Anisotropy in elasticity and thermodynamic properties of zirconium tetraboride under high pressure

Abstract

The recently predicted ZrB₄ with an Amm2 orthorhombic structure has great scientific and technical significance owing to its novel B–Zr–B “sandwich” layer bonding and evaluated high hardness. To better understand the performance of Amm2-ZrB₄, its elastic and thermodynamic properties under pressure and temperature are studied here by taking advantage of first principles calculations in combination with the quasi-harmonic Debye model. It is found that ZrB₄ keeps brittleness and mechanical stability up to 100 GPa, possessing pronounced elastic anisotropy demonstrated by the elastic anisotropy factors, the direction-dependent Young's modulus, shear modulus and Poisson's ratio. The pressure and temperature dependences of the thermodynamics parameters including normalized volume V/V₀, bulk modulus, specific heat, Debye temperature, thermal expansion coefficient and Grüneisen parameter in wide temperature (0–1000 K) and pressure (0–50 GPa) ranges are obtained and discussed in detail.

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I. Introduction

In recent decades, transition metal borides (TMBs) have drawn considerable attention as candidate (super) hard materials, and a number of them have been widely used in high-temperature environments, cutting tools, and hard coatings owing to their superior properties such as high strength, high hardness, ultra-incompressibility and good thermal stability.^1–4 Recently, novel transition metal (e.g., Fe,^5–7 W,^8–10 and Cr,³) borides have been successfully synthesized under ambient or high pressure (65 GPa for Pnnm-FeB₄, 46.2 GPa for P63/mmc-WB₄ and 48 GPa for Pnnm-CrB₄) and extensive experimental and theoretical investigations have been carried out on these borides, which identifies their superhardness and leads to the low-cost synthesis of superhard materials. For the Zr–B system, there are three identified phases (ZrB, ZrB₂, and ZrB₁₂) according to the phase diagram.¹¹ The relatively high hardness of ZrB₂ and ZrB₁₂ naturally leads us to wonder if there are any (super)hard zirconium tetraborides. Inspired by such a hypothesis, our group successfully predicted two new orthorhombic phases of ZrB₄ and estimated their hardness as 42.8 GPa and 42.6 GPa for Cmcm and Amm2 structure, respectively.¹² Both phases exhibit an interesting B–Zr–B sandwiches stacking order along the c and a-axis, and the sandwiches are connected by strong covalent bond (B–B bond). The three-dimensional networks of high atomic density consequently explained the occurrence of superhardness in ZrB₄. The two structures are similar, but we find that Amm2-ZrB₄ might be more easily obtained due to its lower formation enthalpy. These facts stimulate us to conduct a detailed investigation on its fundamental properties, such as elastic constants, elastic anisotropy and thermodynamic properties which are crucial to its practical applications and synthesis. For example, the elastic constants of a solid give important information concerning the nature of the forces operating in the solid^13,14 and help us to understand its mechanical behaviors in practical application, such as anisotropy, phase transformation, elastic instability, plastic deformation and fracture, precipitation, dislocation dynamics, crack and so on.¹⁵ On the other hand, the understanding of thermodynamic properties of solids (such as heat capacity, thermal expansion coefficient, Grüneisen parameters, and Debye temperature) will be beneficial to their synthesis and practical applications.¹⁶ Therefore, in this paper, elastic properties of ZrB₄ coupled with thermodynamic properties at various temperatures and pressures are investigated systematically through the first principles calculations and quasi-harmonic Debye model.¹⁷

This paper proceeds as follows: the details of the calculation methods and theoretical model are described in Section II, followed by the calculated results and analysis in Section III. Conclusions are summarized in Section IV.

