⇦Chinese Physics Letters, 2023, Vol. 40, No. 9, Article code 097404 Superconductivity Modulated by Carbonization and Hydrogenation in Two-Dimensional MXenes $M_{2}$N ($M$ = Mo, W) Xin-Zhu Yin (尹新竹)1, Hao Wang (王浩)1, Qiu-Hao Wang (王秋皓)1, Na Jiao (焦娜)1*, Mei-Yan Ni (倪美燕)1, Meng-Meng Zheng (郑萌萌)1, Hong-Yan Lu (路洪艳)1*, and Ping Zhang (张平)1,2* Affiliations 1School of Physics and Physical Engineering, Qufu Normal University, Qufu 273165, China 2Institute of Applied Physics and Computational Mathematics, Beijing 100088, China Received 17 July 2023; accepted manuscript online 22 August 2023; published online 31 August 2023 *Corresponding authors. Email: j_n2013@126.com; hylu@qfnu.edu.cn; zhang_ping@iapcm.ac.cn Citation Text: Yin X Z, Wang H, Wang Q H et al. 2023 Chin. Phys. Lett. 40 097404    Abstract The superconductivity of two-dimensional (2D) materials has extremely important research significance. To date, superconducting transition temperatures ($T_{\rm c}$) of 2D superconductors are still far from practical applications. Previously, 2D MXene Mo$_2$N has been successfully synthesized [Urbankowski et al. Nanoscale 9 17722, (2017)]. We systematically investigate the effects of carbonization and further hydrogenation on the stability, electronic property and superconductivity of 1T- and 2H-$M_{2}$N ($M$ = Mo, W) based on first-principles calculations. The results show that the 1T-$M_{2}$N and 2H-$M_{2}$N ($M$ = Mo, W) are all dynamically and thermodynamically stable after carbonization and further hydrogenation. After carbonization, $T_{\rm c}$'s of 1T-$M_{2}$NC$_{2}$ ($M$ = Mo, W) are all increased, while $T_{\rm c}$'s of 2H-$M_{2}$NC$_{2}$ ($M$ = Mo, W) are all decreased. By further hydrogenation, the $T_{\rm c}$'s of 1T- and 2H-$M_{2}$NC$_{2}$H$_{2}$ are all increased. Among all of these structures, $T_{\rm c}$ of 1T-Mo$_2$NC$_2$H$_2$ is the highest one, reaching 42.7 K, and the corresponding electron-phonon coupling strength $\lambda$ is 2.27. Therefore, hydrogenation is an effective method to modulate $T_{\rm c}$'s of 2D $M_{2}$NC$_{2}$ ($M$ = Mo, W) materials.
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 DOI:10.1088/0256-307X/40/9/097404 © 2023 Chinese Physics Society Article Text Superconductivity is one of the most fascinating macroscopic quantum phenomena in condensed matter physics. How to increase $T_{\rm c}$'s of 2D materials is widely concerned by researchers. In recent years, many 2D materials with superconductivity have been reported, such as Li-decorated graphene,[1] magic angle bilayer graphene,[2] FeSe-SrTiO$_3$,[3] single-layer NbSe$_2$,[4] Ba$_2$N,[5] B$_3$N,[6] $M$B$_{3}$ ($M$ = Be, Ca, Sr)[7] and hole-doped MoS$_2$.[8] These 2D materials with superconductivity exhibit rich physical properties. In addition, 2D superconducting materials can be used in chip manufacturing, biomedicine, sensor design, power transmission, high-energy physics, and other fields. Therefore, 2D superconductors have become a very interesting frontier field due to their rich physical properties and potential applications valuable. At present, hydrogenation, electron/hole doping and strain are considered as effective methods to modulate the superconductivity of 2D superconductors. The $T_{\rm c}$ of hole-doped graphane with 5% concentration is above 90 K.[9] At the experimental accessible $4 \times 10^{14}$ cm$^{-2}$ hole doping and 16% tensile biaxial strain levels, the $T_{\rm c}$ of graphene is theoretically estimated to be as high as 30 K.[10] Phosphorene exhibits superconductivity with $T_{\rm c}$ of 16 K[11] at the typical doping concentration $3.0 \times 10^{14}$ cm$^{-2}$ when the biaxial strain reaches 4%. Though monolayer MoS$_2$ is a semiconductor, it can be transformed into a superconductor with $T_{\rm c}$ of 22 K at 30% hole doping and $-4$% biaxial compressive strain.[8] The $T_{\rm c}$ of monolayer MgB$_2$ is 20 K, which increases to more than 50 K at about 4% biaxial tensile strain.[12] After hydrogenation, the $T_{\rm c}$ of monolayer MgB$_2$ can reach 67 K.[13] More encouragingly, biaxial tensile strain can further modulate the $T_{\rm c}$ of hydrogenated MgB$_2$ as high as 100 K.[13] Furthermore, the $T_{\rm c}$ in NbSe$_2$ is gradually suppressed due to Li ionic intercalation and being subject to electric gating.[14] Monolayer PC$_3$ transforms from semiconductor to metal after hydrogenation with predicted $T_{\rm c}$ of 31 K.[15] The 2D metal hydride Janus MoSH monolayer is synthesized experimentally in realizing Janus transition metal dichalcogenides.[16,17] Theoretically, the $T_{\rm c}$ of MoSH is 28.58 K, and it can be boosted to 37.31 K by further aligning the Fermi level via electron doping.