## Abstract

The performance of transistors based on two-dimensional (2D) materials is affected largely by the contact resistance due to high Schottky barriers at the metal-2D-material interface. In this work, we incorporate the effect of surrounding dielectrics and image-force barrier-lowering in calculating the resistance of Schottky edge-contacts between a metal and a transition-metal dichalcogenide (TMD) thin layer. The electrostatic potential is computed by solving the Poisson equation numerically. The transmission probability is computed using the Wentzel–Kramers–Brillouin (WKB) approximation using the full-band density of states obtained from density functional theory (DFT). The effect of the image force is obtained analytically using the Coulomb kernel of a point charge with boundary conditions appropriate to the geometry we have considered. We find that the image-force barrier-lowering (IFBL) in edge-contacts is determined mainly by the dielectric permittivity of the surrounding oxide. We find that low-*κ* surrounding dielectrics are crucial for obtaining low resistance monolayer-TMD edge-contacts. Our results show metal-to-n(p)-type MoS_{2} (WSe_{2}) edge-contacts with SiO_{2} as top and bottom insulators, a doping concentration > 1 × 10^{13}cm^{−2} and a metal work-function < 5.1 eV ( > 4.6 eV) result in a contact resistance as low as 50 Ω ⋅ μm.

## Introduction

In scaled Si-based field-effect transistors (FETs), reducing the thickness of the Si body is accompanied by severe short-channel effects which degrade device performance. In order to extend the complementary metal-oxide-semiconductor (CMOS) technology road-map beyond Moore’s law, atomically thin 2D materials, such as graphene, hexagonal boron nitride (h-BN), transition metal dichalcogenides (TMDs), such as MoS_{2}, WS_{2}, MoSe_{2}, WSe_{2}, and MoTe_{2}, silicene, phosphorene^{1,2,3,4,5}, are being studied extensively as candidates for nanoelectronic and optoelectronic applications^{2,6,7,8,9,10,11}.

A low contact resistance is essential for a high on-current in FETs. However, metal-TMD contacts are characterized by high Schottky barriers^{7,12,13,14,15,16} which make it difficult to achieve silicon-like contact resistance (<0.2 kΩ ⋅ μm) and severely limit the drive current in such devices. Semiconducting MoS_{2}, one of the widely studied TMDs considered as channels in 2D FETs, shows contact resistance values usually higher than 1 kΩ ⋅ μm^{12,13,14,15}. One of the lowest contact resistance values reported for multilayer MoS_{2} FETs is 0.54 kΩ ⋅ μm with an on-current of 830 μA/μm at 300 K^{17}. Phase-engineered metallic 1T MoS_{2}, when used as electrode, demonstrated significantly low contact resistance such as 0.2 − 0.3 kΩ ⋅ μm with an on-current of 100 μA/μm for multilayer, and 0.2 kΩ ⋅ μm with on current of 110 μA/μm for monolayer MoS_{2} channel^{18,19}. n-doping of MoS_{2} by AlO_{x} has also resulted in FETs with Au contacts and monolayer MoS_{2} as channel, exhibiting a resistance as low as 0.48 kΩ ⋅ μm with an on-current 700 μA/μm ^{20}. To date, the lowest contact resistance reported is 0.13 kΩ ⋅ μm, which was obtained by using semi-metal contact (semi-metallic bismuth on MoS_{2})^{21}. However, low melting temperature (250°C) limits its application. Another TMD that has been of interest due to its p-type behavior is WSe_{2} which has been shown to exhibit contact resistance as low as 0.3 kΩ ⋅ μm with an on-current of 300 μA/μm ^{22}. In these studies, the top-contact configuration had been used where a 3D bulk metal sits directly on top of the TMD semiconducting layer. Apart from top contacts, edge-contact TMDs are also being studied. Cui et al. demonstrated h-BN encapsulated MoS_{2} thin layers in contact with graphene-metal edge contacts^{23}. Choi et al. studied a multilayer hBN-encapsulated MoS_{2} edge-contact on a SiO_{2}/Si substrate^{24}, whereas Yang et al. showed fermi-level depinning in plasma-etched MoS_{2} metal edge-contacts^{25}. Cheng et al. demonstrated how they utilized in-situ etching for contact metal deposition and obtained a contact resistance of 30 kΩ ⋅ μm at a channel electron density of 1.2 × 10^{12} cm^{−2}^{26}. Recently, a contact resistance as low as 27.8 kΩ ⋅ μm has been reported for edge-contacts with contact metal Ti-Au and monolayers of MoS_{2} encapsulated by h-BN, while retaining an on-current comparable to what is obtained using top-contacts (>50 μA/μm)^{27}.

