Structural stability and compressive behavior of ZrH₂ under hydrostatic pressure and nonhydrostatic pressure

Abstract

The important transition metal dihydride ZrH₂ has been characterized using in situ synchrotron X-ray diffraction combined with diamond anvil cell techniques and ab initio calculations. The effect of a pressure-transmitting medium on the structural stability and compressive behavior was investigated under both hydrostatic pressure and nonhydrostatic pressure conditions. The ambient I4/mmm structure is stable under both nonhydrostatic and hydrostatic compressions. The supplementary theoretical calculations have proposed that the I4/mmm structure transformed into the P4/nmm structure above 100 GPa confirming the stability of the I4/mmm structure during the experimental runs. The difference in the volume reduction between the two compressions becomes larger with increasing pressure. Up to about 50 GPa, the volume collapse of nonhydrostatic compression is 6% relative to the hydrostatic compression. The present study offers a new approach with nonhydrostatic compression or shear stress for finding higher volumetric density hydrogen structures in metal hydrides.

---

Introduction

Nowadays, new energy development in today's world is imminent since the sources of energy such as oil, natural gas, and coal are non-renewable resources. Hydrogen is widely regarded as a promising renewable and clean energy, but a major challenge in a future “hydrogen economy” is to develop a safe, compact and efficient means of hydrogen storage for mobile applications.^1–3 Much effort has been made in the past few years in the discovery of new materials, which will store hydrogen at a high gravimetric and volumetric density.^4–6 The new hydrogen storage materials are usually discovered by experimental synthesis approach, but the synthesis approach have some disadvantages such as time-consuming and inefficient due to the numerous possible reaction pathways and slow reaction kinetics.^7,8 It is well known that high pressure can substantially change the microstructures of condensed materials. The application of high pressure is an effective way to synthesize hydrogen storage materials with high volumetric hydrogen densities, because compressions on materials are usually accompanied by volume collapse.^9,10 A diamond anvil cell (DAC) can generate the highest static pressures. When no pressure transmitting medium is used or the sample bridges with the diamond anvils, the shear stress along the axial direction of the sample is much larger than the radial direction. Therefore, many novel phenomena arising from the differences between hydrostatic and nonhydrostatic compression have been extensively investigated in both theory and experiment. K. E. Lipinska-Kalita et al. have found that the transition pressure points were affected by the pressure-transmitting medium.¹¹ Recently, the dependence of superconducting transition temperature T_c on hydrostatic and nonhydrostatic pressures in iodine have been studied by Duan et al. and the changing symmetry of phase IV under anisotropic stresses was firstly reported.¹²

Binary transition metal hydrides have been considered as ideal candidates of storing hydrogen such as RhH₂, which shows a very high volumetric hydrogen density of 163.7 g L⁻¹ and is one of the highest among all known hydrides.¹³ Besides, binary transition metal hydrides can act as active species to catalyze the reversible hydrogenation and dehydrogenation of such materials like carbon nanotubes and complex hydrides.^14–16 As one of important binary transition metal hydrides, zirconium(II) hydride (ZrH₂) has drawn great attentions due to its widely applications in many fields such as hydrogen storage and fusion technology applications.^17–20 In order to improve the performance of hydrogen storage, a full understanding of the behaviors for binary transition metal hydrides especially for ZrH₂ under high pressure is considered as essential. Zirconium hydride is known to undergo a phase transition as a function of hydrogen concentration.²¹ Recently, Zhu et al. presents a detailed study on the electronic structure, mechanical properties, phase stability, and thermodynamic properties of four polymorphs of crystalline zirconium hydride by density functional theory.²² Up to now, few experimental data have explored the structural stability of ZrH₂ under high pressure, which is limited to lower pressure or quasi-hydrostatic pressure conditions.^23,24

In this paper, we have explored the structural stability and compressive behavior of ZrH₂ under both hydrostatic pressure and nonhydrostatic pressure based on synchrotron X-ray diffraction (XRD) measurements. The difference of the volume reduction between the nonhydrostatic and hydrostatic compression becomes larger with increasing pressure. It is proposed that nonhydrostatic compression or shear stress is more favourable for obtaining higher volumetric hydrogen density hydrogen storage materials.

