Fundamentals of Low-Resistive 2D-Semiconductor Metal Contacts: An Ab-initio NEGF Study

Metal contacts form one of the main limitations for the introduction of 2D materials in next-generation scaled devices. Through ab-initio simulation techniques, we shed light on the fundamental physics and screen several 2D and 3D top and side contact metals. Our findings highlight that a low semiconducting-metal contact resistance can be achieved. By selecting an appropriate 2D metal, we demonstrate both ohmic or small Schottky barrier top and side contacts. This leads to a contact resistance below 100 Ωμm and good device drive performance with currents in ON state up to 1400 μA/μm, i.e., reduced by a mere 25% compared to a reference with perfect ohmic contacts, provided a sufficiently high doping concentration of 1.8×1013 cm−2 is used. Additionally, we show that this doping concentration can be achieved through electrostatic doping with a gate. Finally, we perform a screening of possible 2D–3D top contacts. Finding an ohmic 2D–3D contact without a Schottky barrier has proven difficult, but it is shown that for the case of intermediate interaction strength and a limited Schottky barrier, contact resistances below 100 Ωμm can be achieved.


Introduction
Transistors made of novel 2D semiconducting materials, 1 i.e., an atomically thin layer of material that does not create strong atomic bonds in the 3rd dimension, such as transition-metal dichalcogenides (TMD) 2,3,4,5 are being actively investigated as future replacement of Si as channel materials. Finding a metal with a low Schottky barrier to achieve a low contact resistance is one of the key challenges to address towards 2D-material CMOS. Two main schemes of contacting are possible: Top-and Side contacts (TC and SC respectively).
TC are typically used in Si CMOS technologies as they enable a larger contact area, hence a lower resistivity. Due to their speci c nature, this poses a special challenge for 2D material channel materials.
The proximity of a 3D metal with strong a nity and binding energy may affect the chemical nature of the underlying 2D material. This typically results in an important density of interfacial traps (DIT) that pins the Fermi level (E F ) at the semiconductor-metal interface. As a result, a high Schottky barrier heigh (SB H ), hence a high contact resistance is achieved. Using a low-binding-energy metal, such as a 2D metal, on the other hand, 2D-metal/2D semiconductor van der Waals (vdW) contacts have been shown to be an interesting option, as they may be free of Fermi-level pinning. 6 In such a contact, the expected SB H should be in accordance with the Schottky-Mott theory, dictated by parameters like the metal work function and the semiconductor electron a nity, and low SB H could be achieved by a proper material selection. Due to limited out-of-plane bonding, injection of carriers into the device with metals deposited on top of the 2D material may be limited by vdW tunneling. SC are not limited by a vdW gap for carrier injection. However, fabrication of SC has proven to be more challenging leading to higher device variation 6 . Additionally, covalent bonding can again give rise to higher SB H , even in the case of 2D-2D interfaces. Using ab-initio techniques, based on advanced Density Functional Theory (DFT) and Non-equilibrium Green's function (NEGF) transport simulations, we perform, here, using our atomistic solver ATOMOS, 7 a thorough DFT-NEGF theoretical study of 2D metal -2D semiconductor contacted transistors to explore and understand their physics and fundamental performance limit. Additionally, we perform an initial screening of 3D metal -2D semiconductor interfaces, focusing on interfaces demonstrating intermediate behavior, i.e., an interaction strength high enough to limit the in uence of the vdW gap, but low enough to limit the Fermilevel pinning.

