Figure 1. Schematic illustration of one of the four n-Si(100)|aTiSi models used in this study. The Si and Ti atoms are colored in yellow and gray, respectively. 

The disordered nature of amorphous metals such as TiSi makes the simulation of their interfaces with silicon complex. Indeed, the degrees of freedom at the interface in terms of local bond lengths, coordination number and composition make the development of an interface model challenging. In an attempt to capture these effects, we investigated the contact resistance of four different Si(100)-aTiSi interface models, in which the Ti and Si concentrations set in contact with the silicon substrate, their local coordination and bond lengths were varied. An atomistic model corresponding to the most stable energetic configuration is illustrated in Figure 1. All models lead to a current versus voltage (I.V.) characteristic curves similar to the one reported in Figure 2a, where a linear evolution of the current versus the applied voltage is obtained up to a bias of ∼0.1 V, after which the systems display a non-ideal linear behavior. The linear regime evolution was then used to extract the intrinsic contact resistance of the modeled interface. For instance, the intrinsic contact resistance of Figure 1 corresponds to 9.5 × 10 −8 .cm 2 for a doping concentration of 1 × 10 20 |e|/cm 3 and reflects the needs for the injected electrons to cross the interface potential set by the tail states 19 present at the metal-semiconductor interface (Figure 2b). Interestingly, this model leads to contact resistances values relatively close to what has been recently measured experimentally by Hao et al., 15 as illustrated in Figure 3a. As expected, the contact resistance evolves linearly with the doping concentration to reach values as low as 1.8 × 10 −10 .cm 2 for an active doping concentration of 1 × 10 22 |e|/cm 3 . These findings are consistent with the recent demonstration of a sub 10 −9 .cm 2 con- tact resistance for PMOS devices. 20 Remarkably, the first-principles simulations also suggest that beyond this value the contact resistance should saturate, even for a completely unrealistic active dopant con- centration of 1 × 10 23 |e|/cm 3 .

The disordered nature of amorphous metals such as TiSi makes the simulation of their interfaces with silicon complex. Indeed, the degrees of freedom at the interface in terms of local bond lengths, coordination number and composition make the development of an interface model challenging. In an attempt to capture these effects, we investigated the contact resistance of four different Si(100)-aTiSi interface models, in which the Ti and Si concentrations set in contact with the silicon substrate, their local coordination and bond lengths were varied. An atomistic model corresponding to the most stable energetic configuration is illustrated in Figure 1. All models lead to a current versus voltage (I.V.) characteristic curves similar to the one reported in Figure 2a, where a linear evolution of the current versus the applied voltage is obtained up to a bias of ∼0.1 V, after which the systems display a non-ideal linear behavior. The linear regime evolution was then used to extract the intrinsic contact resistance of the modeled interface. For instance, the intrinsic contact resistance of Figure 1 corresponds to 9.5 × 10 −8 .cm 2 for a doping concentration of 1 × 10 20 |e|/cm 3 and reflects the needs for the injected electrons to cross the interface potential set by the tail states 19 present at the metal-semiconductor interface (Figure 2b). Interestingly, this model leads to contact resistances values relatively close to what has been recently measured experimentally by Hao et al., 15 as illustrated in Figure 3a. As expected, the contact resistance evolves linearly with the doping concentration to reach values as low as 1.8 × 10 −10 .cm 2 for an active doping concentration of 1 × 10 22 |e|/cm 3 . These findings are consistent with the recent demonstration of a sub 10 −9 .cm 2 con- tact resistance for PMOS devices. 20 Remarkably, the first-principles simulations also suggest that beyond this value the contact resistance should saturate, even for a completely unrealistic active dopant con- centration of 1 × 10 23 |e|/cm 3 .

