MC₂ (M = Y, Zr, Nb, and Mo) monolayers containing C₂ dimers: prediction of anode materials for high-performance sodium ion batteries

Abstract

Seeking novel high performance anode materials for sodium ion batteries (SIBs) is an attractive theme in developing energy storage devices. In this work, by means of density functional theory computations, we predicted a family of MC₂ (M = Y, Zr, Nb, and Mo) monolayers containing C₂ dimers to be promising anode materials for SIBs. The stability, electronic structure, and adsorption/diffusion/storage behavior of sodium atoms in MC₂ (M = Y, Zr, Nb, and Mo) monolayers were explored. Our computations revealed that Na adsorbed MC₂ (M = Y, Zr, Nb, and Mo) monolayers show metallic characteristics that give rise to excellent electrical conductivity and Na mobility with low activation energies for diffusion (0.21, 0.04, 0.20, and 0.22 eV, respectively) in these materials, indicative of a high charge/discharge rate. In addition, the theoretical capacities of Na-adsorbed on YC₂, ZrC₂, NbC₂, and MoC₂ monolayers are 478, 697, 687, and 675 mA h g⁻¹, respectively, higher than that of commercial graphite (284 mA h g⁻¹), and the open-circuit voltages are moderate (0.11–0.25 V). Our results suggest that MC₂ (M = Y, Zr, Nb, and Mo) monolayers have great potential to serve as anode materials for SIBs.

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1. Introduction

The global demand for advanced energy storage and conversion technologies, especially electrochemical energy technologies, is growing rapidly. Among them, rechargeable batteries, such as lithium-ion batteries (LIBs), are widely used in mobile electronic and electrified vehicles.^1–3 However, LIBs can meet the tough demands of energy storage and conversion applications, yet the limited abundance (only 20 ppm) and expensiveness of lithium are not the best choices for application in large-scale energy storage systems and sustainability.^4,5 Fortunately, sodium-ion batteries (SIBs) were confirmed as low-cost alternatives to lithium-ion batteries (LIBs) in energy storage applications, on account of the natural abundance, high energy density and similar intercalation chemistry to lithium, suggesting that some mature technologies in LIBs can be translated to SIBs, and SIBs are promising candidates for next generation rechargeable batteries.⁶

Cathode materials with high electrochemical performance for SIBs have been extensively studied experimentally and theoretically, such as layered transition metal oxide compounds, polyanionic compounds, Prussian blue analogues, and sodium-based layered materials.^7,8 However, anode materials matching well LIB systems are not suitable for SIBs due to the bigger ionic radius and the higher redox potential of Na⁺ ions vs. Li⁺. For instance, graphite is not available in SIBs, because the interlayer distance of graphite is too small to accommodate sodium, leading to a larger mismatched layer spacing and poor capacity.^9,10 Thus, the design and exploration of suitable electrode materials for SIBs are greatly desired.

In recent years, two-dimensional (2D) materials as electrodes have attracted great attention due to their excellent properties, including good mechanical flexibleness, rich active sites, large exposed surface area for electrochemical processes, and expanded interlayer spacing.^11,12 For example, transition metal chalcogenides (TMDs),^13–15 oxides (TMOs),^16,17 borides (MBenes),^18,19 transition metal carbides or nitrides (MXenes),^20,21 and main group binary compounds (B₂S, BN₂, SiS, SnP₃, etc.)^22–25 have been theoretically and experimentally explored as anode materials. Among these materials, 2D MXenes have experienced rapid development as anode materials after the successful exfoliation of Ti₃C₂ nanosheets in 2011,²⁶ due to their superior electrical conductivity, high performance rate, and high specific capacities.^27–30 So far, metal-rich 2D MXene-based materials (i.e. Ti₂C,³¹ V₂C,³² Nb₂C,³³ Mo₂C,³⁴ Cr₂C,³⁵ Mn₂C,³⁶etc.) have been extensively studied as anode materials for LIBs and SIBs. However, the major problems with MXenes are surface functionalization and metal atoms exposed on the surface commonly to suppress sodium adsorption, which largely reduce the storage capacity and the performance of batteries.²¹

In this regard, Wang et al.³⁷ proposed a novel carbon-rich 2D metal carbide (MC), namely TiC₂ monolayer, which consists of unique C₂ dimers (Fig. 1a) and exhibits high structural stability. The Li capacity, open-circuit voltage (OCV), and diffusion barrier were calculated to be 622 mA h g⁻¹, 0.96 V, and 0.11 eV, respectively, indicating that the TiC₂ monolayer is a promising anode material for LIBs. Subsequently, other transition metal carbide (VC₂,^38,39 TaC₂,⁴⁰ MnC₂,⁴¹ ScC₂,⁴² and WC₂ (ref. 43)) monolayers with a similar configuration were also theoretically investigated as LIB and SIB anode materials. In particular, Zhao et al.⁴⁴ found that an FeC₂ monolayer containing isolated C₂ dimers with ferromagnetic and half-metallic properties was possible to be synthesized by chemically exfoliating the bulk counterpart of ThFeC₂.

Fig. 1 (a) The geometric structure of the 2 × 2 × 1 supercell of the MC₂ (M = Y, Zr, Nb, and Mo) monolayers (top and side views); (b) the unit cell is highlighted by the black dashed lines; (c) the Brillouin zone of the MC₂ monolayers and the edge-shared MC₆ octahedral structure.