Although there has been an extensive experimental investigation of the contact geometry in 2D FETs, theoretical studies are limited. Some of the studies have employed first-principles and transport calculations using non-equilibrium Green’s functions, to investigate the electronic structure at the interface of both top and edge-contacts and to extract the Schottky barrier heights^{28,29,30}, whereas other study investigated dominant current mechanisms in vertical transport through TMD hetero-structures^{31}. A study incorporating ab initio quantum-transport simulations to predict the influence of transfer length and interfacial oxide on the carrier injection process through metal-TMD contacts has also been recently reported^{32}. Other theoretical studies on the resistance of metal-TMD contacts calculate the electrical resistance for a simplified problem without incorporating the effect of the Schottky barrier^{33,34}. However, all these studies exclude the effect of the surrounding dielectrics, an effect which is shown to be important in such 2D geometry^{35,36}. Another critical effect that controls the properties of contacts is the barrier lowering caused by the image force which has not received any attention for 2D materials.

In this work, we intend to focus on the importance of the surrounding dielectric and image-force barrier-lowering on the contact resistance of metal-TMD edge-contacts. Our model assumes an ideal Schottky interface ignoring a possible change of the atomic morphology of the interface and treats the injection mechanism in a simplified way. We have preferred this geometry for the computational simplicity it affords and for its presumable advantage of not involving tunneling across a van der Waals gap. Despite this restriction, our results regarding the beneficial role of high doping and low-*κ* insulators should apply to more general contact geometries, such as top-contacts. We use the full-band density of states of the TMDs obtained from density functional theory (DFT), and the WKB approximation to calculate the transmission probability through the Schottky barrier at the metal-TMD interface, including also the important effect of image-force barrier-lowering on the TMD potential. However, our ultimate goal consists in emphasizing the major effects played by the Schottky barrier, by its reduction due to the image potential and, most notably, by the surrounding dielectrics (due to the monolayer nature of the semiconductor), effects that remain important, if not even dominant, regardless of the models employed to treat the interface. Specifically, as a simple consequence of the Poisson equation, our results emphasize the role of a high doping concentration and a low surrounding dielectric permittivity to obtain the best contact resistance.

We organize the paper as follows: We first show our results, discussing the values of the contact resistance we have obtained, and the role played by the different choices of the parameters we have used. Next, we describe the numerical and mathematical approach we have used. Finally, we draw our conclusions.

## Results and discussion

### Edge-contact geometry and calculation of contact resistance

In Fig. 1a, b, we illustrate the geometry we consider: A semi-infinite TMD monolayer “sandwiched” between very thick top and bottom dielectrics (*t*_{dielectric} >> *t*_{2D}), with a metal contact on the side. We consider two types of monolayer TMDs namely MoS_{2} and WSe_{2} as the channel material. We first present *n*-type monolayer MoS_{2} to shed light on our main findings. We consider an infinitely wide device and assume translational invariance along the *y* direction. The transport is along the *x* direction. We consider the same dielectric material as top and bottom insulators, either SiO_{2} or HfO_{2}. We use anisotropic dielectric permittivity for the TMDs^{37}. We consider a piece-wise homogeneous isotropic dielectric permittivity for the top and bottom insulators, 3.9*ϵ*_{0} and 25*ϵ*_{0} for SiO_{2} and HfO_{2}, respectively.

We have calculated the transmission probability through the Schottky barrier invoking the WKB approximation with an electrostatic potential calculated accounting also for the effect of the image-force barrier-lowering. To obtain the electrostatic potential in the TMD, we solve self-consistently the Poisson equation using the finite elements-based package, FEniCS^{38,39} over the 2D cross-section shown in Fig. 1b. We assume that the contact clamps the potential at the metal-TMD/metal-oxide interface; that is, the metal is assumed to be a region of constant potential. More details on the computational approach for calculating the contact resistance is available in the “Methods” section.