Experimental and theoretical methods

Commercially available ZrH₂ powder with purity >99% was obtained from Sigma Aldrich Products (with size −325 mesh). Before loading the sample into the diamond anvil cell, the sample was reground finely. Pressure inside the DAC was determined by the standard ruby fluorescence method.²⁵ The first compression run was carried out on neat sample up to 43.8 GPa without any pressure medium to create a nonhydrostatic pressure atmosphere. The angle-dispersive synchrotron XRD experiments for the first nonhydrostatic compression run were performed at the wiggler beamline X17C of the National Synchrotron Light Source (NSLS) in Brookhaven National Laboratory (BNL). The wavelength was 0.4062 Å. The X-ray beam size is about 15 (H) × 20 (V) μm² Two-dimensional patterns were radially integrated using the software FIT2D.²⁶

The second hydrostatic compression was carried out up to 34.9 GPa with neon as pressure transmitting medium. Before loading the powder sample into the hole, we firstly compressed the powder into thin flakes by using the DAC. One flake of the sample was loaded into the medium of the hole and then charged with neon fluid by gas-loading equipment. The gas-loading process was finished in GSECARS of the High Pressure Collaborative Access Team (HPCAT) at Advanced Photon Source (APS) and the encapsulated pressure is about 1.0 GPa. Under this condition, the flake of the sample was suspended in the neon surroundings. Fig. 1 shows the sample chamber at about 1.0 GPa and the highest pressure of 34.9 GPa, respectively. Hence, the pressures are hydrostatic up to the highest pressure in the present experiment.²⁷ In the second compression, high pressure angle-dispersive XRD experiments with a focused 6 (H) × 13 (V) μm² X-ray beam (λ = 0.4133 Å) were performed at beamline 16-BM-D of HPCAT at APS. For the two compression runs, the time interval between a pressure increase and the subsequent X-ray measurement was kept at about 10 min in order to allow for the pressure inside the sample chamber to equilibrate. The XRD patterns were fitted by Rietveld profile matching using the GSAS+EXPGUI programs.²⁸ During each refinement cycle, the fractional coordinates, scale factor, background parameters, isotropic thermal parameters, profile function, and lattice constants were optimized.

Fig. 1 Micro-photographs of the sample chamber with neon as pressure transmitting medium (a) at about 1.0 GPa just after loading the neon liquid and (b) at the highest pressure of 34.9 GPa.

The first-principles calculations described here were performed with the VASP code,²⁹ based on density functional theory (DFT). The optimizations under high pressure were performed within the Perdew–Burke–Ernzerh (PBE) of the generalized gradient approximation (GGA).³⁰ The all-electron projector-augmented wave (PAW) method³¹ was adopted with 4d²5s² and 1s¹ as valence electrons for Zr and H atoms, respectively. Convergence tests give a kinetic energy cutoff E_cutoff as 600 eV and Brillouin zone sampling with the Monkhorst–Pack grid³² of spacing 2π × 0.03 Å⁻¹ for the crystal structures.

