Electric-field-induced strong enhancement of electroluminescence in multilayer molybdenum disulfide

The layered transition metal dichalcogenides have attracted considerable interest for their unique electronic and optical properties. While the monolayer MoS2 exhibits a direct bandgap, the multilayer MoS2 is an indirect bandgap semiconductor and generally optically inactive. Here we report electric-field-induced strong electroluminescence in multilayer MoS2. We show that GaN–Al2O3–MoS2 and GaN–Al2O3–MoS2–Al2O3-graphene vertical heterojunctions can be created with excellent rectification behaviour. Electroluminescence studies demonstrate prominent direct bandgap excitonic emission in multilayer MoS2 over the entire vertical junction area. Importantly, the electroluminescence efficiency observed in multilayer MoS2 is comparable to or higher than that in monolayers. This strong electroluminescence can be attributed to electric-field-induced carrier redistribution from the lowest energy points (indirect bandgap) to higher energy points (direct bandgap) in k-space. The electric-field-induced electroluminescence is general for other layered materials including WSe2 and can open up a new pathway towards transition metal dichalcogenide-based optoelectronic devices.

where Furthermore, we can obtain valley population fraction for electrons (n 2 /(n 1 +n 2 )) and for holes p 2 /(p 1 +p 2 ), as shown in Figure 3  Supplementary Figure 6 displays the calculated electron and hole temperature.
As the applied electric field increases, the electron and hole temperature increases monotonously, which gives rise to the redistribution of electrons and holes among different energy valleys. Finally, the EL efficiency η can be derived from the electron population fraction n 2 /(n 1 +n 2 ), hole population fraction p 2 /(p 1 +p 2 ) and radiative recombination rate B (T) as where T is the temperature and I inj is the injection current. Here we assume the injected electron equals to the hole density, that is n 1 +n 2 = p 1 +p 2 = I inj /2q. Since the thickness of the MoS 2 flake is thick enough to be considered as bulk, the radiative recombination rate can be expressed as a function of the temperature as 8 13 Here B(300) is the radiative recombination rate at 300 K and we take it as a constant for all flakes we investigated in our calculations. Under the thermal equilibrium condition, the electron and hole temperature is the same as environment temperature. Here under the applied electric field, the electron and hole temperature is greatly increased and much greater than the temperature of the environment. Therefore, we used the average temperature of electron and hole temperature in the calculation. Based on Supplementary Equation (7), we obtained the electric field dependent EL efficiency ( Fig.   4(h) in the main text).

The evaluation of the out-plane mobility of MoS 2
To evaluate the out-plane mobility of MoS 2 , we have fabricated the vertical devices with electrodes located on the top and bottom surfaces of MoS 2 plates. Under the condition that the contacts can supply adequate carriers, the current density follows Ohm's law at the low applied voltage or low injected current density. With increasing injection current density, the current density would be mainly governed by Mott-Gurney law (space charge limited current) when the injected carrier density exceeds the intrinsic carrier density 9-12 . The total current density J can expressed as 13 By applying Supplementary Equation (9) to fit the J-V curve, we estimated the outplane electron mobility of MoS 2 to be 2.02 × 10 -2 cm 2 /Vs, which is three orders of magnitude smaller than the in-plane mobility 17 , consistent with the reported conductivity measurement [18][19][20][21] .

The evaluation of the applied electric field
The electric field inside MoS 2 flakes are calculated based on the applied bias voltage and thickness of MoS 2 and Al 2 O 3 . We assume that the average electric field is uniformly distributed in the MoS 2 (Supplementary Fig. 8b), the potential drop in MoS 2 is around 30% for the 90 nm MoS 2 flake device.
The dielectric model is reasonable for field calculation at low-injection current regime, and but can not precisely evaluate the voltage distribution and electrical field in MoS 2 at large injected current regime, although it has previously been used to calculate the electric field in the graphene-BN-graphene tunnelling transistors 26 . To take the influence of the injected current into account, we have further developed a new model to calculate the electric field based on the current density and carrier mobility in the space charge limited current region. Since the injected carrier density surpasses the intrinsic carrier density in MoS 2 , the current density is space-charge limited at the high injected current regime 9-12 , as evidenced by the quadratic dependence of current intensity on the applied voltage ( Supplementary Fig. 5a). Under such case, the average electric field avg E in MoS 2 can be calculated using 10 : where J is the current density,  Fig. 5b).

EL extraction efficiency and effective thickness
The EL from inside of the samples has to propagate to the top surface to be Therefore, the total intensity detected is The EL extraction efficiency can be expressed as We have calculated effective layer number and EL extraction efficiency as a function of the actual layer number. The absorption coefficient was taken as 11.5 µm -1 and the thickness of each layer is 0.65 nm. The effective layer number increases first and reaches a stable value around 120 nm even if the actual layer number is much larger.
Based on Supplementary Equation (18), we obtained the calculation results in Figure 5b in the main text.

EL from GaN-Al 2 O 3 -WSe 2 -Al 2 O 3 -graphene heterostructures
We have also created GaN-Al 2 O 3 -WSe 2 -Al 2 O 3 -Graphene heterostructures with various thickness WSe 2 flakes, in which an n-type GaN was used to inject electrons into p-type WSe 2 flakes. The output characteristic of the vertically stacked heterostructures shows the expected rectification behaviour (Supplementary Fig. 9a). The EL spectrum of an 80-layer WSe 2 device is shown in Supplementary Figure 9b, along with the PL spectra of n-GaN and WSe 2 . Close comparison of EL spectrum and PL spectra, we can assign the EL peaks at 808 nm to the exciton A and at 890 nm to indirect bandgap emission of multi-layer WSe 2 28,29 by taking account into the redshift of the EL emission peaks due to the self-heating effect 30 . The emission peak located around 600 nm can be attributed to the defect emission of Si-doped n-GaN 31 . Similar to vertically stacked heterostructures with MoS 2 , stronger EL was also observed in the thicker WSe 2 flake than thinner one ( Supplementary Fig. 9c).