Exploring high pressure structural transformations, electronic properties and superconducting properties of MH2 (M = Nb, Ta)

Jinquan Zhang; Yanqi Wang; Libiao Tang; Juyi Duan; Jingjing Wang; Song Li; Meng Ju; Weiguo Sun; Yuanyuan Jin; Chuanzhao Zhang

doi:10.1016/j.arabjc.2022.104347

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Original article

12 2022

:15;

104347

doi:

10.1016/j.arabjc.2022.104347

Exploring high pressure structural transformations, electronic properties and superconducting properties of MH₂ (M = Nb, Ta)

Jinquan Zhang^{^a}, Yanqi Wang^{^a}, Libiao Tang^{^a}, Juyi Duan^{^a}, Jingjing Wang^{^b}, Song Li^{^a}, Meng Ju^{^c}, Weiguo Sun^{^d}, Yuanyuan Jin^{^a,⁎}, Chuanzhao Zhang^{^a,⁎}

a Department of Physics and Optoelectronic Engineering, Yangtze University, Jingzhou 434023, China

b College of Computer and Information Engineering, Hubei Normal University, Huangshi 435002, China

c School of Physical Science and Technology, Southwest University, Chongqing 400715, China

d College of Physics and Electronic Information, Luoyang Normal University, Luoyang 471022, China

⁎Corresponding authors. scujyy@163.com (Yuanyuan Jin), zcz19870517@163.com (Chuanzhao Zhang) jyy@163.com (Chuanzhao Zhang)

Received: 2022-8-4, Accepted: 2022-10-9,

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Peer review under responsibility of King Saud University.

Abstract

The high-pressure structures and properties of MH₂ (M = Nb, Ta) are explored through an ab initio evolutionary algorithm for crystal structure prediction and first-principles calculations. It is found that NbH₂ undergoes a phase transition from a cubic Fm $\bar{3}$ m structure with regular NbH₈ cubes to an orthorhombic Pnma structure with fascinating distorted NbH₉ tetrakaidecahedrons at 48.8 GPa, while the phase transition pressure of TaH₂ from a hexagonal P6₃mc phase with slightly distorted TaH₇ decahedron to an orthorhombic Pnma phase with attractive distorted TaH₉ tetrakaidecahedrons is about 90.0 GPa. Besides, the calculated electronic band structure and density of states demonstrate that all of these structures are metallic. The Poisson’s ratio, electron localization function, and Bader charge analysis suggest that these phases possess dominant ionic bonding character with the effective charges transferring from the metal atom to H. From our electron–phonon calculations, the calculated superconducting critical temperature T_c of the Pnma-NbH₂ is 6.903 K at 50 GPa. Finally, via the quasi-harmonic approximation method, the phase diagrams at pressure up to 300 GPa and temperature up to 1000 K of MH₂ (M = Nb, Ta) are established, where the transition pressure of Fm $\bar{3}$ m-NbH₂ → Pnma-NbH₂ and P6₃mc-TaH₂ → Pnma-TaH₂ were found to decrease with increasing temperature.

Keywords

Metal hydrides

First-principles calculation

High pressure

Electron localization function

Superconductivity

Pressure–temperature phase diagram

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1 Introduction

Metal hydrides, especially under high pressure, have attracted wide attention because of their unique physical properties and attractive applications, such as potential high-temperature superconductivity and clean energy with the ability to replace fossil fuels as hydrogen storage materials. For example, some hydrogen-rich compounds, including YH₆ (Troyan et al., 2021; Kong et al., 2021), YH₉ (Snider et al., 2021), CeH₉ (Salke et al., 2019; Li et al., 2019), CeH₁₀ (Chen et al., 2021), ThH₉ (Semenok et al., 2020), ThH₁₀ (Semenok et al., 2020), LaH₁₀ (Drozdov et al., 2019), SnH₁₂ (Hong et al., 2021; Talantsev, 2021); UH₇ (Kruglov et al., 2018); and CaH₆ (Ma et al., 2022; Li et al., 2022), have been proved to have good superconductivity with high superconducting temperatures ranging from 44 to 262 K at high pressures and a hydrogen-rich compound rhodium trihydride (RhH₃) with a high volumetric hydrogen density of 212.5 g H₂/L could be used as a hydrogen storage material (Shao et al., 2021).

