Work Function Tuning in Two-Dimensional MoS2 Field-Effect-Transistors with Graphene and Titanium Source-Drain Contacts

Based on the first principles calculation, we investigate the electronic band structures of graphene-MoS2 and Ti-MoS2 heterojunctions under gate-voltages. By simultaneous control of external electric fields and carrier charging concentrations, we show that the graphene’s Dirac point position inside the MoS2 bandgap is easily modulated with respect to the co-varying Fermi level, while keeping the graphene’s linear band structure around the Dirac point. The easy modulation of graphene bands is not confined to the special cases where the conduction-band-minimum point of MoS2 and the Dirac point of graphene are matched up in reciprocal space, but is generalized to their dislocated cases. This flexibility caused by the strong decoupling between graphene and MoS2 bands enhances the gate-controlled switching performance in MoS2-graphene hybrid stacking-device.

In FETs, an ohmic contact is usually desired for easy current flows between semiconductor and S/D electrodes. To fulfill this requirement, relevant contact searching is primarily focused on avoiding materials with the rectifying responses to forward and backward biases. Reported S/D electrodes on MoS 2 channel are pure metals such as Ti 5,6,8,[40][41][42][43][44] , Au 40,[45][46][47][48][49][50] , and Sc 42,43 , and the most popular material amongst them tends to become Au/Ti deposition followed by a post-annealing process 44 . This choice follows from a prior estimation based on the Schottky-Mott rule [51][52][53] , which states that the potential barrier height at the interface is given by the energy difference between the semiconductor electron affinity and the metal work function. Because the electron affinity of MoS 2 is reported as 4.0 eV and the work functions of graphene and Ti are known as 4.5 and 4.3 eV, respectively, MoS 2 -Ti contact is expected to form a smaller Schottky barrier, leading to a more ohmic behavior. Contrary to this expectation, however, graphene-contacted MoS 2 shows higher on-current and lower off-state behaviors, displaying an on/off ratio over ~7.5 × 10 6 29 . To understand this seemingly anomalous feature, we first constructed graphene/MoS 2 and Ti/MoS 2 stacking structures, and investigated their electronic band structures by using the first-principles density functional method 54 . In this work, we explicitly show the absence of graphene-induced gap-states, which otherwise would cause a Fermi level (E F ) pinning 55 within the bandgap. We then attribute the sensitive variations of E F and work function under gate-voltages to the strong decoupling between graphene and MoS 2 bands in their heterojunction.

Computational Methods
To investigate the gate-voltage tuning effects on the graphene(S/D)-MoS 2 and Ti(S/D)-MoS 2 FETs, we have performed self-consistent density functional calculations using the SIESTA code 54 . Exchange and correlation were treated with the local density approximation (LDA) 56 . Core electrons were replaced by standard norm-conserving pseudo-potentials 57 as transformed into fully nonlocal Kleinman-Bylander form 58 . Valence states were described by numerical atomic orbitals of double-ζ plus polarization basis-set to account for the deformation density induced by bond formations. Electronic wavefunctions and charge densities were projected onto a real space grid with an equivalent energy cutoff of 500 Ry. We used 24 × 24 k-grid sampling in the full Brillouin zone (BZ) for the slab systems. To describe naturally n-doped MoS 2 , we performed the virtual crystal approximation (VCA) by replacing 1.3 × 10 −4 atomic % of sulfur atoms with chlorine atoms. This corresponds to 5.0 × 10 16 cm −3 electron doping in bulk MoS 2 as consistent with experimental observations 40,[59][60][61][62] . After the VCA, additional electron charging effects under gate-voltages were simulated by direct electron addition or subtraction methods. For the graphene stacking on MoS 2 , we considered two different heterostructures with commensurability conditions: (i) × 7 7 graphene sheet (6.51 Å) was stacked on 2 × 2 MoS 2 monolayer (6.32 Å) which includes 2.9% lattice contraction of graphene. (ii) 4 × 4 graphene sheet (9.84 Å) was adjusted to 3 × 3 MoS 2 monolayer (9.48 Å),  Table 1. Work function variations in positive and negative gate-voltages (V G ) in graphene/MoS 2 . V G variations are simulated with the external electric field of E = ± 0.01 V/Å and the electron charging concentration of ± 8.7 × 10 12 cm −2 , which corresponds to ± 0.03 electron charging in the × 7 7 -graphene/2 × 2-MoS 2 supercell. E C , E D , and E F represent the CBM energy of MoS 2 , the Dirac point energy of graphene, and the Fermi level of the whole system, respectively. resulting in 3.7% lattice contraction of graphene. For the Ti stacking on MoS 2 , 4-layered Ti lattice (a = 2.95 Å) was placed on 1 × 1 MoS 2 monolayer (3.16 Å) which includes 6.7% lattice expansion of Ti.