### Importance of the surrounding dielectric

Figure 2 shows our main result, the calculated contact resistance as a function of doping concentration, in bulk MoS_{2} and edge-contact geometry of MoS_{2} monolayers (n-doped) with either SiO_{2} (MoS_{2}/SiO_{2}) or HfO_{2} (MoS_{2}/HfO_{2}) as top and bottom insulators. We see that increasing the doping concentration reduces the contact resistance, a trend that is consistent with what is observed in metal-bulk-semiconductor contacts. Interestingly, we find that in the presence of a low-*κ* surrounding dielectric, such as SiO_{2}, monolayer MoS_{2} presents a lower contact resistance than its bulk counterpart. Contact resistances calculated excluding the effect of IFBL are shown for each case of the surrounding dielectric in Supplementary Fig. 1 of the Supplementary Information. We show a similar plot with the contact resistance of p-WSe_{2} in Supplementary Fig. 2 of the Supplementary Information. However, it should be noted that barrier-lowering due to image force, is a physical phenomenon, arising when a metal encounters a semiconductor, an effect that is always present in metal contacts. The improvement in contact resistance in monolayer MoS_{2} (and WSe_{2}) due to barrier-lowering is the largest, ~40 times (~30 times), when the surrounding dielectric is SiO_{2} compared to HfO_{2}, which gives ~5 times improvement, as illustrated in Supplementary Figs. 1 and 2 of the Supplementary Information. We also find that to achieve a contact resistance of 50 Ω ⋅ μm, monolayer MoS_{2} and WSe_{2} require to be doped at a concentration >1 × 10^{13} cm^{−2}. Thus, from Fig. 2 it is apparent that the surrounding dielectrics affect very strongly the electrostatic behavior of such 2D geometries.

While a direct comparison of our results to experiments is beyond the scope of this work, below we discuss some of the experimental studies. Values of the contact resistances reported below were extracted and reported by Parto et al.^{30}. A multilayer hBN-encapsulated MoS_{2} edge contact on a SiO_{2}/Si substrate^{39} was studied by Choi et al. where a contact resistance of 6 × 10^{2} kΩ ⋅ μm was obtained at a channel electron density of 2 × 10^{12} cm^{−2} with Mn as contact metal^{30}. Chai et al. studied a monolayer MoS_{2} edge-contact (encapsulated by hBN and SiO_{2} as gate dielectric)^{40}, obtaining a contact resistance of 3 × 10^{3} kΩ ⋅ μm at a channel electronic density of 4 × 10^{12} cm^{−2} and Sc/Ni contact metal^{30}. h-BN encapsulated monolayer MoS_{2} edge-contact on Si/SiO_{2} substrate studied by Moon et al.^{41} had a contact resistance of 10^{3} kΩ ⋅ μm at an electron density of 4 × 10^{12} cm^{−2}^{30}. Furthermore, from Fig. 2 in ref. ^{30}, we find that the values of the edge-contact resistance in monolayer MoS_{2} range from 40 kΩ ⋅ μm to 4 × 10^{3} kΩ ⋅ μm as the electron density spans the range 10^{12} cm^{−2} to 2 × 10^{13} cm^{−2}. These values are larger than our calculated values. Note that in our model we ignore the complexities of the interface between the metal and TMD, neglecting, for example, the nature of the chemical bond, the effects of disorder, of interface polarization, of changes in the band offset, of Fermi-pinning, and of phonon scattering. This allows us to treat the contact as an abrupt heterostructure and adopt the idealized mesoscopic approximation of assuming that the band structure of the bulk metal changes to the TMD band-structure at the point of contact. More advanced calculations are therefore needed to predict accurately the absolute contact resistance. Despite these limitations, our work highlights the importance of the image-force barrier lowering and the surrounding dielectric, effects that are largely ignored in more advanced calculations. We will show that these electrostatic effects, which rely only on the Coulomb interaction and geometry, have a surprisingly strong influence on the contact resistance and even drive the design of the dielectric layers around the metal-TMD contact.

Considering the physics that governs the contact, the dielectric response of the surrounding material enters in two ways. First, the length of the depletion region in the 2D material is affected by the surrounding dielectric. Second, the image-force barrier-lowering effect, resulting from the attractive force emerging from the metal, becomes stronger with a surrounding dielectric material having a lower dielectric constant. The first effect is illustrated in Fig. 3 and the second in Fig. 4.

