Theoretical prediction of superconductivity in two-dimensional MXenes of molybdenum carbides

Abstract

Theoretically and experimentally, MXenes consisting of Mo and C have aroused much interest due to superconductivity in their films and even monolayer forms. Here, based on first-principles calculations, we systematically calculate the electronic structure, phonon dispersion, and electron–phonon coupling (EPC) of monolayer Mo₂C (both T- and H-phases), Mo₃C₂, and Mo₃C₃. The results show that H-Mo_xC_y (x = 2 or 3, y = 1–3) always have lower total energies than their corresponding T phase and other configurations. All these two-dimensional (2D) molybdenum carbides are metals and some of them display weak phonon-mediated superconductivity at different superconducting transition temperatures (T_c). The Mo 4d-orbitals play a critical role in their electronic properties and the Mo atomic vibrations play a dominant role in their low-frequency phonons, EPC, and superconductivity. By comparison, we find that increasing the Mo content can enhance the EPC and T_c. Besides, we further explore the impact of strain engineering on their superconducting related physical quantities. With increasing biaxial stretching, the phonon dispersions are gradually softened to form some soft modes, which can trigger some peaks of α²F(ω) in the low-frequency region and evidently increase the EPC λ. The T_c of H-Mo₂C can be increased up to 11.79 K. Upon further biaxial stretching, charge density waves may appear in T-Mo₂C, H-Mo₃C₂, and H-Mo₃C₃.

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1 Introduction

Since the successful fabrication of graphene¹ and monolayers of transition metal dichalcogenides (TMDs),² two-dimensional (2D) materials have attracted much attention due to their fascinating and exotic electronic, optical, mechanical, magnetic, and thermal properties. Among them, 2D transition metal carbides, carbonitrides, and nitrides (MXenes) have expanded rapidly since the discovery of Ti₃C₂ in 2011.³ The mechanical flexibility, easy coating capability, superb carrier anisotropic mobility, hydrophilic nature, and high conductivity enable their good applications in energy storage,⁴ electromagnetic interference shielding,⁵ (photo)electrocatalysis,⁶etc.

In condensed matter physics, quantum states of superconductivity⁷ and charge density waves (CDW)⁸ in 2D materials are fascinating subjects. In recent years, due to the development of the low-dimensional nanostructure technology, such as chemical vapor deposition (CVD)⁹ and molecular beam epitaxy,¹⁰ superconductivity has been observed in ultrathin films of Mo₂C^9,11,12 and Nb₂ C MXenes.^13,14 In spite of the fact that experimental observation of superconductivity is still difficult for materials in the monolayer limit,¹⁵ in theoretical aspects, many 2D MXenes have been predicted to be superconductors. For example, the T phase Mo₂C monolayer has been predicted as an electron–phonon-mediated superconductor with a critical temperature T_c of about 6 K and its electron–phonon coupling (EPC) can be enhanced by bromine absorption.¹⁶ Monolayers W₂C, Sc₂C, Mo₂N, and Ta₂N have also been predicted to be superconducting with the highest T_c of 16 K.¹⁷ Superconductivity in Nb₂CT₂ (T = O, S, Se, or Te) MXenes was also theoretically analyzed and the T_c can be enhanced by biaxial tensile strain.¹⁸ In addition, multigap superconductivity in Mo₂ N was reported by applying strain¹⁹ and superconductivity in Mo₂ N, W₂N,²⁰ and 2H-Mo₂C₃²¹ can be greatly enhanced through hydrogenation. As for the CDW state, it has been theoretically analyzed in MXene W₂N by comparing phonon spectra using different broadening factors of the Fermi–Dirac smearing function.¹⁷ A commensurate CDW phase was reported in V₂CO₂.²²

Along with further development of nanotechnology, more and more high-quality and large-area MXenes should be synthesized and abundant quantum states should also be detected. In fact, large scale synthesis of 2D Mo₂CT_x has been achieved by selectively etching gallium from Mo₂Ga₂C.^23,24 Moreover, a large-area and uniform Mo₂C film can be grown on graphene in one step by ambient pressure CVD.²⁵ By salt-assisted CVD, a lattice spacing of 0.23 nm, which is very close to the thickness of the monolayer T-Mo₂C (2.32 Å) according to our present calculations, corresponding to the (101) plane of hexagonal Mo₂C and a clean surface without carbon deposition were observed.²⁶. Theoretically, one more energetically stable structure of Mo₂C (H-Mo₂C) was found and extended to many other MXenes.^27,28 Growing Mo₂C crystals consisting of mixed polymorphs and polytypes and explaining phase transitions by the gliding of the Mo layer along the Mo-armchair (〈101̄0〉) direction were reported by combination of experiment and theoretical simulations.²⁹ Structurally, both T and H phases of Mo₂C are constructed using a ‘Mo–C–Mo’ sandwich. For multilayer Mo₂C, the possible stacking styles are –Mo₂CX₂–Mo₂CX₂ – and Mo–C–Mo–C–Mo in experiments.^30–34 The former case has been investigated widely and the latter case has been less studied.