Results
In this study, we focus on two options for the semiconducting TMD: WS 2 (in its most stable 2H phase) and HfS 2 (in its most stable 1T phase). WS 2 is one of the most studied 2D materials and shows great promise with reasonably high theoretical predictions for both its n-type and p-type mobility. 8 HfS 2 is a relatively less known material, with fewer experimental results, but is predicted to have an exceptionally high mobility and a higher drive current, while maintaining good scalability down to 5 nm gate lengths. 8 We investigate several device con gurations depending on the type of metal-semiconductor contacts. For 2D-2D TC and SC we consider a single independent contact as well as a full transistor con guration. For the full transistor, both a dual-gate (DG) MOSFET and a Dynamically-Doped Field-Effect Transistor (D 2 -FET) device, i.e., an individually back-gated transistor that does not require a spacer and allows for dynamically doping the source and drain extension with its gate (Fig. 1.b and d), 8 are simulated. For 2D-3D contacts, we are limited to a single contact con guration as the signi cant computational cost prohibits a full transistor simulation. All device structures simulated in ATOMOS are shown in Fig. 1. In all transistor simulations, the gate length is L = 14 nm. The gate oxide has a relative permittivity ε R = 15.6, corresponding to HfO 2 , and thickness of 2 nm, resulting in an equivalent oxide thickness EOT = 0.5 nm.
The work function of the metal gate is typically adjusted to shift the threshold voltage and achieve a xed I OFF value at a gate voltage bias V GS = 0 V. The source-drain bias is set at V D = 0.6 V, unless speci ed otherwise. For single contact simulations, the source and drain are ill-de ned and V D and V GS denote respectively the bias applied over the contact and the potential difference between the doping gate, if present, and the metallic part of the system. The metal contacts and source-and drain-(S&D) extensions are surrounded by a low-K spacer oxide with ε R = 4.

2D-2D Top-contact con guration
For HfS 2 , HfTe 2 (1T) was found to be an interesting n-type contact candidate with a low Schottky barrier.
The results for a single contact and for the DG device with chemical doping are summarized in Fig. 2. Figure 2 (a) demonstrates the in uence of the doping concentration on the contact resistance in a single TC con guration. The contact resistances for doping concentrations of 3×10 20 cm -3 and 5×10 20 cm -3 are respectively 90 Ωµm and 50 Ωµm and are largely independent of the bias. These values are comparable with the quantum limit of 20-30 Ωµm at these doping concentrations 9 . When the doping concentration is reduced, the value of the contact resistance rapidly increases. An average value of 370 Ωµm is observed at 1×10 20 cm -3 and R C ~10 kΩµm at 1×10 19 cm -3 . Also the dependency of the contact resistance on the bias increases signi cantly. Figure 2 (b) shows the in uence of the contact overlap length, L C . The contact resistance appears largely independent of L C , implying that injection happens through edge injection despite the low Schottky barrier height. Figure 2 (c) shows the current for a DG-MOSFET in comparison with a device with perfect ohmic contacts and a device with highly-doped HfS 2 second layer regions acting as metallic TC. We nd that the introduction of HfTe 2 TC reduces the current by about 60% compared to the device with perfect ohmic contacts. From the density of states (DOS) of the device in equilibrium without doping, we can extract an estimate of the Schottky barrier height of nSB H = 40 meV. This SB H is further reduced at the source side under operating conditions, owing to the Fermi-level degeneracy in the conduction band induced by the high doping concentration, as can be seen in Fig. 2 (f) and (g). Hence, the reduction in current can mostly be attributed to the vdW gap. This is con rmed by the results on the HfS 2 DG-nMOSFET with highly-doped HfS 2 TC, that mimic ohmic vdW contacts. This reference case does not have a Schottky barrier but demonstrates a similar current reduction. Figure 2  Increasing the doping concentration beyond this value increases I ON , but the bene ts appear less signi cant than for the single contact and the effect saturates at N SD = 5×10 20 cm -3 . I ON shows a peak around L C = 4.5 nm, but little dependency on L C for higher values of L C , a rming that injection happens through edge injection. We thus nd that the conditions under which the transport simulations were performed were close to ideal. Consequently, doping concentration and contact overlap length do not provide a means to signi cantly alleviate the 60% reduction in I ON imposed by the vdW contact.