Therefore, the proposal of C-rich MC₂ monolayers may offer a new strategy for the rational design of 2D anode materials for SIBs. In this work, four C₂-containing MC₂ (M = Y, Zr, Nb and Mo) monolayers were systematically explored by means of DFT computations. According to our calculations of cohesive energy, phonon dispersion, first-principles molecular dynamics (FPMD) stimulation, mechanical parameters, and sodium adsorption/diffusion/capacity, the four examined monolayers were confirmed to exhibit great thermodynamic, dynamic, thermal, and mechanical stability, excellent electronic conductivity and outstanding electrode performance as a SIB anode material. In particular, the ZrC₂ monolayer exhibits superior electrochemical performance with high theoretical storage capacity (699 mA h g⁻¹) and small diffusion energy barriers (0.04 eV) for Na atoms.

2. Computational methods

In this study, the structural relaxation and performance calculations were based on density functional theory (DFT) methods within the Vienna ab initio simulation package (VASP).^45–47 The general gradient approximation (GGA) in the form of Perdew–Burke–Ernzerhof (PBE) exchange and correlation functional and a plane wave basis set with an energy cut-off of 500 eV were employed.^48,49 Monkhorst–Pack meshes⁵⁰ of 11 × 11 × 1 and 5 × 5 × 1 k-points were used to sample the two-dimensional Brillouin zone for the unit cell and supercell relaxation and electronic structure calculations, respectively. An inter-layer vacuum space of 20 Å was used to avoid the interactions of mirror images in adjacent layers. All the structures are fully relaxed until the convergence tolerances for the energy and force on each atom were less than 1.0 × 10⁻⁵ eV and 0.02 eV Å⁻¹, respectively. All the atomic structures and charge density distributions were visualized by using the VESTA package.⁵¹

The dynamic stability of MC₂ (M = Y, Zr, Nb, and Mo) monolayers was evaluated by recording the phonon spectrum with the PHONOPY code⁵² on the basis of density functional perturbation theory (DFPT). Thermal stabilities were assessed by performing first-principles molecular dynamics (FPMD) simulations (a 4 × 4 × 1 supercell, 96 atoms) in the NVT ensemble lasting for 5 ps with a time step of 1 fs at room temperature (300 K), and the temperature was controlled by the Nosé–Hoover method.⁵³ By using the climbing-image nudged elastic band (CI-NEB) approach,^54,55 the diffusion energy barrier and the minimum energy pathways of Na atom diffusing on MC₂ (M = Y, Zr, Nb, and Mo) monolayers were calculated. The charge distribution and transfer were analyzed by the Bader charge method.⁵⁶

3. Results and discussion

3.1 Geometric structure of the MC₂ (M = Y, Zr, Nb, and Mo) monolayers

The optimized structures of MC₂ (M = Y, Zr, Nb, and Mo) monolayers are displayed in Fig. 1a. From the top view, these monolayers exhibit a rectangular lattice with the space group Pmmn symmetry, where carbon atoms are in the form of C₂ dimers, and each unit cell contains two M atoms and four C atoms (Fig. 1b). From the side view, the crystal structure of the MC₂ (M = Y, Zr, Nb, and Mo) monolayers possesses a tetra-layer structure with two M layers sandwiched between two C layers, in which each M atom bonds with six neighboring C atoms, while each C atom is shared by one C of the same C₂ dimer and three adjacent M atoms, forming an edge-shared MC₆ octahedral structure without M atoms exposed on the surface (Fig. 1c).

Table 1 summarizes the optimized structural parameters of MC₂ (M = Y, Zr, Nb, and Mo) monolayers, including the lattice constants (a and b), bond lengths of C–C (bond I), M–C (bond II along the x axis and bond-III along the y axis) and thickness (h), and the evolution trend was plotted for different MC₂ monolayers with transition metal atoms (M) from Y → Mo (Fig. S1 and S2†). The lattice constants (4.84 ≤ a ≤ 5.80 Å and 3.01 ≤ b ≤ 4.28 Å) are close to those of VC₂ (a = 4.62 Å and b = 3.28 Å),³⁸ FeC₂ (4.54 Å and 2.89 Å),⁴⁴ and ScC₂ (5.39 Å and 4.12 Å).⁴² The remarkable feature we can see is that structural parameters, such as lattice constants (a and b), bond lengths M–C (d_M–C) and bond angle (∠θ_C–M–C), decrease from 5.77 to 4.84 Å (4.45 to 3.03 Å), 2.39 to 2.03 Å (2.51 to 2.26 Å) along the x (y)-direction and 138.14° to 119.28° from Y → Mo (Fig. S3†), mainly originating from the decrease of the atomic radius from Y to Mo. However the others, including bond lengths C–C (d_C–C) and thickness (h), increase (1.30 to 1.37 Å and 1.81 to 2.60 Å) from Y → Mo, respectively. The latter increasing trend could be understood by the electronegativity difference between the transition metal (Y → Mo) and carbon, which are strongly correlated with the structural properties.^57,58 YC₂ has the highest electronegativity difference (1.33 on the Pauling scale), while MoC₂ possesses the lowest (0.39 on the Pauling scale), the angles of M–C–M (∠θ_M–C–M) gradually decrease, resulting in an increase in the d_C–C and thickness of 2D MC₂ from Y to Mo (Fig. S3†). Note that the C–C distances of C₂ dimers are in the range of 1.30–1.37 Å, which are shorter than that in the Ti₈C₁₂ cage and MXenes with the C–C distances in the range of 1.40–3.35 Å,^59–61 implying stronger C–C bonding in C₂ dimers.