In the present work, based on first-principles density functional theory (DFT), we calculate the energies of the T and H phases of Mo₂C and also few-atom-thick Mo–C systems (Mo₃C₂ and Mo₃C₃). The stacking of H-Mo₂C is found to be always stable for different proportions of Mo : C. We then carefully calculate and investigate the electronic structures, phonon dynamics, and EPC of monolayer T(H)-Mo₂C and H phase of Mo₃C₂ and Mo₃C₃. The results indicate that the Mo 4d orbitals play an important role in their metallic features. T-Mo₂C exhibits stronger EPC than H-Mo₂C. The Mo atomic vibrations in the low-frequency region are mainly responsible for their EPC and superconductivity. After increasing the C content, contribution of the C-2p orbitals at the Fermi energy level is increased while the superconductivity is suppressed. Fortunately, superconductivity can be modulated by applying biaxial tensile strain in H-Mo₃C₂ and H-Mo₃C₃, and H-Mo₂C can be changed to a strong conventional superconductor. Upon further applying tensile strain, possible CDW states may emerge in T-Mo₂C, H-Mo₃C₂, and H-Mo₃C₃.

2 Computational details

We performed our DFT calculations using the well-established Quantum-ESPRESSO package.³⁵ The Perdew–Burke–Ernzerhof (PBE)³⁶ parameterized generalized-gradient approximation (GGA)^37,38 was used to describe the exchange-correlation interaction. The projector-augmented wave (PAW) potentials were utilized to describe the interaction between electrons and ionic cores. To avoid coupling within the periodic images, a sufficient vacuum space of 25 Å along the z direction was adopted in the structural models. The plane-wave kinetic-energy cutoff and the energy cutoff for charge density were set as 80 and 800 Ry, respectively, which have been tested for convergence. The convergence criteria of force and energy were set as 10⁻⁶ and 10⁻⁷ Ry a.u.⁻¹, respectively. In our calculations, the Gaussian smearing method with σ = 0.02 Ry was used. The density functional perturbation theory (DFPT)³⁹ was used to calculate the phonon dispersions. A 12 × 12 × 1 q-point grid and a denser 48 × 48 × 1 k-point, which have been carefully tested, were set for the EPC calculation.

The phonon linewidth that is extremely related to the strength of the EPC can be defined as

where g_qν(k,i,j) is the EPC matrix and could be obtained fromHere, V_SCF is the Kohn–Sham potential, ψ is the wave function, [small xi, Greek, circumflex] is the phonon eigenvector, and û is atomic displacement.

We can obtain the EPC constant λ(ω) from the Eliashberg spectral function

where α²F(ω) is the Eliashberg spectral function, ω_qν is the phonon frequency of the ν th phonon mode with wave vector q and N(E_F) is the density of state (DOS) at the Fermi level. Moreover, the total λ can also be obtained by summarizing λ_qνwhere λ_qν can be calculated by

The superconducting transition temperature T_c can be estimated by the Allen–Dynes-modified McMillan formula:^40,41

Here, μ* is the Coulomb pseudopotential and can be generally set as a typical value of 0.1.^42,43 The logarithmically averaged phonon frequency ω_log is defined as

3 Results and discussion

3.1 Crystal structures

Monolayers T- and H-Mo₂C possess a honeycomb-like lattice with space groups of P3̄m1 (No. 164) and P6̄m2 (No. 187), respectively [see Fig. 1(a)–(d)]. Similarly, these Mo₂C systems consist of three atomic layers within a hexagonal cell: two Mo layers and one C layer. Each C atom has six nearest-neighbor Mo atoms and each Mo atom has three nearest-neighbor C atoms. In a unit cell, there are two Mo atoms and one C atom, which is shown as black dashed lines in Fig. 1(a)–(d). Meanwhile, there is only one kind of Mo–C bond length with the same value in each Mo₂C. Differently, the lattice constants of T- and H-Mo₂C are 3.00 and 2.85 Å,²⁷ respectively. The Mo–C bond lengths in T- and H-Mo₂C are 2.09 and 2.14 Å, respectively. The perpendicular distances between two Mo layers on the different sides in T- and H-Mo₂C are 2.32 and 2.72 Å, respectively. Moreover, the total energy per atom for H-Mo₂C is lower by about 67 meV than that of T-Mo₂C, consistent with the previous DFT results.²⁷

Fig. 1 Top and side views of (a and b) T-Mo₂C, (c and d) H-Mo₂C, (e and f) H-Mo₃C₂, and (g and h) H-Mo₃C₃. Lavender and brown balls represent Mo and C atoms, respectively. The 2D BZ and high-symmetry path of Γ–M–K–Γ are also shown.

In addition to pristine T- and H-Mo₂C, we also study two different proportions of Mo and C atoms: Mo₃C₂ and Mo₃C₃. We consider different stacking ways of these few-atom-thick Mo–C systems, which are shown in the ESI,†⁴⁴ and find that the most stable configuration of these Mo–C systems are stacked by H-Mo₂C, in stead of T-Mo₂C. The H phase of Mo₃C₂ and Mo₃C₃ has the lowest energy and there are no imaginary phonon modes for them. The structures are shown in Fig. 1(e)–(h). Although many MXenes have been synthesized in the T phase,^30–34 the functionalized derivatives of the monolayer Mo₂C are energetically more stable in the H phase.²⁸ Of course, our present results still need to be further verified via experiments.

