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Cite This: J. Phys. Chem. C XXXX, XXX, XXX−XXX

Mechanistic Quantification of Thermodynamic Stability and Mechanical Strength for Two-Dimensional Transition-Metal Carbides Zhongheng Fu,†,‡ Hang Zhang,†,‡ Chen Si,†,‡ Dominik Legut,§ Timothy C. Germann,∥ Qianfan Zhang,†,‡ Shiyu Du,⊥ Joseph S. Francisco,# and Ruifeng Zhang*,†,‡ †

School of Materials Science and Engineering and ‡Center for Integrated Computational Materials Engineering, International Research Institute for Multidisciplinary Science, Beihang University, Beijing 100191, P. R. China § IT4Innovations Center, VSB-Technical University of Ostrava, CZ-70833 Ostrava, Czech Republic ∥ Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, United States ⊥ Engineering Laboratory of Specialty Fibers and Nuclear Energy Materials, Ningbo Institute of Materials Technology and Engineering, Chinese Academy of Sciences, Ningbo, Zhejiang 315201, P. R. China # Departments of Chemistry and Earth and Atmospheric Science, Purdue University, West Lafayette, Indiana 47906, United States S Supporting Information *

ABSTRACT: Recently, two-dimensional (2D) materials with superior mechanical properties, unique electronic structures, and specific functionalities have stimulated considerable interest in designing novel flexible devices and multifunctional nanocomposites. However, high-throughput experiments and calculations, which are desirable for identifying those promising candidates with excellent strengths and flexibilities, remain a great challenge due to their difficulty and complexity. In the present work, a systematic investigation has been performed on the oxygen-functionalized 2D transition-metal carbides M2CO2 (M = Sc, Ti, V, Cr, Y, Zr, Nb, Mo, Hf, Ta, and W) to identify those with excellent thermodynamic stabilities and mechanical behaviors via highthroughput first-principle calculations. Our results suggest that the position and bonding/antibonding character of metallic d-band electrons play a vital role in stabilizing M2CO2, whose formation energy is below 0.2 eV/atom, a generally considered threshold observed for freestanding 2D materials, except for Sc2CO2, Y2CO2, and Cr2CO2. The synthetic effect from the surface stacking geometry and the delocalization character of d electrons provides a mechanistic quantification for periodic variation of elastic moduli and ideal strengths for M2CO2, whereas the strain-induced premature dynamic instabilities in different modes may intrinsically limit their achievable strengths, e.g., zonecenter optical phonon instability for Hf2CO2 versus elastic instability for W2CO2. Detailed electronic structure analyses reveal that strong M−C bonds endow M2CO2 with excellent in-plane mechanical strengths but the appearance of different phonon instabilities when M changes from group IVB to group VIB may be attributed to the different filling characters of specific metaldxz orbital or metal-dz2 orbital. These findings resolve an apparent discrepancy for the preferred adsorption sites of the functional group and shed a novel view on the electronic origin of distinct mechanical strengths and flexibilities observed for different M2CO2.

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INTRODUCTION In designing novel flexible devices and multifunctional nanocomposites, light-weight two-dimensional (2D) materials with desirable mechanical properties have brought considerable interest for their potential applications,1 especially with the emergence of novel MXenes, i.e., 2D transition-metal carbides and/or nitrides synthesized by etching “A” layers of MAX phases, where M represents early transition metals (such as Sc, Y, Ti, Zr, Hf, V, Nb, Ta, Cr, Mo, W, etc.), A belongs to the elements in group IIIA or IVA, and X is carbon and/or nitrogen.2−8 MXenes show metallic electrical conductivities,8−12 outstanding electrochemical properties, 2−4,13,14 unique topological properties,15−19 and superior mechanical properties, such as high flexibilities and strengths.20,21 Up to © XXXX American Chemical Society

now, more than 10 MXenes have been synthesized, including Ti3C2, Ti2C, V2C, Nb2C, Ta4C3, Nb4C3, Mo2C, Zr3C2, Ti4N3, Ti3CN, (Ti0.5Nb0.5)2C, (V0.5Cr0.5)3C2, Mo2TiC2, Mo2Ti2C3, and Cr2TiC2, among others.9,11,22−27 As selectively etched in different solution environments, e.g., aqueous hydrofluoric acid (HF), HF-containing etchant such as ammonium bifluoride (NH4HF2) salt, and so on,2,22,28,29 MXenes are always functionalized by various functional groups, including F, O, and OH, among others. Thus, the functionalized ones are generally expressed with a formula of Mn+1XnT2, where T is the Received: January 5, 2018 Revised: February 5, 2018 Published: February 7, 2018 A

DOI: 10.1021/acs.jpcc.8b00142 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

and the delocalization character of d electrons are responsible for different mechanical properties of M2CO2. Different straininduced phonon instability modes are found for different M2CO2, which consequently limit their achievable strengths due to the different filling character of specific metal-dxz orbital or metal-dz2 orbital. This work builds a solid foundation for further experimental exploration on appropriate candidates for flexible devices with target properties or the nanocomposites reinforced by MXenes.