Table 1 The lattice constants (a and b, in Å), C–C bond length (bond-I, in Å), M–C bond length (bond-II along the x axis and bond-III along the y axis, in Å), bond angle (∠θ_C–M–C), thickness (h, Å) and cohesive energy (E_coh, in eV per atom) of the MC₂ (M = Y, Zr, Nb, and Mo) monolayers

System	a	b	Bond-I	Bond-II	Bond-III	∠θC–M–C	h	E coh
YC2	5.77	4.45	1.30	2.39	2.51	138.14°	1.81	−7.40
ZrC2	5.31	3.92	1.34	2.21	2.33	128.23°	2.07	−7.99
NbC2	4.96	3.54	1.34	2.08	2.26	119.74°	2.26	−7.98
MoC2	4.84	3.03	1.37	2.03	2.27	119.28°	2.60	−7.48

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3.2 Stabilities of the MC₂ (M = Y, Zr, Nb, and Mo) monolayers

Prior to exploring the physical properties and evaluating the probability of experimental synthesis of the MC₂ (M = Y, Zr, Nb, and Mo) monolayers, we firstly verified their thermodynamic, dynamic, thermal, and mechanical stabilities and investigated the underlying mechanisms for their stabilities.

First, we examined the thermodynamic stability by calculating their cohesive energy (E_coh), which is defined as E_coh = (E_MC2 − nE_M − mE_c)/(n + m), in which E_MC2 and E_M/E_c are the energies of the MC₂ monolayer and a free M/C atom, respectively, and n/m is the number of M/C atoms in one unit cell. According to this definition, a more negative E_coh value indicates a better thermodynamic stability of the structure. The computed E_coh values for MC₂ (M = Y, Zr, Nb, and Mo) monolayers are −7.40, −7.99, −7.98, and −7.48 eV per atom, respectively, lower than that of the TiC₂ monolayer (−6.59 eV),³⁷ VC₂ monolayer (−7.15 eV),³⁸ ScC₂ monolayer (−6.33 eV),⁴² and the synthesized Ti₂C⁶² (6.18 eV per atom) MXene, all of which are demonstrated to be structurally and thermodynamically stable according to the first principles calculations or experimentally. Thus, the four MC₂ (M = Y, Zr, Nb, and Mo) monolayers have solid thermodynamic stability.

Second, the dynamic stability of the MC₂ (M = Y, Zr, Nb, and Mo) monolayers was evaluated by calculating their phonon dispersions, which are shown in Fig. S4.† All of them are dynamically stable as confirmed by the absence of imaginary phonon modes in the entire Brillouin zone. In particular, the highest frequencies for the dynamically stable monolayers (40.25–49.59 THz) are higher or comparable to those of the ScC₂/ScN₂ monolayer (50/38 THz),⁴² VC₂ (∼43 THz),³⁸ and the recently proposed TaC₂ monolayer (42 THz),⁴⁰ suggesting strong M–C bonding in these MC₂ nanosheets.

Third, FPMD simulations were performed to assess their thermal stability at room temperature (300 K), and the structures after annealing at 300 K for 5 ps are presented in Fig. S5a–d.† Neither bond breakage nor structural distortion can be noted, and then we conclude that the YC₂, ZrC₂, NbC₂ and MoC₂ monolayers possess excellent thermal stabilities at room temperature.

Finally, we assessed the mechanical stability of the MC₂ (M = Y, Zr, Nb, and Mo) monolayers by calculating their elastic constants using the finite distortion method.⁶³ Mechanically, a stable 2D structure should meet the Born–Huang criteria^64,65 given by C₁₁C₂₂ − C₁₂² > 0 and C₆₆ > 0. The calculated values of elastic constants fully satisfy these criteria (shown in Table 2), demonstrating that the MC₂ (M = Y, Zr, Nb, and Mo) monolayers are also mechanically stable.

Table 2 The elastic constants C₁₁ and C₂₂ (in N m⁻¹), in-plane Young's modulus Y_x and Y_y (in N m⁻¹), and Poisson's ratio of the TiC₂, YC₂, ZrC₂, NbC₂, and MoC₂ sheets

System	C 11	C 12	C 22	C 66	Y x	Y y	ν x	ν y
YC2	79.33	22.17	32.21	16.51	64.07	26.01	0.69	0.28
ZrC2	107.71	34.95	49.15	27.00	82.86	37.81	0.71	0.32
NbC2	158.97	40.90	55.50	42.85	128.82	44.98	0.74	0.26
MoC2	180.44	43.76	82.01	47.97	157.09	71.40	0.53	0.24
TiC2 (ref. 37)	140.58	25.10	70.52	16.19	131.17	66.04	0.59	0.21

