screening into the screened Coulomb (W) interaction. Since high-κ E dielectrics such as HfO 2 (ε 0 ≅ 26) usually have large ionic contribution (ε 0 − ε ∝ ≅ 21) to the dielectric screening, the renormalization of electrons in the ML MoS 2 would be further modified by the ionic screening of the surrounding materials. However, this ionic screening effect is neglected in the GW calculations of the band structures shown in Fig. 1n-r. In order to include such an ionic contribution of screening explicitly, the GW plus the lattice polarization effect (LPE) can be applied to the calculation of the bandgap renormalization. In some ionic solids, the inclusion of the LPE has been reported to lead to a large shrinkage of the bandgap [20][21][22] . Compared to full GW + LPE, effective dielectric medium method provide an alternative efficient approach to include full dielectric effects on the MoS 2 bandgap.
The calculated GW bandgaps of ML MoS 2 with an effective dielectric constant (κ E ) of the environments are plotted as a function of the κ E in Fig. 2a. With κ E = 1, the ML MoS 2 represents the freestanding isolated one in vacuum, and the GW bandgap is found to be 2.8 eV at the K valley. With increasing the dielectric constant, κ E , the GW bandgap is found to drop rapidly down. With the one-side dielectric, in such a case of the supported ML MoS 2 on a substrate, the GW bandgap is found to reduce down to about 2.2 eV with a ultrahigh-κ E dielectric (at κ E = 30). With the both-side dielectric, as in a typical top-gate FET structure, the GW bandgap of the ML MoS 2 is found to be smaller down to about 1.9 eV with a ultrahigh-κ E dielectric of κ E = 30. It is notable that the bandgap reduction is very rapid in the range 1 < κ E < 5, and most of the bandgap reduction, about 80%, occurs with κ E = 5. Thus, the presence of a moderate dielectric material in vicinity of a ML MoS 2 , can strongly affect the bandgap renormalization in the ML MoS 2 , even though it is not an ultrahigh-κ E dielectric.
The experimentally measured bandgap of 2.15 eV in STS 16 for the ML MoS 2 on graphite substrate is close to the obtained asymptotic value of 2.2 eV for the one-side dielectric, and the STS bandgap of 2.16 eV for the ML MoS 2 on a bilayer graphene substrate 17 is also close the value. The measured ARPES bandgap of 1.86 eV for the K-intercalated MoS 2 (ref. 14) is close to the obtained bandgap of ML MoS 2 embedded in high-κ E dielectric (1.9 eV). The measured ARPES bandgap of 1.39 eV for the ML MoS 2 on Au substrate 15 indicates a rather strong interaction at the interface, as shown in our previous metal-MoS 2 interface study 18 . The measured bandgap of 1.8 eV for the ML MoS 2 in the Al 2 O 3 /MoS 2 /SiO 2 stacked top-gate FET 19 is closer to the asymptotic value of 1.9 eV obtained for the both-side dielectric system. An additional reduction of the bandgap may be possible by the carrier-induced renormalization of bandgap 20 in n-type ML MoS 2 FET.

Absolute band edge levels of ML MoS 2 with environments.
We now investigate the absolute band edge levels (relative to vacuum level) of ML MoS 2 with including the EDS effect. Figure 2b shows the band edge levels of ML MoS 2 in LDA and those in GW with various κ E . The E V and E C in LDA are found to be − 6.16 and − 4.31 eV, respectively, which are close to the previous LDA calculations (− 5.98 and − 4.29 eV) 23 . The absolute GW band edge levels are obtained using the bandgap center alignment scheme 23 , and the obtained E V and E C in GW with κ E = 1 are − 6.64 and − 3.83 eV, respectively, in good agreement with the previous GW calculations (− 6.50 and − 3.74 eV, respectively) 23 . The calculated GW band edge levels with κ E plotted in Fig. 2b show that the E V increases up and the E C decreases down monotonically with increasing κ E , approaching the LDA values of E V and E C with ultrahigh-κ E .