functional group at surface. The chemical diversity of MXenes provides an opportunity to perform a comprehensive search for an appropriate candidate for flexible devices with tailored properties or strengthening components in a multifunctional nanocomposite coupled with polymer matrix13 or others for structural components. Generally, the mechanical performance of the flexible materials and the nanocomposites composed of MXenes is determined by several critical factors, including stability, stiffness, flexibility, and strength, among which there are also strong mutual couplings that complicate the design process. On the basis of the excellent elastic properties and flexibilities of M2C and M3C2,20,21,30−32 the strain modification of various properties of specific M2C and M2CO2 is widely investigated.33−36 A remarkable research is that a direct and continuous strain control of catalysts by tuning electrode materials was realized in well-controlled experiments,37 which further demonstrated the practical feasibility of strain modification. Besides, strain might modify materials with improved performance, e.g., the strain-enhanced oxygen reduction catalysis capability.38 In addition, the fabricated Ti3C2Tx/polymer composites possess remarkable mechanical flexibilities and electrical conductivities due to the excellent intrinsic properties of MXenes, which are found to be dramatically enhanced as compared with both of the pure substances.13 Nevertheless, narrow searching space, expensive experiments, and technical challenges have far prevented an efficient search in high-performance materials based on experiments alone. Moreover, most recent studies have focused primarily on specific MXenes, lacking the prediction capability for other MXene candidates with potentially more prominent mechanical performance. With the rapid development of theoretical methods, the scalability of computations has made it possible to predict the stabilities and properties of potential hypothetical candidate materials.39 Several successful examples include the Li-ion battery design,40 the electrocatalytic material prediction for hydrogen evolution,41 the gap prediction,42 and others, which make high-throughput screening an important tool in designing new materials. In recent years, some preliminary explorations on the electrochemical properties for a rich variety of MXenes have been performed by high-throughput screening.43,44 However, a systematic theoretical investigation on the stabilization, strengths, and flexibilities has been challenging due to the lack of an efficient high-throughput computational scheme for automatically deriving specified mechanical quantities.32 Although some sporadic works have been done on the stabilization,30,45,46 elastic properties,20,31,47 strengths, and flexibilities21,32 for some specific MXenes, to the best of our knowledge, none of them have yet been done to explore systematically the stabilization, mechanical properties, and strain-induced dynamic stabilities of MXenes and the relevant atomistic mechanism and electronic origin in searching for the optimal candidate for strong MXenes. In this work, by means of a high-throughput first-principles technique, a global investigation has been performed on the oxygen-functionalized 2D transition-metal carbides M2CO2 (M = Sc, Ti, V, Cr, Y, Zr, Nb, Mo, Hf, Ta, and W) to identify the candidates with excellent thermodynamic stabilities, superior flexibilities, and mechanical strengths. Our results suggest that the position and bonding/antibonding character of metallic dband electrons determine the relative stability of M2CO2 in different configurations, whereas the surface stacking geometry

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COMPUTATIONAL DETAILS The density functional theory computations were carried out on the basis of the generalized gradient approximation (GGA) designed by Perdew, Burke, and Ernzerhof (PBE),48 with a plane wave kinetic energy cutoff of 600 eV, as implemented in Vienna ab initio simulation package (VASP).49 Core electrons were treated using the projector augmented wave pseudopotentials.50 Full relaxations were done to optimize lattice parameters and atomic coordinates, with an energy convergence of 10−6 eV/cell and with force convergence of 10−3 eV/Å. To avoid the artificial interaction between the layers and their periodic images, the vacuum layer thickness was more than 20 Å. The k-mesh grid of 18 × 18 × 1 was adopted to sample the first Brillouin zone in the relaxation process. The phonon calculations were performed by PHONOPY code on the basis of the nonvanishing Hellman−Feynman forces within the harmonic approximation.51 Crystal orbital Hamilton population (COHP) analysis was done by employing the Lobster code.52,53 Aiming at the possible magnetic ground states in M2C and M2CO2, every configuration was relaxed fully with an initial setup of ferromagnetic (FM) and antiferromagnetic (AFM) states. Two different spin channels were applied in the two M atoms in a M2C/M2CO2 cell to get an AFM state. The results show that AFM states are determined to be energetically more favorable for specific M2C (M = Sc, Ti, Cr, Y, Zr, and Hf) and there exist magnetic states in certain M2CO2 (M = Sc, Cr, and Y) (use configuration I for comparison) using PBE (shown in Table S1). In addition, considering the potential strong electron correction effect in transition-metal carbides, structure relaxations and property calculations were performed by a spin-dependent GGA plus Hubbard U (GGA + U),54 as a comparison to the PBE results. Herein, we tested U values of transition metals by comparing magnetic moments with the ones adopted by Heyd−Scuseria−Ernzerhof (HSE) screened hybrid functional55 (shown in Table S2). As a result, the onsite Hubbard U values were set as 3 eV for the transitionmetal d electrons. We calculated some parameters (e.g., lattice constants, bond lengths, etc.) of widely studied Ti2CO2 and Mo2CO2 in GGA, GGA + U, and HSE (shown in Table S3), in excellent agreement with the previous investigations,45,56−59 which demonstrates the rationality of our computational method. The adsorption energy Eb of oxygen at the surface of M2C was calculated on the basis of the following equation E b = (E M 2CO2 − E M 2C − EO2)/2