Assuming the case that all metals have the same Fermi energy, the computation of T p (Eq. 1) at k sZ = 0 reveals that high effective mass metals helps in increasing the transmission probability at the Fermi level. For instance, going from a metal with a low effective mass such as W (m m = 0.73 m e ) to Sc (m m = 29.43 m e ), enhances the transmission probability by about 60% at 0.05 eV in the conduc- tion band of silicon. This is translated into a reduction of the intrinsic contact resistance in the ohmic regime whenever the Fermi energy is allowed to vary from metal to metal (Figure 6). There is how- ever a saturation effect as evidenced by the transmission probability (Figures 5a and 5b) and a strong dependency on i) the orientation of the Si substrate and ii) on confinement effects. 21 For instance, the benefit of using a high effective mass metal is more pronounced for heavier transport effective mass (Figures 6 and 10). This is for instance obtained whenever the orientation of the channel is switched, going from a [100] transport direction (m s = 0.23 m e ) 21 to a [111] one (m s = 0.41 m e ) 21 or by narrowing the diameter of the Si channel, as reported in Reference 21 for highly confined [111] nanowires (m s = 0.53 m e ). In these cases, the contact resistivity further benefits from the increase in metal effective mass thanks to the enhanced transmission prob- ability, as pointed out in Figure 10b. At higher energies, the trans- mission switches and consequently the contact resistivity increases for a higher effective mass metal. This effect is more pronounced for smaller effective mass semiconductors as shown in Figures 6 and 10. This observation holds true for 2D semiconductors as well (see Figures 7 to 9). Finally, these results suggest that contacting a faceted substrate, as it is the case in a Fin-FET technology, leads to an orientation dependent contact resistance. We extended the exercise to the case of the metal-2D semicon- ductor interface. Given that the real nature of this interface is still not clearly understood, 27 we used a simple step barrier transmission model to account for it and to evaluate the resistivity limits of side contacted 2D semiconductors in their ohmic regime. Equations 1 and 2 were used to calculate the contact resistivity of MoS 2 and Figure 7 provides a comparison of the resistivity of MoS 2 in contact with different metals with the Fermi energies as reported in Figure 4. The contact resistivi- ties are calculated at a n-type concentration of 1 × 10 21 |e|/cm 3 , which is roughly equivalent to a 2D doping of 6 × 10 13 |e|/cm 2 assuming a monolayer thickness of ∼ 6 Å (i.e. the thickness of the 2D material and of the van der Waals gap) for all the 2D materials considered. This value is chosen as being the ultimate limit of the doping concentration that can be used within this formalism before breaking the bound- aries of the validity of the approximations used in Equations 1 and 2. The use of higher doping concentrations would introduce unrealistic offsets between the Fermi-level and conduction band in degenerate semiconductors.

Assuming the case that all metals have the same Fermi energy, the computation of T p (Eq. 1) at k sZ = 0 reveals that high effective mass metals helps in increasing the transmission probability at the Fermi level. For instance, going from a metal with a low effective mass such as W (m m = 0.73 m e ) to Sc (m m = 29.43 m e ), enhances the transmission probability by about 60% at 0.05 eV in the conduc- tion band of silicon. This is translated into a reduction of the intrinsic contact resistance in the ohmic regime whenever the Fermi energy is allowed to vary from metal to metal (Figure 6). There is how- ever a saturation effect as evidenced by the transmission probability (Figures 5a and 5b) and a strong dependency on i) the orientation of the Si substrate and ii) on confinement effects. 21 For instance, the benefit of using a high effective mass metal is more pronounced for heavier transport effective mass (Figures 6 and 10). This is for instance obtained whenever the orientation of the channel is switched, going from a [100] transport direction (m s = 0.23 m e ) 21 to a [111] one (m s = 0.41 m e ) 21 or by narrowing the diameter of the Si channel, as reported in Reference 21 for highly confined [111] nanowires (m s = 0.53 m e ). In these cases, the contact resistivity further benefits from the increase in metal effective mass thanks to the enhanced transmission prob- ability, as pointed out in Figure 10b. At higher energies, the trans- mission switches and consequently the contact resistivity increases for a higher effective mass metal. This effect is more pronounced for smaller effective mass semiconductors as shown in Figures 6 and 10. This observation holds true for 2D semiconductors as well (see Figures 7 to 9). Finally, these results suggest that contacting a faceted substrate, as it is the case in a Fin-FET technology, leads to an orientation dependent contact resistance. We extended the exercise to the case of the metal-2D semicon- ductor interface. Given that the real nature of this interface is still not clearly understood, 27 we used a simple step barrier transmission model to account for it and to evaluate the resistivity limits of side contacted 2D semiconductors in their ohmic regime. Equations 1 and 2 were used to calculate the contact resistivity of MoS 2 and Figure 7 provides a comparison of the resistivity of MoS 2 in contact with different metals with the Fermi energies as reported in Figure 4. The contact resistivi- ties are calculated at a n-type concentration of 1 × 10 21 |e|/cm 3 , which is roughly equivalent to a 2D doping of 6 × 10 13 |e|/cm 2 assuming a monolayer thickness of ∼ 6 Å (i.e. the thickness of the 2D material and of the van der Waals gap) for all the 2D materials considered. This value is chosen as being the ultimate limit of the doping concentration that can be used within this formalism before breaking the bound- aries of the validity of the approximations used in Equations 1 and 2. The use of higher doping concentrations would introduce unrealistic offsets between the Fermi-level and conduction band in degenerate semiconductors.