II. Methods of calculation

The ab initio calculations were performed using density functional theory within the generalized gradient approximation (GGA),¹⁸ as implemented in the Vienna ab initio simulation package (VASP).¹⁹ The exchange and correlation potential was treated by the generalized gradient approximation in the scheme of Perdew–Burke–Ernzerhof (PBE).²⁰ The all-electron projector augmented wave (PAW) method²¹ was employed with a plane-wave cutoff energy of 600 eV. The k-point grid in the Brillouin zone were generated using the Monkhorst–Pack scheme with the separation of 0.03 Å⁻¹. The total energy convergence tests showed that convergence to within 1 meV/atom was achieved with the above calculation parameters. Single crystal elastic constants were calculated via a strain–stress approach. i.e., by applying a small strain to the equilibrium lattice of orthorhombic unit cell and fitting the dependence of the resulting change in stress on the strain. The bulk modulus, shear modulus, Young's modulus, and Poisson's ratio were determined by using the Voigt–Reuss–Hill approximation,²² in which the Voigt and Reuss expressions represent the upper and lower limit of the polycrystalline modulus. The formulae for orthorhombic structure are:

The Young's modulus E and the Poisson's ratio υ are then calculated from the elastic moduli using the following relations:

III. Results and discussion

3.1 Elastic properties

The lately predicted crystal structure of ZrB₄ is orthorhombic with space group Amm2, no. 38, with 20 atoms per conventional unit cell consisting of four ZrB₄ f.u. in a unit cell, in which one Zr and three B atoms occupy the Wyckoff position 8f (0.3461, 0.6689, 0.2655), 4c (0.8289, 0.5, 0.7469), 4e (0.5, 0.3374, 0.9792), and 4d (0, 0.3332, 0.2486) respectively. The equilibrium lattice constants, volume per formula unit, density, bulk modulus and its pressure derivative are all listed in Table 1, together with available experimental and theoretical results of ZrB₂ and ZrB₁₂ for comparison. In the elastic range, due to the symmetry of the crystal, there are nine independent components in the elastic tensor for ZrB₄, i.e., C₁₁, C₂₂, C₃₃, C₄₄, C₅₅, C₆₆, C₁₂, C₁₃ and C₂₃. Elastic constants play important roles in providing a deeper insight into mechanical stability and stiffness of materials.²³ The pressure dependences of the elastic constants up to 100 GPa are illustrated in Fig. 1. It can be seen that all elastic constants increase monotonically with pressure and all C_ij satisfy the well-known Born stability criteria²⁴ up to 100 GPa, which indicates that ZrB₄ is still mechanically stable at high pressure of 100 GPa.

Table 1 The calculated equilibrium lattice constants a₀, b₀, c₀ (Å) and equilibrium volume per formula unit V₀ (ų), density ρ, EOS fitted bulk modulus B₀ (GPa), and its pressure derivative B′₀ for the orthorhombic ZrB₄ at 0 K and 0 GPa

		a0	b0	c0	V0	ρ	B0	B′0
a Vinet universal EOS.^26b Birch–Murnaghan 3rd-order EOS.^27c Murnaghan EOS.^28d Ref. 29.e Ref. 30.f Ref. 31.g Ref. 32.
ZrB4	This work	10.3120	5.41307	3.17999	300	5.03	239^a, 238^b, 235^c	3.84^a, 3.86^b, 3.90^c
ZrB2	Theo.	3.1768^d	—	3.559^d	31.1^d	—	355^d	4.2^d
	Exp.	3.170^d	—	3.532^d	30.74^d	—	317^f, 245^g	—
ZrB12	Theo.	7.415^e	—	—	407.69^e	—	—	—
	Exp.	7.4077^e	—	—	406.49^e	—	—	—

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Fig. 1 Pressure dependence of the elastic constants (C_ij) of ZrB₄ at 0 K.

Unfortunately, there are no experimental data available for comparison present, therefore, our results could be a reference for future studies and applications under high pressures of ZrB₄.

In general, the large value of shear modulus is an indication of more pronounced directional bonding between atoms, and the Poisson's ratio is a factor that measures the stability of a crystal against shear and Young's modulus provides a measure of stiffness of a solid. The calculated elastic constants, elastic moduli (B, G and E), Poisson's ratio υ and the B/G ratio of ZrB₄ under pressure are given in Table 2, along with the theoretical values of other transition metal tetraborides (WB₄, CrB₄, FeB₄). It is shown in Table 2 that all of the B, G, E, υ and B/G increase substantially with pressure and the calculated bulk and shear moduli of Amm2-ZrB₄ are comparable to those of WB₄, CrB₄, FeB₄, indicating their strong ability to resist volume deformation. According to Pugh's criterion,²⁵ a low (high) B/G value is associated with brittleness (ductility), and the ductile and brittle materials are separated by the critical value (1.75). The B/G of ZrB₄ reaches 1.47 at 100 GPa, implying that ZrB₄ is a brittle and mechanically stable phase within the range of pressures.