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To gain insights into the bonding nature and stabilizing mechanism, we calculated the electron localization functions (ELFs) of MC₂ (M = Y, Zr, Nb, and Mo) monolayers. The ELF is an effective tool for characterizing chemical bonding and can reflect the electron localization in the system.⁶⁶ Generally, the ELF value near zero (blue) denotes a low electron density area, whereas the ELF values of 1.0 (red) and 0.5 (green) represent the perfect localization and the free electron-gas, respectively. As shown in Fig. S6,† in the ELF maps of the (001) and (010) slices we can see that the region between M and C atoms in all the MC₂ monolayers is in the green color corresponding to an ELF of around 0.5, indicating fully delocalized electron distribution, which well shared and delocalized electrons make robust connection between the M and C atoms, and the Bader charge analysis revealed that the each M (M = Y, Zr, Nb, and Mo) atom transfers 1.85, 1.81, 1.73, and 1.15 e⁻ per atom to the C₂ dimer, respectively, similar to the value of 1.80/1.50 e⁻ for TiN₂/C₂,³⁷ 1.74/1.71 e⁻ for ScC₂/N₂,⁴² and 1.85 e⁻ for YN₂ (ref. 67) monolayers, indicating ionic interactions between M and C atom. While, the C₂ dimer exhibits strong covalent bond characteristics, the results are in agreement with previous studies that the formation of the C₂ dimer plays an important role in stabilizing metal 2D structures. As a consequence, the coexistence of ionic and covalent bonding is mainly responsible for the structural stability.

3.3 Mechanical and electronic properties of the MC₂ (M = Y, Zr, Nb, and Mo) monolayers

Having confirmed the stability of the MC₂ (M = Y, Zr, Nb, and Mo) monolayers, the mechanical properties, including Young's modulus Y(θ) and Poisson's ratio ν(θ), were investigated. The angle-dependent in-plane Y(θ) and ν(θ) were estimated by using the following equation:⁶⁸where A = (C₁₁C₂₂ − C₁₂²)/C₆₆ − 2C₁₂ and B = C₁₁ + C₁₂ − (C₁₁C₂₂ − C₁₂²)/C₆₆.

Based on the above equations, we plotted Y(θ) and ν(θ) with the variation of angle θ (Fig. 2). The Y(θ) of Y/Zr/Nb/MoC₂ monolayers decreases from a maximum value of 64.07/82.86/128.82/157.09 N m⁻¹ along the x direction to a minimum value of 26.01/37.81/44.98/71.40 N m⁻¹ along the y direction, reflecting their high mechanical anisotropy. However, the Y(θ) of the TiC₂ monolayer decreases from a maximum value of 131.17 N m⁻¹ in the x direction to a minimum value of 55 N m⁻¹ at about θ = 46°, which is consistent with a previous report.³⁷ Compared with other well-known 2D materials, strikingly, our calculated Y along x and y directions of these C₂-containing monolayers are comparable to the value of MoS₂ (122 N m⁻¹),⁶⁹ silicene (61.7 N m⁻¹) and germanene (44 N m⁻¹),⁷⁰ but smaller than that of graphene (341.60 N m⁻¹)⁷¹ and hexagonal h-BN (271 N m⁻¹),⁷² implying high flexibility for these monolayers. The modest mechanical properties are beneficial for the charge–discharge process of rechargeable ion batteries.

Fig. 2 (a) Young's modulus and (b) Poisson's ratio of the MC₂ (M = Y, Zr, Nb, Mo and Ti) monolayers as a function of the in-plane angle θ, and θ = 0° corresponds to the x-axis.

To analyze the electronic properties of the MC₂ (M = Y, Zr, Nb, and Mo) monolayers, the atom-projected electronic band structure and partial density of states (PDOS) were carefully investigated (as shown in Fig. 3 and S7†). The results showed that all considered monolayers exhibit metallic character, as indicated by the partially occupied bands crossing the Fermi level in the Brillouin zone. The orbital analysis suggested that the bands near the Fermi level are dominated by the M_d orbitals, while the C_p orbitals make minor contribution by hybridizing with the M–d states, which is in line with the electrons transferred from M to C atoms based on the Bader charge analysis, as shown in Fig. S6.† The metallic nature of MC₂ (M = Y, Zr, Nb, and Mo) monolayers indicates the advantage in electrical conductivity, which is highly desirable for high performance of electrode materials.

Fig. 3 The electronic band structures of the (a) YC₂, (b) ZrC₂, (c) NbC₂, and (d) MoC₂ monolayers. The red and blue curves denote the d orbits of the metal atoms and the p orbits of C atoms, respectively. The Fermi energy is denoted by a dashed line at 0 eV.

3.4 Na adsorption and diffusion on the MC₂ (M = Y, Zr, Nb, and Mo) monolayers

The development of sodium ion batteries (SIBs) is often hindered by the limited choice of anode materials. The modest mechanical properties and metallic feature of the examined MC₂ (M = Y, Zr, Nb, and Mo) monolayers facilitate our investigation on their potential application as SIB anode materials.

The adsorption behavior between metal atoms and 2D materials closely correlates with the anode performance. Thus, we firstly examined the energy-preferred adsorption sites for a single Na atom on the surface of MC₂ (M = Y, Zr, Nb, and Mo) monolayers with a 2 × 2 × 1 supercell. Four high symmetry adsorption sites were considered, namely A₁ (where Na sits over the midpoint of the bottom C–C bond), A₂ (where Na sits over the midpoint of the top C–C bond), A₃ (where Na sits over the M atom and interacts with four top C atoms) and A₄ (where Na sits over the M atom and interacts with two top C atoms) sites, as shown in Fig. 1a. After full relaxation, the energetically most favorable adsorption sites for Na on MC₂ (M = Y, Zr, Nb, and Mo) monolayers were identified by calculating their adsorption energy (E_ads), which is defined as:

E_ads = E_tot − E_MC2 − E_Na (3)

where E_tot and E_MC2 are the energies of the MC₂ supercell with and without a single Na atom, respectively, and E_Na represents the energy of an isolated Na atom. According to this definition, a more negative adsorption energy indicates a more favorable exothermic reaction between the MC₂ monolayer and Na atom.