Results and discussion

At ambient conditions, ZrH₂ has a tetragonal (bct) crystal lattice with space group I4/mmm. In this structure, Zr atoms form a body-centered tetragonal sublattice and H atoms are located on the planes, forming a one dimensional chain. Before loading the sample into the DAC, we have confirmed the purity of the sample by using conventional XRD measurements. The obtained unit cell parameters of the ambient tetragonal phase are a = b = 3.5215 Å, c = 4.4492 Å with unit cell volume V = 55.17 ų, which are in good agreement with the previously reported experimental results.³³ In the first series of synchrotron XRD experiments, ZrH₂ was studied on compression from 0.1 GPa to 43.8 GPa without pressure transmitting medium. Selected diffraction patterns from this experimental run are shown in Fig. 2a. It is shown that the sample was compressed smoothly up to the highest pressure without any phase transitions. In the second compression run, the XRD patterns of ZrH₂ were collected upon compression up to 34.9 GPa using neon as the hydrostatic pressure transmitting medium. The selected XRD patterns of ZrH₂ are illustrated in Fig. 2b. Upon compression, all diffraction peaks shift toward higher 2θ angles indicating the decrease of interplanar distance of crystal planes. The original XRD peaks exist up to the highest pressure 34.9 GPa without appearances of new peaks, which indicates no occurrence of the phase transitions. Moreover, in Fig. 3a and b, the Rietveld refinement of the XRD patterns for the two compressions shows good agreement with the ambient phase (I4/mmm), confirming the stability of the ambient structure up to the highest pressure of 43.8 GPa and 34.9 GPa, respectively. As a consequence of the preferred crystallite orientation, the intensities of diffraction lines can sometimes be altered. In addition, in the present experiments the crystallite size was relatively big as compared to the beam size, which further reduced the statistics of the measurement as well as contributed to altering intensities in the diffraction peaks due to the mentioned preferred orientation effects. The same situation is also reported in other high pressure synchrotron XRD experiments. For example, in a recent synchrotron XRD experiment of TiH₂ under high pressure by Kalita et al.,³⁴ the mismatch in intensities between the observed and the calculated X-ray patterns is also observed. So the mismatch in intensities is attributed to the large grain size of the sample and the very small X-ray beam size, both of which contribute to reduce the statistics.

Fig. 2 Selected angle-dispersive synchrotron XRD patterns collected in the DAC (a) without using pressure medium in the first experimental run up to 43.8 GPa measured with the X-ray wavelength of λ = 0.4062 Å and (b) with neon as pressure transmitting medium upon second compression up to 34.9 GPa. The wavelength of the incident X-ray is λ = 0.4133 Å. The peaks with marked symbols represent the bragg peaks from solid neon.

Fig. 3 The Rietveld full-profile refinements of the diffraction patterns collected with (a) nonhydrostatic compression at about 43.8 GPa, and (b) hydrostatic compression at about 34.9 GPa. The mismatch in intensities is attributed to the large grain size of the sample and the very small X-ray beam size, both of which contribute to reduce the statistics.

The present X-ray results have confirmed the stability of the ambient structure up to the highest pressure for the two compressions. However, for one of the IVB hydrides, the recent theoretical and experimental results have reported the phase transitions of TiH₂ under high pressure. The synchrotron XRD experiments of TiH₂ at room temperature and high pressure by Kalita et al. have revealed a phase transition from fcc to I4/mmm at 0.6 GPa, which was concluded to remain stable up to 90 GPa.³⁴ Gao et al. have found a further phase transition from I4/mmm to P4/nmm at 63 GPa by ab initio total-energy calculations.³⁵ In case of ZrH₂, the ambient structure is with space group I4/mmm. By means of replacing Ti atom with Zr atom, we have obtained the crystal structure P4/nmm for ZrH₂, as shown in Fig. 4. The P4/nmm structure contains 2 formula units for unit cell and the atomic positions of Zr at 2c (0.5, 0, 0.26608), and H at 2a (0, 0, 0) and 2c (0, 0.5, 0.33139) sites. The enthalpy value was obtained from the optimization results for the two structures I4/mmm and P4/nmm under high pressure. The enthalpy of I4/mmm structure is taken as reference energy, the relative enthalpy per formula unit of P4/nmm structure as a function of pressure for ZrH₂ was shown in Fig. 4. In Fig. 4, it is seen that the enthalpy of P4/nmm structure is lower than I4/mmm structure above 100 GPa, and the actual pressure point is about 103 GPa according to the curve. So it is concluded that the I4/mmm structure transformed into P4/nmm at about 103 GPa by theoretical calculations, which is far beyond the current experimental pressure range.