### Depletion width

Figure 3a–c shows the contour maps of the 2D potential energy with electric field lines whereas Fig. 3d shows the potential energy, obtained as 1D cut of the 2D potential energy, along the center of the MoS_{2} channel. From Fig. 3a–c, we find that the electric field is screened more effectively in bulk, compared to 2D MoS_{2} monolayers. Monolayer TMDs are subject to strong fringing fields permeating the surrounding dielectric. As a result, a high-*κ* dielectric such as HfO_{2} reduces screening. A direct consequence of weaker screening is a larger depletion width as shown in Fig. 3d. The potential energy in bulk MoS_{2} falls sharply with the smallest depletion width. 2D MoS_{2} surrounded by HfO_{2} exhibits a larger depletion width in contrast to SiO_{2}. A larger depletion width results in a thicker tunneling barrier and, therefore reduces the transmission probability through the Schottky barrier at the metal-TMD interface. This is one of the reasons that we see higher contact resistance in MoS_{2} surrounded by high-*κ* dielectric such as HfO_{2}.

### Image-force barrier lowering

The second effect, the image-force barrier-lowering (IFBL), consists in the reduction of the effective barrier height due to the presence of image charges inside the metal contact. The metal region must maintain a constant potential as an electron tunnels through the Schottky barrier. The role played by the IFBL is illustrated in Fig. 4 which shows the potential energy as in Fig. 3, but now accounting for the IFBL at a fixed doping concentration in the two extreme cases of SiO_{2} (Fig. 4a) and HfO_{2} (Fig. 4b). The magnitude of barrier lowering is the difference between the peak of the no-IFBL and IFBL potential energies (shown by the arrow in Fig. 4). We observe that at a low doping concentration (*N*_{D} = 1 × 10^{12} cm^{−2}), the conventional (as used in bulk semiconductor contacts) barrier-lowered potential, 1/(16*π**ϵ**x*) ^{42}, *ϵ* being the permittivity of the surrounding dielectric, describes the barrier lowering in the contact quite well (it only underestimates the barrier by 0.0039 eV and 0.0005 eV in MoS_{2}/SiO_{2} and MoS_{2}/HfO_{2}, respectively.) However, as the doping concentration increases and tunneling occurs closer to the contact, the conventional bulk model fails to capture the barrier lowering accurately (see “Image potential” in “Methods” section for more details). The thickness of the barrier plays a particularly strong role, and it is also strongly modulated by the dielectric constant of the surrounding insulator. In Fig. 4, we observe that barrier narrowing is larger in MoS_{2}/SiO_{2} than MoS_{2}/HfO_{2}, in addition to barrier lowering. These improve the tunneling and thermionic emission, respectively, and overall lead to a decrease in contact resistance.

### Monolayer n-MoS_{2} and p-WSe_{2} edge-contact

In, Fig. 5a, b, we show the map of contact resistance as a function of metal work-function, Schottky barrier-height and doping concentration, for n-type MoS_{2} and p-type WSe_{2} with SiO_{2} as top and bottom insulators. The *x* axis denotes the doping concentration whereas the left and right *y* axes denote the metal work-function and the Schottky barrier-height, respectively. The electron affinities (*χ*) of MoS_{2} and WSe_{2}, determined from DFT calculations are 3.96 eV and 3.36 eV, respectively. Schottky barrier-height in n-type semiconductor is calculated as *ϕ*_{Bn} = *ϕ*_{M} − *χ* and in p-type as \({\phi }_{{{{\rm{Bp}}}}}=\chi +\frac{{E}_{{{{\rm{g}}}}}}{{{{\rm{e}}}}}-{\phi }_{{{{\rm{M}}}}}\), where *ϕ*_{M} is the metal work-function and *E*_{g} is the energy band-gap which is found to be 1.76 eV in monolayer WSe_{2} from DFT calculation. Therefore, as the metal work-function increases, the Schottky barrier-height increases in n-type and decreases in p-type materials.

In order to obtain a contact resistance as low as 50 Ω μm, both structures require a doping concentration > 1 × 10^{13} cm^{−2}. However, n-type MoS_{2} and p-type WSe_{2} require metals with work-function < 5.1 eV and > 4.6 eV, respectively. These contour plots can be viewed as a guideline for the selection of contact metal and doping concentration for various edge-contact 2D materials, thereby narrowing down the material and design selection space for expensive experimental device fabrication.