Almost all the investigated MC₂ monolayers can effectively adsorb Na atoms, as indicated by the negative E_ads values (Fig. 4). Except for the YC₂ monolayer, on which the most stable adsorption site of Na is the A₃ site (E_ads = −1.55 eV), the Na's energy-favorable site on the other three MC₂ (M = Zr, Nb, and Mo) monolayers is A₁ with an E_ads of −1.67, −2.45 and −2.95 eV, respectively, in agreement with the preferred adsorption site on TiC₂ (A₁),³⁷ VC₂ (A₁),³⁸ and ScC₂ (A₃).⁴² In addition, the A₁ (A₃ and A₄) site is the Na's second favorable adsorption site on the YC₂ (ZrC₂/NbC₂ and MoC₂) monolayer. The difference of the most preferred Na adsorption site is due to the shorter Na–C bond lengths (d_Na–C) formed between Na and C at the A₃ (A₁) site than those at the A₁ (A₃) site on YC₂ (ZrC₂, NbC₂, and MoC₂): 2.51 vs. 3.54/3.70 Å (2.53 vs. 2.61, 2.59 vs. 2.62, and 2.78 vs. 2.81 Å). Among the four examined monolayers, Na has the strongest/weakest interaction with the MoC₂/YC₂ monolayer, which can be understood by the gradually increasing distance between the adsorbed Na and the MC₂ monolayer (from 1.49 Å on MoC₂ to 1.64 Å on YC₂, as shown in Fig. 5).

Fig. 4 The adsorption energies of a single Na atom on the (M = Y, Zr, Nb, and Mo) monolayer at the four examined adsorptions sites A₁, A₂, A₃ and A₄.

Fig. 5 Top and side views of the most stable Na-adsorbed configurations of YC₂ (a), ZrC₂ (b), NbC₂ (c), and MoC₂ (d) monolayers; right plane corresponds to the charge density difference. The isosurface level is set to 0.002 e Å⁻³. Blue and yellow regions indicate electron depletion and accumulation, respectively.

Fig. 5 presents the fully optimized structures for the Na-adsorbed YC₂ monolayer at the site A₃ and MC₂ (M = Zr, Nb, and Mo) monolayers at the site A₁, respectively. For Na-adsorbed Y/Zr/Nb/MoC₂ monolayers, the Na–C bond length is 2.51/2.53/2.59/2.78 Å, and the adsorption height (h), i.e., the vertical distance between the top-layer carbon and Na atom is 1.64/1.58/1.54/1.49 Å. Obviously, the MC₂ monolayer with a larger atomic number (smaller atomic radii) from Y → Mo has a lower adsorption height (h); however, the Na–C bond length and adsorption energy (E_ads) are inversely proportional to the atomic radii.

To better understand the physical mechanism of the adsorption process, we investigated the charge transfer process between a single Na atom adsorbed on the MC₂ (M = Y, Zr, Nb, and Mo) monolayers by using the charge density difference using the following formula,

Δρ = ρ_tot − ρ_MC2 − ρ_Na (4)

where ρ_tot and ρ_MC2 denote the total electron densities of the relaxed MC₂ monolayer with and without Na, respectively, and ρ_Na is the total electron density of an isolated Na atom. Fig. 5 right panel presents charge density difference plots for Na adsorbed YC₂ at site A₃ and Zr/Nb/MoC₂ at site A₁, in which the electron accumulation (yellow) is located between the Na atom and MC₂ monolayer, while the electron depletion (blue) is around the Na atoms, suggesting that the Na donates electrons and the MC₂ monolayer accepts electrons, which originates from the smaller electronegativity of Na than that of C and metal atoms M (M = Y, Zr, Nb, and Mo). The Bader charge analysis showed that the electrons transferred from Na to the MC₂ (M = Y, Zr, Nb, and Mo) monolayers are 0.84, 0.82, 0.84, and 0.85 e⁻, respectively. It can also be observed from the PDOS of Na-adsorbed MC₂ (M = Y, Zr, Nb, and Mo) shown in Fig. S8† that hybridization exists between the Na atom and the C atom of the MC₂ (M = Y, Zr, Nb, and Mo) monolayer. At the same time, the adsorption of Na only slightly influences the electronic structure of the pristine MC₂ monolayer, and the metallic features are maintained after the adsorption of Na according to Fig. S8,† which is rather beneficial to their application as anode materials.

Metal mobility was another significant factor in the design and selection of effective anode materials, which are directly linked to the charge–discharge rates of rechargeable batteries. Thus, we further investigated the diffusion of Na on the MC₂ (M = Y, Zr, Nb, and Mo) monolayers (Fig. 6). For the YC₂ monolayer (Fig. 6b), since A₃ is the energetically most favorable site among the four adsorption sites (A₁, A₂, A₃, and A₄), three representative possible diffusion pathways were considered: path-I (A₃–A₁–A₃), where Na migrates from an A₃ to another A₃ by crossing an A₁ site, marked by red arrows; path-II (A₃–A₄–A₃), where Na migrates from an A₃ to another A₃ by crossing an A₄ site, marked by blue arrows; path-III (A₃–A₂–A₃), where Na migrates from an A₃ to another A₃ by crossing an A₂ site, marked by black arrows. For the Zr/Nb/MoC₂ monolayers (Fig. 6d, f and h), similarly, there are three diffusion pathways with both starting and ending sites at A₁: path-I (A₁–A₃–A₁), path-II (A₁–A₂–A₁), and path-III (A₁–A₄–A₁), marked by red, blue and black arrows, respectively.

