Probing the Electron States and Metal-Insulator Transition Mechanisms in Atomically Thin MoS2 Based on Vertical Heterostructures

The metal-insulator transition (MIT) is one of the remarkable electrical transport properties of atomically thin molybdenum disulphide (MoS2). Although the theory of electron-electron interactions has been used in modeling the MIT phenomena in MoS2, the underlying mechanism and detailed MIT process still remain largely unexplored. Here, we demonstrate that the vertical metal-insulator-semiconductor (MIS) heterostructures built from atomically thin MoS2 (monolayers and multilayers) are ideal capacitor structures for probing the electron states in MoS2. The vertical configuration of MIS heterostructures offers the added advantage of eliminating the influence of large impedance at the band tails and allows the observation of fully excited electron states near the surface of MoS2 over a wide excitation frequency (100 Hz-1 MHz) and temperature range (2 K- 300 K). By combining capacitance and transport measurements, we have observed a percolation-type MIT, driven by density inhomogeneities of electron states, in the vertical heterostructures built from monolayer and multilayer MoS2. In addition, the valence band of thin MoS2 layers and their intrinsic properties such as thickness-dependence screening abilities and band gap widths can be easily accessed and precisely determined through the vertical heterostructures.

3 properties of MoS 2 , the capacitance spectroscopy 14 recently applied to the characterization of MoS 2 FET structures has been demonstrated as one of the most convenient and powerful method for studying the electron states in MoS 2 at room temperature. At low temperatures, however, the information obtained by this technique is limited due to the large impedance near the band edge of MoS 2 . Different from that in graphene quantum capacitors 21-27 , the slow charge carrier mobility in MoS 2 capacitors often leads to incompletely charged states, mainly due to the localization near the band edge. The incompletely charged capacitance confuses the effect of charge traps.
Here, we show an approach to address these problems by introducing a MoS 2 -based vertical metal-insulator-semiconductor-metal (MIS-M) heterostructure suitable for probing electron states using capacitance measurements. Unlike conventional FET structures 14 , our approach eliminates the impedance effects and can directly access the intrinsic characteristics of thin-layer MoS 2 over a wide frequency (100 Hz-1 MHz) and temperature range (2 K-300 K). By combing capacitance and transport measurements, we show that the MIT observed in monolayer and multilayer MoS 2 is consistent with the physical picture of a percolation [28][29][30][31][32][33][34][35]  coated with a SiO 2 thin layer (300 nm). Exfoliated natural crystals of monolayer or multilayer MoS 2 were first transferred onto a BN sheet, serving as an ultra-smooth and disorder-free gate dielectric 37 . A Ti/Au (10 nm/20 nm) local gate sits underneath the BN sheet. The critical step in achieving MIS-M structure is to have the MoS 2 sheet fully covered by a top electrode (Ti/Au: 10 nm/50 nm). The equivalent circuit of this device geometry is shown in Fig. 1c probably benefits from its small MoS  . In fact, the decrease in the optical phonon mode 1 2 g E observed by a Raman spectroscopy study of few-layer and bulk MoS 2 41 is also due to the strong dielectric screening effects.
The valence band of multilayer MoS 2 is also accessed by detecting the inversion layer of holes using low excitation frequencies at sufficiently high temperatures ( Fig. 3a and b). However, the inversion layer is invisible when using high frequencies at low temperatures (T<100 K). This is due to the presence of the Schottky-barrier between the Ti/Au contact and the valence band 7 .

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Holes must form through thermal excitations or minute current leakage into the contacts. This process often requires a long time from ~ms to seconds. In the 12 nm-thick MoS 2 capacitance device, the majority of hole carriers have been relaxed around 20 kHz at 300 K, as confirmed by the phase information of the device, which is defined by , where G is the conductance and V is the excitation voltage. As shown in Fig. 3c Fig. 3d, yielding a band gap around 1.14 eV, which is close to the reported value of 1.2 eV 42 .