For WS 2 , nding an adequate n-type TC has proven challenging. Three metallic TMDs were selected as possible TC candidates: WTe 2 (1T'), MoTe 2 (1T') and NbS 2 (2H). The results for a single contact are shown in Fig. 3 (a) and (b). L C was set to 4.5 nm based on our ndings on the HfS 2 -HfTe 2 TC. The contact resistances for n-type doped WS 2 are all extremely high, even for the high doping concentration of N SD = 5×10 20 cm -3 . As a reference, also the contact resistance of a highly-doped WS 2 layer as TC is shown, demonstrating that contact resistances as low as 45 Ωµm could be achieved for n-type WS 2 by nding a vdW metal with the correct work function. For p-type contacts, NbS 2 is found to be an interesting  000, when compared to the reference case with perfect ohmic contacts of Fig. 3 (c). To provide an additional reference, we discuss a TC con guration with highly doped WS 2 for the metal, which is also characterized by a vdW gap but does not have a Schottky barrier. As a result, I ON is only reduced by 25%, complying with the low contact resistance found for these TC. WS 2 -NbS 2 is characterized by a negative value for pSB H and, hence, provides an ohmic p-type TC. Despite this ohmic contact, I ON is reduced by 95% under normal operating conditions of V D = 0.6 V. When V D is lowered to 0.15 V, the reduction is found to be only 50%. An explanation of this phenomenon can be found in the DOS and current spectrum of the device. NbS 2 is a cold metal with no high energy carriers, which is denoted by the gap in the DOS of the left-most and right-most parts of Fig. 3 (g) and (i). At low bias, this has little in uence on the current. However, at high bias, the energy at which carriers are injected at the source is similar to the energy of the drain-side gap. Therefore, carriers cannot be ballistically extracted at the drain side and the current is reduced. A more in-depth discussion is provided in ref. 10. It should be noted that such behavior emerges only in full device simulations when both contacts are made of the same cold metal and not in the simulation of a single contact, or if a more complex device scheme with asymmetric source and drain contacts would be used. 10 This shows the importance of full device simulations, where possible, as the extraction of contact resistances alone can neglect physics important for device performance. It is interesting to note that the contact resistances for the WS 2 -NbS 2 TC are larger than the ones found for the HfS 2 -HfTe 2 TC. However, the reduction of I ON is slightly less severe for the WS 2 device with NbS 2 TC than for the HfS 2 device with HfTe 2 TC. This may be attributed to the better transport property of HfS 2 . This results in a lower channel resistance in serie with R C , and hence a greater in uence of R C despite its lower value. A second thing to note is that HfS 2 with HfTe 2 TC showed a similar reduction in I ON as the HfS 2 reference case with highly doped HfS 2 TC, implying that the contact resistance is mostly the results of the vdW gap. The WS 2 -WS 2 (n++) TC and WS 2 -NbS 2 TC are also characterized by similar vdW gaps. Indeed, the interlayer coupling is found to be almost identical. Despite this, WS 2 -NbS 2 shows a much greater reduction of I ON than WS 2 -WS 2 (n++). A possible explanation can be found in the necessity of k-matching. This means that ballistic transmission through the contact or even the full device, does not only require states at the same energy at injection and extraction, but also requires states at the same k-point. This requirement is not exclusive to cold-metal based devices but can be more relevant for such transistors as the band structure of cold metals often consists of one band at the Fermi level. Hence, at a speci c energy, states are only available at certain k-points. Additionally, in contrast to the cold-metal behavior discussed above which only arises in full devices, the requirement for k-matching is also relevant for single contacts. This explains the larger contact resistance for the WS 2 -NbS 2 TC than for the WS 2 -WS 2 (n++) or HfS 2 -HfTe 2 TC. A more in-depth discussion is provided in the supplementary material.

