Efficient Ohmic Contact in Monolayer CrX2N4 (X = C, Si) Based Field‐Effect Transistors

Developing Ohmic contact systems or achieving low contact resistance is significant for high‐performance semiconductor devices. This work comprehensively investigates the interfacial properties of CrX2N4 (X = C, Si) based field‐effect transistors (FETs) with different metal (Ag, Au, Cu, Ni, Pd, Pt, Ti, and graphene) electrodes by using electronic structure calculations and quantum transport simulations. It is highlighted that the stronger interlayer coupling allows CrC2N4 to form an n‐type Ohmic contact with Ti electrode in the vertical direction. Furthermore, the absence of tunneling barrier at the CrC2N4–Ti interface greatly improves the electron injection efficiency. On the other hand, the studied metals form Schottky contact with CrC2N4 at the lateral interface due to Fermi level pinning (FLP) effects. Surprisingly, the strong FLP effects restrict the Schottky barrier heights of CrSi2N4‐metal contacts to a narrow range. Where Ag, Au, Ni, Pd, Pt, Ti electrodes and Ag, Ti electrodes form ideal ohmic contact with CrSi2N4 in the vertical and lateral directions, respectively, while the other metals form quasi‐ohmic contact. Ti exhibits the highest contact performance as the electrode in both CrC2N4 and CrSi2N4 based FETs. The findings may provide fundamental understanding for designing high‐performance and energy‐efficient FETs based on CrX2N4.


Introduction
Over the past few decades, the size of integrated circuits has been gradually reduced due to the rapid development of the information technology industry, which may approach the limit of Moore's law. [1] As a result, the urgent demand for nanoscale electronic devices face huge challenges. [2] However, the cost of overcoming this challenge in conventional silicon-based technologies is inevitable due to the fatal short-channel effects of silicon-based transistors. [3] Two-dimensional (2D) semiconducting materials have attracted much attentions due to their atomically thickness (enhancing gate control) and smooth surfaces without dangling bonds (facilitating carrier transport), showing great potential as next generation channel materials. [4][5][6][7][8] The family of 2D semiconductor materials such as black phosphorus (BP), [9] transition metal dichalcogenides (TMDs), [10,11] transition metal carbides/nitrides (MXenes), [12,13] and III-VI compounds. [14][15][16] However, the absence of a band gap in graphene, the instability of BP under environmental conditions, and the relatively low mobility (≈200 cm 2 V −1 s −1 ) of TMDs limit their application in high performance devices. [9,17,18] Therefore, building novel field-effect transistors (FETs) based on novel 2D semiconductors is of great interest and importance.
In the ongoing research on novel 2D semiconductors, a large-area MoSi 2 N 4 monolayer has been fabricated by adding silicon during the growth of molybdenum nitride, which is regarded as one of the most recent groundbreaking accomplishments in the field of 2D materials. [19] This novel 2D system was found to be a semiconductor with significant tensile strength and high carrier mobility, which hold great promises in novel 2D semiconductor material device applications. [19,20] MoSi 2 N 4 monolayer presents particularly 2D layered structure consisting two Si-N layers sandwiched by a Mo-N layer. [21] Interestingly, the septuple-layer structure of MoSi 2 N 4 monolayer in which the electronic states are protected by the outer Si-N sublayer without external influence. [22] This unusual builtin atomic layer protection mechanism enables MoSi 2 N 4 with great environmental stability and forms efficient ohmic contact without severely affecting its electronic properties. [23] Recently, Developing Ohmic contact systems or achieving low contact resistance is significant for high-performance semiconductor devices. This work comprehensively investigates the interfacial properties of CrX 2 N 4 (X = C, Si) based field-effect transistors (FETs) with different metal (Ag, Au, Cu, Ni, Pd, Pt, Ti, and graphene) electrodes by using electronic structure calculations and quantum transport simulations. It is highlighted that the stronger interlayer coupling allows CrC 2 N 4 to form an n-type Ohmic contact with Ti electrode in the vertical direction. Furthermore, the absence of tunneling barrier at the CrC 2 N 4 -Ti interface greatly improves the electron injection efficiency. On the other hand, the studied metals form Schottky contact with CrC 2 N 4 at the lateral interface due to Fermi level pinning (FLP) effects. Surprisingly, the strong FLP effects restrict the Schottky barrier heights of CrSi 2 N 4 -metal contacts to a narrow range. Where Ag, Au, Ni, Pd, Pt, Ti electrodes and Ag, Ti electrodes form ideal ohmic contact with CrSi 2 N 4 in the vertical and lateral directions, respectively, while the other metals form quasi-ohmic contact. Ti exhibits the highest contact performance as the electrode in both CrC 2 N 4 and CrSi 2 N 4 based FETs. The findings may provide fundamental understanding for designing high-performance and energy-efficient FETs based on CrX 2 N 4 .