In recent years, transition metal hydrides have also been extensively studied under high pressure, which can be summarized as the following two aspects. On the one hand, the late transition metals with higher and comparable electronegativity values (e.g., 2.2 for Ru, Pd, Os, and Ir; 2.28 for Pt; 2.36 for W; 2.54 for Au) than that of hydrogen (2.2) are unfavorable react with hydrogen at ambient pressure. However, fortunately, high pressure can make the chemical potential of hydrogen rise sharply, thus effectively stimulating the chemical reactivity of elements, which makes the reaction between hydrogen and post-transition metals become a reality (Miao et al., 2020). Previous high-pressure experiments have shown that high pressure makes it possible to synthesize Rh, Ir, and Pt hydrides. For example, a potential hydrogen source material rhodium dihydride RhH₂ (Li et al., 2011) with a volume hydrogen density of 163.7 g H₂/L was discovered at room temperature at 8 GPa, and iridium trihydride IrH₃ (Scheler et al., 2013) was successfully synthesized above 65 GPa at room temperature by in-situ X-ray diffraction measurement. Besides, PtH is a superconductor with T_c = 10–25 K at high pressure (Kim et al., 2011). On the other hand, as for the early transition metals, although it is found that they are easy to form hydrides under environmental conditions, such as ScH₂/ScH₃, YH₂/YH₃, TiH₂, ZrH₂, HfH₂, VH₂, and NbH₂, after compression, some transition metal hydrides show attractive characteristics and undergo complex phase transitions that cannot be found under normal conditions (Miwa and Fukumoto, 2002; Quijano et al., 2009; Kanagaprabha et al., 2013; Smithson et al., 2002; Purans et al., 2021 ). For example, theoretical studies show that TiH₂ undergoes a structural transition of I4/mmm → P4/nmm → P2₁/m under high pressure at the corresponding transition pressures of 63 GPa and 294 GPa, respectively, and the phase transition from I4/mmm to P4/nmm at 63 GPa is more suitable for the experimental XRD pattern and the resulting unit cell parameters (Gao et al., 2013; Kalita et al., 2010). A subsequent experiment on the structures and properties of ZrH₂ were also investigated under high pressure, which provided a new method for finding hydrogen structures with higher bulk density in transition metal hydrides without static pressure or shear stress (Huang et al., 2014). Subsequently, the theoretical study of HfH₂ revealed that it undergoes pressure-induced structural phase transition I4/mmm → Cmma → P2₁/m at 180 GPa and 250 GPa, respectively, and the T_c of P2₁/m-HfH₂ at 260 GPa is 10.62–12.8 K (Liu et al., 2015). Moreover, Ye et al. reported that the structural sequence of ScH₂ should be Fm $\bar{3}$ m → C2/m, with transformation pressures of 65 GPa (Ye et al., 2015). Recent ab initio calculations predicted that Fm $\bar{3}$ m-ScH₂ (Wei et al., 2016) with T_c up to 38.11 K at 30 GPa is close to that of MgB₂ (39 K) (Choi et al., 2002; Bohnen et al., 2001). This motivated us to find its new structures and explore the high-temperature superconductive property by compressing compounds of early transition metals with hydrogen.

With regard to group 5 element dihydrides, the high-pressure structures and related structural and electronic properties of VH₂ and NbH₂ have also been deeply explored under high pressure. The two groups predicted that VH₂ transforms from cubic Fm $\bar{3}$ m structure to orthogonal Pnma phase at about 50 GPa and the T_c estimate of two stable VH₂ structures is only several Kelvins (Chen et al., 2014; Wang et al., 2020). As for NbH₂, Gao et al. report a phase transition sequence of Fm $\bar{3}$ m → Pnma at 50 GPa (Gao et al., 2013; Liu et al., 2017). However, Chen et al. used an evolutionary algorithm to detect their stability in the pressure range of 0–100 GPa and found that the ground state Fm $\bar{3}$ m-NbH₂ transformed to P6₃mc-NbH₂ at about 43 GPa (Chen et al., 2014). However, the obtained phase transition sequences of NbH₂ were not in agreement with each other. In recent experiments, the tantalum dihydride phase was synthesized at about 5 GPa. Despite large amounts of experimental research on TaH₂ (Saitoh et al., 2021; Kuzovnikov et al., 2017; Ying et al., 2019; Kuzovnikov et al., 2020 ), there are few theoretical studies on the high-pressure phase transition of TaH₂ and its mechanical properties, chemical bonding nature, dynamical properties, and superconductivity under high pressures. Therefore, this stimulates us to explore of high-pressure structures of MH₂ (M = Nb, Ta) and completely understand the structural characters, mechanical properties, electronic properties, and superconductivity of both dihydrides.

In this paper, we examine in detail the crystal structures of MH₂ (M = Nb, Ta) within the pressure range of 0–300 GPa by employing an ab initio evolutionary algorithm in combination with first-principles calculations. Interestingly, our calculated results show that NbH₂ and TaH₂ adopt the Fm $\bar{3}$ m and P6₃mc structure, respectively, at low pressures. However, at high pressures, NbH₂ and TaH₂ possess the same Pnma phases. Band structures and density of states indicate that these structures are metallic, while the estimated T_c is 0.945–1.836 K for Fm $\bar{3}$ m-NbH₂ at 1 atm, 0.775–1.528 K for P6₃mc-TaH₂ at 1 atm, 4.485–6.795 K for Pnma-NbH₂ at 100 GPa, and 3.586–5.542 K for Pnma-TaH₂ at 100 GPa, respectively. In addition, the analyses of the Poisson’s ratio, electron localization function (ELF), and Bader charge show that these phases are ionic crystals with the charges transferring from the metal atom to H. Furthermore, the calculation results of the pressure–temperature phase diagram of MH₂ (M = Nb, Ta) based on the quasi-harmonic approximation method indicate that high temperature and low pressure are more favorable to synthesizing the high-pressure phase of MH₂ (M = Nb, Ta). Our present research attempts to better understand the behavior and properties of MH₂ (M = Nb, Ta) under pressure and temperature, and promote further experimental and theoretical research on transition metal hydrides.