As shown in Fig. 1(e) and (f), H-Mo₃C₂ possesses a honeycomb lattice in the P6̄m2 (No. 187) space group. After structural optimizing, the perpendicular distances between adjacent Mo and C layers are 2.79 and 2.89 Å, respectively. The Mo–C bond lengths on the side and in the middle are 2.13 and 2.20 Å, respectively. As shown in Fig. 1(g) and (h), H-Mo₃C₃ possesses a honeycomb lattice with the P3m1 (No. 156) space group. The perpendicular distances between Mo layers from top to bottom are 2.79 and 2.87 Å, respectively. The perpendicular distances between the C layers from top to bottom are 2.83 and 2.60 Å, respectively. The Mo–C bond lengths from top to bottom are 2.14, 2.24, 2.16, 2.27, and 2.01 Å, respectively.

3.2 Electronic structure

The electronic orbital-resolved band structures of T- and H-Mo₂C are plotted in Fig. 2. It is clearly seen that both the T and H phases of Mo₂C are metallic, with several bands crossing the Fermi level. As shown in Fig. 2(a) and (b), there are three bands crossing the E_F: one at the middle along the Γ–M and K–Γ directions, another at the middle along the Γ–M and M–K directions, and the last one near the Γ point along the K–Γ direction. Near the Fermi level, the Mo-4d orbitals are dominant with limited contributions from the C-p orbitals.¹⁶ As for H-Mo₂C, there are four bands crossing the E_F: two around the M point forming the hole pockets and two others around the Γ and K points forming the electron pockets. Similar to T-Mo₂C, the Mo-4d orbitals are dominant near the Fermi level.²⁸

Fig. 2 Electronic orbital-resolved band structure along high-symmetry lines of Γ–M–K–Γ for (a and b) T-Mo₂C and (c and d) H-Mo₂C. The Fermi level indicated by the dotted line is set at 0 eV.

The total and projected density of states (TDOS and PDOSs) of T- and H-Mo₂C are plotted in Fig. 3. Consistent with the results of the band structures, the Mo-4d orbitals are dominant near the Fermi level and play a critical role in their metalic properties.^16,28 The contributions from C-sp are negligible. For T-Mo₂C, the Mo-4d_z²d_zxd_zy orbitals contribute the most, while for H-Mo₂C, Mo-4d_zxd_zyd_x²−y²d_xy contribute dominantly. We calculate the Fermi surfaces (FSs) of T- and H-Mo₂C and plot them in the first Brillouin zone (BZ) in Fig. 4. From the FSs, one can see clearly the electronic states at the Fermi level. For T-Mo₂C, there are three categories of FSs: one is the large regular Γ-centered circular electron pocket originating mainly from the Mo-4d_z² orbital; one is the M-centered hole pocket originating from Mo-4d_zxd_zy orbitals; and the last one is the six small circular electron pockets originating from Mo-4d_xyd_xz orbitals along the path from K to Γ. For H-Mo₂C, there are five categories of FSs: one consists of a regular Γ-centered circular electron pocket originating from Mo-4d_x²−y²d_xy orbitals; one is the horseshoed K-centered electron pocket originating from Mo-4d_zxd_zy orbitals; two hole pockets with different sizes appear around the M-point contributed mainly from C-p_z and C-s orbitals; and the last one is a small horseshoed electron pocket originating from Mo-4d_zxd_zyd_x²−y²d_xy along the path from K to Γ.

Fig. 3 PDOS and TDOSs for (a and b) T-Mo₂C and (c and d) H-Mo₂C.

Fig. 4 The FSs of (a) T- and (b) H-Mo₂C with the colour drawn proportional to the magnitude of the Fermi velocity ν_F.

Comparing the electronic structures of H-Mo₂C, the results of H-Mo₃C₂ and H-Mo₃C₃, as shown in Fig. 5–7, exhibit some similarities and also some differences. The electronic structures of H-Mo₃C₂ are almost same as those of H-Mo₂C. Increasing the content of C only increases slightly the C-p orbital occupation at the Fermi level, mainly at the hole pockets near the M point. The electronic states near the Fermi level are still governed by the Mo-4d_zxd_zyd_x²−y²d_xy orbitals. There are still four bands crossing the Fermi level. The electron pockets are still near the K point and around the Γ point. After further increasing the C content, in the case of H-Mo₃C₃, the metallic properties, as indicated by N(E_F), are further decreased. The C-p orbitals occupation is dominant, both at the hole and electron pockets. The electronic states near the Fermi level are mainly contributed by Mo-4d_zxd_zyd_x²−y²d_xy as well as the C-2p_z orbitals. One hole pocket near the M point is lowered down in the valence bands and one electron pocket around the Γ point is raised up in the conduction bands. There are only two bands crossing the Fermi level. A new hole pocket or FS appears near the Γ point. The total Fermi velocity ν_F is decreased to some extent.

Fig. 5 Electronic orbital-resolved band structures of (a and b) H-Mo₃C₂ and (c and d) H-Mo₃C₃ along the high-symmetry line of Γ–M–K–Γ. The Fermi level indicated by the dotted line is set to 0 eV.

Fig. 6 PDOS and TDOSs for (a and b) H-Mo₃C₂ and (c and d) H-Mo₃C₃.

Fig. 7 The FSs of (a) H-Mo₃C₂ and (b) H-Mo₃C₃ with the colour drawn proportional to the magnitude of the Fermi velocity ν_F.