theoretical studies have confirmed that CrSi 2 N 4 monolayer is a semiconductor with high carrier mobility, and capable of exhibiting extremely high mechanical strength, thermal conductivity and piezoelectricity. [24,25] On the other hand, CrC 2 N 4 monolayer is theoretically predicted as a direct bandgap semiconductor with considerable carrier mobility (≈1.2 × 10 3 cm 2 V −1 s −1 ) and large light absorption coefficient, which exhibits extremely high elastic modulus and tensile strength, lattice thermal conductivity, and piezoelectric coefficient as well. [26] These features indicate that CrC 2 N 4 and CrSi 2 N 4 monolayers have exceptional performance in the applications of nanoelectronic and optoelectronic devices.
In the line of continuous development of semiconductor technology, metal-semiconductor junctions (MSJs) have become an increasingly significant aspect in electronic devices, where the interface charge carrier transport and Schottky barrier height (SBH) play vital roles in the performance of device. [27,28] In particular, the Schottky barrier is essential for evaluating the charge transport properties of the metal-semiconductor contacts, [29,30] which hinder the carrier transport at the interface to some extent. The high contact resistance brought by the Schottky barrier directly affects the device transmission performance, power loss, and life. [31,32] Conversely, the small or absence of Schottky barriers mean low or no contact resistance, improving of the carrier transfer and injection efficiency, which is necessary for the design of high-performance nanoelectronic devices. [33] Since the majority of MSJs are essentially Schottky type, various intrinsic and extrinsic constraints make the realization of Ohmic contact extremely difficult. [34] At present, in order to decrease or eliminate the Schottky barrier, quasi-Ohmic contact has been widely studied. [35][36][37] Theoretically, the Schottky barrier can be controlled by the metal work function to obtain an ideal MSJ. [38] However, in practice, due to the presence of multiple non-ideal factors at the metal-semiconductor interfaces, such as metal/defect-induced gap states (MIGS/DIGS), mid-gap states and the formation of interface dipoles, which result in Fermi level pinning (FLP) and make the Schottky-Mott (SM) rule no longer applicable. [38][39][40] Interestingly, the studies show that FLP can be effectively avoided and approach the ideal SM limit at defect-free weakly interacting van der Waals (vdW) contacts. [41][42][43] However, the FLP in 2D semiconductors contacts with 3D metal remains to be resolved due to the strong interfacial interactions between 3D metals and the 2D semiconductors that induces FLP. [29,44] Furthermore, strong FLP is beneficial for creating consistently small Schottky barrier, resulting in Ohmic contact. [45] Therefore, from the perspective of realizing highperformance FETs, it is of great worthwhile to find strong FLP that can provide highly reliable metal-semiconductor contacts with small Schottky barrier or natural Ohmic contact.
In this work, we have systematically investigated the interfacial properties of CrX 2 N 4 -metal (X = C, Si) contact and CrX 2 N 4 based FETs with Ag, Au, Cu, Ni, Pd, Pt, Ti, and graphene electrodes using first-principles calculations and nonequilibrium Green's function simulations. For all the CrX 2 N 4 -metal contacts, the band structure nature of CrX 2 N 4 monolayer is well preserved. The obtained vertical and lateral SBHs show that the CrC 2 N 4 -Ti contact forms n-type Ohmic contact in the vertical direction and n-type Schottky contact with a smaller barrier in the lateral direction. For the other CrC 2 N 4 -metal interfaces, smaller vertical and lateral SBHs are formed. It is noted that ideal Ohmic contact and ultra-low barrier quasi-Ohmic contact are formed in both vertical and lateral directions at all the CrSi 2 N 4 -metal interfaces. Strong FLP is found in CrX 2 N 4 based FETs due to MIGS and the interaction between the channel CrX 2 N 4 and the electrodes. The schematic band structures of CrX 2 N 4 based FETs with different metal electrodes are depicted and discussed in light of these results.