For the (strained) Au metal, the work function (5.4 eV) level is located at E V + 0.8 eV or E C − 1.1 eV with the LDA band edge levels. They agree with those obtained in our atomistic GW calculations (E V + 0.8 eV or E C − 1.0 eV) (Fig. 1j,k). In the atomistic GW calculations, the E F is found to be located at E V + 1.0 eV or E C − 1.3 eV in the Au supported structure and at E V + 0.9 eV or E C − 1.2 eV in the Au sandwiched structure (Fig. 1p,q). If we use the GW band edge levels without EDS (κ E = 1), the Au work function level is located at E V + 1.2 eV or E C − 1.6 eV, which is far from the atomistic GW calculations (E V + 0.9 eV or E C − 1.2 eV, as shown in Fig. 1q). When we use the GW band edge levels with ultrahigh-κ E (says κ E = 30), the Au work function level of E V + 0.8 eV or E C − 1.1 eV is close to the atomistic GW results (E V + 0.9 eV or E C − 1.2 eV) (Fig. 1q). Note that within the GW, the Au 5d states are found to emerge inside the bandgap of ML MoS 2 (Fig. 1p,q), which is in contrast to the LDA results (Fig. 1j,k). Although both the LDA and GW results indicate that the Au work function level is located deep inside the bandgap of ML MoS 2 , and some experiments have shown that Au produces n-type Schottky contacts to  (refs 24,25). Au has been typically used as a n-type contact metal to MoS 2 (refs 1,26,27), which may be due to the Fermi level pinning at the interface near to the E C of MoS 2 (refs 18,27-29).
In our calculations, the Au(111) slab is strained by + 9.6% (tensile) to match in lattice to the 1 × 1 ML MoS 2 . In order to match the lattice constant of Au with the MoS 2 within a few % of strain, a larger supercell, for example 2 × 2 Au(111) slab and √ 3 × √ 3R30° MoS 2 with 5.0% of compressive strain, is required with the number of atoms exceeding 33, with which the GW calculation is computationally demanding. One effect of the tensile strain on the Au(111) slab is lowering the Fermi level with smaller band dispersions, and the work function of the strained Au(111) is 5.4 eV, while that of the unstrained Au(111) is 5.1 eV, as indicated in Fig. 2b. In both cases, the Fermi level crosses the Au(111) 6s-bands and is located inside the MoS 2 bandgap. Both are metallic with the 6s Fermi electrons, having the infinite static dielectric constants. Since the quasi-particle bandgap dependence on the dielectric constant is very weak with the large static dielectric constant, less than 0.1 eV when κ E > 10 (see Fig. 2a), the quasi-particle bandgap of the ML MoS 2 is not expected to be significantly altered by the applied strain on the Au(111). We check another metallic slab, Ag(111), to test the quasi-particle bandgap of ML MoS 2 with metallic screening. The Ag(111) slab is strained by 9.4% (tensile) to match with the 1 × 1 ML MoS 2 . The calculated band structures of the ML MoS 2 with Ag(111) in the sandwich structure are shown in Fig. 1l (LDA) and Fig. 1r (GW). The obtained GW bandgap of the ML MoS 2 is 2.11 eV with the metallic Ag(111) environment, which is nearly the same to the GW bandgap (2.09 eV) of the ML MoS 2 with the metallic Au(111). Figure 3 illustrates the effect of EDS on the bandgap of ML MoS 2 in various environments. In an isolated freestanding ML MoS 2 , the strong Coulomb interaction (through the free space) between electrons makes the QP renormalization of electrons huge leading to a large bandgap (Fig. 3a). However, with a dielectric environment, the Coulomb interaction between electrons in the ML MoS 2 is additionally screened, and the QP bandgap of the ML MoS 2 is correspondingly reduced (Fig. 3b representatively for the both-side HfO 2 dielectrics). In a typical top-gate FET device structure, a channel is located between a substrate and a gate dielectric. The channel is also connected with metallic contacts in the source and drain regions. Such typical device structure is shown in Fig. 3c with the metallic contact regions (representatively both-side Au) at both ends and the channel region with nearby dielectrics (representatively both-side HfO 2 ) in the central region.  0.2 eV from the source to the channel, and a hole transport barrier (Δ E V ) of 0.1 eV, even though the ML MoS 2 channel itself is homogeneous atomically, due to the different EDS effects (see bottom in Fig. 3c). This type of environmentally induced barriers has not been known, and can be important in low-dimensional electronic devices. Especially, they can play a role of suppressing off-state leakage current in FET by blocking the minority carrier transport.

Bandgaps of ML MoS 2 in FET.