2D-2D Side-contact con guration
For the 2D-2D SC con guration, we limit ourselves to HfS 2 -HfTe 2 and WS 2 -NbS 2 as these provided low Schottky-barrier heights for the ideal case of vdW contacts. The results for the single contact simulations are shown in Fig. 4 (a) and (b). For both material combinations, the results strongly depend on the doping concentration. HfS 2 -HfTe 2 SC show signi cantly lower contact resistances than the WS 2 -NbS 2 SC. For doping concentrations of respectively 3×10 20 cm -3 and 5×10 20 cm -3 , the contact resistance of the HfS 2 -HfTe 2 SC is largely independent of the bias and has values of respectively 75 Ωµm and 38 Ωµm. For a doping concentration of 1×10 20 cm -3 , both the average value of the contact resistance and its susceptibility to the bias increase signi cantly, with an average value of around 500 Ωµm. Note that for low doping concentrations, the SC has a higher contact resistance than the TC, while for a high doping concentration the SC has a lower contact resistance. For WS 2 -NbS 2 SC, the average value as well as the susceptibility to the bias is large, even in the case of high doping concentrations.

Dynamic doping
The discussion on TC showed that su cient doping concentration is required to allow for tunneling through the vdW gap. A high doping concentration is known to increase the number of available carriers and the electric eld, and hence to promote tunneling. 11 For SC, there is no vdW gap, but there is a signi cant SB H . Su cient doping is required to thin the Schottky barrier. Figure 1 showed how a gate can be used to achieve dynamic doping 8,12 in D 2 -FETs and single contacts. For both 2D-2D TC and 2D-2D SC, HfS 2 -HfTe 2 contacts provided the lowest contact resistance and demonstrated good device performance.  Figure 5 (a) and (c) show how the carrier concentration increases, and hence, the contact resistance decreases, as the gate potential is increased. However, for the single contact case, the carrier concentration depends not on V GS, but on the difference between the gate potential and the potential in the TMD, i.e., V GS -V D . The contact resistance thus depends strongly on the bias even in the case of high carrier concentrations.
Additionally, Fig. 5 (a) and (c) show that carrier concentrations of ~ 3×10 20 cm -3 can be reached. The contact resistances can reach values as low as 50 Ωµm for TC and 55 Ωµm for SC. Note that these values are lower than the respective R C values obtained for the single contacts with chemical doping concentrations of 3×10 20 cm -3 . In addition to providing the required carrier concentration, the doping gate thus lowers the contact resistance through other methods, presumably through creating additional electric elds which are bene cial for tunnelling. The effect appears most pronounced for the TC as the R C value reached are even lower than the values found for N SD = 5×10 20 cm -3 . Fig. S2 in the supplementary material shows that these carrier concentrations and contact resistances are achieved for a bias of V GS -V D = 0.9 V. For a moderate bias of V GS -V D = 0.6 V, we nd a carrier concentration of ~ 1.8×10 20 cm -3 and contact resistances of 105 Ωµm and 115 Ωµm for respectively TC and SC. Figure 5 (b) and (d) demonstrate the importance of the overlap length of the doping gate, ΔL. For both SC and TC, the contact resistance deteriorates when the doping gate does not fully reach the metal contact, i.e., ΔL < 0. For the TC, it is found that the doping gate best extends beyond the metal, with R C decreasing up to ΔL = 2.5 nm. For the SC, the importance of ΔL is less severe, and a small extension below the metal of ΔL = 0.5 nm is found to be su cient. The discussion above showed that for both SC and TC, the contact resistance is very large for such low doping concentrations. Additionally, even with perfect contacts, a source extension with such low doping concentration would also suffer from source starvation and reduced I ON values. 8 For both types of contacts, the D 2 -FET manages to supply the required carrier concentration to lower the contact resistance and restore the current in ON state. It is interesting to note that for the TC, the D 2 -FET also shows signi cantly better performance than the DG-MOSFET for N SD = 3×10 20 cm -3 . However, our results in Fig. 2  cm -3 . The explanation for this discrepancy is linked to the additional contact resistance lowering by the doping gate. Our results in Fig. 5 showed that the doping gate lowers the contact resistance through other methods than supplying the required carrier concentration and that this effect is more pronounced for TC. This explains why the D 2 -FET outperforms the DG-MOSFET for the TC con guration despite a higher doping concentration providing little bene t, and it explains why this is not true for the SC con guration. The reason for the different in uence of the doping gate on TC and SC is linked to the different mechanism limiting the current, i.e., vdW tunneling for the top contact which is more sensitive to electric eld enhancement and Schottky barrier tunneling for the side-contact which is more sensitive to doping though thinning of the SB H .