Results and Discussion
The atomic structures of CrC 2 N 4 and CrSi 2 N 4 monolayers with the space group of P-6m2 are illustrated in Figure 1a,c, respectively. Due to the existence of Cr atom, the possible magnetic properties of CrX 2 N 4 monolayer are carefully examined. To examine the possibility of magnetism in the CrX 2 N 4 monolayer, we investigated three different states: namely nonmagnetic (NM), ferromagnetic (FM), and antiferromagnetic (AF). The calculation results indicate that both CrC 2 N 4 and CrSi 2 N 4 monolayers exhibit NM characteristics. The optimized lattice constants a for CrC 2 N 4 and CrSi 2 N 4 monolayers are 2.51 and 2.85 Å, respectively. The CrX 2 N 4 monolayer comprises two C-N (Si-N) sandwiched with a Cr-N layer. Figure 1b

www.advelectronicmat.de
both CrC 2 N 4 and CrSi 2 N 4 monolayers are very minimal, which can be further confirmed by the PDOS in Figure 1b,d. These results are consistent with the previous theoretical results. [26,46] Furthermore, the valence band of CrX 2 N 4 in the band structure is closer to the Fermi level, indicating a p-type semiconductor.
Next, we investigated the contact properties of CrX 2 N 4 monolayer based MSJs using Ag, Au, Cu, Ni, Pd, Pt, Ti, and graphene as the electrodes with work function values ranging from 4.40 to 5.82 eV. The optimized geometrical structures of all CrX 2 N 4 -metal interfaces are shown in Figure S1 (Supporting Information). In the MSJs formed between these metals and CrX 2 N 4 , the distance between the N atoms at the bottom of CrX 2 N 4 and the atoms at the top of the metal is defined as the equilibrium interlayer distance (d z ). As shown in Figure 2a, the optimized d z of all CrX 2 N 4 -metal contacts are in the range from 2.05 to 3.35 Å (Tables S1 and S2, Supporting Information). To further understand the interface binding, we calculated the binding energy (E b ) [47] defined as: where CrX N 2 4 E and M E are the total energy of isolated CrX 2 N 4 monolayer and clean metal surface, respectively. A is the interface area of the MSJ supercell. The calculated E b is illustrated in Figure 2b. The binding energy of 3D metal contacts lies in the range of 0.37 to 0.62 eV Å −2 , while for graphene contact, smaller binding energies of 0.17 eV Å −2 for CrSi 2 N 4 and 0.19 eV Å −2 for CrC 2 N 4 are formed. The interlayer distance and binding energy features indicate all the metal-semiconductor contacts are bonded by vdW type interactions. From the interlayer distance and binding energy result, relatively large d z (2.93 to 3.35Å) and low E b (0.17 to 0.56 eV Å −2 ) are formed in the CrC 2 N 4 (CrSi 2 N 4 ) contacts with Ag, Au, graphene (Au, Ti, graphene), indicating the vdW interactions in these cases are weak. As for the CrC 2 N 4 -Cu, −Ni, −Pd, −Pt and CrSi 2 N 4 -Ag, −Cu, −Ni, −Pd, −Pt systems, the interlayer distances (2.05-2.79 Å) are relatively smaller and binding energies (0.45-0.62 eV Å −2 ) are relatively large, suggesting the vdW interactions in these contacts are strong. Similar results have also been found in MoS 2 based MSJs. [48] To further understand the interface properties, we constructed a two-probe CrX 2 N 4 based FET device model with a channel length of 5 nm, as illustrated in Figure 3. The injection of carriers in the studied CrX 2 N 4 based FETs involves two representative interfaces: the interface B between the metal electrode A and the bottom CrX 2 N 4 region C, and the interface D between the source/drain region and channel region E. The CrX 2 N 4 based FETs contain vertical and lateral Schottky barriers in the interface B and D, which represent the energy barriers that need to be overcame for carriers to inject from the metal into the underlying CrX 2 N 4 monolayer (from A to C in Figure 3) and the source/drain into the channel (from C to E in Figure 3), as expressed as Φ V and Φ L , respectively. Herein, it is of great significance to determine the height of vertical and lateral Schottky barriers at interfaces B and D, since the Schottky barrier is a key parameter in metal-semiconductor contacts.
