Introduction

Rechargeable Li-ion batteries are very successful in commercial energy storage, however, the scarcity (0.0017 wt% in earth’s crust) and uneven geographical distribution of Li lead to high cost and cannot meet the quickly increasing demand1,2,3,4,5. Therefore, it is of great importance to explore battery systems based on abundant elements. Na-ion batteries (NIBs) and K-ion batteries (KIBs) are considered to be sustainable alternatives to Li-ion batteries because of the similarity of the physicochemical properties of Na/K to those of Li and the abundant availability of Na/K (2.3/1.5 wt% in earth’s crust)6,7,8,9. While NIBs/KIBs share the operating mechanism with Li-ion batteries, the ionic radius of Na/K (1.02/1.38 Å) is much larger than that of Li (0.76 Å). Thus, most conventional anode materials of Li-ion batteries cannot be used for NIBs/KIBs10. For instance, graphite provides a Li specific capacity of 372 mA h g−1, but only a Na/K specific capacity of 284/273 mA h g−111,12,13. Identifying high performance anode materials for NIBs/KIBs therefore is a critical issue.

Two-dimensional materials are promising options for the anode of NIBs/KIBs because they provide a large active surface and interlayer gaps to accommodate Na/K, enabling high energy density and high ion mobility14,15,16,17. Phosphorene, a well-known two-dimensional material, can form (Na/K)3P with a high specific capacity of 2596/1300 mA h g−118,19. However, the large ionic radius of Na/K causes a huge volume change during the charge-discharge process, leading to structure deterioration and rapid capacity decay19. Metal phosphides, on the other hand, can combine high electronic conductivity and small volume changes during the charge-discharge process with high specific capacities and suitable open-circuit voltages20,21. For this reason, we execute in the following an ab initio evolutionary search for metal phosphide monolayers suitable for application as anode of NIBs and KIBs.

Results and discussion

The relaxed structure of the MoP2 monolayer is shown in Fig. 1a and the corresponding structural information is given in Table 1. There are four Mo atoms and eight P atoms in the unit cell, forming a 3.10 Å thick layer with the P atoms exposed at the surfaces, each P atom is bonded to one P and three Mo atoms, and each Mo atom is bonded to six P atoms. The Mo-P bond lengths of 2.43 and 2.47 Å are comparable to those in bulk MoP (2.48 Å)22, orthorhombic MoP2 (2.50 Å)23, and two-dimensional structures with other Mo:P ratios22,24. The P-P bond length of 2.23 Å is comparable to those in black phosphorus (2.22 and 2.26 Å)25. The electron localization function is utilized to analyze the bonding character, with values of 1.0 and 0.5 representing localized and free electrons, respectively, whereas a value close to zero represents low electron density26. According to Fig. 1b, the electrons are more localized around the P atoms than around the Mo atoms. Bader charge analysis indicates that each Mo atom transfers 0.76 electrons to the P atoms. We obtain for the cohesive energy Ecoh = (4EMo + 8EP − EMoP2)/12, where EMo, EP, and EMoP2 denote the total energies of a Mo atom, a P atom, and the MoP2 monolayer (4 formula units per unit cell), respectively, a value of 4.79 eV per atom, which exceeds those of Cu2P (3.21 eV per atom)27, phosphorene (3.30 eV per atom)28, BeP2 (3.53 eV per atom)29, FeP3 (4.13 eV per atom)30, and FeP2 (4.39 eV per atom)30. Absence of negative phonon frequencies in Fig. 1c demonstrates dynamical stability of the MoP2 monolayer. In addition, the results of our AIMD simulations, see Fig. 1d–h, exhibit no signs of structural disruption or phase transition at 300 K, demonstrating thermal stability. The structure is still stable at 1800 K but not at 1900 K, implying that the melting point of the MoP2 monolayer falls into this range.