If the κ E of the gate dielectric can be controlled externally, both the electron and hole transport barriers can be controlled, as schematically illustrated in Fig. 3d. In this tunable bandgap FET, both the hole and electron transport barriers can be controlled by the gate, which is the main difference from the conventional FET that controls only the band offsets between the source/drain and the channel (and thus the transport barrier of only one type of carrier). Such tunable bandgap is highly desirable to optimize and design a novel electronic device, and bilayer graphene has been utilized to realize the tunable bandgap FET, in which the bandgap is varied by an external electric field to break the inversion symmetry of the bilayer graphene 30 . While the bilayer graphene system is highly restrictive in symmetry, the bandgap tuning by EDS can be applied generally to low dimensional semiconductors without any symmetry requirements. A challenge to realize the tunable bandgap FET by EDS is on controlling the environmental (gate) dielectric constant (κ E ) externally. Recently, the electrically controlled dielectric materials utilizing ferroelectric properties have been suggested 31 , and such ferroelectric materials [32][33][34][35] can be promising as gate dielectric materials in the EDS-based tunable bandgap FET. Polar instability at the phase transition 36 can be also utilized to vary the dielectric constant, and distance control from the gate dielectric to the 2D semiconductor channel can be another way to control the EDS externally, which can function as an electromechanical device.

Conclusions
Electronic bandgap of a 2D semiconductor, ML MoS 2 , depends on the nearby dielectric environments, through the screened QP renormalization of electrons in the ML MoS 2 . The bandgap tends to reduce with increasing the environmental dielectric constant. In a ML MoS 2 FET, the vicinity of metallic contacts gives smaller bandgap than that of the gate dielectric, and there should be valence and conduction band offsets between the regions. The band offsets can play a role of barriers to electron and hole transports through the channel. Utilizing the environment-dependent property of the bandgap, a tunable bandgap FET is suggested, which operates with the bandgap controlled by an external source to control the electron and hole transport barriers.

Methods
Density-Functional Theory and GW Calculations. The mean-field density-functional theory (DFT) calculations were performed with the Quantum-Espresso code with the local density approximation (LDA) 37 . The kinetic energy cutoff for the plane-wave basis expansion of the wave-functions was 40 Ry. The 24 × 24 × 1 k-point sampling in the hexagonal Brillouin zone (BZ) of the 1 × 1 ML MoS 2 was used. The GW calculations were performed with the BerkeleyGW code 38,39 . The kinetic energy cutoff for the plane-wave basis expansion of the dielectric matrix was 6 Ry. The number of conduction bands used in the calculations of the static irreducible random phase approximation (RPA) polarizability and the Coulomb-hole self-energy was 100 for the free-standing 1 × 1 ML MoS 2 . For the 1 × 1 ML MoS 2 with the HfO 2 and Au environments, the number of conduction bands used was around 700. The limited number of conduction bands can affect the GW eigenvalues at the M point in the hexagonal BZ 40 , but those at the K point converge fast with respect to the number of conduction bands. When we used more number of conduction bands, and the GW bandgaps at K were not significantly affected. The generalized plasmon pole (GPP) approximation was used for the frequency dependence of the dielectric matrix. We applied slab truncation scheme for the Coulomb interaction to minimize the supercell interaction for the free-standing ML MoS 2 and the ML MoS 2 with the HfO 2 environments.
Model Atomic Structures. The 1 × 1 hexagonal unit cell for the ML MoS 2 was used. The lattice constant was fixed to 3.16 Å, which is the LDA optimized value for the free-standing ML MoS 2 . With the 1 × 1 in-plane periodicity of the ML MoS 2 , the HfO 2 was modeled by the O-terminated 1 × 1 HfO 2 (111) slab, and the Au was modeled by the 1 × 1 Au(111) slab. With these interface structures, the HfO 2 and Au slabs are hydrostatically strained by − 8.5% (compressive) and + 9.6% (tensile), respectively, to match their lattice constants to that of the ML MoS 2 (3.16 Å). The atomistic environments are only model systems that represent dielectric and metallic environments. Six Hf atomic layers were used for the HfO 2 dielectric slab, and six Au atomic layers were used for the Au metallic slab, as shown in Fig. 1b-e in the main article. For all the interfaces, we used the interface spacing of 2.975 Å (between the atomic layers), which was chosen arbitrary as the same to the interlayer spacing between the MoS 2 layers in bulk 2H-MoS 2 . The vacuum thickness of about 12 Å in the supercell was used. The ideal (as-cleaved) atomic structures for the HfO 2 and Au slabs were used to see only the electronic EDS effect.