[18] In addition, the $T_{\rm c}$ of hydrogenated CaB$_3$, i.e., HCaB$_3$, is raised from 22.4 K to 39.3 K.[7] It can be seen that hydrogenation has significant effects on the superconductivity of 2D materials. MXene, which consists of carbon or nitrogen (X) sandwiched between transition metal (M) layers, can be easily prepared experimentally from their three-dimensional counterpart. Bekaert et al. predicted the superconductivity of Mo$_2$C, W$_2$C, Sc$_2$C, Mo$_2$N, W$_2$N and Ta$_2$N, among which the $T_{\rm c}$ (16 K) of Mo$_2$N is the highest one.[19] In recent years, the superconductivity of MXenes can also be modulated by hydrogenation and strain. The $T_{\rm c}$ of 2H-Mo$_2$C can be increased from 3.2 K to 12.6 K by hydrogenation.[20] Under 3% biaxial tensile stress, the $T_{\rm c}$ of 1T-Mo$_2$N decreases slightly from 24.7 K to 21 K.[21] Furthermore, functionalization is also a pathway towards robust superconductivity in MXenes. The $T_{\rm c}$'s of hydrogenated 1T-Mo$_2$N and 1T-W$_2$N are 32.4 K and 30.7 K, respectively.[22] Nb$_2$C is not superconducting in pristine form, but functionalized monolayers Nb$_2$CCl$_2$ and Nb$_2$CS$_2$ are superconductors, whose $T_{\rm c}$'s are 9.6 K and 10.7 K,[23] respectively. Recently, W$_2$N$_3$ has been experimentally synthesized[24] and the theoretical $T_{\rm c}$ of W$_2$N$_3$ has been reported to be 38 K,[25] which is close to the McMillan limit (39 K).[26] In addition, we find that the carbonized and hydrogenated 2H-Mo$_2$C possesses superconductivity with $T_{\rm c}$ of 53 K.[27] Thus, it is necessary to further modulate the superconductivity of MXenes. In this Letter, we systematically study the crystal structures, electronic structure, phonon dispersions, electron-phonon coupling (EPC), and superconducting properties of carbonized and hydrogenated MXenes $M _2$N ($M$ = Mo, W) by performing first-principles calculations. Based on the Eliashberg equation, it is found that these 2D metal materials are all phonon-mediated superconductors with $T_{\rm c}$ in the range of 6.7–42.7 K. Likewise, the $T_{\rm c}$ (42.7 K) of 1T-Mo$_2$NC$_2$H$_2$ is the highest one among these structures. The underlying mechanism of its superconductivity is carefully analyzed. Computational Details. The structural and electronic properties of carbonized and hydrogenated MXenes $M _2$N ($M$ = Mo, W) are studied based on the density functional theory calculations with the projector augmented wave (PAW) method,[28,29] as implemented in the VASP package.[30] The Perdew–Burke–Ernzerhof generalized gradient approximation[31] is employed and the electron-ion interaction is described by using the PAW method. The Fermi surface is broadened by the Gaussian smearing method with a width of 0.05 eV. All the geometries are relaxed, and convergence tolerances of force and energy are set to 0.01 eV and $10^{-6}$ eV, respectively. A vacuum separation is set to be more than 20 Å to prevent any interactions between two neighboring monolayers. A $12 \times 12 \times 1$ Monkhorst–Pack $k$-point mesh is used to sample the 2D Brillouin zone (BZ). To investigate the phonon dispersion and EPC, the density functional perturbation theory[32] calculations are performed with the Quantum ESPRESSO (QE) package.[33] The kinetic energy cutoffs of 80 and 800 Ry are chosen for the wave functions and the charge densities, respectively. The Methfessel–Paxton smearing width of 0.02 Ry is used. The BZ $k$-point[34] grids of $48 \times 48 \times 1$ and $24 \times 24 \times 1$ are adopted for the dense and sparse self-consistent electron-density calculations, respectively. The dynamic matrix and EPC matrix elements are calculated on $12 \times 12 \times 1$ $q$-point meshes for $M _2$NC$_2$ ($M$ = Mo, W) monolayers, respectively. The Fermi surfaces (FSs) colored as a function of an arbitrary scalar quantity in this work are drawn by using the FERMISURFER program.[35] The total EPC constant $\lambda$ is obtained via the isotropic Eliashberg function:[26,36,37] \begin{align} &\alpha^{2}F(\omega)=\frac{1}{{2\pi}{N(E_{\scriptscriptstyle{\rm F}})}}\sum_{\boldsymbol{q}\nu}\delta(\omega-\omega_{\boldsymbol{q}\nu})\frac{\gamma_{\boldsymbol{q}\nu}}{\hbar\omega_{\boldsymbol{q}\nu}}, \tag {1}\\ &\lambda=2\int_{0}^{\infty}\frac{\alpha^{2}F(\omega)}{\omega}\,d\omega=\sum_{\boldsymbol{q}\nu}^{}\lambda_{\boldsymbol{q}\nu}, \tag {2} \end{align} where $\alpha^{2} F(\omega)$ is the Eliashberg function, $N(E_{\scriptscriptstyle{\rm F}})$ is the electronic density of states (DOS) at the Fermi level, $\omega_{\boldsymbol{q}\nu}$ is the phonon frequency of the $\nu$th phonon mode with wave vector $\boldsymbol{q}$, and $\gamma_{\boldsymbol{q}\nu}$ is the phonon linewidth.