The vertical SBH Φ V can be estimated by the energy difference between the Fermi level of CrX 2 N 4 -metal contacts and the energy level of CBM or VBM in the CrX 2 N 4 band structure. In order to get the SBH, the projected band structures of CrX 2 N 4metal contacts were calculated ( Figure S2, Supporting Information). The calculated vertical electron and hole SBHs ( V e Φ and V h Φ ) for CrC 2 N 4 -metal and CrSi 2 N 4 -metal contacts are listed in Tables S1 and S2 (Supporting Information), respectively. In the following, we use Au and Ti contacts as examples of Schottky contacts and Ohmic contacts (Figure 4), respectively. The optimized geometrical structures of CrC 2 N 4 -Au, CrC 2 N 4 -Ti, CrSi 2 N 4 -Au, and CrSi 2 N 4 -Ti contacts are shown in Figure 4a, c, b and d, respectively. Interestingly, almost no band hybridization is found in the band structure of all these CrX 2 N 4 -metal contacts. The same conclusion can be achieved from the projected band structures and PDOS in Figure 4 and Figure S2 (Supporting Information) for the other metal contacts. The electron contributions near the CBM and VBM are mainly from the N-p and Cr-d orbitals. The corresponding partial charge density of the VBM and CBM were calculated, as shown in Figure 4. The semiconducting electronic states are penetrated deeply into the inner Cr-N core layer, while only minor parts are sparsely distributed in the Si-N outer layers. To further understand the electronic properties of CrX 2 N 4 -metal contacts, we calculated the plane-averaged charge density difference (Δρ), which is defined as

www.advelectronicmat.de
ρ , and ρ M are charge densities of CrX 2 N 4metal contact, isolated CrX 2 N 4 monolayer, and clean metal surface, respectively. The apparent charge accumulation and charge depletion indicating significant charge transfer at the metal-semiconductor contact interfaces can be observed in Figure 4 as well, which leads to the formation of an interface dipole and a built-in electric field at the interface. [49,50] The above results show that the band structure of semiconductors is well preserved in the metal-semiconductor contacts, and the electronic states of CBM and VBM are concentrated in the inner core layer. The electronic interaction mainly affects the outer C-N and Si-N layers through charge redistribution without penetrating the Cr-N core layer. Consequently, the Si-N layer in the septuple-layered structure of CrX 2 N 4 acts as an outer layer, which protects the semiconducting properties of CrX 2 N 4 . Similar phenomenon has also been confirmed in MoSi 2 N 4 and WSi 2 N 4 . [23] In the following, we analyze the contact types and Schottky barriers in the vertical direction for the CrX 2 N 4 -metals contacts. For CrC 2 N 4 -Au contact in Figure 4a, the CBM of CrC 2 N 4 locates above the Fermi level, which forms an n-type Schottky contact with electron SBH of 0.32 eV in the vertical direction. However, for CrC 2 N 4 -Ti contact in Figure 4c, the CBM of CrC 2 N 4 crosses the Fermi level to form an Ohmic contact. A slight band hybridization occurs at the CBM of CrC 2 N 4 -Ti contact, which corresponds to its lower interlayer distance of 2.31 Å. Similar phenomenon is observed in other CrC 2 N 4metal contacts (the d z of 2.31 Å for Cu and the d z of 2.05 Å for Ni). Although several MIGS appear in the CBM of CrC 2 N 4 , the VBM and the other parts of band structures are still well preserved. On the other hand, it is interesting to note that the CrSi 2 N 4 -metal contacts have obtained ultralow barrier Schottky contacts (namely quasi-Ohmic contacts) or Ohmic contacts in the vertical direction. For CrSi 2 N 4 -Au contact in Figure 4b and CrSi 2 N 4 -graphene contact in Figure S2b (Supporting Information), the CBM of CrC 2 N 4 locates only slightly above the Fermi level. The electron SBHs for CrSi 2 N 4 -Au and CrSi 2 N 4 -graphene contacts are calculated to 0.02 and 0.03 eV, respectively. While for CrSi 2 N 4 -Ag, CrSi 2 N 4 -Cu, CrSi 2 N 4 -Ni, CrSi 2 N 4 -Pd, CrSi 2 N 4 -Pt contacts in Figure S2b  In addition to the vertical barrier, the tunneling barrier (TB) in the metal-semiconductor contact is also an important parameter in the carrier transport process. [51] In the case of Au contact (Figure 4a,b), there are considerable barriers of Φ TB = 4.10 and 3.69 eV for CrC 2 N 4 and CrSi 2 N 4 , which seriously impedes the surface carrier transport. In contrast, in the case of Ti contact (Figure 4c,d), the absence of Φ TB for CrC 2 N 4 and a low barrier of Φ TB = 2.04 eV for CrSi 2 N 4 indicate the potential of Ti as an effective electrode for CrC 2 N 4 and CrSi 2 N 4 with a higher probability of carrier transfer. The Φ TB and corresponding carrier transport probabilities of various metal contacts of CrC 2 N 4 and CrSi 2 N 4 are described in detail below.