Results and Discussion
In FETs, gate-voltage (V G ) always puts into effects of electron charging and electric field variation in a simultaneous way. Once the system reaches a given state of on-current at a specific positive gate-voltage, both direction and strength of current flows are controlled by the drain-voltage (V D ). Figure 1(f) and (g) show the approaching steps from an ungated case to an on-current stage in × 7 7-graphene/2 × 2-MoS 2 heterojuncion. To describe a weak positive (negative) gate-voltage, we have set the external electric field as E = ± 0.01 V/Å and the electron charging concentration as n c = ± 8.7 × 10 12 cm −2 . Here the positive and the negative sign represent the positive and the negative V G , respectively. To describe stronger gate-voltages, we have increased the electric field and the electron charging concentration by 10 times.
In graphene/MoS 2 systems, the Schottky barrier height is defined as the energy difference between the conduction band minimum (CBM) energy (E C ) of MoS 2 in the graphene/MoS 2 system and the Fermi level (E F ) of the whole system. As seen in Table 1, the Schottky barrier (Δ E = E C − E F ) in graphene/MoS 2 sensitively responds to gate-voltages by decreasing from 0.37 eV (ungated) to 0.03 eV (on-current) in the positive V G , and by increasing from 0.37 eV (ungated) to 0.74 eV (off-state) in the negative V G . Easily reducible Schottky barrier in positive V G means easily diminishable contact resistance 33-37 between graphene and MoS 2 , which induces large current flows in drain-voltages. In contrast, a large Schottky barrier in negative V G reduces the unprofitable current leakage in off-states and contributes to the high on/off ratio in itself. The working principles of V D and V G are similar to each other, except that V D additionally controls the chemical potential of graphene with respect to that of MoS 2 . Thus, the high sensitivity to gate-voltages involves a similar sensitivity to drain-voltages, resulting in a fast slope increase in the I D -V D curve and a corresponding high on/off ratio. Meanwhile, as evaluated in Table 1, the system response to gate-voltages is not completely symmetric due to the shape-change of the Dirac cone caused by the electric field. In the positive V G , the Fermi velocity near the Dirac point decreases from 8.  gate-voltages (E = ± 0.1 V/Å and n c = ± 8.7 × 10 13 cm −2 ). Intriguingly, Fig. 1(j) shows a theoretical possibility that the conduction type of real systems may transform from electrons to holes under a strong enough negative V G .
The band structures shown in Fig. 1(d-j) provide two key features on the flexibility of graphene bands: (i) We clearly see the absence of graphene-induced gap-states inside the MoS 2 bandgap, which would have a flat-band form in k-space and absorb electrons from metals on stacking. This failure in forming the gap-states unlocks the Fermi level pinning and preferably makes the graphene bands flexible. As shown in Fig. 2, the absence of graphene-induced gap-states is confirmed by the clean density of states (DOS) without extra peaks near the Fermi level. (ii) The linear shape of graphene bands around the Dirac point remains intact and the linearity extends over one electron-volt from the Dirac point with a negligible band mixing between graphene and MoS 2 . This weak interaction is evidenced from the relatively small binding energy of − 0.59 eV and the large equilibrium distance of ~3.23 Å between MoS 2 and graphene 24,25 . In another way, positive electric fields play a role of shifting up the Dirac point toward a higher energy, so if we increase the electric field over ~0.5 V/Å without the electron charging, the Dirac point moves up above the CBM, making the conical vicinity of the Dirac point empty, which then forces E F to be located above the CBM to compensate this charge depletion. Even in this extreme electric field, the linearity is retained around the Dirac point, indicating the strong decoupling between graphene and MoS 2 bands.