[26,36,37] The $\gamma_{\boldsymbol{q}\nu}$ can be estimated by \begin{align} \gamma_{\boldsymbol{q}\nu}=\,&\frac{2\pi\omega_{\boldsymbol{q}\nu}}{\varOmega_{\rm BZ}} \sum_{\boldsymbol{k},n,m}| g^{\nu}_{\boldsymbol{k}n,\boldsymbol{k}+\boldsymbol{q}m}|^{2}\cdot\delta(\epsilon_{\boldsymbol{k}n}-E_{\scriptscriptstyle{\rm F}})\notag\\ &\cdot\delta(\epsilon_{\boldsymbol{k}+\boldsymbol{q}m}-E_{\scriptscriptstyle{\rm F}}), \tag {3} \end{align} where $\varOmega _{\rm BZ}$ is the volume of the BZ, $\epsilon_{\boldsymbol{k}n}$ and $\epsilon_{\boldsymbol{k}+\boldsymbol{q}m}$ indicate the Kohn–Sham energy, and $g^{\nu}_{\boldsymbol{k}n,\boldsymbol{k}+\boldsymbol{q}m}$ represents the screened electron-phonon matrix element; $\lambda_{\boldsymbol{q}\nu}$ is the EPC constant for phonon mode $\boldsymbol{q}\nu$, which is defined as \begin{align} \lambda_{\boldsymbol{q}\nu}=\frac{\gamma_{\boldsymbol{q}\nu}}{{\pi}\hbar{N(E_{\scriptscriptstyle{\rm F}})}{\omega^2_{\boldsymbol{q}\nu}}}. \tag {4} \end{align} $T_{\rm c}$ is estimated by the McMillan–Allen–Dynes formula:[37] \begin{align} T_{\rm c}=\frac{\omega_{\log}}{1.2}{\exp}\Big[\frac{-1.04(1+\lambda)}{\lambda-\mu^{*}(1+0.62\lambda)}\Big]. \tag {5} \end{align} The hysteretic Coulomb pseudopotential $\mu^{*}$ in Eq. (4) is set to 0.1 and logarithmic average of the phonon frequencies $\omega_{\log}$ is defined as \begin{align} \omega_{\log}={\exp}\Big[\frac{2}{\lambda}\int_{0}^{\omega}\alpha^{2}F(\omega) \frac{{\log}\;\omega}{\omega}d\omega\Big]. \tag {6} \end{align} For the strong EPC cases, i.e., $\lambda \geq 1.5$, $T_{\rm c}$ is estimated by \begin{align} T_{\rm c}=f_{1}f_{2}\frac{\omega_{\log}}{1.2}{\exp}\Big[\frac{-1.04(1+\lambda)}{\lambda-\mu^{*}(1+0.62\lambda)}\Big], \tag {7} \end{align} where $f_{1}$ and $f_{2}$ are the strong-coupling correction factor and the shape correction factor, respectively, with \begin{align} &f_{1}= \Big\{1+\Big[\frac{\lambda}{2.46(1+3.8\mu^{*})}\Big]^{3/2}\Big\}^{1/3}, \tag {8}\\ &f_{2}=1+\frac{(\omega_{2}/\omega_{\log}-1)\lambda^{2}}{\lambda^{2}+3.312(1+6.3\mu^{*})^{2}(\omega_{2}/\omega_{\log})^{2}}, \tag {9} \end{align} in which $\omega_{2}$ is defined as \begin{align} \omega_{2}=\Big[\frac{2}{\lambda}\int_{0}^{\omega_{\max}}\alpha^{2}F(\omega){\omega}d\omega\Big]^{1/2}. \tag {10} \end{align} In addition, we calculate the formation energy per atom of each structure. The corresponding calculation formulas are given as follows: \begin{align} E_{\rm form}(M_{2}{\rm N})=\,&\frac{1}{3}\Big[E(M_{2}{\rm N})-\big(2E(M)+\frac{1}{2}E({\rm N}_{2})\big)\Big], \tag {11} \end{align}% \begin{align} E_{\rm form}(M_{2}{\rm NC}_{2})=\,&\frac{1}{5}\Big[E(M_{2}{\rm NC}_{2})-\big(2E(M)+\frac{1}{2}E({\rm N}_{2})\notag\\ &+E({\rm graphene})\big)\Big], \tag {12} \end{align} \begin{align} E_{\rm form}&(M_{2}{\rm NC}_{2}{\rm H}_{2})=\frac{1}{7}\Big[E(M_{2}{\rm NC}_{2}{\rm H}_{2})-\big(2E(M)\notag\\ &+\frac{1}{2}E({\rm N}_{2})+E({\rm graphene})+E({\rm H}_{2})\big)\Big], \tag {13} \end{align} where $E$($M$), $E$(graphene), $E$(H$_2$), and $E$(N$_2$) are the total energies of free metal element $M$ ($M$ = Mo, W) (space group is $Im\bar{3}m$) in nature, graphene, hydrogen, and nitrogen molecules, respectively. Structure and Stability. Two phases (1T and 2H) of the monolayer $M _2$N ($M$ = Mo, W) have been studied in this work. The top and side views of 1T-$M_2$N ($M$ = Mo, W) are shown in Fig. 1(a). Its space group is $P\bar{3}m1$, where the metal atoms are located in the center of hexagon. The top and side views of 2H-$M_2$N ($M$ = Mo, W) are shown in Fig. 1(b). Its space group is $P\bar{6}m2$. Unlike 1T phase, the metal atoms are all on the vertex of hexagon, and none of the atoms are located in the center of hexagon. According to the different adsorption sites, each phase of $M _2$N ($M$ = Mo, W) has three carbonization structures. Among the structures of these different adsorption sites, we only consider the stable structures as shown in Figs. 1(c) and 1(d). The C, N, and metal atoms are shown in brown, cyan, and gray, respectively. From the side view, we can see that the C atoms are in the outer layer of the metal atoms and just above the N atoms. The 1T and 2H phases of $M _2$NC$_2$ ($M$ = Mo, W) possess different space group symmetries $P\bar{3}m1$ and $P\bar{6}m2$, respectively. The hydrogenated $M _{2}$NC$_{2}$, named as $M _{2}$NC$_{2}$H$_{2}$ ($M$ = Mo, W), are shown in Figs. 1(e) and 1(f), where the H atoms locate just above the C atoms of 1T- and 2H-$M _{2}$NC$_{2}$ ($M$ = Mo, W).