3.3 EPC and superconductivity

In order to investigate the EPC and superconductivity of T- and H-Mo₂C, H-Mo₃C₂, and H-Mo₃C₃, we calculate their phonon dispersions, phonon density of state (PhDOS), Eliashberg spectral function α²F(ω), and λ(ω) and present the results in Fig. 8 and 9. As shown, all these 2D molybdenum carbides are dynamically stable.

Fig. 8 Phonon dispersions weighted by the magnitude of EPC λ_qν, the Eliashberg spectral function α²F(ω), and the cumulative frequency-dependent of EPC λ(ω) for (a) T-Mo₂C and (c) H-Mo₂C. Phonon dispersions weighted by the vibrational modes of Mo, C atoms and the PhDOS for (b) T-Mo₂C and (d) H-Mo₂C.

Fig. 9 Phonon dispersions weighted by the magnitude of EPC λ_qν, the Eliashberg spectral function α²F(ω), and the strength of EPC (λ) for (a) H-Mo₃C₂ and (c) H-Mo₃C₃. Phonon dispersions weighted by the vibrational modes of Mo and C atoms and the PhDOS for (b) H-Mo₃C₂ and (d) H-Mo₃C₃.

For T-Mo₂C, the low-frequency phonons (below 30 meV), mainly associated with the Mo atomic vibrations, account for about 88% of the total EPC (λ = 0.533). These results are consistent with previous DFT work.¹⁶ Specifically, the in-plane vibrations of Mo atoms in frequency at about 20 meV are responsible for the main peak of the Eliashberg spectral function α²F(ω) and PhDOS. The out-of-plane vibrations of Mo atoms at about 29 meV also trigger a high peak of α²F(ω). The high-frequency phonons, in the frequency range of 67 to 85 meV, are mainly contributed by the C atomic vibrations. The out-of-plane vibrations of C atoms are relatively flat. Similar to Cu₂Si⁴⁵ and Pb-intercalated 1T-TiSe₂,⁸ the EPC induced by high-frequency phonons is almost negligible. In addition, there is a large optic–optic phonon gap of about 37 meV between the C- and Mo-dominant vibrations, which accords well with the previously reported result of 300 cm⁻¹.¹⁶ The highest phonon frequency, 85 meV, is larger than that of Cu₂Si (52 meV),⁴⁵ but still much smaller than those of β₁₂-B (149 meV)^46,47 and B₂C (154 meV),⁴⁸ indicating weak ionic or metallic bonding between Mo and C atoms. A total λ of about 0.5 clearly indicates that T-Mo₂C is a weak conventional superconductor. Using a typical value of the effective screened Coulomb repulsion constant μ* = 0.1 as well as our calculated α²F(ω) and λ, based upon the Bardeen–Cooper–Schrieffer (BCS) theory,⁴⁹ we calculate the logarithmic average frequency ω_log and the superconducting transition temperature T_c to be 235.7 and 3.66 K, respectively, which are close to the previously reported results of 250 and 5.9 K.¹⁶

As for H-Mo₂C, its phonon and superconducting properties are similar with those of the T phase. The Mo atomic vibrations dominate the low-frequency phonon below 35 meV and contribute about 88% of the total EPC (λ = 0.418). Contributions from the C atomic vibrations in the high-frequency region of 59–81 meV are limited. The longitudinal optical (LO)–transverse optical (TO) splitting near the Γ point in BZ, which is related to its polarization effect, is more pronounced than that of T-Mo₂C. This feature is consistent with our previous DFT results.²⁷ Such difference originates from their different symmetry and bond lengths. By analytically solving the Allen–Dynes-modified McMillan formula derived from the microscopic theory of BCS, the T_c and ω_log are calculated to be 1.19 and 221.3 K, respectively. H-Mo₂C is also a weak BCS superconductor.

After increasing the C content, from H-Mo₂C to the H phase of Mo₃C₂ and Mo₃C₃, the EPC is evidently decreased to about 0.2 (see Table 1). Such small values of λ (0.244 and 0.197 for Mo₃C₂ and Mo₃C₃, respectively) make them normal metals, not superconductors. Thus, we can conclude that the superconductivity in 2D molybdenum carbides is mainly controlled by Mo atoms. Increasing the C content can effectively suppress the superconductivity. These results are helpful for experiments. Although the superconductivity has never been observed or reported in truly monolayer of Mo₂C,²⁵ only in the film form,¹¹ we wish our present results can provide guidance for superconducting experiments in the monolayer form of 2D molybdenum carbides.²⁶

Table 1 The superconducting parameters of the biaxial stretching (in %), the electronic density of state N(E_F) at the Fermi level E_F (in the unit of states per spin per Ry per cell), the logarithmic average frequency ω_log (in K), the total EPC λ, and the superconducting transition temperature T_c (in K) for T-Mo₂C, H-Mo₂C, H-Mo₃C₂, and H-Mo₃C₃

Comp.	Strain	N(EF)	ω log	λ	T c	Ref.
T-Mo2C	0	19.33	235.7	0.533	3.66	This work
	0	—	250	0.56	<5.9	16
	4	19.76	147.2	0.789	6.71	This work
H-Mo2C	0	14.30	221.3	0.418	1.19	This work
	0	—	—	0.5	3.2	28
	10	22.79	86.8	1.845	11.79	This work
H-Mo3C2	0	14.17	310.7	0.244	0.01	This work
	11	20.00	162.2	0.585	3.43	This work
H-Mo3C3	0	10.45	415.7	0.197	0	This work
	8	22.04	239.8	0.606	5.66	This work