Fig. 1: Structure and stability.
figure 1

a Top and side views of the MoP2 monolayer (orange = P atoms, green = Mo atoms). The dashed lines mark the unit cell. b Electron localization function of the MoP2 monolayer [(010) plane]. c Phonon spectrum of the MoP2 monolayer. d Total energies during the AIMD simulations of the MoP2 monolayer at 300, 1000, 1800, and 1900 K, and (e–h) top and side views of the corresponding final structures.

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Table 1 Structural information of the MoP2 monolayer (space group Pcca, a = b = 5.80 Å, c = 25 Å)
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The calculated elastic constants of the MoP2 monolayer are C11 = C22 = 178.7 N m−1, C12 = 40.6 N m−1, and C66 = 63.2 N m−1, obeying the mechanical stability criteria C11C22 − C122 > 0 and C66 > 0. We show in Fig. 2a the direction dependence (θ = angle with respect to the x-axis) of Young’s modulus,

$$E(\theta )=\frac{{C}_{11}{C}_{22}-{C}_{12}^{2}}{{C}_{11}{\sin }^{4}\theta +[({C}_{11}{C}_{22}-{C}_{12}^{2})/{C}_{66}-2{C}_{12}]{\sin }^{2}\theta {\cos }^{2}\theta +{C}_{22}{\cos }^{4}\theta },$$
(1)

which characterizes the stiffness of a material, and in Fig. 2b the direction dependence of Poisson’s ratio,

$$v(\theta )=\frac{{C}_{12}{\sin }^{4}\theta -[{C}_{11}+{C}_{22}-({C}_{11}{C}_{22}-{C}_{12}^{2})/{C}_{66}]{\sin }^{2}\theta \,{\cos }^{2}\theta +{C}_{12}{\cos }^{4}\theta }{{C}_{11}{\sin }^{4}\theta +[({C}_{11}{C}_{22}-{C}_{12}^{2})/{C}_{66}-2{C}_{12}]{\sin }^{2}\theta {\cos }^{2}\theta +{C}_{22}{\cos }^{4}\theta },$$
(2)

which characterizes the response of a material to external load. According to Fig. 2a, the maximum of Young’s modulus is 169.5 N m−1 at θ = 0°, which is higher than those of phosphorene (92 N m−1 in the zigzag direction; 23 N m−1 in the armchair direction)31 and MoS2 (124 N m−1)32, but smaller than that of graphene (334 N m−1)32. According to Fig. 2b, Poisson’s ratio varies between 0.23 and 0.27. The fracture strain and strength are investigated by calculating the stress-strain curves shown in Fig. 2c. We find fracture strains of 20% (uniaxial) and 17% (biaxial) with corresponding fracture strengths of 5.8 and 11.4 N m−1, which are comparable to those of phosphorene (27% and 10 N m−1 in the zigzag direction; 30% and 4 N m−1 in the armchair direction)33 and MoS2 (23% and 14.8 N m−1 biaxial)34. The electronic band structure in Fig. 2d shows several bands crossing the Fermi energy, reflecting excellent metallicity.

Fig. 2: Mechanical and electronic properties.
figure 2

a Young’s modulus. b Poisson’s ratio. c Stress-strain curves (uniaxial and biaxial). and (d) electronic band structure of the MoP2 monolayer.

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To determine the energetically favorable stacking pattern of the MoP2 bilayer, we calculate the binding energy Eb = (2EMoP2 − Ebilayer)/N for the stacking patterns shown in Fig. 3a–d, where Ebilayer is the total energy of the MoP2 bilayer. With a binding energy of 39 meV per atom and an interlayer distance of 3.31 Å the AB2 stacking pattern is found to be energetically favorable. We further evaluate the n-layer exfoliation energy Eexf (n) = (En – En/2)/S, where En is the total energy of a MoP2 n-layer in the vacuum, E is the total energy of a MoP2 bilayer in the bulk, and S is the surface area. According to Fig. 3e, the exfoliation energy of the MoP2 monolayer is 0.45 J m−2, which is smaller than those of the InP3 monolayer (1.08 J m−2)35 and CaP3 monolayer (1.30 J m−2)36. Consequently, the predicted MoP2 monolayer is not only stable but it is likely that it can be prepared experimentally.