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Fig. 1. (a) The top (upper panel) and side (lower panel) views of 1T-$M _2$N($M$ = Mo, W), (b) 2H-$M _2$N($M$ = Mo, W), (c) 1T-$M _2$NC$_2$, (d) 2H-$M _2$NC$_2$, (e) 1T-$M _2$NC$_2$H$_2$, and (f) 2H-$M _2$NC$_2$H$_2$.
Table 1. Calculated lattice constant $a$ (Å), formation energy ($E _{\rm form}$) (eV/atom), and bond lengths (Å) of N–$M$, $M$–C and C–H, in which $M$ stands for Mo or W atom.
Materials $a$ $E_{\rm form}$ N–$M$ $M$–C C–H
1T-Mo$_2$N 2.78 $-1.02$ 2.14
1T-Mo$_2$NC$_2$ 2.98 $-0.27$ 2.25 2.03
1T-Mo$_2$NC$_2$H$_2$ 2.95 $-1.23$ 2.17 2.13 1.09
2H-Mo$_2$N 2.82 $-1.12$ 2.13
2H-Mo$_2$NC$_2$ 2.97 $-0.30$ 2.25 2.03
2H-Mo$_2$NC$_2$H$_2$ 2.96 $-1.28$ 2.20 2.11 1.10
1T-W$_2$N 2.77 $-0.82$ 2.17
1T-W$_2$NC$_2$ 2.98 $-0.27$ 2.27 2.02
1T-W$_2$NC$_2$H$_2$ 2.97 $-1.22$ 2.19 2.12 1.09
2H-W$_2$N 2.79 $-0.95$ 2.16
2H-W$_2$NC$_2$ 2.97 $-0.31$ 2.27 2.03
2H-W$_2$NC$_2$H$_2$ 2.98 $-1.31$ 2.21 2.10 1.09
Table 1 shows the lattice constants of these 12 structures and the distance between atoms. It can be seen that compared with the intrinsic structure, the length of N–$M$ ($M$ = Mo, W) after carbonization increased by 0.1–0.12 Å, and the length of N–$M$ ($M$ = Mo, W) after further hydrogenation increased by 0.02–0.07 Å. The $M$–C ($M$ = Mo, W) bond length of $M _2$NC$_2$H$_2$ ($M$ = Mo, W) is approximately 0.1 Å longer than that of $M _2$NC$_2$ ($M$ = Mo, W). In addition, the C–H bond lengths of $M _2$NC$_2$H$_2$ ($M$ = Mo, W) are approximately the same. In order to study the stability of $M _2$NC$_2$ and $M _2$NC$_2$H$_2$ ($M$ = Mo, W), we calculate the formation energy and phonon dispersion in various structures. The negative formation energy suggests the high stability of the strongly bonded $M_2$NC$_2$ and $M_2$NC$_2$H$_2$ ($M$ = Mo, W). The calculated formation energies of $M_2$NC$_2$H$_2$ ($M$ =  Mo, W) are comparable to those of the experimentally fabricated 1T-Mo$_2$N and 1T-W$_2$N,[38] which are computed at the same level in this work. The absence of imaginary frequencies in the phonon dispersion indicates that all the structures except for 1T-W$_2$N are dynamically stable, which will be given later and in Fig. S1 (see the Supporting Information). Electronic Structure. We calculate the band structure and DOS of the monolayer $M _2$NC$_2$ ($M$ = Mo, W) along the $\varGamma$–$M$–$K$–$\varGamma$ high symmetry path by first-principles, as shown in Fig. 2. Obviously, it can be found that all of these structures are metal. Since Mo and W are transition metal elements of the same family, the electronic structure and the properties of EPC have many similarities, thus in the main text, we only show the properties of compounds containing Mo, and the properties of compounds containing W are presented in the Supporting Information. From the orbital-projected band (PBAND) structures of Figs. 2(a) and 2(f), it can be seen that both 1T- and 2H-Mo$_2$NC$_2$ have two energy bands crossing the Fermi level. From Figs. 2(a) and 2(b), 2(f) and 2(g), we can see that the energy bands at the Fermi energy