Table 2 The elastic constants C_ij (GPa), bulk modulus B (GPa), shear modulus G (GPa), Young's modulus E (GPa), Poisson's ratio υ, the B/G ratio and the H_v of ZrB₄ under pressure

	P	C11	C22	C33	C44	C55	C66	C12	C13	C23	B	G	E	ν	B/G
a Ref. 10.b Ref. 3.
ZrB4	0	559	578	458	233	243	262	53	118	113	239	232	528	0.134	1.03
	10	618	625	517	256	270	283	75	146	139	275	253	581	0.149	1.09
	20	674	681	572	278	295	301	97	173	166	311	271	631	0.162	1.15
	30	740	750	638	297	318	324	108	194	190	346	294	687	0.169	1.18
	40	790	801	689	314	340	339	128	221	217	379	310	731	0.179	1.22
	50	838	849	736	330	361	353	148	249	245	412	324	770	0.188	1.27
	60	885	896	780	345	381	366	168	275	272	443	338	808	0.196	1.31
	70	929	940	829	359	401	378	188	302	300	475	351	845	0.204	1.35
	80	972	984	875	372	419	389	207	329	327	506	363	879	0.211	1.40
	90	1016	1027	912	384	438	404	226	356	354	536	375	912	0.217	1.43
	100	1057	1069	954	394	455	414	246	382	381	566	385	943	0.223	1.47
WB4^a	0	389.3		437.0	150.7			280.2	224.2		297.0	103.6			2.86
CrB4^b	0	554	880	473	254	282	250	65	107	95	265	261			1.02
FeB4^b	0	381	710	435	218	114	227	137	143	128	253	177			1.43

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Debye temperature θ_D is a fundamental parameter of a compound, which has close relationships with specific heat, melting temperature, and elastic constants. The θ_D can be calculated from elastic constants (),³³ which gives explicit information about the lattice vibrations.³⁴ The Debye temperature of ZrB₄ under pressure is presented in Table 3, showing an increasing trend with pressure. As is generally known, a crystal with a larger Debye temperature corresponds to a stiffer characteristic. This is because the optical phonons have a higher frequency and therefore require greater energy to activate. Pressure typically enhances the interactions between atoms of a crystal and hence stiffers it, which is manifested by increased elastic moduli B and G. Therefore, pressure typically increases Debye temperature of ZrB₄ and this implies stronger interactions between atoms in the system.

Table 3 The calculated density ρ (in g cm⁻³), the longitudinal, transverse and mean elastic wave velocity (υ_l, υ_t and υ_m in m s⁻¹), and the Debye temperature θ_D (in K) of ZrB₄ under pressure

P	ρ	υl	υt	υm	θD
0	5.03	10 457	6800	7457	1073
10	5.23	10 821	6952	7634	1113
20	5.41	11 150	7082	7788	1148
30	5.58	11 499	7260	7989	1189
40	5.73	11 757	7354	8100	1217
50	5.88	11 983	7426	8188	1241
60	6.02	12 188	7494	8270	1263
70	6.15	12 385	7555	8344	1283
80	6.27	12 565	7607	8407	1302
90	6.39	12 731	7658	8469	1320
100	6.51	12 883	7696	8517	1335

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3.2 Elastic anisotropy

Elastic anisotropy is very important in diverse applications of materials, such as phase transformations, precipitation, dislocation dynamics and microcrack formation. The fundamental information about the bonding characteristics between adjacent atomic planes can also be obtained via the elastic anisotropy. Therefore, this property will be crucial for the potential hard material ZrB₄. The shear anisotropy factors (A₁, A₂, A₃), the universal elastic anisotropy index A^U and the directional bulk modulus B_a, B_b and B_c are appropriate measures to quantify the extent of anisotropy.³⁵ The shear anisotropy factor for the {100} shear planes between the <011> and <010> directions is defined asfor the {010} shear planes between the <101> and <001> directions isfor the {001} shear planes between the <110> and <010> directions isfor the universal elastic anisotropy index A^U, defined by Ranganathan and Ostoja-Starzewski from the bulk modulus B and shear modulus G denoted by Voigt and Reuss approaches,³⁵ isand the directional bulk modulus along different crystallographic axis can be defined as³⁶