2 Computational method

We use the evolutionary algorithm USPEX code (Universal Structure Predictor: Evolutionary Xtallograph) to predict crystal structure, so as to extensively explore stable low-energy structures under high pressure with a different number of formula units (f.u.) at zero temperature (Oganov and Glass, 2006; Oganov et al., 2011; Lyakhov et al., 2013 ). A similar method has been successfully applied to discover transition metal dichalcogenides under high pressure with a similar composition to MH₂ (M = Nb, Ta) and provides good guidance for the synthesis of experiments (Saqib et al., 2021; Rahman et al., 2022). Here, we have predicted the crystal structure of MH₂ (M = Nb, Ta) at 1 atm, 50, 100, 150, 200, 250, and 300 GPa, in which the unit cell is as high as 8 formula units. The 50 structures of the first generation are randomly generated by the evolutionary algorithm, 60 % of the structures of each subsequent generation come from the parent generation, and the new structures of each subsequent generation are generated by mutation operator inheritance (60 %), substitution (10 %) and lattice mutation (30 %). The energy calculations, electronic structure calculations, and elastic properties introduced in this paper are carried out in density functional theory within the Perdew-Burke-Ernzerhof (PBE) parameterization of the generalized gradient approximation (GGA) as implemented in the Vienna ab initio simulation software package (VASP) (Kresse and Furthmüller, 1996; Perdew et al., 1996). The Projector-Augmented Wave (PAW) potentials with 13 valence electrons (4s²4p⁶5s¹4d⁴) for Nb, 11 valence electrons (5p⁶6s²5d³) for Ta, and 1 valence electron (1 s¹) for H were applied with a plane wave basis set up to cutoff energy of 600 eV and the k-point grid with 0.2 Å-¹ spacing was used to sample the Brillouin zone to ensure that all enthalpy calculations are well converged to<1 meV/atom (Kresse and Joubert, 1999; Monkhorst and Pack, 1976). We use Bader charge analysis (Tang et al., 2009) and electronic localization function (Becke and Edgecombe, 1990; Savin et al., 1992) to describe electronic properties. In order to determine the dynamic stability and thermodynamic properties of the studied structure, the phonon calculation is carried out by using the supercell method with PHONOPY program (Togo et al., 2008; Togo and Tanaka, 2015). The independent single crystal elastic constants could be determined by applying a stress tensor generated by a small strain to an optimized unit cell. The bulk modulus (B), shear modulus (G), Young’s modulus (E), and Poisson’s ratio (ν) of the material were determined by Voigt-Reuss-Hill (VRH) approximation (Hill, 1952). Electron–phonon coupling (EPC) parameter λ of NbH₂ and TaH₂ is performed using the density functional perturbation theory as implemented in the QUANTUM-ESPRESSO package (Giannozzi et al., 2009). The Projector-Augmented Wave (PAW) potentials were applied with a plane wave basis set up to cutoff energy of 60 Ry and charge density cutoff of 600 Ry. The q-point mesh of EPC calculations adopted 6 × 6 × 6 for Fm $\bar{3}$ m-NbH₂, 2 × 4 × 2 for Pnma-NbH₂, 4 × 4 × 2 for P6₃mc-TaH₂, and 2 × 4 × 2 for Pnma-TaH₂, respectively.

3 Results and discussion

3.1

3.1 Crystal structure searching and phase transition under pressure

Firstly, to extensively explore the stable MH₂ (M = Nb, Ta) under different pressures, we carry out the evolutionary structural search for MH₂ (M = Nb, Ta) using simulation sizes ranging from one to eight MH₂ (M = Nb, Ta) formula units (f.u.) in the unit cell at 0 K with selected pressures of 1 atm, 50, 100, 150, 200, 250, and 300 GPa. Subsequently, the analysis of the predicted structures provides some candidate structures with space groups Pm, C2/m, Pnma, Imma, P4/nmm, P $\bar{3}$ m1, P6₃mc, and Fm $\bar{3}$ m for NbH₂, and Pm, Pnma, P6₃/mmc, P4/nmm, Fm $\bar{3}$ m, P $\bar{3}$ m1, P6₃mc, and P $\bar{6}$ 2m for TaH₂. The enthalpy-pressure (H—P) relations of the candidate structures relative to the cubic Fm $\bar{3}$ m phase under the pressure range of 0–300 GPa for NbH₂ and TaH₂ are presented in Fig. 1a and Fig. 1c. According to our simulation, NbH₂ and TaH₂ at 1 atm adopt the structures of Fm $\bar{3}$ m and P6₃mc, respectively, which is consistent with the experimental observations (Reilly and Wiswall, 1970; Saitoh et al., 2021; Kuzovnikov et al., 2017; Ying et al., 2019; Kuzovnikov et al., 2020 ). The fact that all the experimental and theoretical structures are successfully reconstructed in a specific pressure range verifies the rationality of our method. Under compression, we observed the orthorhombic Pnma structure is favored over the cubic Fm $\bar{3}$ m phase above 48.8 GPa for NbH₂ and the hexagonal P6₃mc phase above 90.0 GPa for TaH₂, which does not include zero point energy (ZPE). Because of the extremely light mass of hydrogen atoms, it is expected that quantum effects will play an important role, and the (ZPE) of hydrogen may be large enough to affect the structural stability of the calculated phase (Pickard and Needs, 2007). We will discuss the influence of zero point energy on phase stability in Section 3.5. The calculated equations of state (EOS) for ambient-pressure structures and the orthorhombic Pnma structures show the volumetric changes are discontinuous at the transition point as plotted in Fig. 1b and Fig. 1d. We observed a volume collapse of 3.7 % occur at 48.8 GPa for NbH₂ and 1.2 % occur at 90.0 GPa for TaH₂, suggesting that the two-phase transitions are the first-order transition.