levels of these two structures are mainly contributed by the $d$ orbitals of Mo or W, and $p$ orbitals of C. Figures 2(c) and 2(h) show the total DOS and the contribution of each element. It can be seen that the DOS at the Fermi level is mainly contributed by Mo element, followed by C and N. The partial densities of states (PDOSs) of 1T- and 2H-Mo$_2$NC$_2$ are shown in Figs. 2(d) and 2(i), which are consistent with the analyses of the PBAND diagram. It can be seen that the contribution of Mo to the DOS at the Fermi level mainly comes from the $d _{z^2}$, $d _{zx}$, and $d _{yz}$ orbitals, followed by $d _{xy}$ and $d _{{x^2}-{y^2}}$ orbitals. The electronic structures at the Fermi level are different for 1T- and 2H-Mo$_2$NC$_2$, although it is mainly contributed by the $d$ orbitals for both of them. There is an electronic pocket near the $\varGamma$ point of 1T phase, but there is no pocket at the $\varGamma$ point of 2H phase. Therefore, the electronic states near the Fermi level of 1T phase are more abundant. The FSs of 1T-Mo$_2$NC$_2$ and 2H-Mo$_2$NC$_2$ exhibit different characteristics. Figure 2(e) shows the FSs of 1T-Mo$_2$NC$_2$, from which it can be seen that at the $\varGamma$ point there is a hexagonal petal-shaped electronic pocket with a circle-shaped electronic pocket, and at each $K$ point there is an electronic pocket around a circular electronic pocket. The FSs of 2H-Mo$_2$NC$_2$ are plotted in Fig. 2(j). The shape of the electronic pocket at each $K$ point is similar to the shape of the electronic pocket at the $K$ point of 1T-Mo$_2$NC$_2$, except that there is no electronic pocket at the $\varGamma$ point of 2H-Mo$_2$NC$_2$, instead a circle-shaped electronic pocket appearing on the $K$ point to the $\varGamma$ point path.
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Fig. 2. Electronic band structures of (a) Mo atoms, (b) N and C atoms of 1T-Mo$_2$NC$_2$. Electronic band structures of (f) Mo atoms, (g) N and C atoms of 2H-Mo$_2$NC$_2$. The total DOS of (c) 1T-Mo$_2$NC$_2$ and (h) 2H-Mo$_2$NC$_2$ and the total DOS of Mo, N and C atoms. The partial densities of states (PDOS) of (d) 1T-Mo$_2$NC$_2$ and (i) 2H-Mo$_2$NC$_2$. The top views of FSs of (e) 1T-Mo$_2$NC$_2$ and (j) 2H-Mo$_2$NC$_2$. The color bar represents Fermi velocity ($v _{\scriptscriptstyle{\rm F}}$), and the color from blue to red represents Fermi velocity from small to large.
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Fig. 3. Electronic band structures on (a) Mo atoms, (b) N and C atoms of 1T-Mo$_2$NC$_2$H$_2$. Electronic band structures on (f) Mo atoms, (g) N and C atoms of 2H-Mo$_2$NC$_2$H$_2$. The total DOS of (c) 1T-Mo$_2$NC$_2$H$_2$ and (h) 2H-Mo$_2$NC$_2$H$_2$ and the total DOS of Mo, N and C atoms. The partial densities of states (PDOS) of (d) 1T-Mo$_2$NC$_2$H$_2$ and (i) 2H-Mo$_2$NC$_2$H$_2$. The top views of FSs of (e) 1T-Mo$_2$NC$_2$H$_2$ and (j) 2H-Mo$_2$NC$_2$H$_2$. The color bar represents Fermi velocity ($v _{\scriptscriptstyle{\rm F}}$), and the colors from blue to red represent Fermi velocity from small to large.