(1)

where EM2CO2, EM2C, and EO2 are the total energy of M2CO2, M2C, and a free oxygen molecule, respectively.60 The formation energies of M2CO2 were calculated as the difference of the energy for M2CO2 with respect to the most stable end-member phases with the same stoichiometry.30 We take Ti2CO2 as an example; the formation energy is obtained by B


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The Journal of Physical Chemistry C Ef = E Ti 2CO2 − E TiC − E TiO2

oxygen-functionalized MXenes present the most promising mechanical property.32 Three different oxygen adsorption configurations (shown in Figure 1b) are accordingly considered: configurations I and II correspond to face-centered cubic and hexagonal close-packed hollow sites, respectively. Configuration III corresponds to a mixture of site I and site II. Accordingly, the configurations I−III can be formed by either triangular prisms or antiprisms that consist of a metal atom and six neighbor nonmetal atoms (shown in Figure 1b). Table 1

(2)

where ETi2CO2, ETiC, and ETiO2 are the total energy of Ti2CO2, bulk TiC, and bulk TiO2, respectively. For a 2D material, the in-plane elastic properties can be derived from Hooke’s law under the condition of plane stress61 ⎡ σxx ⎤ ⎡ c11 c12 0 ⎤⎡ εxx ⎤ ⎥⎢ ⎥ ⎢σ ⎥ ⎢ ε ⎢ yy ⎥ = ⎢ c 21 c 22 0 ⎥⎢ yy ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ σxy ⎥⎦ ⎣ 0 0 c66 ⎦⎣ εxy ⎦

Table 1. Adsorption Energies Eb of M2CO2 at Different Adsorption Sitesa

(3)

To get the in-plane elastic constants, a set of in-plane strains (εxx, εyy, and εxy) ranging from −2 to +2%, with an increment of 0.4%, were applied. The strain energy Es is quadratically dependent on the applied strains and can be expressed as Es =

A0 (c11εxx2 + c 22εyy2 + 2c12εxxεyy + c66εxy2 ) 2

Sc2CO2 Ti2CO2 V2CO2 Cr2CO2 Y2CO2 Zr2CO2 Nb2CO2 Mo2CO2 Hf2CO2 Ta2CO2 W2CO2

(4)

where Es = E(ε) − E0, and E(ε) and E0 are the total energy of the strained and equilibrium system, respectively. c11, c22, c12, and c66 are the in-plane elastic constants. A0 is the area of the simulation cell in the xy plane. Young’s modulus Ex/Ey, in-plane Poisson’s ratio v/v′, and shear modulus GV (V stands for the Voigt average scheme) of a 2D system are derived from elastic constants as c c − c12c 21 c c − c12c 21 c Ex = 11 22 , Ey = 11 22 , ν = 21 , ν′ c 22 c11 c12 c12 = , G V = c66 c11 (5)

a c

configuration I

configuration II

configuration III

−4.40b −4.89c −4.03c −3.37b −4.49b −5.52c −4.68c −3.41c −5.87c −4.87c −3.70c

−4.65c −4.01c −3.74c −3.46c −4.63c −4.50c −4.40c −3.96c −4.77c −4.58c −4.29c

−4.72c −4.50c −3.91c −3.40b −4.71c −5.06c −4.59c −3.61c −5.37c −4.78c −3.85c

The energy unit is eV/adatom. Nonmagnetic ground state.

b

Ferromagnetic ground state.

and Figure 1c illustrate the adsorption energies of oxygen at the surfaces in different configurations. Interestingly, the most thermodynamically stable configurations can be classified in terms of the position of the “M” element in the periodic table (shown in Figure 1a). For elements in group IIIB (Sc and Y), in group IVB (Ti, Zr, and Hf), in group VB (V, Nb, and Ta) and in group VIB (Cr, Mo, and W), the most stable configurations (the adsorption energies in brackets) are III (−4.72 and −4.71 eV), I (−4.89, −5.52, and −5.87 eV), I (−4.03, −4.68, and −4.87 eV), and II (−3.46, −3.96, and −4.29 eV), respectively, in excellent agreement with previous results.19,30,45,56,64 Notably, profound adsorption energy differences are found for Ti2CO2 (0.88 eV), Zr2CO2 (1.02 eV), and Hf2CO2 (1.10 eV) between the respective configurations I and II. Because M2CO2 are generally synthesized from the MAX precursor phases, it is necessary to first explore the stabilities of the precursor phases. However, a high-throughput computation has been comprehensively performed by Ashton et al. on the thermodynamic stability of various relevant MAX phases65 and Bai et al. investigated the dynamic stability of certain MAX phases, 66 which provided a nice guidance for future experimental preparations. Thus, in the following, we shall focus on the thermodynamic stabilities of various M2CO2 by calculating the formation energies with respect to those of the most stable competing phases. For example, Ti2CO2 is metastable against the bulk phases of TiO2 and TiC and the calculated energy difference between M 2 CO2 and the competing bulk phases is defined as the formation energy of M2CO2, which is shown in Figure 1d and listed in Table S4. To be noted that the competing phases for each M2CO2 are chosen on the basis of the data suggested in the Materials Project database.39 In general, a threshold value of 0.2 eV/atom was proposed as the maximum formation energy for freestanding 2D materials67 that were successfully synthesized in experiments, such as Ti2CO2, V2CO2, Nb2CO2, and Mo2CO2.68 With