B_i = i(dP/di) (i = a, b, and c) (8)

Taking advantage of the formulae mentioned above, the parameters about elastic anisotropy (A₁, A₂, A₃, A^U, B_a, B_b and B_c) are calculated and presented in Table 4. In the case of isotropic crystals, A₁, A₂, and A₃ are all equal to 1, while any deviation from one means the amplitude of anisotropy of the crystal. From Table 4, we can see that A₁, A₂, and A₃ are larger than 1 at 0 GPa and all increase with pressure. The shear anisotropy results of ZrB₄ indicate that the elastic anisotropy for the {010} shear planes between the <101> and <001> directions is more obvious than that of the {100} shear planes between the <011> and <010> directions and the {001} shear planes between the <110> and <010> directions, and the value of A₃ also reveals that ZrB₄ is nearly isotropic in {001} shear planes. Because Amm2-ZrB₄ is orthorhombic, the shear anisotropy factors are not adequate to sufficiently describe its elastic anisotropy. Therefore, the universal elastic anisotropy index A^U should also be considered (A^U is zero for isotropic crystals). In Table 4, A^U is 0.09 at 0 GPa, which increases with increasing pressure. Meanwhile, the directional bulk modulus (B_a, B_b, B_c) also increases with pressure and the bulk modulus along the c-axis is larger than that along a-axis and b-axis at 100 GPa.

Table 4 The shear anisotropy factors A₁, A₂, A₃ and elastic anisotropy index A^U and the directional bulk modulus B_a, B_b and B_c of ZrB₄ under pressure

P	A1	A2	A3	A^U	Ba	Bb	Bc
0	1.194	1.173	1.001	0.090	742.4	786.1	655.8
10	1.215	1.249	1.035	0.098	853.7	853.5	775.58
20	1.234	1.280	1.038	0.099	958.3	956.7	885.98
30	1.199	1.260	1.017	0.091	1049.9	1059.1	1002.6
40	1.211	1.288	1.017	0.095	1144.3	1155.6	1112.1
50	1.224	1.319	1.017	0.101	1236.3	1249.5	1222.1
60	1.237	1.348	1.014	0.108	1325.7	1341.3	1322.2
70	1.243	1.369	1.012	0.112	1409.8	1427.7	1438.3
80	1.249	1.392	1.011	0.118	1492.0	1513.1	1553.2
90	1.261	1.425	1.015	0.131	1581.1	1600.7	1646.3
100	1.265	1.443	1.013	0.137	1660.3	1684.5	1755.1

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Although the factors calculated above have already conveyed that the elastic properties of ZrB₄ are anisotropic, it is still necessary to characterize the mechanical anisotropy in a more straightforward way. The shape of the 3D curved surface is sphere for isotropic materials (A^U = 0), but for anisotropic materials, the sphere will deform. The degree of deformation reflects the extent of anisotropy, and the variation of elastic modulus with direction can be demonstrated. Therefore the Young's modulus, Shear modulus, and Poisson's ratio along different directions in three-dimensional (3D) space as well as the projections in (−110) plane and (001) plane at pressures 0 GPa, 50 GPa and 100 GPa have been drawn to denote the elastic anisotropy of ZrB₄ on crystallographic directions, as is shown in Fig. 2 and 3. The direction dependent Young's modulus (E), Shear modulus (G) and Poisson's ratio (υ) for orthorhombic crystals^36,37 can be defined respectively as:

Fig. 2 Direction dependence of Young's modulus E (a), (d), (g), shear modulus G (b), (e), (h) and Poisson's ratio υ (c), (f), (i) under different pressures for ZrB₄, the units are in GPa for E and G.

Fig. 3 The projections of Young's modulus E (a), shear modulus G (b) and Poisson's ratio υ (c) in (−110) plane and (001) plane at pressures 0 GPa, 50 GPa and 100 GPa respectively, the units are in GPa for E and G.

