Pressure-induced isostructural phase transition in CaB₄

Abstract

The structural and electrical properties of CaB₄ under high pressure have been studied by angle dispersive X-ray diffraction, in situ Hall effect measurement, and first-principles calculations. An abnormal change of the c/a ratio was observed around 12 GPa in the experiments and backed by theory calculation, indicating an isostructural phase transition. Unlike other materials, the analysis of the electronic and phonon band structure doesn't reveal an electronic topological transition or phonon softening supporting this phenomenon. While the subtle changes in the electron band dispersions at the Fermi surface shown by the measurements of the resistivity, Hall coefficient, carrier concentration, mobility, and temperature-dependent resistivity can explain the anomaly in the c/a ratio. The study of electrical transport properties provides strong support for the occurrence of the isostructural phase transition in CaB₄.

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Introduction

The metal borides, MB_n (M = rare-earth or alkaline elements, n = 2, 4, 6, and 12), have been studied intensively in the last few decades for their variety of physical and chemical properties, such as high hardness, high melting point, outstanding electrical conductivity, and superior chemical stability.^1–3 As a consequence, they can be applied as cutting tools, abrasives, high-temperature surface protection, super conductivity, and chemically-stable coatings.^3,4 Owing to that most of the metal borides can have simple crystal structures and present extended two-dimensional (2D) or three-dimensional (3D) boron networks, interesting structure-property relationships have been extensively investigated. The most typical example is the superconducting MgB₂, in which the boron atoms are arranged in graphite-like layers and the strong B–B bonds driven metallic by special crystal structure and chemistry.^5,6 Great interests in the divalent hexaborides have been generated recently by the discovery of ferromagnetism in SrB₆, EuB₆ and La-doped CaB₆, as well as La-doped BaB₆.^7–9

Metal borides MB₄ are compounds with potential application value, and they have been noticed mainly because of their mechanical, magnetic, electronic properties, and anisotropic structures. MB₄ (M = Ce, Th, U, Pa, Y and Ho) have been predicted theoretically to be hard materials (bulk modulus 180–240 GPa) with mechanical and thermodynamical stability, within which the B₆ octahedron structural units play an important role in the structure.^10,11 For the alkaline-earth elements below magnesium in group IIA of the periodic table, there has been no experimental report on other borides expecting for hexaborides, and only one vague indication given for the existence of CaB₄ in the ThB₄-type structure.¹² Therefore, the synthesis of pure CaB₄ remains an ongoing controversy and challenging work,^13,14 the investigation of the existence and properties of CaB₄ is an attractive study area. And the understanding of the transition origin under pressure in CaB₄ is greatly motivated in order to help design other superhard materials.

Pressure-induced isostructural phase transition is an unusual phase transition identified with the volume collapse or the anomaly in c/a ratio without any symmetrical change. It has been detected in many compounds and metals, and its origin have excited both theoretical and experimental investigations.^15–18 Recently, metals with hcp structure and unusually large c/a ratio, such as Zn and Cd, have been found having a discontinuous change in c/a ratio under high pressure and attracted quite a lot of interest.^19–25 In general, this pressure induced abnormal change in c/a ratio in Zn, Os, and Cd is usually considered to be originated by the electronic topological transitions (ETT),^19–24 which arises when distortion of the electronic band results in a topological singularity of the Fermi surface. However, it is worth awareness that such anomaly of c/a ratio was just observed in metals with hcp structure and lattice parameter c > a.

However, several problems need further investigation: (i) whether this type of isostructural phase transition can happen in materials with the lattice parameter c < a still remains unknown; (ii) whether the ETT is the origin of this isostructural phase transition still remains uncertain. (iii) Furthermore, the influence of isostructural phase transition on the electrical transport parameters is still untested.

The subtle nature of the transition, without any structural change but associated with c/a ratio anomaly, makes it difficult to detect the phenomenon. According to the theory that the electron transport behaviour is sensitive to the change of crystal structure, the transport parameters are expected to show anomalies as the c/a ratio does. Among them the resistivity, Hall coefficient, carrier concentration, and carrier mobility are the best parameters for identifying the anomaly in c/a ratio. For example, pressure induced resistance change of Zn shows a minimum at 10 GPa where the discontinuous change in c/a ratio occurs.²⁶ Recent evidence suggested that the anomaly of c/a ratio in Os around 25 GPa is not from ETT, and the phonon softening also can be ruled out. Calculations of the Hall coefficient reveal subtle changes in the electron band dispersions at the Fermi surface which could account for the observed anomaly in the c/a ratio.²⁷