In the calculations of stress−strain curves, both in-plane strains under uniaxial stress state and biaxial strain state are used to evaluate the mechanical response of M2CO2. The tensile strain a−a is defined as ε = a 0 , where a0 and a are the equilibrium and 0

strained lattice constants, respectively. To ensure that M2CO2 is under uniaxial stress state, the geometrical relaxation is performed for both the lattice basis vectors and the atomic coordinates by keeping the applied in-plane strain component fixed and relaxing the other in-plane strain component until their conjugate stress components, i.e., Hellmann−Feynman stresses, reach less than 0.1 GPa. In the case of a 2D system, the basis vector normal to the 2D layer is kept constant during the relaxation to keep a sufficient thickness of vacuum. To ensure that the strain path is continuous, the starting position at each strain step is taken from the relaxed coordinates of the previous step. Because the crystal symmetry may decrease under deformation, a high energy cutoff of 600 eV and the verified convergence of the stress−strain calculations with different kpoint grids are adopted. A similar scheme for stress−strain calculation has been described and thoroughly checked in our previous papers, which we refer for further details.32,62 The inplane stress imposed on the cell is rescaled by Z to get an inplane stress, where Z is the dimensional length of the cell along z direction.

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RESULTS AND DISCUSSION Thermodynamic Stability. We focus on M2CO2 in the latter sections with the following three considerations: (i) the surface is mostly terminated by oxygen in the LiF−HCl method;63 (ii) a full surface coverage of oxygen is demonstrated to be thermodynamically more favorable;45 and (iii) the C


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Figure 1. (a) Relevant elements shown in the periodic table; the blue elements represent “M”, and the orange elements represent “X”. Pink squares represent the MXenes corresponding to the “M” synthesized by the experiment; (b) the atomic structures in configuration I−III; (c) the adsorption energies of M2CO2 in different configurations; and (d) the formation energies of M2CO2 in the most stable configuration. The yellow region highlights the general threshold of 0.2 eV/atom observed for freestanding 2D materials, and the green region highlights the 0.118 eV/atom formation energy of the Nb2CO2, the highest one of those that have been synthesized.

Figure 2. Crystal orbital Hamilton populations (COHPs) analyses of Hf2CO2, Ta2CO2, and W2CO2 (by using the lattice constants of Ta2CO2) in (a) configuration I and (b) configuration II, respectively. M_M_i (M_M_o) represents the in-plane (out-of-plane) metal−metal interaction.

stabilization. Figure 3 shows the projected band structures, projected density of states (PDOS), and electron location function (ELF) maps of I-Hf 2 CO 2 (the most stable configuration, the same below), I-Ta2CO2, and II-W2CO2. Under the D3d symmetry group of M2CO2, the d orbitals of the transition-metal atoms split into three groups, i.e., a single dz2 orbital and two groups of doubly degenerate dxy + dx2−y2 and dxz + dyz orbitals. For Hf2CO2 (shown in Figure 3a), all d bands are located above the Fermi level, with the appearance of a distinct band gap. It is additionally observed that the peaks located between −2 and −1 eV correspond mostly to the nonmetal p states, which is envisioned in the ELF image shown in Figure 3b, where the valence electrons are mainly located around the nonmetal atoms, suggesting that the triangular antiprism structure is preferred due to the repulsion interactions among the nonmetal anions. In the case of I-Ta2CO2, partial metallic d bands composed of Ta-dxz + dyz are nearly filled (see Figure 3c), consistent with the decrease of the localization character of the valence electrons in I-Ta2CO2 (see ELF maps shown in Figure 3d). When M changes to d2 system, the d bands consisting of W-dx2−y2, dxy, and dz2 are partially filled and split

this rule as a guidance, one may see that the formation energies of most of M2CO2 are well below this threshold except for Sc2CO2, Y2CO2, and Cr2CO2, in excellent agreement with the previous results.30 To reveal why the most stable configuration changes with different elemental groups in the periodic table, crystal orbital Hamilton populations (COHPs) analyses of Hf2CO2 (d0 system), Ta2CO2 (d1 system), and W2CO2 (d2 system) in configurations I and II are shown in Figure 2a,b, respectively. Because of the appearance of distinct antibonding states nearby the Fermi level, both II-Hf2CO2 and II-Ta2CO2 are not preferred as compared with I-Hf2 CO 2 and I-Ta 2 CO 2 , respectively. In comparison, we found that both the bonding states of in-plane metal−metal (IMM) interactions and the antibonding states of out-of-plane metal−metal (OMM) interactions are located at the Fermi level in both I-W2CO2 and II-W2CO2. However, in I-W2CO2, the value of the antibonding states of OMM interactions is greater than that of the bonding states of IMM interactions, as opposed to IIW2CO2 (see Figure 2a,b), indicating that the strong bonding states of IMM interactions of II-W2CO2 play a critical role in its D


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Figure 3. Projected band structures and projected density of states (PDOS) of (a) I-Hf2CO2, (c) I-Ta2CO2, and (e) II-W2CO2, and the calculated ELF section maps of (b) I-Hf2CO2, (d) I-Ta2CO2, and (f) II-W2CO2, respectively.