As mentioned above, the tunnel barrier formed at the metalsemiconductor interface hinders the carrier injection efficiency. [52] The tunnel barrier height Φ TB and the tunnel barrier width w TB can be determined from the effective electrostatic potential in Figure 4, Figures S3 and S4 (Supporting Information). The specific values of Φ TB and w TB are listed in Tables S3 and S4 (Supporting Information). The tunneling probability P TB [53] can be used to evaluate the transport efficiency of carriers, a small or zero tunneling barrier can improve the efficiency of carrier injection. The tunneling probability can be calculated as follows: where m e is the mass of the free electron, ħ is the reduced Planckʼs constant, Φ TB and w TB represent the tunnel barrier height and width. The tunneling probabilities of various metal contacts are represented in Figure 5a. For the CrC 2 N 4 -Ti contact, the absence of tunneling barrier makes the tunneling probability with maximum P TB (100%), and the same absence of electron SBH, indicating an excellent Ohmic contact with high carrier injection efficiency. The P TB for CrC 2 N 4 -Cu and CrC 2 N 4 -Ni contacts are calculated to be 50.48% and 88.74%, respectively, which are greater than 50%. Whereas in the CrSi 2 N 4 -metal contacts, all metals show small tunneling probability. Interestingly, the vdW-type contact of graphene with interlayer distances greater than 3 Å and exhibits very low tunneling probability P TB < 5%, indicating a low carrier transport efficiency in this vdW interface. www.advelectronicmat.de Figure 4. Side view of the optimized structure configuration, the projected band structures and the partial density of states, the decomposed partial charge density of valence band maximum (VBM) and conduction band minimum (CBM) states, the plane averaged charge density difference and the effective electrostatic potential for a) CrC 2 N 4 -Au, b) CrSi 2 N 4 -Au, c) CrC 2 N 4 -Ti, and d) CrSi 2 N 4 -Ti contacts. In the projected band structure plots, the red and gray circles represent the contributions from CrX 2 N 4 and metal layers, respectively.

www.advelectronicmat.de
In addition to the tunneling probability P TB , the obtained Φ TB and w TB can also be used to estimate the tunneling-specific resistivity (ρ t ), [27,54] which can be obtained from the Simon model as follows where α = 1 under an ideal square potential barrier, q is the charge of the electron. The absence of tunneling barrier leads to the calculated ρ t = 0 for the CrC 2 N 4 -Ti contact. While for the CrC 2 N 4 -Au/Ag/Pd/Pt/graphene and CrSi 2 N 4 -Ag/Au/Ni/graphene contacts, the calculated lies in the range of 0.127 × 10 −9 to 3.226 × 10 −9 Ωcm 2 . However, for metal contacts with small tunneling barrier width, Equation (4) produces an unphysical negative injection current. Therefore, based on the low-bias approximation, the tunnel specific conductivity t ρ * [23] is calculated as For the CrC 2 N 4 -Cu/Ni and CrSi 2 N 4 -Cu/Pd/Pt/Ti contacts, the t ρ * calculated by Equation (5) lies in the range of 0.005 × 10 −9 to 0.091 × 10 −9 Ωcm 2 , which are greater than the recently reported ultralow contact resistance for metal/MoSi 2 N 4 and metal/WSi 2 N 4 contacts (0.1 × 10 −9 to 3.85 × 10 −9 Ωcm 2 ) [23] and Bi/MoS 2 contact (≈1.81 × 10 −9 Ωcm 2 ). [27] Furthermore, t ρ can also be used for the calculation of contact resistance. The calculated t ρ and t ρ * according to Equations (4) and (5) for different metal contacts are listed in Table 1.