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As indicated in Fig. 9, the main contributions to the EPC for H-Mo₃C₂ are still from Mo atomic vibrations, while for H-Mo₃C₃ the C atomic contributions cannot be neglected. The highest phonon frequencies are still in the level of 80–90 meV. The optic–optic phonon gap is decreased from 37 meV to about 17.7 (15.2) meV for H-Mo₃C₂ (H-Mo₃C₃). For H-Mo₃C₂, the T_c and ω_log are calculated to be 0.01 and 310.7 K, respectively, while for H-Mo₃C₃ they are 0 and 415.7 K, respectively. We find that increasing the C content can increase ω_log. It is understandable that light element would generate high phonon frequencies and enable large values of ω_log.⁵⁰

As indicated by the λ_qν-weighted phonon spectra in Fig. 8 and 9 as well as the integrated EPC, which is at a given q, distributions in the plane of q_z = 0 in Fig. 10 for our studied four 2D Mo_xC_y, we can clearly analyze the specific characteristic of EPC and its relation with superconductivity. For T-Mo₂C, the largest EPC region appears at the half way from Γ to the BZ boundary, forming a circle. Such large values of EPC are mainly contributed by the in-plane acoustic vibrations of Mo atoms. The second largest EPC region is around the Γ point, mainly from Mo vibrations in low-frequency acoustic and optic phonons. As for H-Mo₂C, the largest EPC region is around the Γ point, mainly from the out-of-plane optic phonon vibrations of Mo atoms. A snowflake-like distribution of the large EPC is formed in 2D BZ. Only near the M point, the EPC is small. For H-Mo₃C₂ and H-Mo₃C₃, the large EPC regions in the BZ are obviously small, when comparing with those of the T(H)-Mo₂C. The largest EPC region appears at the K point for H-Mo₃C₂ and around Γ and M points for H-Mo₃C₃.

Fig. 10 The integrated EPC distributions of pristine (a) T- and (c) H-Mo₂C, (e) H-Mo₃C₂, and (g) H-Mo₃C₃ in the plane of q_z = 0. The corresponding results for the limited biaxial stretching of 4%, 10%, 11%, and 8% are plotted in (b), (d), (f), and (h), respectively. The high-symmetry path of Γ–M–K–Γ is shown in (e).

3.4 Strain engineering

Inspired by previous successes in modulating the electronic properties as well as the superconductivity of 2D systems by strain engineering,^47,51–57 also considering their experimental synthesis on different substrates,⁴⁵ here we also want to investigate their effects on the superconducting related physical quantities of our four studied 2D Mo_xC_y. By changing the lattice parameters, we apply biaxial tensile strain ε on them, where ε is defined as (a − a₀)/a₀ × 100%. Here a₀ and a are the lattice constants of the unstrained and strained structures.

The maximal biaxial tensile strains ε for T-Mo₂C, H-Mo₂C, H-Mo₃C₂, and H-Mo₃C₃ are 4%, 10%, 11%, and 8%, respectively, over which they are stretched to dynamically unstable. We present their corresponding phonon dispersions, α²F(ω), and λ(ω) in Fig. 11 and 12. Applying biaxial tensile strain can weaken the atomic force constant and soften the phonon modes.^53,54 In our present study, we also find soft acoustic phonon modes in T-Mo₂C (around M point), H-Mo₂C (around Γ K), H-Mo₃C₂ (around K point), and H-Mo₃C₃ (around K point). The soft mode for T(H)-Mo₂C is in its out-of-plane flexural (ZA) acoustic branch, while for H-Mo₃C₂ and H-Mo₃C₃ it is in in-plane transverse acoustic (TA) and ZA branches as well as the TO modes at around 40 meV and near Γ point. The full phonon spectra are wholly softened. As shown, the highest phonon frequencies of H-Mo₂C and H-Mo₃C₂ are lowered down to around 62 meV. The two high-frequency TO branches of H-Mo₂C are greatly lowered.

Fig. 11 Phonon dispersions with the maximum biaxial stretching weighted by the magnitude of EPC λ_qν, the Eliashberg spectral function α²F(ω) and the strength of EPC(λ) for (a) T-Mo₂C at 4% and (b) H-Mo₂C at 10%.

Fig. 12 Phonon dispersions with the maximum biaxial stretching weighted by the magnitude of EPC λ_qν, the Eliashberg spectral function α²F(ω) and the strength of EPC (λ) for (a) H-Mo₃C₂ at 11% and (b) H-Mo₃C₃ at 8%.