Table 2. Young’s Moduli Ex and Ey, Poisson Ratio v and v′, and Shear Moduli GV of M2CO2 at Different Adsorption Sitesa configuration I Sc2CO2 Ti2CO2 V2CO2 Cr2CO2 Y2CO2 Zr2CO2 Nb2CO2 Mo2CO2 Hf2CO2 Ta2CO2 W2CO2 a b

configuration II

configuration III

Ex

Ey

v

v′

GV

Ex

Ey

v

v′

GV

Ex

Ey

v

v′

GV

140b 241c 238c −b 104b 241c 253c 105c 271c 260c 138c

147b 240c 231c −b 106b 241c 250c 105c 270c 257c 139c

0.29b 0.32c 0.37c −b 0.49b 0.30c 0.34c 0.72c 0.28c 0.34c 0.65c

0.31b 0.32c 0.36c −b 0.50b 0.30c 0.33c 0.72c 0.28c 0.34c 0.66c

46b 91c 90c −b 36b 93c 95c 33c 106c 93c 44c

142c 166c 286c 277c 116c 156c 281c 306c 209c 307c 357c

142c 162c 291c 282c 116c 155c 279c 320c 208c 309c 359c

0.53c 0.57c 0.29c 0.32c 0.55c 0.58c 0.37c 0.37c 0.50c 0.36c 0.39c

0.53c 0.56c 0.29c 0.32c 0.55c 0.58c 0.37c 0.38c 0.50c 0.37c 0.39c

46c 47c 110c 104c 37c 53c 103c 117c 71c 116c 129c

121c 228c 264c 229b 99c 219c 250c 276c 253c 281c 289c

121c 227c 263c 227b 99c 218c 250c 279c 252c 281c 287c

0.53c 0.34c 0.31c 0.35b 0.55c 0.36c 0.38c 0.27c 0.34c 0.36c 0.34c

0.53c 0.34c 0.31c 0.35b 0.55c 0.36c 0.38c 0.28c 0.33c 0.36c 0.33c

40c 86c 100c 85b 32c 81c 90c 109c 95c 102c 106c

The unit of N/m is used for the Young’s modulus and shear modulus. The dash symbols “−” in the table represent the elastic instability. Ferromagnetic ground state. cNonmagnetic ground state.

Mechanical Property. Before investigating the mechanical properties of M2CO2, we first validate our computational approach for graphene. The calculated elastic properties and ideal tensile strengths are provided in Tables S5 and S6, in good agreement with previous results.32,69−75 With the same procedure, the mechanical properties of M2CO2 in different configurations are summarized in Figure 4, Tables 2, and 3, including Young’s moduli, uniaxial ideal strengths along X and

into two distinct parts by the appearance of a pseudogap at the Fermi level (see Figure 3e), in agreement with the ELF fact (Figure 3f) that the delocalized valence electrons appear around the W atoms. In brief, the remarkable delocalized dband electrons and filling character of W-dx2−y2, dxy, and dz2 orbitals appearing in II-W2CO2 (see Figure 3e,f) are responsible for its stabilization, in agreement with the aforementioned COHP analyses. E


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The Journal of Physical Chemistry C Table 3. Calculated Ideal Strengths of M2CO2 under Uniaxial Tension and Biaxial Tensiona uniaxial X

tension type

uniaxial Y

biaxial

configuration

I

II

III

I

II

III

I

II

III

Sc2CO2 Ti2CO2 V2CO2 Cr2CO2 Y2CO2 Zr2CO2 Nb2CO2 Mo2CO2 Hf2CO2 Ta2CO2 W2CO2

8.5b 30.7c 29.6c −b 12.8b 31.8c 30.9c 34.2c 37.4c 32.5c 36.6c

12.8c 11.4c 18.9c 26.9c 10.8c 8.1c 25.1c 30.1c 14.8c 29.9c 35.2c

12.6c 19.9c 26.4c 27.3b 9.0c 17.7c 30.2c 27.5c 24.0c 33.4c 30.9c

19.4b 23.5c 25.3c −b 16.2b 23.4c 25.6c 15.4c 26.7c 27.0c 16.0c

15.3c 1.1c 17.5c 27.1c 15.6c 3.9c 22.2c 32.7c 7.2c 23.8c 37.8c

14.4c 5.8c 11.5c 20.4b 12.5c 6.8c 19.2c 21.9c 9.1c 23.3c 23.7c

17.8b 30.3c 27.5c −b 14.5b 28.8c 28.5c 23.5c 31.4c 30.9c 24.9c

24.0c 26.3c 26.4c 23.7c 21.4c 25.9c 34.7c 34.7c 28.0c 39.8c 42.1c

14.9c 26.0c 25.3c 22.2b 13.3c 27.5c 31.1c 23.8c 30.0c 33.8c 26.7c

The unit of ideal strength is N/m. The dash symbols “−” in the table represent the instability. bFerromagnetic ground state. cNonmagnetic ground state.