For TC, the top metal prevents the introduction of a second doping gate. For SC, a top doping gate can be introduced, by using, for instance, the compact doubled-forked (E 2 ) dynamically-doped E 2 D 2 -FET architecture. 12 The device con guration as well as the corresponding contact resistances and device currents are shown in Fig. 7. As expected, the introduction of a second doping gate doubles the carrier concentration to 6×10 20 cm -3 reduces the contact resistance to a minimum value of 25 Ωµm. 2D-3D Top-contact con guration For the 2D-3D TC con guration, we limit ourselves to WS 2 for the TMD and Pt, Ru, Mo, Bi, Sb for the metal. The low melting temperature of Bi (~ 209°C) 13 makes it unsuitable for direct use in fabrication. Therefore, Bi doped with Y and La (YBi and LaBi) are also considered here as alternatives, as their melting temperature is signi cantly increased (2020°C and 1615°C respectively) 13,14 . For 2D-3D systems, transport simulations are characterized by a large computational cost. Combined with the large number of combinations of metal and surface orientation, this makes an initial screening before performing transport simulations indispensable. We consider four parameters for screening: the vdW gap and the binding energy (E B ), giving an indication of the interaction strength, and the n-type and p-type SB H (nSB H and pSB H ). A more thorough discussion is presented in the methods section. The results are shown in  LaBi surface orientations. Only one exception, YBi (10 − 1), with an intermediate interaction strength greater than the WS 2 bilayer, is found. Additionally, it can be seen that, except for a few metal-surface combinations, the estimated Schottky barrier is always several 100 meV's. From the discussion on 2D-2D interfaces, it is known that the combination of the vdW gap in a bilayer and a Schottky barrier of several 100 meV's greatly reduces device performance. Even an ohmic vdW contact introduces a contact resistance that requires a relatively high doping concentration to be mitigated. To limit the additional contact resistance of this vdW gap, the interfaces exhibiting an interaction strength greater than the bilayer may be of interest. However, these strongly interacting interfaces tend to be strongly pinned, resulting in large SB H both for n-type and p-type. Three exceptions were found with adequate values for nSB H and strong to intermediate interaction strength: YBi (10 − 1) and LaBi and YBi (111) terminated on Y/La. Of these three interfaces, YBi (111) is predicted to be ohmic with a negative nSB H . However, YBi (111) shows a strongly corrugated structure, destroying the 2D nature of WS 2 , as shown in the supplementary material. One of the consequences is a severe reduction of the band gap to 1.4 eV and a very strong sensitivity of the SB H to the local strain level and microstructure, which is not desirable in practice, especially for edge-dominated injection, as also discussed just below. We therefore restricted the transport simulations to YBi (10 − 1) and LaBi (111) with respectively nSB H = 190 meV and nSB H = 15 meV. The DOS obtained from the transport simulations provides a second way to extract an estimate for the Schottky barrier height, resulting in respectively nSB H = 260 meV and nSB H = 540 meV. The difference between the estimates extracted from the DFT screening and from the transport simulation is explained as follows. As discussed in further detail in the methods sections, the pure TMD parts of the transport simulation use matrix elements extracted from a pure TMD DFT simulation with relaxed atomic positions. This is done to remove any effect of corrugation due to the metal in the parts that do not have any metal. However, the relaxation process changes the band alignment and, hence, changes the Schottky barrier height. The second estimate extracted from the DOS of the transport simulation corresponds to the Schottky barrier between the metal and the relaxed structure that is not under it, while the rst estimate extracted during DFT screening corresponds to a Schottky barrier value between the metal and the corrugated TMD below the metal. For edge dominated-injection, as encountered here, the second and higher barrier is probably the most relevant. Figure 9  shows that the contact resistance is largely independent of the contact overlap length, implying edge injection. However, Fig. 9 (a) indicates that the dependency on the doping concentration is signi cant.