Assuming the case that all metals have the same Fermi energy, the computation of T p (Eq. 1) at k sZ = 0 reveals that high effective mass metals helps in increasing the transmission probability at the Fermi level. For instance, going from a metal with a low effective mass such as W (m m = 0.73 m e ) to Sc (m m = 29.43 m e ), enhances the transmission probability by about 60% at 0.05 eV in the conduc- tion band of silicon. This is translated into a reduction of the intrinsic contact resistance in the ohmic regime whenever the Fermi energy is allowed to vary from metal to metal (Figure 6). There is how- ever a saturation effect as evidenced by the transmission probability (Figures 5a and 5b) and a strong dependency on i) the orientation of the Si substrate and ii) on confinement effects. 21 For instance, the benefit of using a high effective mass metal is more pronounced for heavier transport effective mass (Figures 6 and 10). This is for instance obtained whenever the orientation of the channel is switched, going from a [100] transport direction (m s = 0.23 m e ) 21 to a [111] one (m s = 0.41 m e ) 21 or by narrowing the diameter of the Si channel, as reported in Reference 21 for highly confined [111] nanowires (m s = 0.53 m e ). In these cases, the contact resistivity further benefits from the increase in metal effective mass thanks to the enhanced transmission prob- ability, as pointed out in Figure 10b. At higher energies, the trans- mission switches and consequently the contact resistivity increases for a higher effective mass metal. This effect is more pronounced for smaller effective mass semiconductors as shown in Figures 6 and 10. This observation holds true for 2D semiconductors as well (see Figures 7 to 9). Finally, these results suggest that contacting a faceted substrate, as it is the case in a Fin-FET technology, leads to an orientation dependent contact resistance. We extended the exercise to the case of the metal-2D semicon- ductor interface. Given that the real nature of this interface is still not clearly understood, 27 we used a simple step barrier transmission model to account for it and to evaluate the resistivity limits of side contacted 2D semiconductors in their ohmic regime. Equations 1 and 2 were used to calculate the contact resistivity of MoS 2 and Figure 7 provides a comparison of the resistivity of MoS 2 in contact with different metals with the Fermi energies as reported in Figure 4. The contact resistivi- ties are calculated at a n-type concentration of 1 × 10 21 |e|/cm 3 , which is roughly equivalent to a 2D doping of 6 × 10 13 |e|/cm 2 assuming a monolayer thickness of ∼ 6 Å (i.e. the thickness of the 2D material and of the van der Waals gap) for all the 2D materials considered. This value is chosen as being the ultimate limit of the doping concentration that can be used within this formalism before breaking the bound- aries of the validity of the approximations used in Equations 1 and 2. The use of higher doping concentrations would introduce unrealistic offsets between the Fermi-level and conduction band in degenerate semiconductors.

As observed previously for Si, the contact resistivity of MoS 2 is reduced with the increase of the metal effective mass and shows a sat- uration effect with the doping concentration. We focus on a 6 × 10 13 |e|/cm 3 doping in Figures 6-9 to probe the contact resistivity in a high doping regime. Note that the Fermi energies do not follow the evolution of the effective masses (Figures 6-9) and we do not see a smooth contact resistivity curve with metal effective masses. The modulation of the semi-conductor effective mass, going from Si to MoS 2 , has a larger impact on the contact resistivity than a variation in m m . Changing the orientation of the channel in Si, when going from [111] to [100], decreases the electron effective mass and leads to a rise of the contact resistivity for metals with heavy m m , as illustrated in Figure 6. This is also observed in MoS 2 (Figure 8), where its rela- tively small effective mass compared to the HfS 2 and ZrS 2 ones, leads to a large contact resistivity (see Figure 10a). In this case, the value of the semiconductor electron effective mass leads to a low trans- mission probability (and thereby to an increase in contact resistivity). Switching to larger values leads to an improved transmission prob- ability (Figure 10b). Intuitively, this can be pictured as the balance achieved by the system between the charge density injected by the density of states of the semiconductor in the metal, and the capability of its electronic structure to accommodate the injected electrons.

