### TiS_{2} contact formation with ML MoS_{2}

The “Methods” section describes the density functional theory (DFT) techniques used for this research. The contact formation between TiS_{2}–MoS_{2} leads to two interfaces as seen in Fig. 2d. To confirm this, two geometries are adopted to investigate the contact, which are shown in Fig. 2a, b. A group of TiS_{2}–MoS_{2} junctions are used to test the electron transport from TiS_{2} contact to the ML MoS_{2} sheet at the lateral interface (interface A in Fig. 2d). Another group of FET-like TiS_{2}–MoS_{2} heterostructures (TiS_{2}–MoS_{2} FET-like junctions) are used to simulate the electron transportation from the contact region to semiconducting channel region at the vertical interface (interface B in Fig. 2d).

For TiS_{2}–MoS_{2} FET-like junction, the PLDOS is calculated for revealing the band structure after the contacting of 1–4 layers of TiS_{2} and ML MoS_{2} sheet. The configuration after geometrical optimization (GO) is shown by Fig. 2d. The overlap of 1–4 layers of TiS_{2} and ML MoS_{2} is 2.5 nm. Due to the limitation of computational capability, the ML-MoS_{2} channel is set as 8 nm. To increase the accuracy in the transportation direction, the *k*-point sampling is changed to 10 × 1 × 40. For each of 1–4 layers of TiS_{2} simulations, The MoS_{2} channel doping is varied to understand the type of electrode formation as 5 × 10^{18} cm^{−3} (3.25 × 10^{11} cm^{−2} for ML MoS_{2}) *n*-type doping, 1 × 10^{19} cm^{−3} (6.5 × 10^{11} cm^{−2} for ML MoS_{2}) *n*-type doping, 5 × 10^{19} cm^{−3} (3.25 × 10^{12} cm^{−2} for ML MoS_{2}) *n*-type doping, and 5 × 10^{18} cm^{−3} (3.25 × 10^{11} cm^{−2} for ML MoS_{2}) *p*-type doping. Thus, we obtain a *n*_{i} value of 1.4 × 10^{4} cm^{−3} for ML MoS_{2}. We chose the doping concentration carefully to meet the calculated upper and lower limits: a large doping concentration will lead to degenerate behavior of the channel and a small doping concentration will cause the depletion length to exceed the length of channel creating artificial results. The overlap region between the TiS_{2} and MoS_{2} is undoped. Only the uncovered ML-MoS_{2} channel is doped with the chosen doping concentrations.

To investigate the charge transport and its mechanism through the interface A in Fig. 2d, the calculation of DOS, ED, average binding energy (EBE), and EDP are conducted after the GO.

After simulating the DOS of the optimized TiS_{2}–MoS_{2} structure, the projected partial DOS (PDOS) is shown in Fig. 3. The PDOS of TiS_{2} after making contact with ML MoS_{2} is shown in Fig. 3a–d for one–four layers of TiS_{2}. It shows that, after making a contact with ML MoS_{2}, the TiS_{2} DOS remains the same as the free-standing TiS_{2}. The only difference is that some trap states are added which shown as spikes in Fig. 3a–d. These added states are created by the tiny displacement of the atoms in TiS_{2} layer, which is of the order of 0.01 Å. This indicates that the presence of ML MoS_{2} has a negligible effect on the TiS_{2} band structure. The situation of MoS_{2} is different, as shown in Fig. 3e–i which show the PDOS of ML MoS_{2} after making contact with one–four layers of TiS_{2}. The plot clearly demonstrates that the TiS_{2} contact modifies the MoS_{2} band structure significantly and adds a large amount of *p*-type doping to the ML MoS_{2}. In Fig. 3e–i, compared with the band structure and DOS of intrinsic ML MoS_{2} on the left side, for all the cases of ML to four-layer TiS_{2} contacts, the PDOS of ML MoS_{2} exhibits Fermi level pinning towards valence band. This shift of Fermi level ranges from 0.40 to 0.45 eV. Even though the three-layer and four-layer TiS_{2} forming a contact with ML MoS_{2} show a larger Fermi level shift, the difference in Fermi level shift with change in number of TiS_{2} layers is very small. The energy difference between the valence band and Fermi level is 0.05–0.1 eV. The energy difference between the Fermi level and valence band is defined by

$$varPhi = E_{mathrm{f}} – E_{mathrm{V}},$$

(1)

where (E_{mathrm{f}}) and (E_{mathrm{V}}) are the Fermi level and the valence band top, respectively. Knowing the Fermi level shift and (varPhi), the *p*-type doping concentration can be determined by calculating the effective DOS in the conduction band ((N_{mathrm{c}})) and the valence band ((N_{mathrm{v}})). As there are not too many reported values of *N*_{c} and *N*_{v} reported, we have calculated these to estimate what doping concentration we should use for our electrode simulations for ML MoS_{2}. Hence, the effective density of state values is required only for estimating the doping concentrations of MoS_{2}. After simulating the DOS, the energy difference between the quasi-Fermi level and the valence band can be calculated. These two well-defined equations can be applied to determine (N_{mathrm{c}}) and (N_{mathrm{v}}):

$$N_{mathrm{c}} = nexp left( {frac{{E_{mathrm{c}} – E_{mathrm{F}}}}{{kT}}} right),$$