a

Figure 4. (a) Young’s moduli, (b−d) ideal strengths in uniaxial tensions along X and Y directions, and biaxial tension in different configurations in M2CO2, respectively.

irreversible response of materials to a finite strain. Although various defects, e.g., voids, dislocations, grain boundaries, cracks, etc. govern the mechanical strength of a real material, the ideal strength gives the upper limit of achievable strength that an infinite perfect single crystal can tolerate.77 The trends of ideal strengths under biaxial tension are found to be generally similar to those of Young’s moduli. Interestingly, when M belongs to group IVB, the most stable configuration corresponds to the largest ideal strengths under all strain states. However, when M is in group VB or VIB, no direct correspondence is found between the most stable configuration and the largest strength. As the valence electron numbers of M increase, the ideal strengths of different II-M2CO2 increase under both uniaxial tensions along X and Y direction. However, the ones in configuration I under uniaxial tension along X (Y) direction remain basically unchanged (decreased), indicating that the surface stacking geometry might influence the strengths

Y directions (corresponding to the zigzag and armchair directions, respectively) and biaxial ideal strengths. M2CO2, where M is a 5d element, exhibits the highest Young’s moduli among their respective groups in the most stable configurations, which is also attributed to the increasing delocalization character from 3d electrons to 5d electrons.76 The most stable configurations correspond to the largest Young’s moduli in M2CO2 when M belongs to groups IVB and VIB. However, when M is in group VB, configuration I has the largest adsorption energy, whereas the highest Young’s modulus is found for configuration II, indicating that the extra delocalization electrons cannot solely explain the trends of the elastic property, in a different manner to the trend of thermodynamic stability. Elastic properties describe the reversible responses of materials to small lattice distortions around equilibrium, whereas plastic properties are essential to describe the F


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Figure 5. Lower frequency phonon dispersions (LFPDs) of Hf2CO2 at different strains in uniaxial tension along (a) X direction and (b) Y direction, the LFPDs of Ta2CO2 at different strains in uniaxial tension along (c) X direction and (d) Y direction, and those of W2CO2 at different strains in uniaxial tension along (e) X direction and (f) Y direction, respectively. The first Brillouin zone with specific symbols is represented in Figure S1. Because the crystal symmetry is broken under uniaxial tension, M* and K* are different from M and K.

brings another critical issue to be further explored because the dynamical instability could intervene along the instability path.32 As representative cases of three groups of M2CO2, the dynamical stabilities of I-Hf2CO2, I-Ta2CO2, and II-W2CO2 are addressed in this section below. The calculated phonon dispersions of Hf2CO2, Ta2CO2, and W2CO2 at equilibrium (Figure S1) indicate that they are dynamically stable because of the absence of imaginary frequency, whereas under straining, the phonon instabilities of Hf2CO2, Ta2CO2, and W2CO2 exhibit very different characteristics. For Hf2CO2, at strain ε = 10 (12)%, the optical (acoustic) soft mode appears at the Brillouin zone center (Brillouin zone boundary) under uniaxial tension along X (Y) direction (shown in Figure 5a,b), whereas the premature phonon instabilities of Ta2CO2 occur at substantially low strain ε = 2 (5)% under uniaxial tension along X (Y) direction (shown in Figure 5c,d). It is interesting to note that the strain-induced phonon instabilities of Hf2CO2

greatly. To further illustrate the topological geometrical factor (i.e., the bonding environment of atoms) derived from the surface stacking order, the bond topologies of Hf2CO2, Ta2CO2, and W2CO2 in configuration I and II are compared (shown in Table S7). In either configuration I or II, the metal− metal and metal−carbon bond lengths decrease as d-electron counting increases, consistent with the decreasing ion radius from Hf to W. However, in configuration I, the metal−oxygen bond length increases from d0 system to d2 system, which might be responsible for the decrease of the strengths in configuration I. The change of the metal−oxygen bond length in configuration II is similar to that of the metal−carbon (metal) bond, corresponding to the increase of the strengths in configuration II from d0 system to d2 system. Instability Mode and Its Physical Origin. Although Figure 4 compares the mechanical strengths of different M2CO2, whether the ideal strengths can be attained or not G


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Figure 6. (a) Topological structure in top view and (c) the calculated stress−strain curves of Hf2CO2 under uniaxial tension along X direction. The PDOS of Hf2CO2 (b) at ε = 10% (at the peak) and (d) at ε = 20% (after instability) under uniaxial tension along X direction, respectively. Points A and B represent the critical strain predicted for the dynamical instability and the lattice instability, respectively. (e) The imaginary eigenvector and (f) the total energy as a function of the eigenvector amplitude corresponding to the zone-center phonon softening at ε = 0.11.