### WKB *vs*. NEGF

We use a finite elements solver to solve the Poisson equation and the WKB approximation to calculate the contact resistance. To understand why we opt for the Poisson-WKB approach instead of the atomistic Non-Equilibrium Green’s function (NEGF) approach used by other recent studies on contacts^{30}, we provide an estimate of the depletion length that needs to be considered. For the lowest doping concentration, 10^{12} cm^{−2}, the highest dielectric constant dielectric, HfO_{2}, and the highest Schottky barrier *ϕ*_{SB} = 0.7 V, the 2D depletion width is estimated as \({\phi }_{{{{\rm{SB}}}}}\pi \epsilon /({{{\rm{\ln }}}}(4)e{N}_{{{{\rm{D}}}}})=219\,{{{\rm{nm}}}}\)^{36}. Even at high doping concentration, a relatively large simulation domain, much larger than typically used in DFT, is still required. Dealing with simulation domains on this order of magnitude makes the use of atomistic NEGF codes^{43,44,45} computationally prohibitive whereas solving the Poisson equation and using the WKB approximation can yield an accurate value of the contact resistance. Moreover, to match our results, one must employ hybrid DFT functionals and include spin-orbit coupling. The large number of *k*-points required to reach convergence for the calculation of the current adds to the computational expense. Moreover, only results obtained ignoring image-force barrier lowering could be compared with the DFT + NEGF code.

To assess the accuracy of the WKB approximation, we implement an effective mass NEGF calculation^{46} and compare with the WKB. In Fig. 6, we show the plot of the contact resistances calculated using both the WKB approximation and the NEGF, at a Schottky barrier height of 0.3 eV for MoS_{2} surrounded by SiO_{2} or HfO_{2}. It should be noted, that the NEGF calculations use a full effective-mass Hamiltonian whereas the WKB calculations are done using the band structure from DFT. We treat both ‘ideal’ and ‘metal’ contact in our NEGF model. The details of the calculation are discussed in the ‘Methods’ section. If we compare the contact resistances obtained from the WKB approximation with those from the ‘ideal’ contact-NEGF formalism, we see that the values are very close, with the WKB approximation resulting in a slight overestimation of the conductance with respect to the NEGF. Moreover, we find that the contact resistances calculated by the ‘metal’ contact-NEGF formalism depends on the self-energy or the coupling strength. The stronger the coupling, the better the contact resistance. However, based on this NEGF estimate, it appears that by using SiO_{2} rather than HfO_{2} results in a more noticeable improvement of the contact resistance than what is obtained by increasing the bond strength by a factor 20. Therefore, the main conclusion of our work (namely: low-*κ* dielectrics are better candidates to obtain low resistance contacts in 2D materials), still holds.

In summary, we have simulated the transmission through metal-2D-materials edge-contacts with Schottky barrier at their interface, using the WKB approximation, and DFT density of states, and obtained the contact resistance. We have shown how the Schottky barrier height, the doping concentration, and the surrounding dielectric environment largely control the electrostatics in 2D devices by considering cases with both low and high-*κ* oxides.

The primary finding of our study is that low-*κ* top and bottom insulators surrounding monolayer TMDs result in a low contact resisitance, outperforming bulk TMD contacts, thanks to a smaller depletion length and a higher image-force barrier-lowering. We find that, contrary to the bulk case, image-force barrier-lowering in edge-contact 2D devices is determined by the dielectric permittivity of both the surrounding oxide and the TMD. We have also compared the contour plots of contact resistance in two of the most widely studied TMD materials, n-type MoS_{2} and p-type WSe_{2} as a function of metal work-function and Schottky barrier-height, which can be considered as guidelines for the fabrication of realistic edge-contact 2D devices.

## Methods

### DFT calculation

We performed DFT calculations for bulk MoS_{2}, monolayer MoS_{2} and monolayer WSe_{2}, using the Vienna Ab initio Simulation Package (VASP)^{47,48,49,50}. We first ran geometry optimization until the maximum force on every atom dropped below 0.01 eV/Å. We employed the generalized gradient approximation (GGA) with the projector-augmented wave (PAW) method^{51} using the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional^{52}. A large vacuum space of 30Å was used along the *z* direction to avoid interaction between successive layers. We used the DFT-D3 dispersion correction of Grimme^{53} to describe van der Waals interactions and accurately calculate the interlayer distance. Thereafter, we used the relaxed structure to perform electronic calculations.