(a) Enthalpy-pressure relation concerning the Fm 3 ¯ m phase and (b) volume-pressure relations for NbH2. (c) Enthalpy-pressure relation concerning the Fm 3 ¯ m phase and (d) volume-pressure relations for TaH2.

To further verify the stability of four structures in comparison with other M−H (M = Nb, Ta) compositions, the typical convex hull diagrams of the M−H system are plotted in Fig. S1 at 1 atm, 100 GPa, 200 GPa, and 300 GPa. (Fig. S1 in the Supplementary Material) It is found that MH₂ (M = Nb, Ta) always lie on the convex hull curve, suggesting that they are thermodynamically stable under both high pressure and low pressure. Besides, we approach the pressure dependence of the MH₂ (M = Nb, Ta) decomposition reaction (see Fig. S2). Fig. S2a shows that NbH₂ can’t decompose to $NbH + \frac{1}{2} H_{2}$ or $\frac{1}{3} N b + \frac{2}{3} N b H_{3}$ . For TaH₂, we consider three substantially different decomposition paths displayed in Fig. S2b: $\frac{1}{2} {Ta}_{2} H + \frac{3}{4} H_{2}$ , $TaH + \frac{1}{2} H_{2}$ , or $\frac{1}{3} T a + \frac{2}{3} T a H_{3}$ . (Fig. S2 in the Supplementary Material) Apparently, TaH₂ can be stabilized in 0–300 GPa. Through the above discussion, we can conclude that MH₂ (M = Nb, Ta) are ground states at 0–300 GPa without phase separation.

Crystal Structures of MH2 (M = Nb, Ta) phases, together with metal coordination polyhedral: (a) Fm 3 ¯ m-NbH2 at 1 atm, (b) Pnma-NbH2 at 100 GPa, (c) P63mc-TaH2 at 1 atm, and (d) Pnma-TaH2 at 100 GPa. Green atoms depict Nb and red atoms depict Ta, while blue atoms present H.

3.2

3.2 Structural features

The crystal structures of the thermodynamically stable phases at the corresponding pressures are illustrated in Fig. 2 and their structural parameters, together with previous experimental data, are presented in Table 1 and Table S1. (Table S1 in the Supplementary Material) The Fm $\bar{3}$ m-NbH₂ phase at 1 atm, shown in Table 1, has lattice parameters of a = b = c = 4.556 Å, which is in good agreement with previous experimental results (a = b = c = 4.53 Å) with a lattice parameter difference of 0.57 % and other theoretical results (Zhang et al., 2017; Gao et al., 2013; Liu et al., 2017; Saitoh et al., 2021 ). For Fm $\bar{3}$ m-NbH₂ at atmospheric pressure, each Nb atom is surrounded by 8H atoms to form a regular NbH₈ cube with an Nb-H separation of 1.982 Å. For Pnma phase at 100 GPa, each Nb atom is surrounded by 9H atoms. The Nb-H distances range from 1.791 Å to 2.077 Å, and the H-Nb-H angle ranges from 57° to 121°, which indicates that the NbH₉ arrangement in Pnma phase is a highly distorted tetrakaidecahedron. Obviously, with the increase of pressure, the Nb-H environment in NbH₂ changes from regular NbH₈ cube to highly distorted NbH₉ tetrakaidecahedron.

Table 1 Compound MH₂ (M = Nb, Ta) unit cell lattice parameters, volume, bulk modulus B₀, and its pressure derivative B₀′ at selected pressure.

	Space group	Pressure	Refs	a (Å)	b (Å)	c (Å)	V (Å³)	B₀	B₀′
NbH₂	Fm $\bar{3}$ m	1 atm	This work	4.556				178	3.92
			Experiment (Reilly and Wiswall, 1970)	4.530
			Calculation (Chen et al., 2014)	4.558
			Calculation (Zhang et al., 2017)	4.577
NbH₂	Pnma	100 GPa	This work	4.682	2.838	5.067
			Calculation (Gao et al., 2013)	4.704	2.860	5.085
			Calculation (Zhang et al., 2017)	4.696	2.862	5.092
TaH₂	P6₃mc	1 atm	This work	3.231	3.231	5.162	23.34	180	4.08
			Calculation (Zhuang et al., 2017)	3.222	3.222	5.153
			Experiment (Kuzovnikov et al., 2020)	3.223	3.223	5.143	23.14	210(20)	4
TaH₂	Pnma	100 GPa	This work	4.709	2.842	5.095	17.04

For the P6₃mc-TaH₂ at ambient pressure, the lattice parameter (a = b = 3.231 Å, c = 5.162 Å) is in excellent agreement with the previous results (a = b = 3.222 Å, c = 5.153 Å) calculated by Zhang et al. (Zhuang et al., 2017) and experimental data (Kuzovnikov et al., 2020) (a = b = 3.223 Å, c = 5.143 Å), with a lattice constant deviation of 0.4 %. The third-order Birch-Murnaghan equation of state (EOS (Birch, 1947) is used to determine the bulk modulus B₀, its pressure derivative B₀′ and the unit cell volume V₀ at ambient conditions of P6₃mc-TaH₂ (see Table 1).