Fig. 3: Stacking.
figure 3

a–d Relaxed structures of four possible stacking patterns of the MoP2 bilayer. e Exfoliation energy per unit cell of a MoP2 n-layer.

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We use a 2 × 2 × 1 supercell of the MoP2 monolayer to determine the energetically favorable adsorption site of Na/K, considering sites S1 to S5 as marked in Fig. 4a. The adsorption energy is given by Ead = EMoP2+Na/K − EMoP2 − ENa/K, where EMoP2+Na/K is the total energy of the MoP2 monolayer with adsorbed Na/K atom and ENa/K is the total energy per atom in bulk Na/K. Negative Ead avoids agglomeration of Na/K atoms (dendrite formation). According to the results in Fig. 4b, adsorption of Na/K is possible on top of the center of the Mo2P3 ring (S1 site), on top of Mo (S2 site), on top of P (S3 site), and on top of the P-P bond (S4 site), whereas Na/K on top of the Mo-P bond (S5 site) relaxes to the S2 site during the structure relaxation. The obtained values of Ead and corresponding adsorption heights are reported in Table 2, demonstrating that the S2 site is energetically favorable. Furthermore, the charge redistributions between Na/K and the MoP2 monolayer are shown in Fig. 4c–j for the S1–S4 sites, demonstrating that Na/K acts as charge donor. Bader charge analysis results in charge transfers of 0.85–0.89 electrons, see Table 2.

Fig. 4: Adsorption.
figure 4

a Considered adsorption sites of Na/K on the MoP2 monolayer. b Adsorption energies of Na/K on the MoP2 monolayer. Na/K at the S5 site relaxes to the S2 site. Side views of the charge redistributions induced by c–f Na at the S1–S4 sites and (g–j) K at the S1–S4 sites. The cyan and yellow isosurfaces (isovalue = 0.008 electrons/Å3) represent charge depletion and accumulation, respectively.

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Table 2 Adsorption energies (eV), adsorption heights (Å), and charge transfers (electrons) of Na/K on the MoP2 monolayer at different sites
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To estimate the charge-discharge rate, a key factor of an anode material, we study the diffusion barrier of Na/K in Fig. 5a, b. Figure 5c–f presents the obtained Na/K diffusion paths between neighboring S2 sites: S2 → S1 → S2 (path 1) and S2 → S1 → S4 → S1 → S2 (path 2). In the case of Na/K, we obtain for path 1 a diffusion barrier of 0.12/0.09 eV and for path 2 a diffusion barrier of 0.32/0.26 eV. Hence, path 1 is energetically favorable for both Na and K. The obtained Na/K diffusion barriers are comparable to those of blue phosphorene (0.11/0.09 eV)37 and lower than those of Ti2CP2 (0.29/0.19 eV)38, GeP3 (0.27/0.29 eV)39, AlP3 (0.41/0.20 eV)40, CoP (0.38/0.32 eV)41, and MoN2 (0.56/0.49 eV)42. A low diffusion barrier ensures mobility of Na/K and, consequently, a high charge-discharge rate.

Fig. 5: Diffusion.
figure 5

a Na and (b) K diffusion barriers on the MoP2 monolayer. The diffusion paths are shown in (c, e) and (d, f), respectively.

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To determine the specific capacity, another key factor of an anode material, we successively add Na/K atoms on both sides of the MoP2 monolayer (2 × 2 × 1 supercell). The maximal adsorption is achieved when the adsorption energy Ead(m) = EMoP2(Na/K)m+1 − EMoP2(Na/K)m − ENa/K switches from negative to positive, where EMoP2 (Na/K)m is the total energy when m Na/K atoms are adsorbed. It turns out that the MoP2 monolayer can accommodate 32 Na and 29 K atoms, which corresponds to stoichiometries of MoP2Na2 and MoP2K1.81, respectively. Therefore, the specific capacity C = mF/M, where F = 26801 mA h mol-1 is the Faraday constant and M = 158 g mol−1 is the relative molecular mass of the MoP2 monolayer, is found to be 339/308 mA h g−1 for Na/K. These values exceed those of the commercial graphite anode (284/273 mA h g−1)12,13. Compared with other two-dimensional anode materials, the value for Na exceeds those of Mo2C (132 mA h g−1)43, MoS2 (146 mA h g−1)44, Mo2P (240 mA h g−1)45, SnP3 (253 mA h g−1)46, and InP3 (258 mA h g−1)47. The value for K exceeds those of Ti3C2 (192 mA h g−1)48, CoP (208 mA h g-1)41, GeS (256 mA h g−1)49, and T-NiSe2 (247 mA h g−1)50.