The contact resistances for N SD = 5×10 20 cm -3 and N SD = 3×10 20 cm -3 , are respectively R C = 50 Ωµm and R C = 95 Ωµm. Doping concentrations lower than N SD = 3×10 20 cm -3 result in severe increases of the value of R C as well as its susceptibility to the bias. As references, contact resistances for a monolayer of WS 2 with pure Schottky-barrier contacts of respectively 190 meV and 260 meV, as well as a WS 2 -bilayer vdWlimited but ohmic contact as a function of doping concentration are shown in Fig. 9   , su cient to thin the Schottky barrier, the contact resistance is found to be 40 Ωµm, i.e., slightly lower than the YBi TC. This can be attributed to the stronger interaction strength for the LaBi TC. However, for lower doping concentrations, the contact resistance increases signi cantly, up to several 100 kΩµm for a doping concentration of N SD = 1×10 20 cm -3 . This is an indication that, despite the low Schottky barrier estimate between metal and the corrugated TMD underneath the metal, there is a signi cant Schottky barrier impeding the current when insu cient doping is provided. This Schottky barrier is present for edge injection between the metal and the relaxed free-standing TMD. A more in-depth discussion of the different types of TMD in the 2D-3D TC simulation is provided in the methods section.

Discussion
In this study, we show that low contact resistances can be achieved for transistors based on 2D materials. More speci cally, we nd that the 2D metal HfTe 2 can provide low n-type contact resistances for HfS 2 devices, both in a top contact and a side contact con guration. For WS 2 we nd that NbS 2 TC can achieve low p-type contact resistances although the cold metal behavior of NbS 2 can degrade device performance depending on the source-drain bias that is applied. For n-type WS 2 , YBi top contacts with the surface corresponding to the (10 − 1) plane and LaBi top contacts with the surface corresponding to a Larich (111) plane are predicted to have low contact resistances. In all cases, a high doping concentration is indispensable to either thin the Schottky barrier or to allow for tunneling through the vdW gap in between the metal and the 2D semiconductor. A minimally required doping concentration of 3×10 20 cm − 3 is found to achieve contact resistances below 100 Ωµm. Increasing the doping concentration beyond this value further reduces the contact resistance, but only slightly. Going towards lower doping concentrations, on the other hand, causes a rapid and sharp increase of the contact resistance. Exceptions are NbS 2 TC which achieves a contact resistance of 150 Ωµm for a doping concentration of 3×10 20 cm − 3 , due to a lack of k-matching, and LaBi which requires a doping concentration of 5×10 20 cm − 3 to thin its signi cant Schottky barrier. Finally, simulations based on HfS 2 with HfTe 2 contacts show that, by the addition of a doping gate, electrostatic doping can achieve a carrier concentration of 3×10 20 cm − 3 , allowing for the required carrier concentration for a low contact resistance. Additionally, the doping gate appears to further introduce additional contact resistance lowering effects, most likely a eld-enhanced tunneling effect, beyond supplying the required carrier concentration. This phenomenon is most pronounced for top contacts as expected by the vdW tunneling mechanism. For a transistor with side contacts, the addition of a second doping gate is possible, increasing the carrier concentration further to values of 6×10 20 cm − 3 and reaching contact resistances as low as 25 Ωµm.

Methods
The methodology in this study generally consists of two parts. In a rst part, Hamiltonian and/or overlap matrix elements are calculated for a metal-semiconductor interface using DFT. Secondly, these matrix elements are used in our quantum-transport solver, ATOMOS, for the calculation of transport properties and the study of device performance.