For monolayer TMDs, we first calculated the band structure on a coarse mesh followed by interpolation on a finer mesh by using Maximally Localized Wannier Functions (MLWF) generated by the Wannier90 code^{54}. The interpolation preserves the accuracy of the DFT calculations at a lower computational cost. We used the Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional^{55} with spin-orbit coupling and an electronic convergence of 10^{−6} eV. To compute the band structure we sampled the Brillouin zone with a Γ-centered 8 × 8 × 1 and 8 × 8 × 4 *k*-mesh in monolayer and bulk TMDs respectively. We used the d and p orbitals of the metal and chalcogenide atoms as Wannier projectors to interpolate the DFT band structure on a denser 100 × 100 × 1 *k*-mesh in monolayer TMDs, which was then utilized in calculating the transmission probability.

### The Poisson equation

We obtained the potential inside the TMD layer by solving the 2D Poisson equation (in the (*x*, *z*) plane) self consistently with the charge determined from the full-band density of states. The 2D Poisson equation in a metal-n type semiconductor contact is:

where *N*_{D}(*x*, *z*) and *n*_{0}(*x*, *z*) are the n-type doping concentration and electron carrier density, respectively, *V*_{dep}(*x*, *z*) is the 2D depletion potential, and \(\overline{\epsilon }(x,z)\) is the dielectric permittivity tensor. The in-plane dielectric constant values used for monolayer MoS_{2}, WSe_{2} and bulk MoS_{2} are 15.5*ϵ*_{0},15.6*ϵ*_{0} and 15.9*ϵ*_{0}^{37}, whereas the out-of plane values are 6.2*ϵ*_{0}, 7.4*ϵ*_{0} and 6.9*ϵ*_{0}, respectively, with *ϵ*_{0} being the vacuum permittivity. We calculated the electron density from the DFT band structure as:

where, *E*_{n}(**k**) denotes the full-band dispersion obtained from DFT, *n* is the band index for the monolayer TMD, and **k** is the two-dimensional wave-vector.

The simulation domain for solving the 2D Poisson equation is a rectangle (see Fig. 1) with a TMD layer “sandwiched” between 50 nm-thick oxide layers. The thickness of the TMD layer was determined from the relaxed geometry obtained with DFT, and was found to be 0.62 nm and 0.65 nm for monolayer MoS_{2} and monolayer WSe_{2}, respectively. We used Dirichlet boundary conditions at the metal contact and Neumann boundary conditions on the other sides (thus ignoring any effect of a gate bias). The size of the simulation domain is 100.62 nm × 400 nm. The top and bottom oxide thickness is substantially greater than the thickness of the 2D layer to ensure minimal impact of the Neumann boundary conditions on the electrostatics of the 2D layer^{36,56}.

We solved Eq. (1) numerically by finite-elements in FEniCS^{38,39}. We used the built-in mesh generator of FEniCS to generate a structured mesh of 1000 × 250 elements throughout the computational domain and further refined it at the metal-semiconductor interface to accurately capture the high electric fields near the metal.

### Contact resistance

Owing to the very high density of states in the metal and high Schottky barrier-height, we assumed that the transmission probability is governed only by the states in the semiconducting monolayer TMD. We also conserve parallel momentum, which results from our assumption of translational invariance along the *y* direction. We calculated the contact resistance for edge-contact 2D monolayers as the inverse of ballistic conductance modulated by the probability of the carriers injected through a Schottky barrier^{57}, using the following:

where *ρ*_{c} denotes the contact resistance, *e* is the electronic charge, *h* is Planck’s constant, *n* is the band index for the monolayer TMD, *k*_{y} is the parallel wave-vector, and *f*(*E*) is the Fermi-Dirac distribution function. *E*_{n}(*k*_{y}) was calculated from the full band DFT energy band-dispersion. *T*(*k*_{y}, *E*) is the transmission probability as a function of energy *E* and *k*_{y}. *M*(*k*_{y}, *E*) denotes the number of conducting channels at the energy of interest, and would result in the ballistic conductance if the WKB integral was not present. We calculated Eq. (4) numerically from the energy band dispersion obtained with DFT.