As observed previously for Si, the contact resistivity of MoS 2 is reduced with the increase of the metal effective mass and shows a sat- uration effect with the doping concentration. We focus on a 6 × 10 13 |e|/cm 3 doping in Figures 6-9 to probe the contact resistivity in a high doping regime. Note that the Fermi energies do not follow the evolution of the effective masses (Figures 6-9) and we do not see a smooth contact resistivity curve with metal effective masses. The modulation of the semi-conductor effective mass, going from Si to MoS 2 , has a larger impact on the contact resistivity than a variation in m m . Changing the orientation of the channel in Si, when going from [111] to [100], decreases the electron effective mass and leads to a rise of the contact resistivity for metals with heavy m m , as illustrated in Figure 6. This is also observed in MoS 2 (Figure 8), where its rela- tively small effective mass compared to the HfS 2 and ZrS 2 ones, leads to a large contact resistivity (see Figure 10a). In this case, the value of the semiconductor electron effective mass leads to a low trans- mission probability (and thereby to an increase in contact resistivity). Switching to larger values leads to an improved transmission prob- ability (Figure 10b). Intuitively, this can be pictured as the balance achieved by the system between the charge density injected by the density of states of the semiconductor in the metal, and the capability of its electronic structure to accommodate the injected electrons.

Finally, four different Si(100)-aTiSi two-probe device models con- taining 260 atoms were built and subsequently relaxed within the QuantumWise Atomistix ToolKit software package. 4,5 The dimen- sion of the silicon body was set to ∼5.5 nm to ensure that the width of the depletion layer generated by the doping concentration (rang- ing from 1 × 10 20 to 10 22 |e|/cm 3 n-Si(100)) is properly captured in the 'central' region of the device. The electronic temperature is set at 300 K. The thickness of the aTiSi layer in the central region is ∼2.3nm and corresponds to 2 repetitive units of disordered aTiSi (2.3 nm). The electrodes were built from ∼1.1 and ∼1.2 nm of bulk Si and aTiSi, respectively. The four models differ in the number of Ti atoms contacting the silicon substrate and their local coordination. Note that, though naive, these models lead to Schottky barrier heights close to the ones reported experimentally Aboelfotoh et al. 29 and H. R. Liauh et al. 30 The approach consists in solving the Kohn-Sham equations to evaluate the Hamiltonian of the system using DFT. The transmission spectrum is calculated using the Non-Equilibrium Green function (NEGF) method, and the current-voltage characteristics of the device are then computed within the framework of the Landauer-Büttiker formula, 18 considering a ballistic transport process. To improve the description of the electronic bandgap, strongly underestimated by PBE, we used the Tran and Blaha exchange correlation functional 13 with a c parameter value (c = 1.08) tuned to reproduce the bandgap of bulk Si combined with a norm-conserving (NC) pseudopotential and single ξ polarized (SZP) localized basis sets. The kinetic energy cutoff for the integration of the wave functions is set to be 75 Ha (2041 eV). We used a 21 × 21 × 100 Monkhorst-Pack integration k-point mesh in the device simulation part. The applied boundary conditions are periodic along the A, B directions and Dirichlet along the C one ( Figure 1). The contact resistivity has been extracted by a linear fit of the current obtained upon the application of a potential bias in the linear regime (i.e. typically up to 0.1 V). The voltage applied spans from 0 to 0.2 V per steps of 0.025 V. Finally, the effective masses of the bulk metals were obtained by the second numerical derivation of their Fermi surface energy using a 60 × 60 × 60 Monkhorst-Pack integration k-point mesh and a Delaunay triangulation approach, as implemented in Paraview. 14 The effective masses of 2D materials were extracted from the band structure computed using GGA-PBE with double-ς polarized basis and DFT-D3 correction, 4,26 using one monolayer (ML), a Bi-Layer (BL) and a Bulk model.