Recently, a special liquid–solid reaction under high pressure and high temperature for pure CaB₄ synthesis was reported.²⁸ The CaB₄ crystals were found to have a tetragonal ThB₄-type structure (c < a) which contains B₆ octahedra and ethylene-like B₂ dumbbells. The B₆ octahedra are directly bonded to each other only in c direction and linked to each other through B₂ units in ab plane. All of these fragments are constructed into a 3D network (Fig. 1).²⁸ Here we report a unique isostructural phase transition of CaB₄ under pressure. An anomaly of c/a ratio without symmetrical change around 12 GPa was observed both theoretically and experimentally. It was notable that such isostructural phase transition has never been found in other MB₄ compounds with tetragonal structure. The electronic and phonon band structure, the Fermi surface, and the transport properties of CaB₄ under high pressure have been studied to reveal the nature of the anomaly in c/a ratio. Meanwhile, the experimental determination of the bulk modulus for CaB₄ was also performed and accords well with the calculation in the present work.

Fig. 1 Crystal structure of CaB₄. Different types of B atoms are marked by different colors.

Experimental and theoretical section

Our samples are synthesized by Liu et al.²⁸ Quantitative analysis of Ca and B contents were performed by means of the measurements of inductively coupled plasma-atomic emission spectrometry (ICP-AES). The ratio of Ca to B is corresponding to Ca_1−xB₄ with x determined to be less than 0.05. The impurities such as Fe, C, Ta, etc. are found to be below the detectable limits of ICP-AES.²⁸ Three in situ angle dispersive X-ray diffraction (ADXRD) experimental runs were carried out at the 4W2 beam line at BSRF (Beijing Synchrotron Radiation Facility) using 0.6199 Å X-ray beam and Mar345 detector (Run 1), at the beam line X17C of National Synchrotron Light Source in Brookhaven National Laboratory with monochromic 0.408511 Å and Mar CCD detector (Run 2), and at the beam line B2 in CHESS (Cornell High Energy Synchrotron Source) with monochromic 0.485946 Å and Mar345 imaging plate (Run 3). The methanol–ethanol (4 : 1) mixture was used as the pressure medium and the platinum powder was covered with sample as an inner pressure standard in Run 1; hydrostaticity was provided by argon and the pressure was determined by the ruby scale²⁹ in the other runs. Two-dimensional XRD images were analyzed using the FIT2D software, yielding one-dimensional intensity versus diffraction angle 2θ patterns. Rietveld analyses were performed using the software MATERIALS STUDIO REFLEX of Accelrys, Inc.

In electrical parameters measurement, a DAC made of nonmagnetic titanium alloy was utilized to generate high pressure; a nonmagnetic rhenium flake was used as the gasket which has no detectable effect on magnetic filed measurement.³⁰ CaB₄ was loaded into the DAC along with chips of ruby which monitor the sample pressure. No pressure medium was loaded in order to avoid the introduction of impurities in measurement of electrical parameters and ensure good electrical contact. Van der Pauw electrodes were integrated on one of the diamond anvils, the manufacture of which has been reported in our previous work.³¹ The magnetic flux density applied to the sample was 0.9 tesla.

The underlying ab initio structural relaxations and the electronic calculations were performed using density functional theory with the Perdew–Burke–Ernzerhof exchange-correlation as implemented in the Vienna ab initio Simulation Package (VASP) code³² and the generalized gradient approximation (GGA)³³ is implemented on a projector augmented wave (PAW) basis. The PAW pseudo-potential was adopted with (3p, 4s) and (2s, 2p) electrons as valence for Ca and B, respectively. Based on atomic orbital information, Ca 3d orbital is unoccupied when the pseudo-potential was generated. Our first-principle calculations have taken account of the contribution from 3d orbital of Ca atom, since the pseudo-potential already provides the information of 3d orbital. A cutoff energy of 1200 eV was adopted and the Monkhorst–Pack k-point was generated with 5 × 5 × 8 mesh parameter grid for this structure, which was shown to give excellent convergence of total energies, energy differences, and structural parameters. The phonon calculation was carried out by a supercell approach³⁴ as implemented in the PHONOPY code.³⁵