Following the WKB approximation, we have the transmission probability, *T*(*k*_{y}, *E*) < 1, for carriers tunneling through the Schottky barrier, and *T*(*k*_{y}, *E*) = 1, for carriers injected over the top of the barrier. The transmission probability was calculated as:

where \({m}_{n}^{* }({k}_{y})\) is the tunneling effective-mass computed numerically using finite difference from the DFT energy band structure, \({x}_{\min }\) and \({x}_{\max }\) denote the start and end of the depletion region, *E* is the energy of the carriers, and \({E}_{n,\min }({k}_{y})\) is the band-edge obtained from the DFT energy band structure. *U*(*x*) is the barrier-lowered potential energy where *U*(*x*) = *U*_{dep}(*x*) + *U*_{image}(*x*). *U*_{dep}(*x*) is the Schottky depletion potential energy which was obtained from the 1D cut of the 2D depletion potential at the center of the monolayer TMD along the transport direction. *U*_{image}(*x*) is image-force potential energy. The potential energy is related to the electrostatic potential by the relation *U*_{dep(image)} = − e*V*_{dep(image)}.

### Image potential

In order to calculate how the image force lowers the height of the Schottky barrier in our 2D geometry, we first computed the Coulomb kernel for a charged particle somewhere in the middle of the monolayer shown in Fig. 7. The effect of the metal was ignored in this first step and accounted for next by the method of images. The equation for the Green’s function of the Poisson equation in our geometry is:

where **r** is the three-dimensional position vector, *V*_{image}(**r**) stands for the image potential, *ϵ*(**r**) is the dielectric permittivity, and *δ*(**r**) is the point charge source.

We considered circular symmetry and applied the Hankel or Fourier-Bessel transform of zeroth order on Eq. (6). We used the relation, \({\hat{V}}_{{{{\rm{image}}}}}(Q)=\int\nolimits_{0}^{\infty }r{V}_{{{{\rm{image}}}}}(r){J}_{0}(Qr){{{\rm{d}}}}r\), (where, *Q* is the transformed Hankel coordinate and is reciprocal to *r*, and *J*_{0} is the Bessel function of the first kind of order zero), to derive the following partial differential equation:

where *ϵ*_{⊥} is the out-of plane dielectric permittivity and *ϵ*_{∥} is the in-plane dielectric permittivity of the 2D layer.

The boundary conditions for the potential are continuity of \({\hat{V}}_{\mathrm{image}}(Q,z)\) and electric displacement \(({\epsilon}_{\perp}\frac{{{{\mathrm{d}}}}{\hat{V}}_{{{{\mathrm{image}}}}}(Q,\,z)}{{{{\mathrm{d}}}}z})\) at *z* = 0, and \(z=\pm a=\frac{{t}_{{{{\mathrm{2D}}}}}}{2}\), \({\hat{V}}_{{{{\mathrm{image}}}}}(Q,z)=0\) at *z* = ± *∞* and a discontinuity due to the source at *z* = 0 for \({\hat{V}}_{{{{\mathrm{image}}}}}^{\,{\prime}}(0+)-{\hat{V}}_{{{{\mathrm{image}}}}}^{\,{\prime}}(0-)=\frac{1}{{\epsilon}_{2{{{\mathrm{D}}}}}Q}\). Using these boundary conditions we obtained six linear equations with six unknown coefficients, whose solution is:

where, *ϵ*_{top} and *ϵ*_{bot} are the top (*z* > *a*) and bottom (*z* < *a*) oxide dielectric permittivity, the thickness of the middle 2D layer (−*a* < *z* < *a*) is 2*a*, \({\epsilon }_{2{{{\rm{D}}}}}=\sqrt{{\epsilon }_{\parallel }{\epsilon }_{\perp }}\) and \(\beta =\sqrt{\frac{{\epsilon }_{\parallel }}{{\epsilon }_{\perp }}}\).

Finally, the real space potential *V*_{image}(*x*) at (*x*, 0) due to the point charge at (*x*, *z* = 0) was obtained numerically^{58} as the 2D Fourier-Bessel or Hankel transform of Eq. (8) and calculated as:

where, *J*_{0} is the Bessel function of the first kind of order zero.