Percolation-induced MIT in monolayer and multilayer MoS 2
Similar to the MIT observed in transport measurements [3][4][5][6] , the capacitance data of the 5.9 nmthick MoS 2 device measured at different temperatures ( Fig. 4m) Fig. 6). More evidence is provided by capacitance measurements of monolayer MoS 2 samples (Fig. 5a). In monolayer MoS 2 , the intersections of the capacitance curves showed obvious temperature-dependent characteristics. At temperatures below 100 K, we observed that the cross-over point was stabilized roughly at The electronic transport of MoS 2 suffers from charge impurities 2, 3 and short-range disorders 12,14,19,20 , such as ripples, dislocation and sulphur vacancies. These disorders result in the insulating transport behaviour of MoS 2 in the low carrier density region, where electrons transport through hopping between localized states (Fig. 4j) and can be well described by the variable-rangehopping model 3,12,20 . In the region where sufficient large carrier densities are introduced, metal behaviour is observed 3 . Here, we propose a percolation-type MIT in MoS 2 , driven by density inhomogeneity of electron states [28][29][30][31][32][33][34][35] which describes the systems in which charge carriers are transported through percolating conductive channels in the disorder landscapes due to the poor screening effect at low carrier densities. When carrier density is low enough, conductive paths are efficiently blocked and MIT occurs. MoS 2 has been proven to be such a disordered system, with impurity concentration ranging from 11 -2 10 cm to 13 -2 10 cm , especially for monolayer MoS 2 , which is more vulnerable to ripples and charge impurities 2,3,12,14,19,20 . Thus, the MIT in MoS 2 is in line with the percolation transition theory in which disorder plays an important role. Moreover, our capacitance and transport data, shown below, provide further evidences to this effect. 8 The evolution of concentration and effective thickness of electron states probed by capacitance measurements can explain the observed MIT in transport measurements fairly well and provide details of the percolation transition process. The percolation transition phenomenon is illustrated in Fig. 4j-k. With increasing carrier densities n (by increasing gate voltage), the localized electron states begin to percolate with each other till a conductive channel occurs at a critical density (Fig. 4k). Further increasing carrier densities will lead to sufficient conductive channels spanning the entire system and result in metal-like transport behaviours (Fig. 4l). On the other hand, at the same carrier density, the effective thickness 9 The percolation transition also suggests an increasing transition density at the cross-over point with increasing impurity concentration [30][31][32] . In our MoS 2 samples the transition density was in the range 12 The trap densities in our monolayer and trilayer MoS 2 samples were in the order of 12 -1 -2 10 eV cm (Fig. 5d). The trap densities in monolayer MoS 2 were apparently large, suggesting that monolayer MoS 2 is more sensitive to disorder. In fact, the trap densities in our samples were underestimated because of the limitation of the excitation frequency ranges. At relatively high temperatures (inset of Fig. 5c), the charge traps were easily excited, and the capacitances measured at low and high frequency show no difference.
At the transition point  (Fig. 1a). The thicknesses of MoS 2 and BN flakes were measured by an atomic force microscope (Veeco-Innova).
Capacitance and transport measurements. Capacitance measurements were carried out using an HP Precision 4284A LCR Meter with a sensitivity of ~0.1 fF in a cryogenic system (2 K-300 K). All wires in the measurement circuits were shielded and the p-Si substrates were also grounded to minimize residual capacitance. The residual capacitance in the measurement setup is at the order of 1 fF (see Supplementary Material). Transport measurements were performed in the same cryogenic system using lock-in techniques.

Capacitance measurements on MoS 2 -based FET structures
Based on standard field-effect transistor (FET) structures, the top electrode is partially covered on MoS 2 as shown in Fig. S1a-  The capacitance measured from the partially-covered MoS 2 devices largely depends on temperatures and frequencies as shown in Fig. S1d-g. The capacitance increases with decreasing excitation frequencies. This is consistent with previously reported results measured at 300 K 1 .
The frequency-dependent behavior of the partially-covered devices becomes serious at low temperatures ( Fig. S1f). At low temperatures and higher frequencies, MoS 2 sheets are normally not fully charged. This is attributed to the charge trapping effect or the huge lateral resistance of MoS 2 near the band edge. Hence, some intrinsic characteristics of MoS 2 are smeared, especially at low temperatures.

Extracting the parallel capacitance C p
To accurately determine the charge trap densities, dielectric constant and quantum capacitance of MoS 2 , the parallel capacitance p C shown in Fig. 1c has to be determined first. p C contains two terms: the residual capacitance To determine ex C , we only need to know the capacitance of BN per unit area, which can be simply obtained through measuring the reference capacitor shown in Fig. 1a in the main text. We have measured the thicknesses of MoS 2 and BN sheets by atomic force microscopy (AFM) in order to determine the dielectric constant of MoS 2 (Fig. S3). The thickness of BN is 14 nm for 5.9 nm-MoS 2 device, 6.0 nm for monolayer-MoS 2 device, 14.9 nm for trilayer-MoS 2 device, and 13.5 nm for 12 nm-MoS 2 device as shown in the main text. The extracted dielectric constant of BN is around 3.1, consistent with previous results 2 .

Determining the residual capacitance r C
The residual capacitance r C is accurately determined by a simple method shown in Fig. S4. In the device structure without any overlap between the top electrode and bottom gate (Fig. S4a), As the electric field in MoS 2 is described by dV E dx  , the integral of Eq. (2) yields:  Here, we present an approximate method to satisfactorily extract the quantum capacitance (     100 kHz (e) for different temperatures. f,g, t C at temperature 2 K (f) and 300 K (g) for different excitation frequencies.