In order to explore the effect of hydrogenation on properties of the electronic structures of 1T- and 2H-$M _{2}$NC$_{2}$ ($M$ = Mo, W), we calculate the electronic bands of these structures, as shown in Fig. 3 and in the Supporting Information. For 1T- and 2H-Mo$_{2}$NC$_{2}$H$_{2}$, the largest contribution at the Fermi level is mainly contributed by the $d _{z^2}$, $d _{zx}$, and $d _{yz}$ orbitals, followed by the $d _{xy}$ and $d _{{x^2}-{y^2}}$ of Mo element, which can be seen in the PBANDS of Figs. 3(a) and 3(b) and Figs. 3(f) and 3(g). The contribution of H element to the electronic band at the Fermi level is small for all these structures, which has been ignored in this work. Figures 3(c) and 3(d), 3(h) and 3(i) are the DOS and PDOS of 1T-Mo$_2$NC$_2$H$_2$ and 2H-Mo$_2$NC$_2$H$_2$, which also show the same results as PBANDs. The FSs of 1T-Mo$_2$NC$_2$H$_2$ and 2H-Mo$_2$NC$_2$H$_2$ are shown in Figs. 3(e) and 3(j), respectively, which are obviously different from the ones of 1T-Mo$_2$NC$_2$ and 2H-Mo$_2$NC$_2$. For 1T-Mo$_2$NC$_2$H$_2$, there are two FSs, which are composed of a circle centered on the $\varGamma$ point and an electronic pocket approximately in the shape of petals. The FSs of 2H-Mo$_2$NC$_2$H$_2$ consist of a circle around the $\varGamma$ point and two electronic pockets around the $M$ and $K$ point, respectively. The electronic pocket near the $\varGamma$ point is more conducive to the formation of Cooper pairs of electrons, thus enhancing the EPC and $T_{\rm c}$. The electronic properties of 1T-W$_2$NC$_2$, 2H-W$_2$NC$_2$, 1T-W$_2$NC$_2$H$_2$, and 2H-W$_2$NC$_2$H$_2$ are similar to those of 1T-Mo$_2$NC$_2$, 2H-Mo$_2$NC$_2$, 1T-Mo$_2$NC$_2$H$_2$, and 2H-Mo$_2$NC$_2$H$_2$, respectively. Electron–Phonon Coupling and Superconductivity. Firstly, we calculate the phonon dispersion of the intrinsic 1T- and 2H-$M _2$N ($M$ = Mo, W), which are shown in Fig. S1 (see the Supporting Information). We can see that the absence of imaginary phonons confirms the dynamical stability of monolayer 1T-Mo$_2$N, 2H-Mo$_2$N, and 2H-W$_2$N. However, 1T-W$_2$N has a CDW phase transition, which is consistent with the previous work.[19] Meanwhile, chemical adsorption can stabilize the structure.[19] For metallic materials, we further investigate their superconductivity based on first-principles calculations. Figures 4(a) and 4(e) show the phonon dispersion with phonon linewidth $\gamma _{{\boldsymbol q}\nu}$ (red hollow circles) for 1T- and 2H-Mo$_2$NC$_2$, respectively. There are 5 atoms in each unit cell. Thus, there will be 15 phonon branches, including 3 acoustic bands and 12 optical bands. It shows a wide range of frequencies extending up to about 700 and 687 cm$^{-1}$ for 1T- and 2H-Mo$_2$NC$_2$, respectively. The absence of imaginary frequency shows the dynamical stability for both the carbonized structures.
cpl-40-9-097404-fig4.png
Fig. 4. Phonon dispersion of (a) 1T-Mo$_2$NC$_2$ and (e) 2H-Mo$_2$NC$_2$. The red hollow circles indicate the phonon linewidth $\gamma _{{\boldsymbol q}\nu}$. The magnitude of $\gamma _{{\boldsymbol q}\nu}$ is shown with an identical scale in all figures for comparison. Phonon dispersion of (b) 1T-Mo$_2$NC$_2$ and (f) 2H-Mo$_2$NC$_2$ weighted by the in-plane and out-plane vibration modes of Mo, N, and C elements. Phonon density of states (PhDOS, including the atom-resolved contributions) of (c) 1T-Mo$_2$NC$_2$ and (g) 2H-Mo$_2$NC$_2$. Eliashberg spectral function $\alpha ^2 F(\omega)$ and cumulative frequency-dependent EPC function $\lambda(\omega)$ of (d) 1T-Mo$_2$NC$_2$ and (h) 2H-Mo$_2$NC$_2$.
As can be seen from Fig. 4(b), the in-plane vibration of N contributes the most in 1T-Mo$_2$NC$_2$, and followed by the out-of-plane vibration of Mo and C to the low-frequency range of 0–276 cm$^{-1}$. In the frequency range of 293–426 cm$^{-1}$, it is mainly contributed by the in-plane vibration of C and the out-of-plane one of N. In the range of 428–700 cm$^{-1}$, it is mainly contributed by the in-plane vibration of N and the out-of-plane one of C to the phonon dispersion. Figure 4(c) shows the total phonon density of states (PhDOS) and the contribution of each element. In the low-frequency range of 0–280 cm$^{-1}$, the contribution of Mo element is the most one to PhDOS. In the frequency range of 288–425 cm$^{-1}$, the N element has the largest contribution to the PhDOS. For frequency range above 425 cm$^{-1}$, Mo element has the largest contribution to PhDOS. Figure 4(d) shows the Eliashberg spectral function $\alpha ^2 F(\omega)$ and cumulative frequency-dependent EPC function. From Fig. 4(d), we can see that the contribution of