We then used the method of images to account for the metal contact and to evaluate the image potential energy. It is defined as the (negative) work done to bring a charge *e* from infinity to a distance *x* from the metal-2D interface (at a distance 2*x* from the image charge) and is given by:

which equals \(\frac{{{{\rm{e}}}}}{2}{V}_{{{{\rm{image}}}}}(2x)\).

Finally, we show in Fig. 8, the plot of \({| 4\pi x{V}_{{{{\rm{image}}}}}(x)| }^{-1}\) with respect to *x*. Here, *x* denotes the distance of the point charge from the metal-TMD interface, and *V*_{image}(*x*) is the calculated image potential for a point charge located in the middle of the MoS_{2} monolayer in the presence of different surrounding dielectrics. Asymptotically, we see that at distances much larger than the layer thickness (*x* >> *t*_{2D}), the quantity \({| 4\pi x{V}_{{{{\rm{image}}}}}(x)| }^{-1}\) matches the dielectric constant of the surrounding oxide, which implies that the barrier-lowered potential behaves as 1/(*ϵ*_{dielectric}*x*). On the other hand, when *x* << *t*_{2D}, the 2D-material dielectric constant dominates, as in the bulk case, and the barrier-lowered potential behaves as 1/(*ϵ*_{2D}*x*). When *x* ≈ *t*_{2D}, a numerical evaluation is required to obtain the correct result. For *x* >> *t*_{2D}, the conventional bulk model^{42} (using oxide permittivity) matches the correct solution but fails when *x* << *t*_{2D}.

### Calculation of contact resistance using NEGF

We used an effective-mass Hamiltonian to compute the transmission with NEGF formalism. We calculated the effective mass of monolayer MoS_{2} along the transport direction *x* from the DFT band structure using finite differences. We solved the retarded Green’s function as a function of *k*_{y} and discretized using Δ*k*_{y} = 2*π**n*/(150 nm). We used the same electrostatic potential as in the WKB calculations. The barrier lowered potential needs to be truncated to avoid the singularity at *x* = 0 that occurs when ignoring the self-consistent many-body effects considered in ref. ^{59}. For simplicity, we just truncated the potential at *U* = − 1 eV. We chose a spacing of 0.4 nm to discretize the effective-mass Hamiltonian and 0.003 eV to discretize the energy range. In the WKB approximation, we assumed ideal metal contacts and ignored any band-structure mismatch. However, under the NEGF formalism, we calculated the transmission by considering both ‘ideal’ and ‘metal’ contacts. In case of ‘ideal’ contacts, the contacts were considered as an extension of the device, and the surface Green’s function was used to compute the self-energy^{46}. On the contrary, the ‘metal’ contacts were treated using the wide-band limit^{60,61,62,63}, and the contact self-energy was represented as purely imaginary and energy-independent, Σ(*E*) = − *i*t_{0}^{61,64,65}, and related to the coupling of the active region (the 2D TMD channel) to the metal via chemical bonds. A stronger coupling strength between the metal and the semiconductor was captured by a higher value of t_{0}. To illustrate this effect on the contact resistance, we had chosen two values of Σ (Σ(*E*) = − *i*0.03 and Σ(*E*) = − *i*0.1), to capture a bond that is ~20 times weaker and a bond that is ~6 times weaker than the bond in the TMD.

## Data availability

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

## Code availability

The codes that are necessary to reproduce the findings of this study are available from the corresponding author upon reasonable request. All DFT calculations were performed by using the Vienna ab initio simulation package (VASP).

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## Acknowledgements

This work has been supported by the Taiwan Semiconductor Manufacturing Company, Ltd. and the NEWLIMITS/nCORE program of the Semiconductor Research Corporation (SRC). We also acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing the high-performance computing resources that have contributed to the research results reported within this paper. http://www.tacc.utexas.edu.

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W.G.V. conceived the project. M.B. developed the code and performed the simulations. M.B., M.L.V. de.P., E.C., M.V.F., and W.G.V. analyzed the obtained results. M.B. wrote the paper with all the authors contributing to the discussion and preparation of the manuscript.

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Brahma, M., Van de Put, M.L., Chen, E. *et al.* The importance of the image forces and dielectric environment in modeling contacts to two-dimensional materials.
*npj 2D Mater Appl* **7**, 14 (2023). https://doi.org/10.1038/s41699-023-00372-6

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DOI: https://doi.org/10.1038/s41699-023-00372-6