## 3 Results and Discussion
Figure 5: The comparison of the on-state current of MoS\({}_{2}\) devices with 4 different metallic electrodes at 3 different types of drain-source voltage of **(a)** 10 V; **(b)** 20 V; **(c)** 30 V.  
drain-source current (I\({}_{DS}\)) is the product of the current density (J) and the average contact area S of the electrode in the device, as shown in Equation (3). We substituted it into Equation (2) to obtain Equation (4). SBHs corresponding to different gate voltages between metals and semiconductors can be calculated with the known constant and I\({}_{DS}\) in the device transfer characteristic curve at V\({}_{DS}\) = 10 V.  
\[J=A*T^{2}\exp(-\frac{q\Phi_{SB}}{k_{0}T})\exp(\frac{qV_{DS}}{k_{0}T}) \tag{2}\]  
\[I_{ds}=J*S \tag{3}\]  
\[\Phi_{SB}=-\ln\left(\frac{I_{DS}}{A*T^{2}*S}\right)*\frac{k_{0}T}{q}+V_{DS} \tag{4}\]  
The effective barrier height (\(\Phi_{ESB}\)), defined as the SBH when Vg equals to flat-band voltage (VFB), can be represented as the longitudinal coordinate value corresponding to an intersection point of the tangent of the descending position and the curve, as shown in Figure 6. It can be seen that \(\Phi_{ESB}\) values of the four types of metal-semiconductor contacts ranged from 10.448 to 10.606 eV with a difference of no more than 0.16 eV. In ideal metal-semiconductor contact, SBH can be well predicted by the Schottky-Mott rule, as shown in Equations (5) and (6) [20], where \(\Phi_{M}\) is the work function of metals; X\({}_{\rm S}\) and I\({}_{\rm S}\) are the electron affinity and ionization potential of semiconductors; \(\Phi_{SB,n}\) and \(\Phi_{SB,p}\) are the SBH of electrons and holes, respectively. The Schottky-Mott rule shows that \(\Phi_{SB}\) is linearly related to the work function of metals, and its relationship curve has a uniform slope, as shown in Equation (7). In Equation (7), if \(S=1\), the metal-semiconductor system conforms to Schottky-Mott rule. However, the calculation based on the experimental data in this paper shows that there is no linear correlation between \(\Phi_{SB}\) and metal work function for \(S=0.11\) in the four metal material systems as shown in Figure 7a. Theoretically, the above results are due to the effect of Fermi-level pinning (FLP). Meanwhile, the results we obtained were concluded from the statistical data based on a certain number of devices that meant the experimental results have high repeatability. Moreover, we present another group of four devices with the calculated \(S\) value of 0.019 in Figure 7b. This result is in good accordance to the former group. We suppose the main causes of FLP are the surface states of semiconductors [20,21] formed by crystal defects [20,22, 23] such as interface reconstruction. If the surface density of states of MoS\({}_{2}\) is very large, the metal semiconductor system will screen the effect of metal contact on the barrier. Here, the barrier height at the metal-semiconductor interface is mainly determined by the surface properties of MoS\({}_{2}\), but not the work function of the metal. In general, due to the difference in the surface density of states, a part of the contact potential difference will fall in the semiconductor with metal-semiconductor contact, so the work function of metal may still impact on the surface barrier, but to a small extent. This is in accordance with our calculated results for the barrier heights of the four devices. \[\Phi_{SB,n}=\Phi_{M}-X_{s} \tag{5}\]  
\[\Phi_{SB,p}=I_{\rm S}-\Phi_{M} \tag{6}\]  
\[S=\left|\frac{d\Phi_{SB}}{d\Phi_{M}}\right| \tag{7}\]