phonons to EPC ($\lambda=1.49$) in the frequency range of 0–280 cm$^{-1}$ reaches 64$\%$, and the contribution of phonons in the frequency range 280–700 cm$^{-1}$ reaches 36$\%$. The phonon dispersion of 2H-Mo$_2$NC$_2$ is shown in Fig. 4(e), with the lowest acoustic branch contributing mostly to $\lambda$, which is also consistent with the distribution of the Eliashberg spectral function $\alpha ^2 F(\omega)$ in Fig. 4(h) and the accumulation of EPC intensity $\lambda({\omega})$. From Fig. 4(f) we can see that the contribution in the low-frequency region 0–275 cm$^{-1}$ mainly comes from the vibrations inside and outside the Mo atomic plane. The out-of-plane vibration of N and the in-plane vibration of C contribute mostly to the intermediate-frequency region of 296–480 cm$^{-1}$. The in-plane and out-of-plane vibrations of C contribute mostly to the high-frequency region 480–686 cm$^{-1}$. The PhDOS of 2H-Mo$_2$NC$_2$ is very similar to the one of 1T-Mo$_2$NC$_2$, as shown in Fig. 4(g). The main contribution of 2H-Mo$_2$NC$_2$ in the low-frequency region of 0–275 cm$^{-1}$ comes from the Mo element. The contribution of the intermediate-frequency band at 296–480 cm$^{-1}$ mainly comes from the N and C elements. The main contribution of the high-frequency region at 480–686 cm$^{-1}$ comes from the C element. This also confirms the previous analysis for phonon dispersion. The contribution of different frequency ranges to EPC can be seen from Fig. 4(h). The contribution of low-frequency (0–275 cm$^{-1}$), intermediate-frequency (296–480 cm$^{-1}$) and high-frequency (480–686 cm$^{-1}$) to EPC ($\lambda=1.09$) is 41%, 22% and 37%, respectively. The phonon linewidth and Elishberg spectral functions of 1T-Mo$_2$NC$_2$H$_2$ are illustrated in Fig. 5(a). The EPC is mainly contributed by the low-frequency parts, which contribute to the EPC ($\lambda=2.27$) of about 72$\%$. Figures 5(b) and 5(d) show that the low-frequency part of 0–254 cm$^{-1}$ is mainly contributed by the in-plane vibration modes of Mo. In the ranges of 254–424 cm$^{-1}$ and 600–650 cm$^{-1}$, the in-plane vibration modes of C element have the most contributions. In addition, the in-plane and out-plane contributions of N are mainly to the intermediate-frequency among 454–556 cm$^{-1}$. Meanwhile, the out-plane vibration modes of H mainly contribute to the high-frequency band of 704–760 cm$^{-1}$ and 3059–3066 cm$^{-1}$ as shown in Fig. 5(c), which is consistent with the results of PhDOS as shown in Fig. 5(d).
cpl-40-9-097404-fig5.png
Fig. 5. Phonon dispersion of (a) 1T-Mo$_2$NC$_2$H$_2$ and (f) 2H-Mo$_2$NC$_2$H$_2$. The red hollow circles indicate the phonon linewidth $\gamma _{{\boldsymbol q}\nu}$. The magnitude of $\gamma _{{\boldsymbol q}\nu}$ is shown with an identical scale in all figures for comparison. Phonon dispersion of 1T-Mo$_2$NC$_2$H$_2$ weighted by the in-plane (b) and out-plane (c) vibration modes of Mo, N, H and C elements. Phonon dispersion of 2H-Mo$_2$NC$_2$H$_2$ weighted by the in-plane (g) and out-plane (h) vibration modes of each element. Phonon density of states (PhDOS, including the atom-resolved contributions) of (d) 1T-Mo$_2$NC$_2$H$_2$ and (i) 2H-Mo$_2$NC$_2$H$_2$. Eliashberg spectral function $\alpha ^2 F(\omega)$ and cumulative frequency-dependent EPC function $\lambda$($\omega$) of (e) 1T-Mo$_2$NC$_2$H$_2$ and (j) 2H-Mo$_2$NC$_2$H$_2$.
The phonon linewidth of 2H-Mo$_2$NC$_2$H$_2$ is shown in Fig. 5(f). The difference of phonon linewidth between 1T- and 2H-Mo$_2$NC$_2$H$_2$ lies in the intermediate-frequency part. According to the analysis of Figs. 5(f) and 5(j), the low-frequency part contributes 41$\%$ to the EPC ($\lambda=0.98$), and the intermediate-frequency part contribute 48$\%$ to the EPC ($\lambda=0.98$). The in-plane and out-of-plane vibrations of phonons and the density of phonon states are shown in Figs. 5(g), 5(h), and 5(i). We can see that Mo contributes mostly to the low-frequency part of 0–300 cm$^{-1}$, C contributes mostly to the part of 309–485 cm$^{-1}$, N contributes mostly to the part of 485–560 cm$^{-1}$, and H contributes mostly to the high-frequency section of 575–714 cm$^{-1}$ and 3045–3051 cm$^{-1}$. According to the analyses of 1T-Mo$_2$NC$_2$H$_2$ and 2H-Mo$_2$NC$_2$H$_2$, although the configurations are different, the contributions to different frequency bands of phonons are similar for each element.
Table 2. Logarithmic averaged phonon frequency $\omega_{\log}$ (K), total EPC constant $\lambda$, strong-coupling correction factor $f_1$, shape correction factor $f_2$, and estimated $T_{\rm c}$ (K) for $M _2$N, $M _2$NC$_2$ and $M _2$NC$_2$H$_2$.