Thus far, all analyses were performed for the case with the CBM of MoS 2 and the Dirac point of graphene coincided at the special k-point K. To extend the validity to more general cases where the CBM point and the Dirac In this new stacking, the CBM point is moved to Γ whereas the Dirac point is still located at K as shown in Fig. 3. We see that all features previously analyzed in Fig. 1 are similarly exhibited in Fig. 3. The band lineups in 4 × 4-graphene/3 × 3-MoS 2 reproduce sensitive and asymmetric responses to gate-voltages. Also, we see that graphene-induced gap-states do not appear, and the linear dispersion around the Dirac point remains intact on stacking and in onward applications of gate-voltages, leading to the strong decoupling between graphene and MoS 2 bands. Thus, Fig. 3 confirms that the flexibility of graphene bands is not confined to the special cases where the CBM and Dirac points are matched up in reciprocal space, but is generalized to their dislocated cases.
Before going to Ti stacking on MoS 2 , we here analyze the position movement of special k-points from the 1 × 1 hexagonal unit cell to a general hexagonal supercell. Figure 4 and =b y are the lattice vectors of 1 × 1 hexagonal unit cell (black lines). In Fig. 4(b), we can see how the special k-points of general supercells (blues lines) are mapped onto those of 1 × 1 unit cell (black lines) in reciprocal space. In k-space, the length of supercell lattice vector is contracted by the factor, The location of special k-point K for a general hexagonal supercell is determined by the conditions, In striking contrast to graphene/MoS 2 , the band offsets in Ti/MoS 2 show very insensitive behaviors to gate-voltages. To see the differences between graphene/MoS 2 and Ti/MoS 2 , we constructed three different Ti/ MoS 2 stacking configurations (T1, T2, and T3) as shown in Fig. 5(a-c). Among them, T1 configuration has the lowest total energy, but all the band structures of T1, T2, and T3 configurations are very similar to one another.   changes under V G (E = ± 0.01 V/Å and n c = ± 8.7 × 10 12 cm −2 ). Even when we increase the gate-voltages by 10 times (E = ± 0.1 V/Å and n ch = ± 8.7 × 10 13 cm −2 ), the relative positions of Ti bands are not significantly altered from those in Fig. 5(g) and (h), implying that the intrinsic small Schottky barrier at the interface remains nearly constant without diminishing or rising under the strong gate-voltage. This insensitivity to the gate-voltage is also found in other ohmic metals such as Au (See Supplementary Information).
In conclusion, by simultaneous control of external electric fields and electron charging concentrations, we have compared the band offsets in graphene (S/D)-MoS 2 with those in Ti(S/D)-MoS 2 . Under gate-voltages, the Dirac point positioning inside the MoS 2 bandgap is easily modulated with respect to the co-varying Fermi level, while the linearity of graphene bands remains intact around the Dirac point. Band lineups in graphene/MoS 2 explicitly confirms the absence of graphene-induced gap-states, which is thought to prevent the Fermi level pinning and preferably make graphene bands flexible. In contrast to Ti-MoS 2 , graphene-MoS 2 interactions are very weak as evidenced from the small binding energy of − 0.59 eV and the large equilibrium distance between graphene and MoS 2 ~3.23 Å, which is more than twice the equilibrium distance ~1.60 Å between Ti and MoS 2 . In effect, the strong decoupling between graphene and MoS 2 bands causes a high sensitivity to gate-voltages. We respectively analyzed the fast increasing on-current and the steadily maintained (or lowered) off-current states, which originate from such a sensitive work function tuning of graphene under positive and negative V G ; such work function tuning, in sum, leads to the high on/off ratio in graphene/MoS 2 .