Young's modulus:

E⁻¹ = s′₁₁ = s₁₁l₁⁴ + s₂₂l₂⁴ + s₃₃l₃⁴ + 2s₁₂l₁²l₂² + 2s₂₃l₂²l₃² + 2s₁₃l₁²l₃² + s₄₄l₂²l₃² + s₅₅l₁²l₃² + s₆₆l₁²l₂² (9)

Shear modulus:

G⁻¹ = 4s₁₁l₁²m₁² + 4s₂₂l₂²m₂² + 4s₃₃l₃²m₃² + 8s₁₂l₁m₁l₂m₂ + 8s₂₃l₂m₂l₃m₃ + 8s₁₃l₁m₁l₃m₃ + s₄₄(l₂m₃ + m₂l₃)² + s₅₅(l₁m₃ + m₁l₃)² + s₆₆(l₁m₂ + m₁l₂)² (10)

Poisson's ratio:

s′₁₂ = l₁²m₁²s₁₁ + (l₁²m₂² + l₂²m₁²)s₁₂ + (l₁²m₃² + l₃²m₁²)s₁₃ + l₂²m₂²s₂₂ + (l₂²m₃² + l₃²m₂²)s₂₃ + l₃²m₃²s₃₃ + l₂l₃m₂m₃s₄₄ + l₁l₃m₁m₃s₅₅ + l₁l₂m₁m₂s₆₆ (11)

where s_ij is the usual elastic compliance constants, l_i is the direction cosines in any arbitrary direction and m_i is the direction cosines in perpendicular direction. From Fig. 2, elastic anisotropy is clearly seen in ZrB₄, and the greater the pressure, the more obvious the anisotropy. In addition, the magnitude of Young's modulus in a specific direction can also be used to indicate the strength of chemical bonds in that direction. In Fig. 2(a), (d) and (g), the maximum of Young's modulus is observed in <111> direction, and the minimum occurs in <001> direction. Because a larger Young's modulus often stands for more covalent feature of a material,^38,39 we can substantiate that the covalent feature of the bonding in <111> direction is more dominant than other directions. The G is remarkably dependent on the stress direction (Fig. 2(b), (e) and (h)) with the highest (lowest) value in the [001] ([111]) direction, and the Poisson's ratio (Fig. 2(c), (f) and (i)) has similar characteristics. Fig. 3(a) and (b) show the orientation dependence of E and G changing from [001] to [110] direction in (−110) plane and from [100] to [010] direction in (001) plane under different pressures, and the shape of the projections in (001) plane at 0 GPa, 50 GPa, 100 GPa is almost round, which illustrates that ZrB₄ is nearly isotropic in (001) plane. This result is consistent with the shear anisotropy factor A₃. Poisson's ratio represents the negative ratio of transverse and longitudinal strains which plays a significant role in mechanical engineering design.⁴⁰ The values of υ in (−110) plane varies in a very large range as shown in Fig. 3(c), the features of υ under 100 GPa are 0.283 < υ < 0.381 in (−110) plane. It means that when the stress direction is perpendicular to the (−110) plane, the maximum strain is noted in the [001] direction and the minimum strain in the [111] direction.

3.3 Thermodynamic properties

The thermodynamic properties of ZrB₄ at various temperatures (0–1000 K) and pressures (0–50 GPa) are systematically calculated. In Fig. 4, we present the normalized volume–pressure and bulk modulus–pressure diagram of ZrB₄ at temperatures 0, 200, 400, 600, 800, and 1000 K, where V₀ is the zero-pressure equilibrium volume. It is easily seen from Fig. 4 that, as pressure increases, the relative volume V/V₀ decreases at a given temperature and the V/V₀ curve becomes steeper with temperature increasing, which implies that ZrB₄ is more easily compressed when temperature increases. Furthermore, it is found that the bulk modulus increases with pressure at a constant temperature and decreases with temperature at a given pressure.

Fig. 4 The calculated normalized volume V/V₀ and bulk modulus of ZrB₄ as a function of pressure at temperatures 0, 200, 400, 600, 800, and 1000 K.