By fitting for P6₃mc-TaH₂, the following parameters are obtained: B₀ = 180 GPa, B₀′ = 4.08, and V₀ = 23.34 Å³. These values are in agreement with previous Energy-dispersive X-ray diffraction studies (B₀ = 210(20) GPa, B₀′ = 4, and V₀ = 23.14 Å³) (Kuzovnikov et al., 2020). As displayed in Fig. 2a, each Ta atom is surrounded by one H atom at a separation of 1.889 Å, three H atoms at a separation of 1.921 Å, and a further three H atoms at a separation of 1.990 Å. In addition, the H-Ta-H angles vary from 68° to 115°. Therefore, the Ta environment in the P6₃mc configuration can be described as distorted TaH₇ decahedrons. In the Pnma phase at 100 GPa (see Fig. 2b), the Ta-H distances range from 1.787 to 2.06 Å and the H-Ta-H angles vary from 58° to 164°. These results describe that the Ta environment in the Pnma phase is strongly distorted TaH₉ tetrakaidecahedron. In short, the structural features of P6₃mc and Pnma TaH₂ can be summarized as slightly distorted TaH₇ decahedrons and strongly distorted TaH₉ tetrakaidecahedron, respectively.

3.3

3.3 Mechanical stability, mechanical properties, and electronic properties

To inspect the mechanical stabilities of MH₂ (M = Nb, Ta) structures, we calculated the elastic constants of MH₂ (M = Nb, Ta) at different pressures by using the strain–stress method, as summarized in Table 2. As given in Table 2, four phases satisfy their respectively mechanical stability standards, which state that four structures are mechanically stable at ambient pressure. Besides, Fig. S3 shows that the Pnma-NbH₂ and Pnma-TaH₂ are dynamically stable at 1 atm, which means that the high-pressure phases Pnma-NbH₂ and Pnma-TaH₂ can be quenched to normal pressure. (Fig. S3 in the Supplementary Material) At the same time, the bulk modulus (B), shear modulus (G), B/G, Young’s modulus (E), and Poisson’s ratio (ν) can be derived from the calculated elastic constants on the basis of the Voigt-Reuss-Hill method. Firstly, the bulk modulus B of Fm $\bar{3}$ m-NbH₂ and P6₃mc-TaH₂ is consistent with the zero-pressure bulk modulus B₀ fitted by EOS (see Table 1), which underlines the accuracies of our calculations. The calculated bulk modulus (B), shear modulus (G), and Young’s modulus (E) of MH₂ (M = Nb, Ta) at 1 atm are all larger than those of MgH₂ and Pnma-TiH₂ hydrogen storage materials, implying that these MH₂ (M = Nb, Ta) dihydrides present good mechanical properties (Junkaew et al., 2014; Pan and Chen, 2020). As we all know, the ductility or brittleness of materials can be estimated by the value of B/G with 1.75 as the critical value (Pugh, 1954). Our results show that the calculated identical ratio values of B/G for Fm $\bar{3}$ m-NbH2, Pnma-NbH₂, P6₃mc-TaH₂, and Pnma-TaH₂, are 2.31, 2.07, 2.20, and 2.27, respectively, indicating their toughness behavior. Poisson’s ratio is an important parameter to describe the type of bonding in a material (Haines et al., 2001). The threshold value of Poisson’s ratio ν for covalent, and ionic materials is 0.25: large ν values (>0.25) correspond to ionic or metallic materials, while smaller ν values (<0.25) correspond to covalent materials. The Poisson’s ratios of Fm $\bar{3}$ m-NbH_2, Pnma-NbH₂, P6₃mc-TaH_2, and Pnma-TaH₂, are all larger than 0.25, suggesting that they can be classified into ionic materials and that the ionic bond component is dominant. Vickers hardness was estimated according to the empirical model proposed by Chen et al (Chen et al., 2011). This model is applied as follows: H_v = 2(K²G)^0.585-3, where K = G/B. The estimated hardness values are 6.6 GPa for Fm $\bar{3}$ m-NbH₂, 7.6 GPa for Pnma-NbH₂, 7.5 GPa for P6₃/mc-TaH₂, and 6.6 GPa for Pnma-TaH₂ at 1 atm, which are lower than that of Pd₂Mo₃N with the strong three-dimensional covalent network (17.1 GPa) (Errandonea et al., 2010). Due to strong ionic bonds, the hardness values of all MH₂ (M = Nb, Ta) phases are lower than 10 GPa.

Table 2 The calculated elastic constants C_ij (GPa), bulk modulus B (GPa), shear modulus G (GPa), Young’s modulus E (GPa), B/G, Poisson’s ratio ν, and hardness H_v (GPa) for TaH₂ and NbH₂ at the selected pressures.