We further study the stability of the structures with adsorbed Na/K atoms by calculating the formation energy of MoP2(Na/K)m with respect to the MoP2 monolayer and MoP2(Na2/K1.81) as

$$E_{f}(m) = E_{{{{\rm{MoP}}2{\rm{Na}}m}}}-(mE_{{\rm{MoP}}2{\rm{Na2}}} + (2 - m)E_{{\rm{MoP}}2})/2$$

or

$$E_{f} (m) = E_{{\rm{MoP}}2{\rm{K}}m} - (mE_{{\rm{MoP}}2{\rm{K}}1.81} + (1.81 - m) E_{{\rm{MoP}}2})/1.81.$$

The results are shown in Fig. 6a, b, with the data points on the convex hull representing the thermodynamically stable structures. Stability is found for m = 0.06, 0.13, 0.19, 0.25, 0.31, 1.00, and 2.00 in the case of Na (Supplementary Fig. 1) and for m = 0.06, 0.13, 0.19, 0.25, 0.31, 0.75, 1.00, 1.38, and 1.81 in the case of K (Supplementary Fig. 2). The open-circuit voltage −Eave/e, where e is the elementary charge, is calculated for the stable structures. According to Fig. 6c, e, it decreases from 0.79 V at m = 0.06 to 0.13 V at m = 2.00 in the case of Na, with an average of 0.48 V, and from 1.38 V at m = 0.06 to 0.04 V at m = 1.81 in the case of K, with an average of 0.70 V. The decreasing trends of the open-circuit voltage to low values at maximal adsorption imply that the working voltage of the battery (when the anode is coupled to the cathode) will increase to high values. The metallicity of the MoP2 monolayer is preserved at maximal Na/K coverage (Fig. 6d, f), as required for the anode to operate.

Fig. 6: Adsorption.
figure 6

Formation energy as function of the (a) Na and (b) K coverage. Open-circuit voltage as function of the (c) Na and (e) K coverage (for the thermodynamically stable structures). Electronic band structures of (d) MoP2Na2 and (f) MoP2K1.81 (Fermi level = energy zero).

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The electron localization functions of the MoP2 monolayer with different amounts of adsorbed Na/K atoms are shown in Fig. 7a−e. Bader charge analysis indicates in the case of one adsorbed layer that each Na/K atom donates 0.64/0.51 electrons to the MoP2 monolayer. In the case of maximal adsorption each atom in the inner Na/K layer donates 0.11/0.19 electrons to the outer Na/K layer and 0.52/0.47 electrons to the MoP2 monolayer. The charge accumulation in the outer Na/K layer, see Fig. 7c, e, prohibits further Na/K adsorption. As volume changes during the charge-discharge process introduce structural defects and, therefore, lead to capacity decay, we turn our attention to the MoP2 bilayer and find that it can accommodate an intercalation of two layers of Na/K atoms with Eave = −0.33/ − 0.55 eV. The intercalation enlarges the interlayer distance to 7.24/9.12 Å, corresponding to a volume expansion of 118%/176%, which is decisively smaller than that of black phosphorus (443% for Na3P and 695% for K3P)19, for example. The electron localization functions of the MoP2 bilayer without and with the intercalated atoms are shown in Fig. 7f−h. Each Na/K atom donates 0.44/0.43 electrons to the MoP2 bilayer according to Bader charge analysis. AIMD simulations of the MoP2(Na2/K1.81) monolayer and Na/K-intercalated MoP2 bilayer, see Fig. 8, show no sign of structural collapse. The stacking of the MoP2 bilayer shifts gradually to an AA pattern in the AIMD simulation while the interlayer distance increases to 7.31/9.28 Å for Na/K-intercalation, which still is a very small volume expansion of only 116%/180%. We finally investigate in Fig. 9a the Na specific capacity as a function of the number of MoP2 layers (n). It turns out that always two layers of Na atoms can be accommodated, see Fig. 9b, corresponding to a stoichiometry of MoP2Na1+1/n. For bulk MoP2Na (n → ∞), see Fig. 9c, we find Eave = −0.34 eV and a Na specific capacity of 170 mA h g−1. The obtained interlayer distance of 7.45 Å corresponds to a volume expansion of 125%.