Materials $\omega_{\log}$ $\lambda$ $f_1$ $f_2$ $T_{\rm c}$
1T-Mo$_2$N 1.20 16 Ref. [19]
Mo$_2$NH$_2$-MD3 2.68 32.4 Ref. [22]
1T-Mo$_2$N 145.10 1.71 1.107 1.033 21.2 This work
1T-Mo$_2$NC$_2$ 254.03 1.49 1.089 1.016 31.7 This work
1T-Mo$_2$NC$_2$H$_2$ 227.22 2.27 1.156 1.041 42.7 This work
2H-Mo$_2$N 149.78 1.69 1.106 1.050 22.1 This work
2H-Mo$_2$NC$_2$ 207.50 1.09 16.4 This work
2H-Mo$_2$NC$_2$H$_2$ 398.02 0.98 27.0 This work
1T-W$_2$N 1.06 9.8 Ref. [19]
W$_2$NH$_2$-MD1 1.91 30.7 Ref. [22]
1T-W$_2$NC$_2$ 220.09 1.02 16.5 This work
1T-W$_2$NC$_2$H$_2$ 180.16 2.31 1.160 1.099 36.2 This work
2H-W$_2$N 153.10 1.69 1.105 1.037 22.2 This work
2H-W$_2$NC$_2$ 340.17 0.57 6.7 This work
2H-W$_2$NC$_2$H$_2$ 388.14 0.72 14.6 This work
cpl-40-9-097404-fig6.png
Fig. 6. Evaluated $T_{\rm c}$'s of 1T- and 2H-$M$NC$_{2}$ ($M$ = Mo, W) and 1T- and 2H-$M$NC$_{2}$H$_{2}$ ($M$ = Mo, W) versus Coulomb pseudopotential $\mu^*$. The vertical line marks the value $\mu^*=0.10$ used in this work.
The superconductivity related parameters of $M _2$N, $M _2$NC$_2$, and $M _2$NC$_2$H$_2$ ($M$ = Mo, W) in the 1T and 2H phases are listed in Table 2. It is seen that the intrinsic $T_{\rm c}$ of 1T phase is generally higher than that of 2H phase. This is because the lambda which represents the strength of electron–phonon coupling of 1T phase is generally higher than that of 2H phase, which can be seen from the figures of Eliashberg spectral function $\alpha ^2 F(\omega)$ and cumulative frequency-dependent EPC function $\lambda$($\omega$). After carbonization, the $T_{\rm c}$ of 2H phase structure is slightly lower than that of 1T phase. We can see that 1T-Mo$_2$NC$_2$H$_2$ has the highest $T_{\rm c}$ after hydrogenation, reaching 42.7 K, followed by 1T-W$_2$NC$_2$H$_2$ with $T_{\rm c}$ of 36.2 K. It should be noted that the $T_{\rm c}$ of 1T-Mo$_2$N shows a 30% change compared to the previous work of Bekaert et al., as also presented in Table 2 for comparison.[19,22] We believe that this change comes from the differences in computational software, and we have considered the $f_1$ and $f_2$ correction factors. From McMillan–Allen–Dynes formulas (5) and (7), we can see that $T_{\rm c}$ has a tendency to monotonically change with Coulomb pseudopotential. In order to verify it, we take a $\mu^*$ value in the range of 0.05–0.15 and calculate its corresponding $T_{\rm c}$, and the result is shown in Fig. 6. We can see from Fig. 6 that $T_{\rm c}$ decreases monotonically as $\mu^*$ increases. In this study, we choose the $T_{\rm c}$ corresponding to the traditional empirical value $\mu^*=0.10$. In addition, we can also see from Fig. 6 that the $T_{\rm c}$'s of 1T- and 2H-Mo$_2$NC$_2$ (Mo$_2$NC$_2$H$_2$) are higher than those of the corresponding phases for W$_2$NC$_2$ (W$_2$NC$_2$H$_2$), respectively. In a word, when the compounds have the different elements but the same phase, the compounds containing Mo have higher $T_{\rm c}$'s than those containing W. When compounds have the different phases but the same elements, the $T_{\rm c}$ of the 1T phase is higher than that of the 2H phase. In conclusion, we have predicted several kinds of conventional phonon-mediated 2D superconductors based on first-principles calculations, systematically. Firstly, the dynamic stable 1T- and 2H-$M _2$NC$_2$ ($M$ = Mo, W) structures are obtained by carbonized 1T- and 2H-$M _2$N ($M$ = Mo, W). The electronic structures show that they are all metal. By further calculating the EPC, we find that the $T_{\rm c}$ of 1T-Mo$_2$NC$_2$ reaches 31.7 K, which is the highest among carbonized $M _2$N ($M$ = Mo, W). The $T_{\rm c}$ increases significantly after hydrogenation, with the highest $T_{\rm c}$ of 42.7 K of 1T-Mo$_2$NC$_2$H$_2$. The calculated results show that the compounds containing Mo have higher $T_{\rm c}$ than the ones containing W, and the 1T phase has a higher $T_{\rm c}$ than the 2H phase. Therefore, the superconductivity of $M _2$N ($M$ = Mo, W) can be modulated by carbonization and hydrogenation. Acknowledgements. This work was supported by the National Natural Science Foundation of China (Grant Nos. 12074213, 11574108, and 12104253), the Major Basic Program of Natural Science Foundation of Shandong Province (Grant No. ZR2021ZD01), the Natural Science Foundation of Shandong Provincial (Grant No. ZR2023MA082), the Project of Introduction and Cultivation for Young Innovative Talents in Colleges and Universities of Shandong Province, and the Texas Center for Superconductivity at University of Houston, the Robert A. Welch Foundation (Grant No. E-1146). 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