An adjustable SBH at the interface is what one would expect. Therefore, understanding the FLP effect is crucial for analyzing interface properties and improving device performance. To investigate the FLP effect at the metal-semiconductor interfaces, we calculated the Fermi level pinning factor S as [41,55] where Φ V/L is the vertical or lateral Schottky barriers of the CrX 2 N 4 -metal contact. The SM limit is restored when S = 1. Conversely, a strong FLP occurs when S = 0. In metal-semiconductor contacts, it is inherently difficult for FLP to reach the SM limit due to the existence of many nonideal factors. Among them, the formation of an interface dipole is one of the main factors affecting Fermi level pinning factor S, which can be defined by the potential step (ΔV) between the metal side and the semiconductor side in the MSJ. [56,57] Herein, ΔV = W contcat − W M , where W contact and W M are the work functions of metal-semiconductor contact and metal. As shown in Figure 5b, www.advelectronicmat.de the increase of work function difference between CrX 2 N 4 and metals lead to the increase of ΔV, and the ΔV value of CrC 2 N 4 is larger than that of CrSi 2 N 4 , indicating the existence of a larger interfacial dipole for CrC 2 N 4 . Therefore, we can modify SBH by V e Φ minus ΔV. As shown in Figure 5c, the pinning factor S of the CrC 2 N 4 -metal contacts is 0.25, while the corrected pinning factor S is 0.86, which is relatively close to the SM limit. Based on the above findings, it can be confirmed that ΔV and MIGS are the main causes of SM limit deviations in the CrC 2 N 4 -metal contacts. It is worth noting that we can find the pinning factor S of the CrSi 2 N 4 -metal contacts is only 0.05 in Figure 5d, indicating that the SBH is insensitive to the variation of the work function of metal electrodes. Furthermore, the corrected pinning factor S of the CrSi 2 N 4 -metal contacts reaches only 0.49, and no apparent MIGS is found at the CrSi 2 N 4 -metal interfaces. It was mentioned in previous work that a narrow bandgap leads to strong FLP and robust SBH. [58] Therefore, we speculate that the reason for such strong FLP of CrSi 2 N 4 -metal contacts due to narrow bandgap (0.49 eV) of the CrSi 2 N 4 monolayer and its intrinsic properties. The work function approximation (WFA) is a regularly used method to calculate the value of lateral SBH, [59] where the SBHs can be obtained by calculating the energy difference between the Fermi level of CrX 2 N 4 -metal contacts and the CBM/VBM of CrX 2 N 4 channel materials. [60,61] The calculated lateral electron and hole SBHs ( , L W e Φ and , L W h Φ ) obtained by the WFA method for CrC 2 N 4 -metal and CrSi 2 N 4 -metal contacts are listed in Tables S1 and S2 (Supporting Information), respectively. Although the WFA method is a simple way to estimate the lateral SBH at interface D, it ignores the coupling interactions between the electrode and channel regions. To obtain a more reliable and comprehensive lateral SBH, we further adopt the quantum transport simulations (QTS) using the constructed two-probe CrX 2 N 4 based FETs model in Figure 3. The calculated lateral electron and hole SBHs ( , obtained by the QTS method for CrC 2 N 4 -metal and CrSi 2 N 4metal contacts are listed in Tables S1 and S2 (Supporting Information) as well, respectively. The local device density of states (LDDOS) of CrX 2 N 4 based FETs with Ti and Cu electrodes under zero-bias and zero-gate voltage are presented in Figure 6. The LDDOS of the other metal electrodes are plotted in Figures S5 and S6 (Supporting Information) for references. As shown in Figure 6a,c, for the CrC 2 N 4 based FETs with Cu Table 1. Tunneling-specific resistivity ρ t or t ρ * (10 −9 Ωcm 2 ) of CrC 2 N 4metal and CrSi 2 N 4 -metal contacts.  Φ calculated by QTS is significantly different from the value obtained by the WFA method, which is attributed to the simplification of WFA. In contrast, an extremely low electronic SBH , L T e Φ of 0.03 eV is observed at interface D for the CrSi 2 N 4 -Cu contact (see Figure 6b), indicating the formation of a quasi-Ohmic contact. For the CrSi 2 N 4 -Ti contact (see Figure 6d), an ideal Ohmic contact with negative electronic SBH of −0.04 eV is formed at interface D. Notably, CrC 2 N 4 based FTEs with all metal electrodes form n-type Schottky contact at interface D (see Figure S5, Supporting Information). While CrSi 2 N 4 based FTEs with Ag electrode obtained barrier-free Ohmic contact, and the electrodes of Au, Ni, Pd, Pt, Ti, and graphene formed quasi-Ohmic contact with very low barrier at interface D (see Figure S6, Supporting Information). Therefore, the required ohmic contact can be achieved by Ag and Ti electrodes in the CrSi 2 N 4 based FETs, which is beneficial for the high-performance of 2D semiconductor based FETs.