The calculated heat capacity of ZrB₄ as a function of temperature (pressure) at given pressure (temperature) is demonstrated in Fig. 5. It is shown in Fig. 5(a) that the heat capacity C_V fits T³ term in their sufficiently low-temperature regions and approximates to absolute zero when the temperature vanishes at the given 0, 10, 20, 30, 40, 50 GPa. This is due to the harmonic approximations of the Debye model used here. At intermediate temperatures, the temperature dependence of C_V is dominated by the details of vibrations of atoms.⁴¹ At high temperatures, the calculated C_V is expected to get close to the Dulong–Petit limit, 3nN_Ak_B (n is the number of atoms in a molecule, N_A is the Avogadro constant and k_B is the Boltzmann constant), which is common to all solids at high temperatures. For ZrB₄, the Dulong–Petit limit is about 120 J mol⁻¹ K⁻¹. The pressure dependence of the heat capacity for ZrB₄ at 100, 200, 400, 600, 800 and 1000 K is presented in Fig. 5(b). It is noted that the calculated C_V decreases with pressure at a constant temperature and increases with temperature at a given pressure. From Fig. 5(a) and (b), we can get that the effect of temperature on C_V is greater than that of pressure.

Fig. 5 (a) Temperature dependence of heat capacity at different pressures and (b) pressure dependence of heat capacity at various temperatures.

The volume thermal expansion coefficient α as a function of temperature (pressure) at different pressures (temperatures) is shown in Fig. 6(a) and (b). Because of the weak dependence of the bulk modulus on temperature and that α is proportional to C_V (, γ is the Gruneisen parameter, and K is the bulk modulus), the trend of the volume thermal expansion coefficient is similar to the heat capacity. As shown in Fig. 6(a), at given pressures, α increases rapidly with temperature at sufficiently low temperatures (α(T) ∼ T³) and gradually turns to a slow increase at high temperatures (T > 400 K). Additionally, it is noted in Fig. 6(b) that α decreases with increasing pressure at a constant temperature, and the trend slows down at high pressures.

Fig. 6 (a) Temperature dependence of thermal expansion coefficient at different pressures. (b) Pressure dependence of the thermal expansion coefficient at various temperatures.

Fig. 7 shows the pressure dependence of the Debye temperature θ_D and Grüneisen parameter γ of ZrB₄ at different temperatures (0, 100, 200, 400, 600, 800, and 1000 K). It is easily seen in Fig. 7(a) that when temperature keeps constant, Debye temperature increases almost linearly with increasing pressure and compared with pressure, the variation of θ_D caused by temperature is very small. Therefore, we can draw a conclusion that the effect of the temperature on θ_D is not as significant as that of pressure. In quasi-harmonic Debye model, Grüneisen parameter γ describes the anharmonic effects of the crystal lattice thermal vibration. From Fig. 7(b), we can see that at fixed temperature, γ decreases sharply with pressure, and as temperature goes higher, γ decreases more rapidly with the increase of pressure.

Fig. 7 Debye temperature θ_D (a) and Grüneisen parameter γ (b) for ZrB₄ as a function of pressure at different temperatures.

IV. Conclusion

In conclusion, we have focused our attention on prediction and detailed analysis of elastic constants, anisotropic properties, and thermodynamic properties under high pressures of Amm2-ZrB₄ by first principles calculations in combination with the quasi-harmonic Debye model in this work. In the light of the Born stability criteria and the Pugh criterion, ZrB₄ (Amm2) is mechanically stable and exhibits brittle nature within the scope of the studied pressure (0–100 GPa). The Debye temperature of ZrB₄ was calculated by taking advantage of the relationship that Θ is proportional to the averaged sound velocity v_m, and it increases with pressure. Young's modulus, shear modulus and Poisson's ratio as a function of crystal orientation have been systematically investigated and analyzed. ZrB₄ exhibits pronounced elastic anisotropy and the extent increases with pressure. Furthermore, the pressure and temperature dependences of calculated normalized volume V/V₀, bulk modulus, volume thermal expansion coefficient, specific heat, Debye temperature, and Grüneisen parameter have also been evaluated in the ranges of 0–50 GPa and 0–1000 K through quasi-harmonic Debye model. The results point out that pressure and temperature have manifest effects on these thermodynamic properties. The present study provides detailed and systematic information for Amm2-ZrB₄, which is of fundamental importance for its industrial application.

Acknowledgements

This work was supported by the NBRPC (grant 2013CB733000), NSFC (grants 51171160/51434008/11447243). S. Z. would like to acknowledge support by the Natural Science Foundation of Hebei province, China (grant E2015203200).

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