DFT-Hamiltonian computation
Three different DFT packages were used for the geometry relaxation as well as the Hamiltonian extraction, depending on the type of interface. For HfS 2 (1T) -HfTe 2 (1T) and WS 2 (2H) -NbS 2 (2H) TC, the top metal layer and bottom semiconducting layer have the same phase as their most stable phase and the TC interface could be achieved with a DFT supercell with small dimensions. For these interfaces, we used QUANTUM ESSPRESSO 15 with the optB86b exchange-correlation functional 16 , ultrasoft pseudopotentials and the Grimme DFT-D3 van der Waals correction. 17 Convergence of the total energy was used for setting the plane-wave cut-off. The convergence criteria were set to 10 − 3 eV for the totalenergy variation between two subsequent iterations and 10 − 3 eV/Å for the forces acting on each ion during geometry relaxation. First a variable cell geometry relaxation was performed on the pure TMD and the metal to extract the lattice constant. Subsequently, both materials were joined in a supercell, straining both equally to achieve commensurate lattices 10 (2.4% and 3.9% for respectively WS 2 -NbS 2 and HfS 2 -HfTe 2 ), after which an additional xed lattice geometry relaxation was performed to relax the atomic positions. Finally, the Bloch wavefunctions were transformed into maximally-localized Wannier functions, typically centered on the ions, using the wannier90 package 18 to extract Hamiltonian matrix elements expressed in a localized orbital basis set as required for transport. 12 For 2D-2D SC interfaces and the 2D-2D TC interfaces with differing phase in the top layer and bottom layer, i.e., WS 2 (2H) -WTe 2 (1T') and WS 2 (2H) -MoTe 2 (1T'), a larger DFT cell is required. For these systems, we used OpenMX 19 with the GGA-PBE exchange-correlation functional, the pseudopotentials provided in ref. 20, the standard basis sets provided in ref. 21 and an energy cutoff of 4081 eV. The convergence criteria were set to 2.7×10 − 5 eV for the self-consistent eld (SCF) step and 5×10 − 3 eV/Å for the forces acting on each ion during geometry relaxation and 2.7×10 − 7 eV for the SCF step during Hamiltonian extraction. The same lattice constants were taken as in the TC con guration for the pure TMD to limit the in uence of the DFT package. Subsequently, a heterojunction was built, again straining both subsystems equally to achieve commensurate lattices. The same values for the strain were obtained as for the 2D-2D TC. However, for the SC the materials were only strained along the orthogonal direction. Along the transport direction, the supercell dimension was relaxed together with the atomic positions in a subsequent relaxation step. The supercell dimension along the transport direction in the SC con guration were set at respectively 132 nm and 114 nm for HfS 2 -HfTe 2 and WS 2 -NbS 2 , i.e., long enough to assume the middle of each TMD part is far enough from the interface to behave bulklike. This is required to obtain matrix elements which can represent bulklike TMD behavior. No further Wannierization process is required as the provided matrix elements are already expressed in a localized orbital basis set.
For 2D-3D TC interfaces, we used CP2K 22 with the GTH-PBE exchange-correlation functional 23 , the DZVP-MOLOPT-SR-GTH basis sets from ref. 24 and the Grimme DFT-D3 van der Waals correction. The energy cutoff was set based on convergence tests on the atomic species in the DFT cell. The convergence criteria were set to 10 − 6 eV for the SCF step and 5.2×10 − 2 eV/Å for the forces acting on each ion during geometry relaxation. The building of TC supercells was performed in a similar fashion as the 2D-2D interfaces, except that here, multiple supercells are possible depending on the surface orientation of the 3D metal at the interface and the thickness of the metal lm. The thickness of the metal lm was set to 10 Å. Surface orientations were selected based on the existence of a supercell with both a limited strain and cell dimensions small enough to achieve realistic simulation times. The upper limit for the strain was set at 1.5% strain in both TMD and metal.