The transport gap is the sum of the lateral electron and hole SBHs, which is calculated as These transmission bandgaps are essentially comparable to the 1.81 and 0.49 eV bandgaps of CrC 2 N 4 and CrSi 2 N 4 monolayers, respectively. Furthermore, Figure 6 and Figures S5 and S6 (Supporting Information) illustrate the transmission spectra of CrX 2 N 4 based FTEs at zero-bias and zero-gate voltage. It is noted that the lateral SBH obtained from the transmission spectrum is significantly larger than that obtained from the LDDOS due to the apparent CBM bending of channel CrX 2 N 4 . Moreover, the lateral SBH obtained from QTS and WFA are compared in Figure S7 (Supporting Information). The type of the lateral Schottky barriers for CrX 2 N 4 based FETs is essentially the same in both methods, but completely opposite in the CrC 2 N 4 -Ag/graphene and CrSi 2 N 4 -Ag/Cu/Pt contacts. Herein, the SBH obtained by these two methods differs, which is common in 2D semiconductor based FETs. [30,53,62] The reason for the difference is that the interaction between the electrode and the channel region is considered in the QTS, but it is ignored in the WFA. Furthermore, the FLP effect is induced at the metal-semiconductor interface due to the interaction between the electrode and the channel. [53,60] The degree of FLP can be assessed by the pinning factor S, which is expressed by the slope of the fitted line of the relationship between the SBH and the metal work function refer to Equation (6). As shown in Figure 7, the values of the pinning factor S obtained by fitting the QTS data are 0.20 and 0.06 for CrC 2 N 4 and CrSi 2 N 4 , respectively, indicating the emergence of a strong FLP between the CrX 2 N 4 monolayer and metal. The strong FLP mainly comes from MIGS and the interaction between the channel CrX 2 N 4 and the metal electrode. Contrastingly, the pinning factors S calculated by WFA are 0.42 and 0.52 for CrC 2 N 4 and CrSi 2 N 4 , respectively. The results show that the pinning factor S calculated by QTS is significantly smaller than that calculated by WFA without coupling into account.
Furthermore, the schematic band structures of CrX 2 N 4 based FETs with different metal electrodes are determined based on the electronic structure and QTS results. Four representative band structure types are selected and described in Figure 8. For the CrC 2 N 4 -Ti contact in Figure 8a, due to the absence of Schottky barrier and tunneling barrier at the interface, electrons can be readily injected from the electrode into the bottom CrC 2 N 4 . But the smaller Schottky barrier (0.16 eV) at interface D hinders the flow of electrons along the channel and reduces the efficiency of electron transport. For the CrC 2 N 4 -Cu contact in Figure 8b, although the n-type Ohmic contact is formed at interface B, the smaller lateral SBH at interface D and the smaller tunneling barrier at interface B still hinder electron injection and reduce the electron injection efficiency. Similar phenomenon is also occurred in the CrC 2 N 4 -Ni contact. For the CrC 2 N 4 -Au contact in Figure 8c, the low SBH at interfaces B and D, but the large tunneling barrier (4.1 eV) at interface B makes the electron injection efficiency drop sharply. Similar results are observed in the CrC 2 N 4 -Pd/Pt/graphene and CrSi 2 N 4 -Au/ Cu/Ni/Pd/Pt/graphene interfaces. For the CrSi 2 N 4 -Ag contact in Figure 8d, the n-type Ohmic contact at interfaces B and www.advelectronicmat.de D can promote electron injection and electron transport to the CrSi 2 N 4 channel. However, the presence of a large tunneling barrier (4.6 eV) at interface B reduces the electron injection efficiency, similar results are found in the CrSi 2 N 4 -Ti contact.
Lastly, our results show that the CrC 2 N 4 based FETs with Ti electrode have the highest electron transport efficiency, while the CrSi 2 N 4 based FETs with Ag and Ti electrodes are free of vertical and lateral Schottky barriers. Interestingly, the tunneling barrier of the Ti electrode is smaller than that of the Ag electrode. Therefore, we believe that Ti metal is a promising electrode material for CrX 2 N 4 based FETs.