ahead of the elastic instabilities resemble those reported previously in Ti3C2O2,32 suggesting a similar instability mode for the same group of M. In contrast with Hf2CO2 and Ta2CO2, an elastic instability occurs for W2CO2 at strain ε = 17% under uniaxial tension along X direction (see Figure 5e), whereas under uniaxial tension along Y direction, the phonon soft mode appears at strain ε = 16%, where an imaginary wave vector occurs between the Γ and M points (close to Γ point) that corresponds to an incommensurate phase transition78 (see Figure 5f). Anyway, the critical strains of phonon instabilities for W2CO2 are pretty close to those of the elastic instabilities, indicating its superior mechanical strengths and flexibilities. The soft modes at Γ point shown in the phonon dispersions (see Figure 5) suggest the appearance of elastic instabilities or internal coordinate instabilities, such as ferroelectric phenomena and Jahn−Teller distortions under the homogeneous deformations.78,79 Taking Hf2CO2 and W2CO2 as two illustrations, it is next worth exploring the instability paths and electronic mechanism. To remove the influence of the constraint of lattice symmetry on the instability paths, a disturbed scheme is introduced in the calculation of stress− strain response for comparison. In the disturbed scheme, every atom is disturbed in a random displacement at each strain step

before the relaxation of atomic coordinates and cell vectors. For Hf2CO2, the strain at the peak stress in the disturbed stress− strain curve is pretty close to the critical strain corresponding to the appearance of the soft mode (shown in Figure 6c), suggesting that the disturbed scheme provides an efficient way to capture the appearance of phonon instability in the stress− strain calculations. To underline the zone-center phonon softening shown in Hf2CO2 at ε = 0.11, the atomic displacements and the related energy profile corresponding to the imaginary eigenvector have been analyzed and shown in Figure 6e,f, respectively. It is seen that the in-plane atomic displacements along X direction (reflect the interaction between Hf−C/O_2 in Figure 6a) are responsible for the phonon instability and the double-well potential energy as a function of the eigenvector amplitude confirms the appearance of Jahn−Teller distortion of Hf-C/O octahedron. The calculated partial density of states (PDOS) shown in Figure 6b,d reveals the variation of electronic structures at the peak and after instability, respectively. During the instability in uniaxial tension along X direction, the band gap increases significantly from 0.88 to 1.87 eV and meanwhile the DOS peak consisting of Hf-dxz states moves upward from −2 to −1 eV (see the arrows in Figure 6b,d), which is accompanied by a H


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Figure 7. (a) Calculated stress−strain curves of W2CO2 under uniaxial tension along Y direction. Points A and B represent the critical strain predicted for the dynamical instability and the lattice instability, respectively. The panel shows the topological structure of W2CO2 in the top view. Notice that the surface oxygen atoms are removed to expose the inner carbon atoms in the red rectangle to show the bonds inside more clearly. The PDOS of W2CO2 (b) at ε = 18% (at the peak) and (d) at ε = 20% (after instability) under uniaxial tension along Y direction. (c) Cross-sectional plots of VCDDs for W2CO2 at equilibrium and after instability. The cyan thick solid and orange thin dotted contours represent positive and negative values, respectively. The charge unit is electron/Bohr3.

similar movement from −2.5 to −1.5 eV for the DOS peak composed of C-px states. Therefore, the hybridization between specific Hf-dxz and C-px orbitals is responsible for the premature phonon instability. For W2CO2, Figure 7b,d shows the electronic structures revealed by partial density of states (PDOS) at the peak and after instability under uniaxial tension along Y direction (the stress−strain curve is shown in Figure 7a). To be noted that the PDOS at the peak and after instability under uniaxial tension along X direction is shown in Figure S2b,d. It can be observed that at the peak as shown in Figure 7b, the sharp peak composed of W-dz2 located at the Fermi level moves above the Fermi level with a pseudogap appearing after instability (see the arrows in Figure 7b,d). A similar instability mode is also observed for W2CO2 under uniaxial tension along X direction (see Figure S2b,d). Therefore, the elastic instability of W2CO2 can be attributed to the movement of specific W-dz2 orbital, differing from the behavior of the electronic structure in Hf2CO2, in which the zone-center optical phonon instability appears. To further reveal the instability process during straining, an analysis of valence charge-density difference (VCDD) before and after instability is performed and shown in Figures 7c and S2c. The distinct depletion region of VCDD between W−C_1 bonds in Figure 7a and W−C_2 bonds in Figure S2a suggests the appearance of bonding weakening there (see the circled regions in the figures). Therefore, the W−C bonds are responsible for the excellent mechanical strength and flexibility of W2CO2. Comparison between M2CO2 and Other Well-Known 2D Materials. Figure 8 compares the calculated mechanical properties of Hf2CO2, Ta2CO2, and W2CO2 with other well-