Phase	NbH₂		TaH₂		MgH₂ (Junkaew et al., 2014)	TiH₂ (Pan and Chen, 2020)
Phase	Fm $\bar{3}$ m	Pnma	P6₃mc	Pnma	P4₂/mnm	Pnma
Pressure	1 atm	1 atm	1 atm	1 atm	1 atm	1 atm
C₁₁	273	273	312	288	74.4	262.6
C₂₂		242		284		273.3
C₃₃		248	313	263	136.0	248.0
C₄₄	84	77	78	74	37.6	45.1
C₅₅		68		70		83.7
C₆₆		81	73	85	53.0	13.4
C₁₂	133	102	166	140	38.8	55.8
C₁₃		89	95	105	31.4	104.8
C₂₃		121		122		75.6
B	180	154	183	174	54.2	139.5
G	78	75	83	77	37.8	51.5
E	204	193	216	200	92.1	51.5
B/G	2.31	2.07	2.20	2.27		2.71
ν	0.31	0.29	0.3	0.31
H_v	6.6	7.6	7.5	6.6

To further gain insight into the electronic properties of four stable MH₂ (M = Nb, Ta) compounds, we calculated the electronic band structures and densities of states (DOS), as displayed in Fig. 3. We can see that they all exhibit metallic characteristics, confirmed by the overlap between the conduction bands and the valence bands, and the finite electronic DOS at the Fermi level (E_f). The metallic behavior of these phases indicates that these materials may be magnetic. We have carried out calculations taking into account the spin polarization. Results suggest that MH₂ (M = Nb, Ta) are all nonmagnetic because the magnetic moment of each mental atom is zero. For these phases, it is found that there is a deep valley below the Fermi level, and the localized hybridization between M-d and H-s is observed in the energy region below the pseudogap, while the contributions of M-d (M = Nb, Ta) states mainly dominate in this energy range above pseudogap, so the metallic properties are mainly due to partially filled Nb-4d or Ta-5d shell. Note that the contribution from H-s states at the Fermi level is almost zero, which is of great significance to superconductivity according to previous studies as will be discussed later.

The electronic band structure and densities of states (DOS) of (a) Fm 3 ¯ m-NbH2 at 1 atm, (b) Pnma-NbH2 at 100 GPa, (c) P63mc-TaH2 at 1 atm, and (d) Pnma-TaH2 at 100 GPa.

To clarify the bonding futures for these phases MH₂ (M = Nb, Ta), we calculated the electronic localization function (ELF), as depicted in Fig. 4. It is worthy pointing out that these four crystals display a common feature of the ELF that there is no electron localization toward the neighboring M−H and H—H connections, thus suggesting that no covalent interaction exists for neighboring M−H and H—H. Moreover, the ELF values between M and the nearest H atom in four crystals are all smaller than 0.5, implying that ionic bonds are present between M and H atoms. This conclusion also demonstrates the validity of the analyses of Poisson’s ratio. In addition, in order to evaluate the strength of ionic bonds, we explicitly calculated the actual charge transfer between MH₂ (M = Nb, Ta) and H atoms, which is summarized in Table 3. Each H atom respectively gains approximately 0.65 e, 0.51 e, 0.59 e, and 0.52 e for Fm $\bar{3}$ m-NbH_2, Pnma-NbH₂, P6₃mc-TaH₂, and Pnma-TaH₂, while the M (M = Nb, Ta) atoms lose approximately 1.30 e, 1.02 e, 1.18 e, and 1.04 e. Therefore, there is a large amount of charge transfer from M (M = Nb, Ta) to the H atoms, demonstrating the ionic character of the M−H bonding in MH₂ (M = Nb, Ta) crystals. Such ionic bonding properties are similar in RhH₂ (Shao et al., 2021), ThH₂ (Zhang et al., 2015), TcH₂ (Li et al., 2016), TiH₂ (Zhuang et al., 2018), HfH₂ (Quijano et al., 2009), and VH₂ (Zhuang et al., 2017; Li and Peng, 2017), but differ from many hydrogen-rich polycrystalline forms (i.e. SiH₄ (Zhang et al., 2015), SnH₄ (Zhang et al., 2016), MgH₁₂ (Lonie et al., 2013), MgH₁₆ (Lonie et al., 2013), AsH₈ (Fu et al., 2016), RuH₆ (Liu et al., 2016); and MoH₁₁ (Du et al., 2021) where H atoms are bonded to the nearest H atoms to form H₂ or H₃ units and/or they are covalently bonded to X (X = Si, Sn) to form X-H bonds. Combination analysis of ELF and Bader charges reveals that MH₂ (M = Nb, Ta) forms an ionic bond between M and H, and MH₂ (M = Nb, Ta) belongs to ionic crystal with the charge transfers from M atom to H atom, which is consistent with Poisson’s ratio analysis mentioned above.

Electron localization function (ELF) of for stable MH2 (M = Nb, Ta) phases: (a) ( −1 1 0) plane for the Fm 3 ¯ m-NbH2 phase at 1 atm, (b) (0 1 0) plane for the Pnma-NbH2 phase at 100 GPa, (c) (1 1 0) plane for the P63mc-TaH2 phase at 1 atm, and (d) (0 1 0) plane for the Pnma-TaH2 phase at 100 GPa.

Table 3 Calculated Bader charges for NbH₂ and TaH₂ at the selected pressures. δ represents the amount of charge transferred from the mental atom to the H atom.