In all cases, spin-orbit coupling was neglected, a vacuum layer of 20 Å was introduced along the out-ofplane direction to prevent spurious interaction of periodic images and convergence of the total energy was used to determine the density of the Monkhorst-Pack k-point grid for the Brillouin-zone integration.
Quantum transport solver ATOMOS enables the simulation of quantum transport in devices using the Green's function formalism 25,26 within the real-and mode-space framework. 8,27 In this work, a real-space description was employed. Transport can be either ballistic or dissipative, including electron-phonon scattering within the self-consistent Born approximation. 28 The Hamiltonian and overlap matrix elements are obtained from DFT as described in the procedure above. For the electron-phonon interaction, we used the DFT-computed parameters provided in ref. 4. The systems in this work are characterized by periodicity in the transversal direction, captured by 10 k-points along ½ the Brillouin zone (the other half is obtained by symmetry) for the 2D-2D systems, and 5 k-points for the relatively larger 2D-3D systems.
DFT-based screening Figure 10 shows the methodology on how several parameters of interest can be extracted from the atomic structure and its corresponding DOS. The binding energy, a measure for the interaction strength, is computed as the difference between the total energy of a DFT simulation of the full stack and the total energies of DFT simulations of the TMD and metallic parts of the stack.
A second estimate for the interaction strength can be found from the size of the vdW gap, which is expected to be lower for stronger interactions. An estimate for the SB H is obtained from the DOS of the  inserted in between the TMD and the metal (see Fig. 10). This vacuum prevents direct interaction between the two surfaces and therefore prevents Fermi-level pinning. The difference between SB H and SB H,id , Δ F, provides an estimate of the Fermi-level pinning. As energy levels of different simulations are compared, we rst shifted all DOS energies to align the lowest energy deep-valence state of the TMD. As these deepvalence states corresponds to core electrons of certain atoms, they are affected only slightly by their surroundings and should provide a good reference to eliminate the arbitrary shift in energies introduced by the DFT simulation.
A third parameter of interest when comparing contacts is the density of interface traps induced by the metal. A DIT estimate can be obtained from the DOS calculation, by integrating the projected DOS (PDOS) of the TMD atoms of the full stack within the bandgap.
Only 2D-3D interfaces which show interesting values for the Schottky barrier height and interaction strength were simulated using the transport solver. For these interfaces, matrix elements were extracted. As in OpenMX, these elements are expressed in a localized orbital basis set and no Wannierization step is required. Due to interactions between metal and TMD in the stack, the atomic structure of the TMD part can be slightly corrugated and the electronic states do not always correspond well with bulk TMD, as was the case for 2D-2D TC. Therefore, the matrix elements for bulk TMD were extracted from a separate DFT simulation containing only the TMD part of the stack. The TMD part was relaxed before extraction, keeping the lattice dimensions xed to preserve a commensurate system.

Author Contributions
A.A. developed the theory, simulation code, and concepts (e.g., the D 2 -FET). R.D. and A.A. performed the 2D-2D metal-semiconductor rst-principle simulations. R.D. and G.P. developed the algorithms used to build the 3D-2D interface models, performed the rst-principle simulations and analyzed the resulting data. R.D. and A.A. performed the transport simulations and the related algorithm optimizations, analyzed the data and wrote the manuscript. M.H, G.P. and A.A. supervised the work. All authors discussed the results and proofread the manuscript.

Data availability
The data that support the ndings of this study are available from the corresponding author upon reasonable request.

Code availability
Access to the code that is used in this study is restricted by imec legal policy.

Competing interests
The authors declare no competing interests.    MOSFET with HfTe 2 SC and TC, as well as the perfect ohmic contact reference case, (d) IV curves for the WS 2 DG-MOSFET with respectively NbS 2 SC and TC, as well as the perfect ohmic contact reference case.

Supplementary Files
This is a list of supplementary les associated with this preprint. Click to download. 2dcontactatomosaryan2022v3SI.docx