Fig. 4 The relative enthalpy per formula unit of P4/nmm structure as a function of pressure for ZrH₂. The enthalpy of I4/mmm structure is taken as reference energy. The inset figures represent the two crystal structures.

Rietveld full profile structural refinements were performed on all XRD patterns collected for these two compressions in order to follow the evolution of the lattice constants and unit cell volume under high pressure. For comparison, the calculated lattice constants of the two compressions were presented in Fig. 5. The lattice constants as a function of pressure are linearly fitted without jumps, which further prove the stability of the ambient tetragonal phase under both nonhydrostatic and hydrostatic compression. Table 1 shows the fitted linear compressibility of the lattice constants for the hydrostatic and nonhydrostatic compressions. It is clearly seen from Table 1 that the lattice constants of the nonhydrostatic compression have a larger linear compressibility, indicating the sample is easier to be compressed under nonhydrostatic compression. For the two compressions, the larger difference of the linear compressibility for lattice constants a and c reflects the larger anisotropic during the compression runs. To compare the volume reduction differences between the two compressions, we present the volume reduction as a function of the pressure under both hydrostatic and nonhydrostatic pressure (Fig. 6). The experimental P − V/V₀ data were fitted for the two compressions by third-order Birch–Murnaghan (BM) equation of state (EOS)³⁶

where V₀ is the volume per formula unit at ambient pressure, V is the volume per formula unit at pressure P given in GPa, B₀ is the isothermal bulk modulus, and B^′₀ is the first pressure derivative of the bulk modulus. Under nonhydrostatic compression, the fitted result yields an isothermal bulk modulus B₀ = 134 ± 4 GPa with its pressure derivative B^′₀ = 2.5 ± 0.2. Under hydrostatic compression, the fitted result yields an isothermal bulk modulus B₀ = 119 ± 6 GPa with its pressure derivative B^′₀ = 7.7 ± 0.9. Remarkably, the volume reduction under hydrostatic pressure shows interesting and different features from that of nonhydrostatic compression.

Fig. 5 The lattice constants as a function of pressure under both hydrostatic and nonhydrostatic compressions. Solid black symbols represent the lattice constants obtained from the nonhydrostatic compression, and solid red ones are from the hydrostatic compression.

Table 1 The fitted linear compressibility of the lattice constants for the hydrostatic and nonhydrostatic compressions

	Hydrostatic compression	Nonhydrostatic compression
Linear compressibility ka (GPa^−1)	0.00264	0.00375
Linear compressibility kc (GPa^−1)	0.0126	0.01459

---

Fig. 6 (a) Volume reduction under hydrostatic and nonhydrostatic compressions. Solid black square: nonhydrostatic compression; and solid sphere: hydrostatic compression, solid line: fitted results of the nonhydrostatic and hydrostatic compressions. (b) The difference of volume reduction at pressures of hydrostatic and nonhydrostatic compressions. The dotted lines are merely guides to the eye.

Having carefully examined the experimental data under hydrostatic compression shown in Fig. 6a, we found that the volume change undergoes two stages. First, the volume reduction is almost same with that of under nonhydrostatic compression below 13.3 GPa. Above 13.3 GPa, the volume reduction under hydrostatic pressure is less than that of nonhydrostatic pressure. Using the fitted equation of state of the nonhydrostatic and hydrostatic experimental data, we calculate the difference in volume reduction under the same pressure of the hydrostatic and nonhydrostatic compressions. Fig. 6b shows the pressure dependence of ΔV/V₀ = (V_h − V_n)/V₀, where V_h, V_n, and V₀ are the volumes under hydrostatic, nonhydrostatic, and atmospheric pressures, respectively. It is found that the volume reduction under hydrostatic compression is smaller than the one of nonhydrostatic compression with increasing pressure. At about 35 GPa, the difference in volume reduction increases to 4%. Up to 50 GPa, the difference of volume reduction is up to 6%. Based on the fitted results, the estimated error of the volume collapse is about 1% at about 50 GPa. In contrast to the large volume collapse during the pressure-induced phase transitions, the present nonhydrostatic compression seems to generate the same effect with the large volume collapse comparing with the hydrostatic compression. So it is proposed that the nonhydrostatic compression offers a new route to higher volumetric hydrogen density structure.