Phase	Space group	Pressure	Atom	Charge value (e)	δ (e)
NbH₂	Fm $\bar{3}$ m	1 atm	Nb	11.70	1.30
	Fm $\bar{3}$ m	1 atm	H	1.65	−0.65
	Pnma	100 GPa	Nb	11.98	1.02
	Pnma	100 GPa	H	1.51	−0.51
TaH₂	P6₃mc	1 atm	Ta	9.82	1.18
	P6₃mc	1 atm	H	1.59	−0.59
	Pnma	100 GPa	Ta	9.96	1.04
	Pnma	100 GPa	H	1.52	−0.52

3.4

3.4 Electron–phonon coupling

For exploring the superconductivity, we calculate the logarithmic average phonon frequency ω_log, the electronic DOS at the Fermi level N(E_f), and the EPC parameter λ as shown in Table 4. It is found that for materials with λ < 1.5, it is more accurate to estimate the superconducting temperature of the obtained structure by using the modified Allen-Dynes McMillan equation $T_{c} = \frac{ω_{\log}}{1.2} \exp [\frac{1.04 (1 + λ)}{λ - μ^{*} (1 + 0.62 λ)}]$ (Allen and Dynes, 1975). The estimated T_c of NbH₂ and TaH₂ with a typical choice of Coulomb pseudopotential μ* = 0.1–0.13 is presented in Table 4, which shows that Pnma-NbH₂ has the highest T_c, and can reach 6.9 K at 50 GPa, higher than the maximal 4 K predicted for Pnma-VH₂ (Chen et al., 2014) at 60 GPa. The estimated T_c of P6₃mc-TaH₂ is 0.775–1.528 K at 1 atm, which is consistent with the experimental measurement of no superconducting transition in the temperature range of 4.4–300 K for TaH₂ (Kuzovnikov et al., 2017). In addition, the calculated T_c of the Pnma-TaH₂ is 3.586–5.542 K at 100 GPa, which is smaller than that of the Pnma-NbH₂ at 100 GPa. The main reason is that the electron DOS of Pnma-TaH₂ at Fermi level N (E_f) is much smaller than that of Pnma-NbH₂.

Table 4 The calculated logarithmic average phonon frequency ω_log, electronic density of states at the Fermi level N(E_f) (states/spin/Ry/Unit cell), electron–phonon coupling parameters λ, and superconducting critical temperatures T_c of Fm

\bar{3}

m-NbH₂, Pnma-NbH₂, P6₃mc-TaH₂, and Pnma-TaH₂ at different pressures.

Phase	Structure	Pressure	ω_log(K)	N(ε_f)	λ	T_c(K)μ* = 0.1	T_c(K)μ* = 0.13
NbH₂	Fm $\bar{3}$ m	1 atm	262.565	7.208	0.441	1.836	0.945
	Pnma	50 GPa	376.899	26.477	0.560	6.903	4.668
		100 GPa	394.013	20.208	0.550	6.795	4.485
		300 GPa	547.141	16.527	0.554	9.694	6.439
TaH₂	P6₃mc	1 atm	229.106	10.402	0.436	1.528	0.775
	P6₃mc	50 GPa	297.577	7.146	0.325	0.299	0.071
	Pnma	100 GPa	348.496	16.927	0.537	5.542	3.586
	Pnma	300 GPa	371.099	14.736	0.532	5.719	3.672

According to Fig. 5, we computed phonon dispersions, phonon density of states (PHDOS), Eliashberg spectral function (α²F(ω)/ω), and EPC integrated (λ(ω)) for the Fm $\bar{3}$ m-NbH₂ and P6₃mc-TaH₂ at 0 GPa, and the Pnma-NbH₂ and Pnma-TaH₂ at 100 GPa. As a result, we found that all structure is dynamically stable because it exhibits the absence of any imaginary frequencies (Zhang et al., 2022; Sun et al., 2020; Zheng et al., 2021; Li et al., 2022; Jin et al., 2022 ). A clear gap between acoustic and optical phonon modes is observed in the theoretical phonon density of states (Fig. 5), which is due to the huge difference in mass between the M (M = Nb, Ta) atom and H atom, thus dividing the phonon density of states into low-frequency vibrations of M (M = Nb, Ta) atom and high-frequency vibrations of H atom. Note that the low-frequency vibrational modes below 10 THz come from M (M = Nb, Ta) atoms, accounting for 90.8 %, 85.5 %, 84.5 %, and 78 % of the total λ, respectively, while the high-frequency vibrational modes above 10 THz come from H atoms, accounting for 9.2 %, 14.5 %, 15.5 %, and 22 % of the total λ, respectively. We, therefore, conclude that the coupling between electron and phonon vibration of M (M = Nb, Ta) is the reason why MH₂ (M = Nb, Ta) shows superconductivity, which is consistent with the dominant DOS of M-d (M = Nb, Ta) shell at Fermi energy. For MH₂ (M = Nb, Ta), contrary to the case of superconducting H₃S (Drozdov et al., 2015; Duan et al., 2014), YH₃ (Kim et al., 2009), and PdH (Gupta, 1982), the electron-optical phonon contribution and consequently to T_c is small, due essentially to the low value of the DOS of H-s states at the Fermi energy, which leads to the low-T_c features of all MH₂ (M = Nb, Ta) phases.

Phonon dispersion relations, projected phonon densities of states (PHDOS), and Eliashberg spectral function for (a) Fm 3 ¯ m-NbH2 at 1 atm, (b) Pnma-NbH2 at 100 GPa, (c) P63mc-TaH2 at 1 atm, and (d) Pnma-TaH2 at 100 GPa.