(2)

$$N_{mathrm{v}} = pexp left( {frac{{E_{mathrm{F}} – E_{mathrm{v}}}}{{kT}}} right),$$

(3)

where *n* and *p* are the electron and hole concentrations, respectively. *k* is Boltzmann constant and *T* is temperature in *K*. From the simulation, (N_{mathrm{c}}) and (N_{mathrm{v}}) for ML MoS_{2} are calculated to be 2 × 10^{19} cm^{−3} (1.38 × 10^{12} cm^{−2}) and 1 × 10^{19} cm^{−3} (6.5 × 10^{12} cm^{−2}), respectively, which are similar to the reported ML and bulk MoS_{2} values^{36,37,38,39}. Using these values, the calculated doping concentration added to the ML MoS_{2} by TiS_{2} contacts with different number of layers ranges from 3.85 × 10^{17} to 2.63 × 10^{18} cm^{−3} (2.50 × 10^{10} to 1.71 × 10^{11} cm^{−2} for ML MoS_{2}). Compared with metal–MoS_{2} contacts^{14,15,16,17,18}, although the ML MoS_{2} is *p*-type doped after forming a contact with TiS_{2}, its bandgap is preserved as shown in Fig. 3e–i; in contrast, the metal contact will metalize the ML MoS_{2} and fill its bandgap with states as shown in Fig. 1a. The TiS_{2}–MoS_{2}(ML) contact makes ML MoS_{2}*p*-type as compared to the *n*-type in graphene–MoS_{2} contact (Fig. 1b).

Thus, the DOS simulation indicates that TiS_{2}–MoS_{2} (ML) contact is a unique contact with less metallization and *p*-type behavior. It is necessary to conduct a comprehensive analysis on its interfacial bonding condition. ED and EBE are two important criteria to evaluate the bonding between TiS_{2} and ML MoS_{2}. The average binding energy (EBE), *E*_{B}, can be defined as

$$E_{mathrm{B}} = left( {E_{mathrm{T}} + E_{mathrm{M}} – E_{{mathrm{T}} – {mathrm{M}}}} right)/N,$$

(4)

where (E_{mathrm{T}}) is the total energy of the free-standing TiS_{2}, (E_{mathrm{M}}) is the total energy of the intrinsic ML MoS_{2}, (E_{{mathrm{T}} – {mathrm{M}}}) is the total energy of the TiS_{2}–MoS_{2} (ML) contact after GO, and *N* can be considered as the number of interfacial sulfur atoms on the MoS_{2} side.

The ED of the TiS_{2}–MoS_{2} junctions are shown by Fig. 4a–d for one–four layers of TiS_{2}. As seen in the plot, there is a clear gap between the TiS_{2} layer and MoS_{2} with no charge distribution in it. The overlap of the electron gas between TiS_{2} and MoS_{2} is very limited, as can be observed from the contour plot. The plot of the projected electron density (ED) in the *y* direction also clearly shows that the ED overlap at the contact interface is very similar to those between TiS_{2} layers, indicating that the bonding between TiS_{2} and MoS_{2} is not much stronger than interfacial bonding within TiS_{2} layers. The buckling distance can also be extracted from both the ED plot and GO. For TiS_{2}–MoS_{2}(ML) junctions, the optimized buckling distance for TiS_{2}–MoS_{2} contacts are: 3.2282 Å for TiS_{2}(ML)–MoS_{2}(ML), 2.8112 Å for TiS_{2}(2L)–MoS_{2}(ML), 2.8069 Å for TiS_{2}(3L)–MoS_{2} (ML), and 2.7381 Å for TiS_{2}(4L)–MoS_{2}(ML). For comparison, the inter-layer distance for TiS_{2} and MoS_{2} is 2.8678 and 2.9754 Å, respectively. It is clear that thicker TiS_{2} has larger attractive force to ML MoS_{2} and leads to smaller inter-layer distance. These ED results reveal that the bonding between the TiS_{2} contact and ML MoS_{2} in TiS_{2}–MoS_{2} (ML) junctions is Van der Waals bonding. Unlike the covalent bond, the delocalization of interfacial electron gas would not exist in these cases.

