Figure 4. (a) Fermi surfaces (at 300K) of copper and ruthenium. The surfaces are colored with respect their direction dependent electron effective masses. (b) Evolution of the averaged effective masses computed for different metals. The empty white boxes correspond to the values reported in Reference 17, while the filled ones correspond to the averaged values computed using first-principles simulations. Since the Fermi surfaces of most metals are highly anisotropic, we report their edge values as error bars. (c) Evolution of the Fermi energy computed for different metals. 

Keeping these aspects in mind, one can nonetheless establish a qualitative ranking of the potential of metals to maximize the trans- mission probability during the electron injection process since it de- pends on the Fermi energy (Figure 4c), on the effective masses in the metal (m m ) (Figure 4b) and in the semiconductor (m s ). Follow- ing this line of thoughts, Equations 1 and 2 offer a possible strategy to further reduce the contact resistance and suggest that metals with a high effective mass could be used to maximize the transmission probability of the electron injection. The crystal symmetry together with the orbital population of the metals lead to a rich modulation of their electronic properties at the Fermi level, their Fermi energy (Figure 4c) and is translated by a pronounced variation of their elec- tron effective masses as evidenced in Figure 4b. Since the shape of the Fermi surface of a metal is set by its crystal symmetry, it ends up to be strongly anisotropic, as illustrated by the distribution of the electron effective masses computed for copper and ruthenium (Figure 4b). Interestingly, some metals such as bismuth can have an averaged effective mass as low as 0.047 m e , while other ones, such as scandium can have an averaged electron effective mass as high as 29.43 m e .

Keeping these aspects in mind, one can nonetheless establish a qualitative ranking of the potential of metals to maximize the trans- mission probability during the electron injection process since it de- pends on the Fermi energy (Figure 4c), on the effective masses in the metal (m m ) (Figure 4b) and in the semiconductor (m s ). Follow- ing this line of thoughts, Equations 1 and 2 offer a possible strategy to further reduce the contact resistance and suggest that metals with a high effective mass could be used to maximize the transmission probability of the electron injection. The crystal symmetry together with the orbital population of the metals lead to a rich modulation of their electronic properties at the Fermi level, their Fermi energy (Figure 4c) and is translated by a pronounced variation of their elec- tron effective masses as evidenced in Figure 4b. Since the shape of the Fermi surface of a metal is set by its crystal symmetry, it ends up to be strongly anisotropic, as illustrated by the distribution of the electron effective masses computed for copper and ruthenium (Figure 4b). Interestingly, some metals such as bismuth can have an averaged effective mass as low as 0.047 m e , while other ones, such as scandium can have an averaged electron effective mass as high as 29.43 m e .

Keeping these aspects in mind, one can nonetheless establish a qualitative ranking of the potential of metals to maximize the trans- mission probability during the electron injection process since it de- pends on the Fermi energy (Figure 4c), on the effective masses in the metal (m m ) (Figure 4b) and in the semiconductor (m s ). Follow- ing this line of thoughts, Equations 1 and 2 offer a possible strategy to further reduce the contact resistance and suggest that metals with a high effective mass could be used to maximize the transmission probability of the electron injection. The crystal symmetry together with the orbital population of the metals lead to a rich modulation of their electronic properties at the Fermi level, their Fermi energy (Figure 4c) and is translated by a pronounced variation of their elec- tron effective masses as evidenced in Figure 4b. Since the shape of the Fermi surface of a metal is set by its crystal symmetry, it ends up to be strongly anisotropic, as illustrated by the distribution of the electron effective masses computed for copper and ruthenium (Figure 4b). Interestingly, some metals such as bismuth can have an averaged effective mass as low as 0.047 m e , while other ones, such as scandium can have an averaged electron effective mass as high as 29.43 m e .

Keeping these aspects in mind, one can nonetheless establish a qualitative ranking of the potential of metals to maximize the trans- mission probability during the electron injection process since it de- pends on the Fermi energy (Figure 4c), on the effective masses in the metal (m m ) (Figure 4b) and in the semiconductor (m s ). Follow- ing this line of thoughts, Equations 1 and 2 offer a possible strategy to further reduce the contact resistance and suggest that metals with a high effective mass could be used to maximize the transmission probability of the electron injection. The crystal symmetry together with the orbital population of the metals lead to a rich modulation of their electronic properties at the Fermi level, their Fermi energy (Figure 4c) and is translated by a pronounced variation of their elec- tron effective masses as evidenced in Figure 4b. Since the shape of the Fermi surface of a metal is set by its crystal symmetry, it ends up to be strongly anisotropic, as illustrated by the distribution of the electron effective masses computed for copper and ruthenium (Figure 4b). Interestingly, some metals such as bismuth can have an averaged effective mass as low as 0.047 m e , while other ones, such as scandium can have an averaged electron effective mass as high as 29.43 m e .

Keeping these aspects in mind, one can nonetheless establish a qualitative ranking of the potential of metals to maximize the trans- mission probability during the electron injection process since it de- pends on the Fermi energy (Figure 4c), on the effective masses in the metal (m m ) (Figure 4b) and in the semiconductor (m s ). Follow- ing this line of thoughts, Equations 1 and 2 offer a possible strategy to further reduce the contact resistance and suggest that metals with a high effective mass could be used to maximize the transmission probability of the electron injection. The crystal symmetry together with the orbital population of the metals lead to a rich modulation of their electronic properties at the Fermi level, their Fermi energy (Figure 4c) and is translated by a pronounced variation of their elec- tron effective masses as evidenced in Figure 4b. Since the shape of the Fermi surface of a metal is set by its crystal symmetry, it ends up to be strongly anisotropic, as illustrated by the distribution of the electron effective masses computed for copper and ruthenium (Figure 4b). Interestingly, some metals such as bismuth can have an averaged effective mass as low as 0.047 m e , while other ones, such as scandium can have an averaged electron effective mass as high as 29.43 m e .

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.