3.5

3.5 Pressure–temperature phase diagram

As discussed earlier, we established a phase transition sequence of MH₂ (M = Nb, Ta) under pressure at 0 K. High pressure and high-temperature technology are very important for the synthesis of new materials. Here, the main function of pressure and temperature is to change the free energy diagram of materials or the reaction barrier between materials, so as to promote phase transition or chemical reaction, which leads to the formation and synthesis of the target phase, which can be stored in environmental conditions. To further discover the high-pressure synthesis conditions and the stable pressure–temperature region of each phase, the phase diagram of MH₂ (M = Nb, Ta) was constructed by using the quasi-harmonic approximation (QHA) method (Togo and Tanaka, 2015). At a given pressure and temperature, the Helmholtz free energy is expressed as:

(2)

F (V, T) = E (V) + F_{vib} (V, T) + F_{el}

where E(V) is the cold static lattice energy with atoms being clamped at their equilibrium positions, F_vib (V, T) is the nonequilibrium vibrational Helmholtz free energy of phonons, and F_el is the electronic thermal excitation free energy. Of course, considering that the contribution of thermoelectrons (F_el) to free energy is negligible at relatively low temperatures, the F (V, T) we obtain does not include the contribution of electrons (Kvashnin et al., 2018; Yan et al., 2016; Rybkovskiy et al., 2020 ). F_vib (V, T) can be obtained from.

(3)

F_{vib} (V, T) = \int_{0}^{\infty} [\frac{h ω}{2} + κ_{B} T \ln (1 - e^{- h ω / κ_{B} T})] g (ω, V) d ω

where κ_B is Boltzmann’s constant and g (ω, V) is the phonon density of state and depends on the crystal geometry. Then, the Gibbs free energy is given by.

(4)

G_{gibbs} = F (V, T) - V {(\frac{\partial F}{\partial V})}_{T}

Therefore, based on the calculated Gibbs free energies of these phases at different temperatures and pressures, we obtain a pressure–temperature phase diagram (Fig. 6), which can determine the possible existence region of MH₂ (M = Nb, Ta) phases. According to Fig. 6, by comparing the transition pressures with and without the influence of temperature, we find that the transition pressure of NbH₂ from cubic Fm $\bar{3}$ m phase to orthorhombic Pnma phase at 0 K is 27.9 GPa, while the transition pressure of TaH₂ from the hexagonal P6₃mc phase to orthorhombic Pnma phase at 0 K is 54.0 GPa. Therefore, ZPE does not change the phase transition series, but only reduces the phase transition pressure, especially for high-pressure transition, which is also observed in other hydrides, such as MgH₆ (Feng et al., 2015), AcH₂ (Jiang et al., 2014); and PtH (Gao et al., 2012). It is obvious that the transition pressures of Fm $\bar{3}$ m-NbH₂ → Pnma-NbH₂ and P6₃mc-TaH₂ → Pnma-TaH₂ decrease obviously with the increase of temperature, which originates mainly from the contributions of the thermal vibration at the atomic position to the high-pressure structures more significantly than the low-pressure structures, thus significantly expanding their high-pressure structural stability fields.

Calculated temperature–pressure phase diagram of (a) NbH2 and (b) TaH2 based on quasi-harmonic approximation.

4 Conclusions

In a word, we have extensively investigated the systematic structure evolution of MH₂ (M = Nb, Ta) compounds in the pressure range of 0 to 300 GPa by ab initio evolutionary simulation. Based on the structural prediction, the phase transition sequence of NbH₂ is Fm $\bar{3}$ m → Pnma at 48.8 GPa, while the TaH₂ is P6₃mc → Pnma at 90.0 GPa, respectively. The calculations of elastic constants and phonon dispersions confirmed that these phases are stable. The analyses of DOS, Poisson’s ratio, ELF, and Bader charge indicate that four MH₂ (M = Nb, Ta) phases are metallic and an ionic crystal with large amounts of charge transferring from M (M = Nb, Ta) atoms to H. The estimated superconducting transition temperature T_c values of Pnma-NbH₂ are 6.903 K at 50 GPa. In addition, it is found that the high-pressure phase transition of MH₂ (M = Nb, Ta) is more favorable upon heating, and this originates mainly from the contributions of the thermal vibration at the atomic position. The current research will stimulate further high-temperature and high-pressure experiments to synthesize these transition metal dihydrides and make structural, mechanical, and superconducting measurements.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 11804031, 11904297 and 11747139), the Scientific Research Project of Education Department of Hubei Province (No. Q20191301) and Talent and High Level Thesis Development Fund of Department of Physics and Optoelectronic Engineering of Yangtze University (C. Z. Z.).

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Appendix A

Supplementary material

Supplementary data to this article can be found online at https://doi.org/10.1016/j.arabjc.2022.104347.

Appendix A

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Introduction

Computational method

Results and discussion

Crystal structure searching and phase transition under pressure
Structural features
Mechanical stability, mechanical properties, and electronic properties
Electron–phonon coupling
Pressure–temperature phase diagram

Conclusions

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Exploring high pressure structural transformations, electronic properties and superconducting properties of MH2 (M = Nb, Ta)

Abstract

Keywords

Metal hydrides

First-principles calculation

High pressure

Electron localization function

Superconductivity

Pressure–temperature phase diagram

1 Introduction

2 Computational method

3 Results and discussion

3.1 Crystal structure searching and phase transition under pressure

3.2 Structural features

3.3 Mechanical stability, mechanical properties, and electronic properties

3.4 Electron–phonon coupling

3.5 Pressure–temperature phase diagram

4 Conclusions

Acknowledgements

Declaration of Competing Interest

References

Appendix A

Supplementary material

Appendix A

Supplementary material

Supplementary data 1

Supplementary data 2

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Exploring high pressure structural transformations, electronic properties and superconducting properties of MH₂ (M = Nb, Ta)