By using this definition, the EBE for TiS_{2}–MoS_{2}(ML) interface is 0.7426 eV, while this value is 0.7157 eV for the Van der Waals bonds of bilayer MoS_{2}. This result further confirms the previous results and demonstrates that TiS_{2}–MoS_{2} (ML) contact is a Van der Waals contact.

The tunneling barrier at the TiS_{2}–MoS_{2} (ML) interface will determine the charge transport through the interface A. To evaluate the impedance to the current transportation added by the tunneling barrier, the electrostatic difference potential (EDP) of the TiS_{2}–MoS_{2} contact is simulated. As shown in Fig. 4e, the potential between the TiS_{2} and MoS_{2} is considered as the tunneling barrier. The shape of the potential barrier can be estimated as a rectangle. The tunneling probability, from the TiS_{2} to the ML MoS_{2} can be defined as, of carriers tunneling through the barrier, *T*_{B}

$$T_{mathrm{B}} = {mathrm{exp}}left( { – 4{pi }frac{{sqrt {2{m}Delta {V}} }}{{h}}W_{mathrm{B}}} right),$$

(5)

where (Delta V) is the barrier height, which is defined by the length of the rectangle, *h* is the Planck constant, and (W_{mathrm{B}}) is the barrier width, which is defined as the half width of the rectangle. (Delta V) and (W_{mathrm{B}}) can be directly extracted from Fig. 4e: (Delta V) is 0.404620 eV and (W_{mathrm{B}}) is 0.35412 Å. By using these parameters and Eq. (4), we calculate (T_{mathrm{B}}) as 79.4%. A large (T_{mathrm{B}}) indicate small impedance and a higher charge injection. It is very clear that even though the tunneling barrier at the interface A resists the charge transport vertically through the interface, the possibility of tunneling is high for carriers because of a very tiny barrier width (0.35412 Å). For this reason, the charge injection at the lateral interface can be ignored. However, the interfacial tunneling barrier within the TiS_{2} contact may scatter the electrons and holes. This scattering effect can be reduced by using a thinner TiS_{2} film. Thus, we also address scaling down the thickness of TiS_{2} contacts after meeting the stability requirement.

By analyzing the ED, EBE, and GO for configuration for interface A, it can be concluded that TiS_{2} tends to form Van der Waals bond with ML MoS_{2}, with thicker TiS_{2} creating stronger bonding. The DOS in Fig. 3 shows that less metallization is created due to the weak Van der Waals bonding and the bandgap of ML MoS_{2} is preserved. DOS results also show that TiS_{2} will add *p*-type doping to the ML MoS_{2}. The EDP results in Fig. 4 show that the tunneling barrier is small at interface A, which means that the current injection would not be seriously impeded.

To understand the source-to-channel/channel-to-drain working mechanism for the TiS_{2}–MoS_{2} (ML) contacts applied in 2D FET, the PLDOS is adopted to sketch out the framework of the band structure at the interface B of the TiS_{2}–MoS_{2} FET-like junctions. The PLDOS uses a contour plot to map the DOS projected onto the *c*-axis, which is the transportation direction. The band structure can be determined by plotting the boundary between the states-filled region (bright region) and the no-states region (dark region). The extraction of the barrier height and contact type of TiS_{2}–MoS_{2} contact can be achieved by evaluating the PLDOS. Two reasons let us dope the ML MoS_{2}. First, the intrinsic carrier concentration of ML MoS_{2} is small; therefore, it is a good choice to dope the MoS_{2} channel to improve the conductivity. For this reason, it is of great interest to investigate the contact consisting of doped ML MoS_{2}. Also, the channel length is set to 8 nm based on computational considerations, as mentioned earlier. An adequate amount of doping concentration is required to make sure that the depletion length is smaller than the channel length of the simulated configuration. Although the barrier height and contact type for the contact of intrinsic ML MoS_{2} and TiS_{2} cannot be extracted directly from the PLDOS, but the barrier height and contact type of doped cases can still be extracted accurately.

### MoS_{2} doping-dependent contact formation with TiS_{2}

Four different doping concentrations: 5 × 10^{18}, 1 × 10^{19}, and 5 × 10^{19} cm^{−3}*n*-type doping and 5 × 10^{18} cm^{−3}*p*-type doping is tested. The simulation results are shown in Table 1. Figure 5 shows only the results of the TiS_{2}(4L)–MoS_{2}(ML) cases and the variation of the band structures with different doping concentrations.