The most striking result of the present work is the different compressive behavior of ZrH₂ depending on the hydrostatic conditions. Generally, there are two possible reasons for the abnormal compressive behavior: one is isostructural electronic phase transitions, the other one is the shear stress due to the nonhydrostatic pressure. Firstly, isostructural phase transitions with abnormal lattice constants or large volume collapses are reported to related with a change of electronic state, as in the cases of Ce and SmS.^37,38 The present linearly fitted lattice constants in Fig. 5 provide a statistically valid description of the data, which is different from the reported isostructural transitions. So it is proposed that the abnormal behavior in this work do not belong to the reported isostructural electronic transitions. Secondly, a quantitative analysis of volume differences between the two compressions is presented from the point view of energy. When applying pressure to the sample, Helmholtz free energy F increases due to the work done by compression. Hence, the difference in free energy increase between the hydrostatic and nonhydrostatic compressions will reveal certain information at microscopic level.³⁹ In an isothermal compression process, the change of the Helmholtz free energy F equals to the work done by external pressure, i. e., F − F₀ = PΔV, where F₀ is the Helmholtz free energy at ambient pressure, P is the pressure in the system, ΔV = V₀ − V is the volume reduction, and V and V₀ are the volumes at pressure P and ambient pressure. Therefore, the difference of free energy increase of the sample between the nonhydrostatic and hydrostatic compressions is ΔF = F_n − F_h = P(ΔV_n − ΔV_h), where F_n and F_h are the free energies at the nonhydrostatic and hydrostatic pressures P, and ΔV_n, ΔV_h are the corresponding volume reductions under nonhydrostatic and hydrostatic pressure P. We utilize the data presented in Fig. 6a to calculate ΔF as a function of pressure, which is illustrated in Fig. 7, and turned out to be expressed as a polynomial curve:

ΔF = 0.424 − 0.1P + 0.006P²

Fig. 7 The difference in Helmholtz free energy on compressing ZrH₂ under the same pressure with the two compressions (hydrostatic and nonhydrostatic conditions). The error bar indicates the uncertainties. The solid red line represents the fitted polynomial curve.

It is clearly seen from Fig. 7 that the difference of free energy between the two compressions increases with increasing pressure. Under the nonhydrostatic compression, shear stress becomes much stronger with increasing pressure contributing to the increasing energy. So the larger differences of free energy between the two compressions induce the different compressive behaviors.

More importantly, it is likely that hydrogen can be stored more efficiently in the tetragonal ZrH₂ under nonhydrostatic pressure owing to its larger volume shrinkage. Previous studies have found that some hydrogen storage materials were observed volume collapse during the phase transition under high pressure. For example, the ambient structure of LiBH₄ with Pnma symmetry transforms into a tetragonal phase with Ama2 symmetry at 1.2 GPa, which shows a remarkable volume collapse by 6.6%.⁹ Vajeeston et al. has predicted that there is a huge volume collapse during the phase transition by theoretical calculations, so the β-LiAlH₄ phase was considered as a potential hydrogen storage material.¹⁰ Our recent published work have found that pressure-induced phase transition in LiNH₂ is with much larger volume collapse.⁴⁰ The present study offers another method for finding the structure with higher volumetric hydrogen density. The volume collapse between the nonhydrostatic and hydrostatic compression become larger with increasing pressure. Up to about 50 GPa, the volume collapse is about 6%. Similarly, we suggest that the nonhydrostatic compression or shear stress is more favorable for obtaining improved hydrogen-storage materials.