Fig. 6 The energy profile (left plane) and diffusion pathways (right plane) of Na diffusing on theYC₂ (a and b), ZrC₂ (c and d), NbC₂ (e and f), and MoC₂ (g and h) monolayers.

The energy profiles for Na diffusion over MC₂ (M = Y, Zr, Nb, and Mo) monolayers along the examined diffusion pathways are illustrated in Fig. 6a, c, e and g. Obviously, Na diffusing along path-I has the lowest diffusion barrier (E_d) on the YC₂ (E_d = 0.21 eV), ZrC₂ (E_d = 0.04 eV), and NbC₂ (E_d = 0.20 eV) monolayers, while on the MoC₂ monolayer, path-III has the smallest diffusion barrier of 0.22 eV. Actually, the energy-favored diffusion route is Na migration from the most energy-preferred site to another crossing the second energy-preferred site on the four monolayers. Since the second lowest adsorption site on MoC₂ is A₄, the Na diffusion path with the smallest barrier is path-III instead of path-I on the other three monolayers. The corresponding diffusion lengths of Na along the minimum energy pathways on the four MC₂ monolayers are given in Fig. S9,† and the diffusion length of Na on MC₂ monotonically increases with the lattice constant from the MoC₂ to the YC₂ monolayer. Note that the Na diffusion barriers on the four MC₂ monolayers are all lower than that on borophene (0.32 eV),⁷³ Ti₃C₄ (0.35)⁷⁴ and MoS₂ (0.28 eV),⁷⁵ and the lowest E_d value of 0.04 eV on the ZrC₂ monolayer is comparable with the E_d on ScC₂ (0.05 eV)⁴² and Ti₃C₂ (0.096),⁷⁶ as shown in Table 3, indicating their excellent charge–discharge rate. Compared with the reported TiC₂ monolayer as a promising anode for LIBs,³⁷ the Na diffusing along path-I has the lowest diffusion barrier on TiC₂ (E_d = 0.19 eV, Fig. S10†), and the E_d is comparable with the values on Y/Nb/MoC₂ monolayers (0.20–0.22 eV), but higher than that on the ZrC₂ monolayer (0.04 eV). Thus, as for SIB anode materials, compared to the TiC₂ monolayer, ZrC₂ distinguishes itself among the four examined MC₂ (M = Y, Zr, Nb, and Mo) monolayers, since it has the lowest Na diffusion barrier and high Na capacity comparable to that of TiC₂. Overall, from the perspective of Na diffusion behavior, anode materials with low energy barriers and short diffusion pathway lengths are suitable for SIBs.

Table 3 Summary of the theoretical specific capacity (in mA h g⁻¹) and diffusion barriers (in eV) of some widely investigated promising electrode materials for SIBs

	Theoretical specific capacity	Diffusion barrier	References
YC2	478	0.21	This work
ZrC2	699	0.04	This work
NbC2	687	0.20	This work
MoC2	675	0.22	This work
TiC2	727	0.19	This work
Ti3C2	352	0.096	78
Ti3C4	560	0.35	74
ScC2	777	0.05	42
Sc2C	362	0.012	81
Nb2C	271	0.015	33
Mo2C	132	0.015	34
MoC	284.2	0.09	82
MoS2	146	0.28	75
Phosphorene	315.52	0.30	80
Na2Mn3O7	166.1	0.16	7

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3.5 Theoretical storage capacity and open-circuit voltage

2D materials as anode materials display a lot of attractive properties; besides mechanical flexibility, excellent electrical conductivity, and lower diffusion energies, they also have the ability to adsorb multilayer Na atoms and exhibit high capacity.^77,78 Inspired by these properties, we expect that the predicted MC₂ (M = Y, Zr, Nb, and Mo) monolayers can be used as anodes for SIBs with high storage capacity. Here, the two important descriptors, namely storage capacity (C) and average open-circuit voltage (OCV), were calculated to determine the performance of the SIBs based on the charge–discharge mechanism.

The OCV can be evaluated from the following common half-cell reaction of the charge/discharge process:

MC₂ + xNaⁿ⁺ + xne⁻ ↔ Na_xMC₂ (5)

when the volume and entropy effects during the sodiation process were neglected, the open-circuit voltage (OCV) can be derived from the average adsorption energy (E_ave) as:

OCV = −E_ave/xne (6a)

E_ave = (E_NaxMC2 − E_MC2 − xE_Na-bulk)/x (6b)

where E_MC2, E_NaxMC2, and E_Na-bulk are the energy of the pristine MC₂ monolayer, the total energy of Na adsorbed MC₂, and the energy per Na atom in the sodium bcc bulk, respectively, n is the number of valence electrons (n = 1 for Na), and x is the chemical content of sodium atoms; ideally, the anode material can keep adsorbing Na atoms if E_ave is smaller than 0 eV per atom. The corresponding maximum capacity were calculated by using the following equation:

C = x_max × F/M_MC2 (7)

where x_max is the highest concentration of Na in Na_xMC₂, F is the Faraday constant (26 801 mA h mol⁻¹), and M_MC2 is the molar weight of MC₂ (M = Y, Zr, Nb, and Mo).