The softening of phonon is beneficial to increasing the EPC λ and T_c. As shown, the peaks of α²F(ω) are red shifted and low-frequency proportion becomes more important. The EPC is redistributed in BZ (see Fig. 10). The large values of EPC are still mainly in the low-frequency region. At ε = 4%, the λ and T_c of T-Mo₂C are enhanced to 0.789 and 6.71 K, respectively, although ω_log is decreased to 147.2 K due to the softening of phonons. The T_c is almost doubled by biaxial tensile strain. As for H-Mo₂C, since the maximal biaxial tensile strain ε is up to 10%, its λ and T_c are greatly increased up to 1.845 and 11.79 K, respectively, making it a strong BCS superconductor. Its ω_log is lowered down to 86.8 K. Such evident modulations of the superconducting-related properties by stain engineering have also been reported in graphene,⁵³ monolayer h-BN,⁵⁴ aluminum-deposited graphene AlC₈,⁵⁶ and HPC₃.⁵⁷ For H-Mo₃C₂ (H-Mo₃C₃), its λ and T_c are increased to 0.585 (0.606) and 3.43 (5.66) K, respectively, at a strain of 11% (8%). They are transformed from normal metals to weak BCS superconductors. Their ω_log are decreased to 162.2 and 239.8 K, respectively. The detail data can be found in Table 1. In addition, we show the strain-dependent behaviors of ω_log, N(E_F), T_c, and λ in Fig. S3 in the ESI.†⁴⁴ As shown, N(E_F), T_c, and λ increase steadily under biaxial tensile strains, while only ω_log exhibits decreasing behavior. H-Mo₂C can only be tuned as a strong BCS superconductor at ε = 10%, lower strains which all the λ are smaller than 1. As for H-Mo₃C₂ and H-Mo₃C₃, they can be tuned from normal metals to superconductors at ε beyond 6% and 3%, respectively. These strain-dependent behaviors of the superconducting-related properties clearly indicate that T_c is tightly related to EPC and N(E_F). In order to increasing T_c, one must increase N(E_F) first, such as modulating the Fermi level to a van Hove singularity.⁵⁸

By further applying biaxial tensile strains ε beyond their maximal amplitudes, H-Mo₂C becomes dynamically unstable, while T-Mo₂C, H-Mo₃C₂, and H-Mo₃C₃ exhibit characteristics of CDW.^44,59 As shown in Fig. S4 in the ESI,†⁴⁴ similar to the normal structure of the monolayer 1T-TiSe₂,⁸ an imaginary phonon mode at vector q_M for the acoustic mode is observed for T-Mo₂C under ε = 5%. Thus, the superlattice of its CDW state may similar to that of 1T-TiSe₂. Similarly, unstable phonon mode at q_M and possible CDW have also been reported and discussed in T phase MXene W₂N.¹⁷ As for H-Mo₃C₂ and H-Mo₃C₃, their imaginary phonon mode is at the vector of q_K. The CDW superlattices of them may be more complicated than that of 1T-TiSe₂.⁶⁰ We may study carefully the CDW states of these 2D Mo_xC_y in our future studies.

4 Conclusion

In summary, we have calculated the electronic structures, phonon dispersions, and EPC of the T and H phases of monolayer Mo₂C and have checked the stacking style of the 2D Mo–C systems in Mo:C ratios of 3 : 2 and 3 : 3. We find that the most stable configurations of the Mo–C systems are stacked by the component of H-Mo₂C, which is energetically more stable than T-Mo₂C. Although many MXenes have been synthesized in the T phase,^30–34 the H phase of 2D Mo_xC_y may be synthesized in the future. According to our calculations, similar to our recent study of the MoSH monolayer⁶¹ and MoP₂,⁶² the Mo 4d orbitals are dominant at the Fermi level, while the C2p orbitals are negligible for most our studied 2D molybdenum carbides. Only in H-Mo₃C₃, when the C content is increased, the C 2p_z orbital also contributes dominantly. From Mo₂C to Mo₃C₃, the metallic properties are weakened and the superconductivity is suppressed by increasing the C content. The EPC in these systems is mainly controlled by the Mo atomic vibrations, in the low-frequency region. The pristine T(H)-Mo₂C is a weak conventional superconductor while pristine H-Mo₃C₂ and H-Mo₃C₃ are normal metals. By applying biaxial tensile strain, H-Mo₂C can be modulated to a strong EPC superconductor with a large λ of about 1.85 and H-Mo₃C₂ and H-Mo₃C₃ can be changed to weak BCS superconductors. By further applying a biaxial tensile strain, CDW states may appear in T-Mo₂C, H-Mo₃C₂, and H-Mo₃C₃. Our present study can provide guidance for superconducting experiments in the monolayer form of 2D molybdenum carbides and is also beneficial for further theoretical and experimental studies of electronic structures and superconductivity in 2D forms of metal carbides.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 12074213, 12074381, 11574108), the Major Basic Program of Natural Science Foundation of Shandong Province (Grant No. ZR2021ZD01), Guangdong Basic and Applied Basic Research Foundation (Grant No. 2021A1515110587), and the Project of Introduction and Cultivation for Young Innovative Talents in Colleges and Universities of Shandong Province.