For *n*-type doping concentrations smaller than (N_{mathrm{c}}), the Schottky barrier height of the contacts are large: 1.03–1.07 eV for 5 × 10^{18} cm^{−3}, and 0.88–0.95 eV for 1 × 10^{19} cm^{−3}, as shown in Table 1 and Fig. 5b, c. Table 1 also clearly shows that for the cases with ML TiS_{2}, ({varPhi }_{p}) of the contact is larger than ({varPhi} _{p}) of 2L–4L cases with the same doping concentration. For the *n*-type doping concentration of 5 × 10^{18} cm^{−3}, ({varPhi}_{p}) is 0.67 eV for ML but 0.21–0.25 eV for 2–4L. For *n*-type doping concentration of 1 × 10^{19} cm^{−3}, ({varPhi}_{p}) is 0.7 eV for ML but 0.40–0.44 eV for 2–4L. By summing ({varPhi}_{n}) and ({varPhi}_{p}), the modified (E_{mathrm{g}}) is 1.70 eV for the ML case doped *n*-type with concentration 5 × 10^{18} cm^{−3}, and for 2–4L cases doped with this concentration, ({E}_{mathrm{g}}) ranges from 1.26 to 1.32 eV. For *n*-type doping of concentration 1 × 10^{19} cm^{−3}, (E_{mathrm{g}}) is 1.65 eV for ML and 1.32 eV for 2–4L.

For a larger *n*-type doping concentration (5 × 10^{19} cm^{−3}), as shown in both Table 1 and Fig. 5a, the Schottky barrier height is 0.30–0.45 eV, which is much smaller than the cases with *n*-type doping concentrations of 5 × 10^{18} and 1 × 10^{19} cm^{−3}. This reduction in barrier height may be brought about by multiple causes. One possible reason is the imaging force created by the larger amount of excess charge, as a result of the larger doping concentration. It is observed that ({varPhi}_{p}) increases compared with lower doping concentrations. ({varPhi}_{p}) is 0.92–0.97 eV for the 1–4L cases. (E_{mathrm{g}}) for this concentration is 1.27–1.37 eV.

The simulation results and mapped band structures in Fig. 5a–c for *n*-type-doped TiS_{2}–MoS_{2}(ML) contacts show that, normally, *n*-type-doped TiS_{2}–MoS_{2}(ML) contacts show a large barrier height, which is around 1.0 eV below a degenerate doping. Even though Fig. 5 shows that a larger doping concentration will reduce the barrier height for *n*-type carriers, it is obvious that, even for a doping concentration very close to degeneration, the Schottky barrier height is still larger for TiS_{2}–MoS_{2}(ML) contacts. The inference from Fig. 5 is that since the majority carriers in these *n*-type-doped contacts are always faced with a large Schottky barrier height, the *n*-type-doped TiS_{2}–ML MoS_{2} Schottky diodes can probably act as high-power switches or Schottky barrier MOSFETs based on tunneling.

Unlike the *n*-type contacts, the *p*-type-doped TiS_{2}–MoS_{2}(ML) contacts shows zero barrier height when the *p*-type doping concentration reaches 5 × 10^{18} cm^{−3}. In Fig. 5d, it is observed that, at this doping concentration, the depletion width vanishes and the band is flat. The Schottky barrier at the interface shows both a small barrier height and a small built-in potential, which indicates the contact is an ohmic contact for ML MoS_{2-}doped *p*-type at a concentration of 5 × 10^{18} cm^{−3}.

For the cases with *p*-type doping concentration of 5 × 10^{18} cm^{−3}, only the ML and 2L cases show a very small barrier height ({varPhi}_{p}) (0.13 eV for ML and 0.05 eV for 2L). For the 3L and 4L cases, there is zero barrier height. ({varPhi}_{n}) is 1.35 eV for ML and 1.18 eV for 2–4L, as shown in Table 1. For the cases with *p*-type doping concentration of 5 × 10^{18} cm^{−3}, the bandgap shrinks to 1.48 eV for ML TiS_{2} and 1.18 eV for 2–4L.

As inferred from Fig. 5d, when the doping is *p*-type with a concentration of 5 × 10^{18} cm^{−3}, the contact is ohmic for *p*-type carriers while the barrier for *n*-type carriers is still large. As mentioned earlier, the doping concentration (P_{mathrm{c}}) was added to the ML MoS_{2} within the contact region ranges from 3.85 × 10^{17} to 2.63 × 10^{18} cm^{−3}. When the *p*-type doping of channel MoS_{2} reaches the value of (P_{mathrm{c}}), the band will become flat, and the contact becomes an ohmic contact. Thus, TiS_{2} can be used as an either an ohmic or Schottky contact depending on the doping of the ML MoS_{2}.