Then the maximum storage capacity of Na atoms on MC₂ (M = Y, Zr, Nb, and Mo) monolayers was evaluated by adding Na atoms to the surface gradually. As mentioned above, the most stable adsorption sites for Na on the four monolayers have been determined: the first layer of Na interaction at the A₃ site for YC₂ and the A₁ site for Zr/Nb/MoC₂, respectively; when the adsorption positions of the first layer are completely occupied, the subsequently added Na atoms form a second and a third adsorption layer. In addition, for the different sodium concentrations on the MC₂ monolayer, the most stable configurations for each concentration, we considered by calculating their formation energy (E_f) of Na_xMC₂ (M = Y, Zr, Nb, and Mo) compounds with respect to the MC₂ monolayer and Na bulk phase as the references (Fig. S11†).⁷⁹

Here, based on the MC₂ monolayer, taking the process of successive adsorption of Na atoms on the NbC₂ monolayer as an example, adding a Na atom at site A₁ on one surface of NbC₂ results in a low-limit chemical formula Na_xNbC₂ (x = 0.125, Fig. 5c). Our computations showed that Na atoms tend to bond to both surfaces of the NbC₂ monolayer when occupying the A₁ site (Fig. S12–S14†). When the A₁ sites of both surfaces were all occupied, Na atoms start to occupy the A₃ sites in the second layer (Fig. S15†). Thus, the several possible configurations with both surfaces exposed to Na atoms were carefully considered for Na_xNbC₂ systems with higher x values (x = 0.25, 0.375, 0.5, 1, 2, 3), and the most stable configurations for each concentration are shown in Fig. 7a–f. The same approach was used to construct Na_xYC₂, Na_xZrC₂, and Na_xMoC₂ systems.

Fig. 7 Top and side views of the optimized structures of Na_0.25NbC₂ (a), Na_0.375NbC₂ (b), Na_0.5NbC₂ (c), NaNbC₂ (d), Na₂NbC₂ (e), and Na₃NbC₂ (f). The average adsorption energy (g) and open circuit voltage profile (h) of Na_xYC₂, Na_xZrC₂, Na_xNbC₂, and Na_xMoC₂ as a function of Na concentration.

The average adsorption energy decreases with the increase of Na concentration (x), indicating the decreased thermodynamic stability of Na_xMC₂ (Fig. 7g), which can be attributed to the repulsive interaction between Na atoms, thus making more adsorption more difficult. For the Zr/Nb/MoC₂ monolayer, the first Na-adsorption layer is formed by placing Na at the most favorable sites (A₁) on both sides with an E_ave of −0.41/−0.77/−0.65 eV (the composite is NaZrC₂/NaNbC₂/NaMoC₂), suggesting the feasibility of adsorbing the second Na layer. The E_ave of the second Na layer with Na adsorbing at site A₃ (A₄ for MoC₂) is −0.27/−0.32/−0.28 eV and the corresponding configurations are Na₂ZrC₂, Na₂NbC₂, and Na₂MoC₂. When adding the third Na-layer on both sides with site A₄ (A₃ for MoC₂), the E_ave is −0.14/−0.25/−0.24 eV and the composites are Na₃ZrC₂, Na₃ZrC₂, and Na₃MoC₂. However, for the YC₂ monolayer, in the first Na-layer Na atoms are adsorbed at the most favorable sites (A₃) on both sides, and the E_ave is −0.21 eV (the chemical formula is NaYC₂). The E_ave of the second layer with Na atoms locating at site A₁ is −0.11 eV and the corresponding configuration is Na₂YC₂. The third Na-layer with Na atom adsorption at site A₄ on both sides has an E_ave of 0.06 eV, which implies that only the two adsorption layers of Na atoms are stable on the YC₂ monolayer. Thus, we can conclude that the three (two) layers of Na atoms can be stably adsorbed on the Zr/Nb/MoC₂ (YC₂) monolayer.

Based on the above results, we estimated the maximum capacity (C) and open-circuit voltage (OCV) of Na atoms adsorbed on the MC₂ (M = Y, Zr, Nb, and Mo) monolayers with the chemical stoichiometry of Na₂YC₂, Na₃ZrC₂, Na₃NbC₂, and Na₃MoC₂ (Fig. 7h). The corresponding values of maximum capacity (C) for SIBs are 478, 699, 687, and 675 mA h g⁻¹, respectively. The MC₂ (M = Y, Zr, Nb, and Mo) monolayers have much larger theoretical specific capacity in comparison to other electrode materials for SIBs, such as Ti₃C₂ (352 mA h g⁻¹),⁷⁶ phosphorene (315.52 mA h g⁻¹),⁸⁰ Sc₂C (362 mA h g⁻¹),⁸¹ Nb₂C (271 mA h g⁻¹),³³ MoS₂ (146 mA h g⁻¹),⁷⁵ MoC (248.2 mA h g⁻¹)⁸² and Na-based layered materials (Na₂Mn₃O₇, 166.1 mA h g⁻¹),⁷ as shown in Table 3. We also estimated the Na capacity on the TiC₂ monolayer and found that the TiC₂ monolayer can adsorb two Na layers with the final formula of Na₂TiC₂, and the corresponding maximum specific capacity was then determined to be 727.29 mA h g⁻¹, higher than that of the Y/Zr/Nb/MoC₂ monolayer. Thus, as for SIB anode materials, compared to the TiC₂ monolayer, ZrC₂ distinguishes itself among the four examined MC₂ (M = Y, Zr, Nb, and Mo) monolayers, since it has the lowest Na diffusion barrier and high Na capacity comparable to that of TiC₂.