Conclusion

In summary, the structural stability and compressive behavior of ZrH₂ have been studied by in situ synchrotron XRD under nonhydrostatic and hydrostatic compressions, respectively. We have found that the tetragonal structure of ZrH₂ is stable under both nonhydrostatic and hydrostatic compressions. The supplementary theoretical calculations have proposed that the I4/mmm structure transformed into the P4/nmm structure at about 103 GPa confirming the stability of the I4/mmm structure below 100 GPa. The difference of the volume reduction between the nonhydrostatic and hydrostatic compression becomes larger with increasing pressure. Up to about 50 GPa, the volume collapse of nonhydrostatic compression is up to 6% relative to the hydrostatic compression. The remarkable phenomenon is attributed to the shear stress generating the differences of the free energy in ZrH₂. These findings are important guidance and interesting phenomena for the future study of hydrogen storage hydrides, providing a new route to generating hydrides with higher volumetric hydrogen density.

Acknowledgements

The authors are grateful to Sergey N. Tkachev for his technical support at APS and thank Zhiqiang Chen for his help at X17C, NSLS. HPCAT is supported by CIW, CDAC, UNLV and LLNL through funding from DOE-NNSA, DOE-BES and NSF. APS is supported by DOE-BES, under Contract no. DE-AC02-06CH11357. This work was also supported by the National Basic Research Program of China (no. 2011CB808200), Program for Changjiang Scholars and Innovative Research Team in University (no. IRT1132), National Natural Science Foundation of China (no. 51032001, 11074090, 10979001, 51025206, 11274137, 11004074, 11204100), and National Found for Fostering Talents of basic Science (no. J1103202).