Notes and references

1. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva and A. A. Firsov, Science, 2004, 306, 666–669.
2. K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V. Khotkevich, S. V. Morozov and A. K. Geim, Proc. Natl. Acad. Sci. U. S. A., 2005, 102, 10451–10453.
3. M. Naguib, M. Kurtoglu, V. Presser, J. Lu, J. Niu, M. Heon, L. Hultman, Y. Gogotsi and M. W. Barsoum, Adv. Mater., 2011, 23, 4248–4253.
4. B. Anasori, M. R. Lukatskaya and Y. Gogotsi, Nat. Rev. Mater., 2017, 2, 16098.
5. F. Shahzad, M. Alhabeb, C. B. Hatter, B. Anasori, S. M. Hong, C. M. Koo and Y. Gogotsi, Science, 2016, 353, 1137–1140.
6. J. Peng, X. Chen, W.-J. Ong, X. Zhao and N. Li, Chem, 2019, 5, 18–50.
7. X.-H. Tu, P.-F. Liu, W. Yin, J.-R. Zhang, P. Zhang and B.-T. Wang, Mater. Today Phys., 2022, 24, 100674.
8. B.-T. Wang, P.-F. Liu, J.-J. Zheng, W. Yin and F. Wang, Phys. Rev. B, 2018, 98, 014514.
9. D. Geng, X. Zhao, L. Li, P. Song, B. Tian, W. Liu, J. Chen, D. Shi, M. Lin, W. Zhou and K. P. Loh, 2D Mater., 2016, 4, 011012.
10. K. Chang, P. Deng, T. Zhang, H.-C. Lin, K. Zhao, S.-H. Ji, L.-L. Wang, K. He, X.-C. Ma, X. Chen and Q.-K. Xue, EPL, 2015, 109, 28003.
11. C. Xu, L. Wang, Z. Liu, L. Chen, J. Guo, N. Kang, X.-L. Ma, H.-M. Cheng and W. Ren, Nat. Mater., 2015, 14, 1135–1141.
12. L. Wang, C. Xu, Z. Liu, L. Chen, X. Ma, H.-M. Cheng, W. Ren and N. Kang, ACS Nano, 2016, 10, 4504–4510.
13. V. Kamysbayev, A. S. Filatov, H. Hu, X. Rui, F. Lagunas, D. Wang, R. F. Klie and D. V. Talapin, Science, 2020, 369, 979–983.
14. K. Wang, H. Jin, H. Li, Z. Mao, L. Tang, D. Huang, J.-H. Liao and J. Zhang, Surf. Interfaces, 2022, 29, 101711.
15. M. Liao, Y. Zang, Z. Guan, H. Li, Y. Gong, K. Zhu, X.-P. Hu, D. Zhang, Y. Xu, Y.-Y. Wang, K. He, X.-C. Ma, S.-C. Zhang and Q.-K. Xue, Nat. Phys., 2018, 14, 344–348.
16. J.-J. Zhang and S. Dong, J. Chem. Phys., 2017, 146, 034705.
17. J. Bekaert, C. Sevik and M. V. Milošević, Nanoscale, 2020, 12, 17354–17361.
18. S.-Y. Wang, C. Pan, H. Tang, H.-Y. Wu, G.-Y. Shi, K. Cao, H. Jiang, Y.-H. Su, C. Zhang, K.-M. Ho and C.-Z. Wang, J. Phys. Chem. C, 2022, 126, 3727–3735.
19. Z. S. Pereira, G. M. Faccin and E. Z. da Silva, Nanoscale, 2022, 14, 8594–8600.
20. J. Bekaert, C. Sevik and M. V. Milošević, Nanoscale, 2022, 14, 9918–9924.
21. N. Jiao, H.-D. Liu, L. Yang, Y.-P. Li, M. Zheng, H.-Y. Lu and P. Zhang, Europhys. Lett., 2022, 138, 46002.
22. S. Bae, Y.-G. Kang, M. Khazaei, K. Ohno, Y.-H. Kim, M. Han, K. Chang and H. Raebiger, Mater. Today Adv., 2021, 9, 100118.
23. R. Meshkian, L.-Å. Näslund, J. Halim, J. Lu, M. W. Barsoum and J. Rosen, Scr. Mater., 2015, 108, 147–150.
24. J. Halim, S. Kota, M. R. Lukatskaya, M. Naguib, M.-Q. Zhao, E. J. Moon, J. Pitock, J. Nanda, S. J. May, Y. Gogotsi and M. W. Barsoum, Adv. Funct. Mater., 2016, 26, 3118–3127.
25. D. Geng, X. Zhao, Z. Chen, W. Sun, W. Fu, J. Chen, W. Liu, W. Zhou and K. P. Loh, Adv. Mater., 2017, 29, 1700072.
26. M. Qiang, X. Zhang, H. Song, C. Pi, X. Wang, B. Gao, Y. Zheng, X. Peng, P. K. Chu and K. Huo, Carbon, 2022, 197, 238–245.
27. W. Sun, Y. Li, B. Wang, X. Jiang, M. I. Katsnelson, P. Korzhavyi, O. Eriksson and I. D. Marco, Nanoscale, 2016, 8, 15753–15762.
28. J. Lei, A. Kutana and B. I. Yakobson, J. Mater. Chem. C, 2017, 5, 3438–3444.
29. X. Zhao, W. Sun, D. Geng, W. Fu, J. Dan, Y. Xie, P. R. C. Kent, W. Zhou, S. J. Pennycook and K. P. Loh, Adv. Mater., 2019, 31, 1808343.
30. L. Gao, C. Li, W. Huang, S. Mei, H. Lin, Q. Ou, Y. Zhang, J. Guo, F. Zhang, S. Xu and H. Zhang, Chem. Mater., 2020, 32, 1703–1747.
31. C. J. Zhang, Y. Ma, X. Zhang, S. Abdolhosseinzadeh, H. Sheng, W. Lan, A. Pakdel, J. Heier and F. Nüesch, Energy Environ. Mater., 2020, 3, 29–55.
32. F. Ming, H. Liang, G. Huang, Z. Bayhan and H. N. Alshareef, Adv. Mater., 2020, 33, 2004039.