In addition, the OCV values are 0.11, 0.18, 0.25 and 0.24 V, which can avoid the dendrite formation of Na metal during the charge–discharge process due to satisfying the requirement of 0.1–1.0 eV for SIB anode materials.⁸³ At the same time, the values are rather low and comparable to or slighter larger than that of 2D materials Mo₂C (0.166 V),³⁴ Ti₃C₂ (0.137 V),⁷⁸ and NiC₃ (0.17 V).⁸⁴ The OCV values are low for Na atoms to be adsorbed on MC₂ (M = Y, Zr, Nb, and Mo) nanosheets, which are beneficial for increasing and operating voltage and safety in practical applications.

Then, we examined the geometries of the MC₂ (M = Y, Zr, Nb, and Mo) monolayers upon Na adsorption to evaluate the expansion in volume. The changes of the lattice constant and volume for the host sheets with interacted Na atoms are listed in Table S1.† Compared with the bare sheets, the expansions of the lattice and volume for the maximum Na concentrations are less than 8% and 10%, respectively, comparable to or much smaller than the lattice changes of the FeSe monolayer (5.50%)⁷¹ and the volume expansion of graphite (10%)⁸⁵ and Na-based bulk electrode material NaTiF₄ (13%),⁷ implying good cycling stability.

Finally, we explored the stability of the examined four C₂-containing MC₂ (M = Y, Zr, Nb, and Mo) monolayers under high concentration Na adsorption, by carrying out the FPMD stimulations of Na₂YC₂, Na₃ZrC₂, Na₃NbC₂, and Na₃MoC₂ monolayers for 5 ps at 300 and 500 K, respectively. No bond breakage or atomic structure reconstruction was observed, indicating that the 2D MC₂ (M = Y, Zr, Nb, and Mo) materials are still stable under the maximum coverage of sodium. Overall, we conclude that the good structural stability of MC₂ (M = Y, Zr, Nb, and Mo) sheets endows them with great potential as high-performance anode materials for SIBs.

4. Conclusions

In summary, based on first-principles calculations, we proposed four C₂-containing monolayers (MC₂, M = Y, Zr, Nb, and Mo) and systematically investigated the structural stability, mechanical and electronic properties, as well as the performance as anode materials for SIBs. Our results show that the MC₂ (M = Y, Zr, Nb, and Mo) monolayers possess high cohesive energy and good dynamic, thermal and mechanical stability, and have metallic properties with excellent electronic conductivities. When used as battery electrodes for adsorbing Na atoms, the YC₂ and Zr/Nb/MoC₂ monolayers can adsorb two and three layers of Na atoms with the final formula of Na₂YC₂ and Na₃ZrC₂/Na₃NbC₂/Na₃MoC₂, respectively, and the corresponding maximum specific capacities (open circuit-voltage) were calculated to be 478 (0.11) and 699 (0.18)/687 (0.25)/675 (0.24) mA h g⁻¹ (V), respectively. Taking advantage of containing C₂ dimers, the MC₂ (M = Y, Zr, Nb, and Mo) monolayers show superior stability and have high Na capacity due to the higher content of C. In addition, Na atoms show fast diffusion on the MC₂ (M = Y, Zr, Nb, and Mo) monolayers with low diffusion barriers (0.21, 0.04, 0.20 and 0.22, respectively), indicating excellent rate capability. In view of the superior properties, such as good electronic conductivity, fast Na diffusion, low OCV, and high theoretical Na storage capacity, MC₂ (M = Y, Zr, Nb, and Mo) monolayers become promising anode materials for SIBs.

Conflicts of interest

There are no conflicts of interest to declare.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (11964024), the “Grassland Talents” project of Inner Mongolia autonomous region (12000-12102613), the Young Scientific Talent Development Project (10000-21221505) and the Training Program of Innovation and Entrepreneurship for Undergraduates of Inner Mongolia University (201812236).

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Footnote

† Electronic supplementary information (ESI) available: Lattice constants of the MC₂ monolayers; bond length and thickness of the MC₂ monolayers; the calculated phonon spectra of the MC₂ monolayers; the FPMD simulations at the end of 5 ps at 300 K; the electronic localization function (ELF) of the MC₂ monolayers; the partial density of states (TDOS and PDOS) of the MC₂ monolayers; the partial density of states (PDOS) of Na-adsorbed MC₂ monolayers; the variation of the path-I diffusion lengths of Na on the MC₂ monolayers with the lattice constants of MC₂; the energy profile (left plane) and diffusion pathways (right plane) of Na diffusing on the TiC₂ monolayer; the formation energy (E_f) of Na_xMC₂ with respect to MC₂ monolayers and Na bulk bcc metal; the configurations and their optimized adsorption energies (eV) for Na_xNbC₂; the side views of the Na-intercalated MC₂ monolayer at the end of 5 ps FPMD simulations at 300 and 500 K, respectively. See DOI: 10.1039/d1na00422k

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