known 2D materials, including the elastic properties, critical strains, and in-plane ideal strengths. Note that an excellent agreement is obtained between this work and the previous studies for some typical 2D materials (see Tables S5 and S632,69−75,80−83). A significant isotropic Young’s modulus (357 N/m) of W2CO2 is comparable to that of graphene (324 N/ m), followed by Hf2CO2 (260 N/m) and Ta2CO2 (271 N/m). It should be noted that Ex and Ey of M2CO2 and graphene are identical because of the hexagonal symmetry. Although borophene presents an unexpectedly high Young’s modulus along Y direction (391 N/m), its profound anisotropic feature of in-plane Young’s modulus (Ex/Ey = 2.62) rules it out from ultrastiff 2D materials, in agreement with the reported in-plane elastic anisotropy (Ex/Ey = 2.34 in ref 81). The in-plane shear modulus of W2CO2 (129 N/m) approaches nearly that of graphene (132 N/m), further supporting that it is ultrastiff. As the valence electrons of the metal atoms increase, the Poisson ratio of different groups of M2CO2 increases gradually. All of their ratios are larger than that of graphene (0.224). Totally, the calculated ideal strengths and critical strains (in brackets) of Hf2CO2 (37.4 N/m (0.29), 26.7 N/m (0.30), and 31.4 N/m (0.18) in uniaxial tension along X, Y directions and biaxial tension, respectively), Ta2CO2 (32.5 N/m (0.40), 27.0 N/m (0.20), and 30.9 N/m (0.16)), and W2CO2 (35.2 N/m (0.21), 37.8 N/m (0.18), and 42.1 N/m (0.16)) are comparable to those of graphene (38.1 N/m (0.23), 34.4 N/m (0.19), and 32.1 N/m (0.21)). Considering that the premature dynamical stability for Hf2CO2 and Ta2CO2 occurs far below their ideal strength, W2CO2 is identified as a promising candidate among all of the studied M2CO2 from a mechanical perspective. Nevertheless, the possible pathways for experimental synthesis I


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Figure 8. (a) In-plane Young’s moduli Ey and Ex in direction Y and X, shear moduli GV, (b) Poisson ratio v and v′, (c) critical strains, and (d) ideal strengths of different 2D materials in uniaxial tensions along X and Y directions and biaxial tensions. The bars in dark colors correspond to the values for the appearance of dynamical instability. Note that the Y direction corresponds to armchair direction for graphene, silicene, germanene, stanene, hexagonal boron nitride, and MoS2 and the Y direction corresponds to direction b for borophene.81

of M2CO2 from the related MAX phases or other routines would be desirable for both theoretical calculations and experiments in the future. Note that the in-plane elastic moduli and ideal strengths of M2CO2 are obtained by normalizing the strain energy density by area, which provides the intrinsic properties of the single layer without dependence on layer thickness, as suggested by Lee et al.69 A higher value of in-plane strength of M2CO2 (in unit of N/m) is not surprising if the contribution from the multiple stacking chemical bond framework to the scaled in-plane stress is involved. All of these suggest that M2CO2 have unique combined mechanical strengths and flexibilities that may surpass those of other 2D materials, thus providing more choices for appropriate candidates for flexible devices or the MXene-reinforced nanocomposites. Last but not the least, a further investigation of the correlation effect of d electron on the adsorption and mechanical property is performed by GGA + U method and compared with the normal GGA method (refer to Tables S8− S10, Figures S3 and S4). Our results confirm further that the trends of the calculated adsorption energy, Young’s moduli, and ideal strengths of different M2CO2 show similarities to the results by normal GGA method.

(1) The absorption stabilities of M2CO2 differ in different groups of M when M changes from group IVB to group VIB, which are found to be attributed to the variation of position and bonding/antibonding character of metallic d-band electrons. (2) Most of M 2 CO 2 are found to possess superior mechanical strengths and flexibilities that may even surpass those of other 2D materials, providing valuable guidance for the choice of appropriate candidates for flexible devices or suitable reinforcement in nanocomposites. However, the strain-induced dynamic instabilities may intrinsically limit the achievable strength of some M2CO2, which suggests a design rule in strengthening MXenes by suppressing premature phonon instability of MXenes, such as the modification of functional group and/or the hybridization of different M elements. (3) The superior mechanical strengths and flexibilities of M2CO2 can be attributed to the synthetic effect from the delocalization character of specific d electrons and surface stacking geometry, thus showing distinct features for different M. For instance, the hybridization between specific Hf-dxz orbital and C-px orbital is responsible for the premature phonon instability of Hf2CO2, whereas the movement of specific W-dz2 orbital under straining is responsible for the elastic instability of W2CO2 without appearance of premature phonon instability. Such results indicate that W2CO2 would be a promising potential candidate for flexible devices or strengthening components in nanocomposites.

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CONCLUSIONS In summary, by means of high-throughput first-principles methods, we have systematically investigated mechanical properties of M2CO2 and provided a mechanistic quantification for the physical origin of the excellent thermodynamic stabilities and superior mechanical strengths. The key findings are summarized as below: J


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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.8b00142. Formation energies, bond lengths, and phonon dispersions of M2CO2; mechanical properties of some selected 2D materials; adsorption energies and mechanical properties of M2CO2 in GGA+U; test of U values; and deformed electronic structures of W2CO2 (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Zhongheng Fu: 0000-0001-8779-9086 Shiyu Du: 0000-0001-6707-3915 Ruifeng Zhang: 0000-0002-9905-7271 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported by the National Key Research and Development Program of China (No. 2017YFB0702100), National Natural Science Foundation of China (NFSC) with No. 51672015, and National Thousand Young Talents Program of China. D.L. was supported by project IT4Innovations-path to exascale No.CZ.02.1.01/0.0/0.0/16_013/ 0001791 and grants No. 17-27790S of the Czech Science Foundation and No. LQ1602 of the National Programme of Sustainability. We appreciate the support from the key technology of nuclear energy, 2014, CAS Interdisciplinary Innovation Team and ITaP at Purdue University for computing resources. We would also like to thank Prof. G. Kresse for valuable advice for the application of VASP.

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