Conclusion
In conclusion, we have comprehensively investigated the interfacial properties of CrX 2 N 4 based FETs with Ag, Au, Cu, Ni, Pd, Pt, Ti, and graphene electrodes using electronic structure calculations and quantum transport simulations. The results show that CrC 2 N 4 based FETs with Ti electrode form n-type Ohmic contact in the vertical direction, while the other metals form n-type Schottky contact with smaller barriers. Furthermore, the presence of 100% tunneling probability at the CrC 2 N 4 -Ti interface promotes electron injection. Combined with quantum transport simulations, it is found that all the CrC 2 N 4 -metal contacts form n-type Schottky contact with smaller barriers. Surprisingly, in both vertical and lateral directions, all the studied metals form ultra-low-barrier quasi-Ohmic contact or barrier-free Ohmic contact with CrSi 2 N 4 . Where the vertical and lateral SBHs of Ag and Ti electrodes are negative, indicating an Ohmic contact is formed. A strong FLP is observed in the vertical and lateral interfaces of CrX 2 N 4 based FETs, which is mainly due to the MIGS and the coupling between the electrode and the channel CrX 2 N 4 . These results indicate that the Ti electrode is the best candidate electrode material for CrX 2 N 4 based FETs, exhibiting the highest contact performance in the CrC 2 N 4 and CrSi 2 N 4 . Our study has instructive implications for designing high-performance FETs based on CrX 2 N 4 .

Computational Details
All calculations based on density functional theory (DFT) [63] were performed using the Vienna Ab initio Simulation Package (VASP) [64] with projector augmented wave (PAW) [65] method and adopted using the ALKEMIE platform. [66] The generalized gradient approximation (GGA) [67] with the Perdew-Burke-Ernzerhof (PBE) [68] were used to describe the exchangecorrelation functional. The plane wave basis was set with a cutoff energy of 500 eV. The positions of all atoms were relaxed until the maximum residual force per atom less than 0.01 eV Å −1 and the energy difference smaller than 10 −5 eV. For geometry relaxation and electronic properties calculations, the Monkhorst-Pack k-point mesh densities of 0.025 and 0.02 Å −1 were sampled in the Brillouin zone, [69] respectively. A vacuum space of 20 Å was set in the Z direction to avoid the spurious interaction. The DFT-D3 Method [70] was used to describe van der Waals interaction at interfaces.
We chose eight materials as the contact electrode, including Au, Ag, Cu, Ni, Pd, Pt, Ti, and graphene, which span a large work function range. Herein, except for graphene, five-layer metal atoms were used to simulate the bulk metals. And the top two layers were fixed to simulate the interior of bulk metals. The lattice parameters of bulk metal are tuned to match the supercells of CrC 2 N 4 and CrSi 2 N 4 monolayers. The 2 × 2 supercell of CrC 2 N 4 match the supercell of CrSi 2 N 4 match the 1 × 1 Cu (111)/Ni (111)/Graphene. As shown in Tables S1 and S2 (Supporting Information), the mismatches of lattice constant are less than 5% in all the metal-semiconductor interfaces, which is smaller than the limitation for lattice mismatch of heterostructures. [71] The nonequilibrium Green's function (NEGF) simulations [72] implemented by the Quantum Atomistix ToolKit (QuantumATK) 2022 package [73] were used for quantum transport calculations . In the device calculations, the linear combination of the atomic orbits (LCAO) basis set with the double-ζ plus polarization www.advelectronicmat.de (DZP) form was considered, and the GGA of PBE method was adopted to describe the exchange-correlation functional. The realspace mesh cutoff energy and electrode temperature were set as 75 Hartree and 300 K, respectively. The Monkhorst-Pack k-point meshes were sampled in the device with 1 × 50 × 50 in the electrode region and 1 × 50 × 1 in the channel region, respectively.
At the semiconductor-metal interface, the 2D semiconductor is heavily doped with carriers in contact with the metal electrode, resulting in a high shielding of electron-electron interactions by the doped carriers. Therefore, the DFT-GGA method based on the single-electron approximation can accurately estimate the bandgap and SBH of FETs. [62,74] For instance, the bandgap of 1.53 eV for doped MoSe 2 monolayer was calculated by the DFT-GGA method, which is consistent with the experimental value of 1.58 eV and the renormalized band gap of 1.59 eV calculated by the GW approximation. [75] In particular, the electron (hole) SBH of the monolayer/bilayer/trilayer black phosphorene  [76][77][78]

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.