Fig. 7: Localization.
figure 7

Electron localization function of the MoP2 monolayer [(010) plane] (a) without adsorbed atoms, (b) with one layer of adsorbed Na atoms, (c) with maximal number of adsorbed Na atoms, (d) with one layer of adsorbed K atoms, and (e) with maximal number of adsorbed K atoms. Electron localization function of the MoP2 bilayer [(010) plane] (f) without adsorbed atoms, (g) with maximal number of intercalated Na atoms, and (h) with maximal number of intercalated K atoms.

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Fig. 8: Adsorption and intercalation.
figure 8

Top and side views of the MoP2 monolayer with maximal number of adsorbed (a) Na and (b) K atoms after the AIMD simulations at 300 K. Top and side views of the MoP2 bilayer with two layers of intercalated (c) Na and (d) K atoms after the AIMD simulations at 300 K.

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Fig. 9: Adsorption and intercalation.
figure 9

a Na specific capacity as function of the number of MoP2 layers. Side view of (b) the MoP2 bilayer with maximal Na loading and (c) bulk MoP2Na.

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Our ab initio evolutionary search for metal phosphide monolayers points to the existence of a MoP2 monolayer that is dynamically, thermally, and mechanically stable according to the calculated phonon spectrum, AIMD simulations at room and elevated temperatures, and the calculated elastic constants, respectively. We find that this MoP2 monolayer provides excellent properties for application as anode material for NIBs/KIBs. Specifically, for Na it combines a high specific capacity of 339 mA h g−1, a low diffusion barrier of 0.12 eV, and a low average open-circuit voltage of 0.48 V with a small volume expansion of only 125%.

Methods

We employ the USPEX51,52,53 code to perform a global structure search for different Mo:P ratios, limited to a maximum of 16 atoms in the primitive unit cell. The population size is set to 100 and the number of generations is set to 50. We adopt density functional theory (Vienna Ab-initio Simulation Package54) to perform structural relaxations and electronic structure calculations with the exchange-correlation functionals of Perdew-Burke-Ernzerhof and Heyd-Scuseria-Ernzerhof, respectively, where the van der Waals interaction is taken into account by the DFT-D3 method55 for the MoP2 monolayer and by the optB88-vdW method56 for the MoP2 bilayer to achieve reliable results. The plane wave cutoff energy is set to 600 eV, the total energy is converged to 10−6 eV, and the atomic forces are converged to 10−3 eV/Å. A Monkhorst-Pack k-sampling with 0.015 Å−1 spacing is used. Each simulation cell contains a vacuum slab of at least 20 Å thickness. The Phonopy code is adopted to calculate the phonon band structures using a 4 × 4 × 1 supercell of the MoP2 monolayer57. Ab initio molecular dynamics (AIMD) simulations are carried out for 10 ps based on a canonical ensemble and a Nosé-Hoover temperature control58. They are conducted at 300, 1000, 1800, and 1900 K for a 4 × 4 × 1 supercell of the MoP2 monolayer and at 300 K for a 4 × 2 × 1 supercell of the MoP2 bilayer. A time step of 1 fs is used. The Na/K diffusion barriers and pathways are derived by means of the climbing-image nudged elastic band method59.