Notes and references

1. L. Schlapbach and A. Züttel, Nature, 2001, 414, 353–358.
2. M. Felderhoff, C. Weidenthaler, R. von Helmolt and U. Eberle, Phys. Chem. Chem. Phys., 2007, 9, 2643–2653.
3. W. Grochala and P. P. Edwards, Chem. Rev., 2004, 104, 1283–1316.
4. A. M. Seayad and D. M. Antonelli, Adv. Mater., 2004, 16, 765–777.
5. C. Liu, Y. Y. Fan, M. Liu, H. T. Cong, H. M. Cheng and M. S. Dresselhaus, Science, 1999, 286, 1127–1129.
6. C. Wolverton, D. J. Siegel, A. R. Akbarzadeh and V. Ozolinš, J. Phys.: Condens. Matter, 2008, 20, 064228.
7. J. Yang, A. Sudik, C. Wolvertonb and D. J. Siegelw, Chem. Soc. Rev., 2010, 39, 656–675.
8. K. Chłopek, C. Frommen, A. Léon, O. Zabara and M. Fichtner, J. Mater. Chem., 2007, 17, 3496–3503.
9. Y. Filinchuk, D. Chernyshov, A. Nevidomskyy and V. Dmitriev, Angew. Chem., Int. Ed., 2008, 47, 529–532.
10. P. Vajeeston, P. Ravindran, R. Vidya, H. Fjellvåg and A. Kjekshus, Phys. Rev. B: Condens. Matter Mater. Phys., 2003, 68, 212101.
11. T. Kenichi, Phys. Rev. B: Condens. Matter Mater. Phys., 2004, 70, 012101.
12. D. Duan, X. Jin, Y. Ma, T. Cui, B. Liu and G. Zou, Phys. Rev. B: Condens. Matter Mater. Phys., 2009, 79, 064518.
13. B. Li, Y. Ding, D. Y. Kim, R. Ahuja, G. Zou and H. K. Mao, Proc. Natl. Acad. Sci. U. S. A., 2011, 105, 18618.
14. J. Íñiguez, T. Yildirim, T. J. Udovic, M. Sulic and C. M. Jensen, Phys. Rev. B: Condens. Matter Mater. Phys., 2004, 70, 060101(R).
15. P. Wang, X. D. Kang and H. M. Cheng, J. Phys. Chem. B, 2005, 109, 20131–20136.
16. T. Yildirim and S. Ciraci, Phys. Rev. Lett., 2005, 94, 175501.
17. P. W. Bickel and T. G. Berlincourt, Phys. Rev. B: Solid State, 1970, 2, 4807.
18. L. Holliger, A. Legris and R. Besson, Phys. Rev. B: Condens. Matter Mater. Phys., 2009, 80, 094111.
19. J. Blomqvist, J. Olofsson, A.-M. Alvarez, and C. Bjerkén, Structure and Thermodynamical Properties of Zirconium Hydrides from First Principle, in 15th International Conference on Environmental Degradation of Materials in Nuclear Power Systems-Water Reactors, John Wiley & Sons, Inc., 2012, pp. 671–679.
20. P. Zhang, B.-T. Wang, C.-H. He and P. Zhang, Comput. Mater. Sci., 2011, 50, 3297–3302.
21. E. Zuzek, J. P. Abriata, A. San-Martin and F. D. Manchester, Bull. Alloy Phase Diagrams, 1990, 11, 385–395.
22. W. Zhu, R. Wang, G. Shu, P. Wu and H. Xiao, J. Phys. Chem. C, 2010, 114, 22361–22368.
23. Z. Bao, Z. Zhang and X. Wu, Chin. Sci. Bull., 1995, 40, 287–290.
24. Y.-H. Kim, J. Mineral. Soc. Korea, 1996, 9, 35–42.
25. H. K. Mao, J. Xu and P. M. Bell, J. Geophys. Res., 1986, 91, 4673–4676.
26. A. P. Hammersley, S. O. Svensson, M. Hanfland, A. N. Fitch and D. Hausermann, High Pressure Res., 1996, 14, 235–248.
27. A. Jayaraman, Rev. Mod. Phys., 1983, 55, 65–108.
28. B. H. Toby, J. Appl. Crystallogr., 2001, 34, 210–213.
29. G. Kresse and J. Furthm€uller, Phys. Rev. B: Condens. Matter Mater. Phys., 1996, 54, 11169.
30. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 1996, 77, 3865.
31. G. Kresse and D. Joubert, Phys. Rev. B: Condens. Matter Mater. Phys., 1999, 59, 1758.
32. H. J. Monkhorst and J. D. Pack, Phys. Rev. B: Solid State, 1976, 13, 5188.
33. R. E. Rundle, C. G. Shull and E. O. Wollan, Acta Crystallogr., 1952, 5, 22–26.
34. P. E. Kalita, S. V. Sinogeikin, K. Lipinska-Kalita, T. Hartmann, X. Z. Ke, C. F. Chen and A. Cornelius, J. Appl. Phys., 2010, 108, 043511.
35. G. Gao, A. Bergara, G. Liu and Y. Ma, J. Appl. Phys., 2013, 113, 103512.
36. F. Birch, J. Appl. Phys., 1938, 9, 279–288.
37. H. T. Hall, L. Merrill and J. D. Barnett, Science, 1964, 146, 1297–1299.
38. M. B. Maple and D. Wohlleben, Phys. Rev. Lett., 1971, 27, 511.
39. Y. Ma, J. Liu, C. Gao, W. N. Mei, A. D. White and J. Rasty, Appl. Phys. Lett., 2006, 88, 191903.
40. X. Huang, D. Li, F. Li, X. Jin, S. Jiang, W. Li, X. Yang, Q. Zhou, B. Zou, Q. Cui, B. Liu and T. Cui, J. Phys. Chem. C, 2012, 116, 9744–9749.

---