33. J. Chen, Y. Ding, D. Yan, J. Huang and S. Peng, SusMat, 2022, 2, 293–318.
34. X. Wu, S. Xiao, Y. Long, T. Ma, W. Shao, S. Cao, X. Xiang, L. Ma, L. Qiu, C. Cheng and C. Zhao, Small, 2022, 18, 2105831.
35. P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. D. Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. P. Seitsonen, A. Smogunov, P. Umari and R. M. Wentzcovitch, J. Phys.: Condens. Matter, 2009, 21, 395502.
36. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 1996, 77, 3865–3868.
37. P. E. Blöchl, Phys. Rev. B: Condens. Matter Mater. Phys., 1994, 50, 17953–17979.
38. P. E. Blöchl, O. Jepsen and O. K. Andersen, Phys. Rev. B: Condens. Matter Mater. Phys., 1994, 49, 16223–16233.
39. S. Baroni, S. de Gironcoli, A. Dal Corso and P. Giannozzi, Rev. Mod. Phys., 2001, 73, 515–562.
40. R. Dynes, Solid State Commun., 1972, 10, 615–618.
41. P. B. Allen and R. C. Dynes, Phys. Rev. B: Solid State, 1975, 12, 905–922.
42. P. B. Allen and B. Mitrović, Solid State Physics, Elsevier, 1983, pp. 1–92.
43. J. Kortus, I. I. Mazin, K. D. Belashchenko, V. P. Antropov and L. L. Boyer, Phys. Rev. Lett., 2001, 86, 4656–4659.
44. See ESI† at xxx for (I) the structures and energy comparison of Mo3C2 and Mo3C3 with different configurations, (II) superconductive parameters of N(EF), ωlog, λ, and Tc under different biaxial tensile strains, and (III) phonon spectra of 2D MoxCy at and beyond the maximal biaxial tensile strains.
45. L. Yan, P.-F. Liu, T. Bo, J. Zhang, M.-H. Tang, Y.-G. Xiao and B.-T. Wang, J. Mater. Chem. C, 2019, 7, 10926–10932.
46. M. Gao, Q.-Z. Li, X.-W. Yan and J. Wang, Phys. Rev. B, 2017, 95, 024505.
47. C. Cheng, J.-T. Sun, H. Liu, H.-X. Fu, J. Zhang, X.-R. Chen and S. Meng, 2D Mater., 2017, 4, 025032.
48. J. Dai, Z. Li, J. Yang and J. Hou, Nanoscale, 2012, 4, 3032.
49. J. Bardeen, L. N. Cooper and J. R. Schrieffer, Phys. Rev., 1957, 108, 1175–1204.
50. D. Duan, Y. Liu, Y. Ma, Z. Shao, B. Liu and T. Cui, Natl. Sci. Rev., 2016, 4, 121–135.
51. G. Li, Y. Zhao, S. Zeng, M. Zulfiqar and J. Ni, J. Phys. Chem. C, 2018, 122, 16916–16924.
52. D. Campi, S. Kumari and N. Marzari, Nano Lett., 2021, 21, 3435–3442.
53. C. Si, Z. Liu, W. Duan and F. Liu, Phys. Rev. Lett., 2013, 111, 196802.
54. X.-T. Jin, X.-W. Yan and M. Gao, Phys. Rev. B, 2020, 101, 134518.
55. T. Bo, P.-F. Liu, L. Yan and B.-T. Wang, Phys. Rev. Materials, 2020, 4, 114802.
56. H.-Y. Lu, Y. Yang, L. Hao, W.-S. Wang, L. Geng, M. Zheng, Y. Li, N. Jiao, P. Zhang and C. S. Ting, Phys. Rev. B, 2020, 101, 214514.
57. Y.-P. Li, L. Yang, H.-D. Liu, N. Jiao, M.-Y. Ni, N. Hao, H.-Y. Lu and P. Zhang, Phys. Chem. Chem. Phys., 2022, 24, 9256–9262.
58. E. R. Margine and F. Giustino, Phys. Rev. B: Condens. Matter Mater. Phys., 2014, 90, 014518.
59. J.-G. Si, W.-J. Lu, Y.-P. Sun, P.-F. Liu and B.-T. Wang, Phys. Rev. B, 2022, 105, 024517.
60. P. Chen, Y. H. Chan, X. Y. Fang, Y. Zhang, M. Y. Chou, S. K. Mo, Z. Hussain, A. V. Fedorov and T. C. Chiang, Nat. Commun., 2015, 6, 8943.
61. P.-F. Liu, F. Zheng, J. Li, J.-G. Si, L. Wei, J. Zhang and B.-T. Wang, Phys. Rev. B, 2022, 105, 245420.
62. X.-H. Tu, T. Bo, P.-F. Liu, W. Yin, N. Hao and B.-T. Wang, Phys. Chem. Chem. Phys., 2022, 24, 7893–7900.

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Footnote

† Electronic supplementary information (ESI) available: (I) The structures and energy comparison of Mo₃C₂ and Mo₃C₃ with different configurations, (II) superconductive parameters of N(E_F), ω_log, λ, and T_c under different biaxial tensile strains, and (III) phonon spectra of 2D Mo_xC_y at and beyond the maximal biaxial tensile strain. See DOI: https://doi.org/10.1039/d2cp04306h

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