Results and discussion

In ambient condition, all diffraction peaks were resolved and indexed to the tetragonal structure, P4/mbm (no. 127) symmetry. Fig. 2 shows representative XRD patterns which are similar at each pressure point. No new features is observed in the diffraction patterns under pressure in all three runs, indicating that the crystal structure of the sample stays in the tetragonal form throughout the whole pressure range. The compression curve is reported in Fig. 3a. Since the pressure medium used in Run 1 (methanol–ethanol mixture) provides hydrostatic pressure below 10 GPa only, the lattice parameters and the volume value in Run 1 are higher than that of the two following runs in which liquid argon was used as the pressure medium. In all three runs, the lattice parameters and the volume monotonously decrease with increasing pressure and no obvious discontinuity was observed. By fitting the experimental data to a second-order Birch–Murnaghan equation of state (B^′₀ = 4.0), the isothermal bulk modulus (300 K) is determined to be B₀ = 206.4 ± 2.9 GPa for Run 1, which is much higher than the other experimental measurements of 145.3 ± 1.1 GPa (Run 2) and 143.2 ± 1.0 GPa (Run 3). The bulk moduli of Run 2 and Run 3 reveal a good agreement with the calculated bulk modulus B₀ = 140.4 GPa, which also reflects the pressure medium influence on experiment results. However, the bulk moduli of Run 2, Run 3 and theoretical calculation are rather small than the expectation modulus of the other MB₄ (M = Ce, Th, U, Pa, Y and Ho), which indicates that CaB₄ is not as hard as expected. The reason is that CaB₄ has a relatively longer B3–B3 bond (Fig. 1) which makes it much easier to be compressed under pressure compared to the other tetraborides.

Fig. 2 XRD patterns of CaB₄ (in Run 3) under high pressure. The red line represents the Rietveld fit for the lattice and observed data. The solid line at the bottom is the residual intensity and the vertical bars indicate the peak positions.

Fig. 3 (a) Compression curve of CaB₄ at room temperature. The solid line is a second-order Birch–Murnaghan EOS fits to the experimental data. Inset: pressure dependence of the lattice parameters. (b) Pressure dependence of the axial ratio c/a.

The optimized lattice parameters a and c under pressure and the comparison of c/a versus pressure among the theoretical result and the three sets of experimental data are presented in Fig. 3a and b, respectively. It can be observed in Fig. 3 that the experimental observation of pressure-induced anomaly in c/a is correctly reproduced by theoretical calculation. A detailed analysis of the c/a ratio as a function of pressure (Fig. 3b) shows that two distinct compression regimes can be identified below and above 12 GPa, respectively. Hence, it suggests that around 12 GPa CaB₄ undergoes an isostructural phase transition which is very similar to those observed in metals Zn, Cd, and Os.^19–25 Among all the MB₄ compounds with tetragonal structure, this type of phase transition is only observed in CaB₄.

Comparing with other tetraboride, CaB₄ is deficient by one valence electron per metal atom, which plays an important role in determining the physical properties of this system. The band structure analysis also shows that the energy bands are dominated by orbital contributions from B atoms. Orbital contributions of Ca atoms play only a minor role in the region around the Fermi level because their empty 4s orbitals have higher energies. As a divalent cation, Ca hardly takes part in covalent interactions with the boron network which is in clear contrast to the electronic structure of GdB₄, where the d orbitals of Gd interact with orbitals of the boron network.¹³ As a consequence, isostructural phase transition can be observed only in CaB₄.

Discontinuities of c/a during the pressure evolution have been experimentally observed and theoretically predicted in many hcp metals (e.g., Zn and Cd) and are often considered to be motivated by ETT^19–24 which affects the screening of the ionic cores and modifying the lattice dynamics. To explore the origin of the c/a anomaly in CaB₄, the variation of the energy band structure and the Fermi surface under compression was examined. Fig. 4 shows the band structure and 3D Fermi surface of CaB₄ along several high symmetry directions in the Brillouin zone at 10 GPa and 14 GPa, respectively. It is clear that no noticeable change in the electron topology of the Fermi surface is found as pressure increases to 14 GPa. Therefore, the possibility of the existence of an ETT in CaB₄ has been excluded, unlike Zn and Cd.^20–24

Fig. 4 Calculated band structures of CaB₄ and three-dimensional Fermi surface at (a) 10 GPa and (b) 14 GPa.

Another common explanation to this c/a anomaly is phonon softening. In order to testify the validity of phonon softening in this phase transition, the pressure dependent lattice dynamics was examined. The phonon dispersion curves of CaB₄ at 10 GPa and 14 GPa are shown in Fig. 5. According to Fig. 5, no obvious change of phonon dispersion is observed. For example, the top value of Γ point is 38.1 THz at 10 GPa, and increases to 39.4 THz at 14 GPa. No unusual softening in the phonon dispersions that is associated with the occurrence of structural instability under pressure up to 14 GPa is observed.

Fig. 5 Calculated phonon dispersion curves of CaB₄ at (a) 10 GPa and (b) 14 GPa.

It has been demonstrated in many cases that the subtle change of structure consequentially will influence the transport parameters. For example, the anomaly in c/a for Os accompanies a small variation in the Hall coefficient.²⁷ And a very small change in the Hall coefficient of Nb reflects the a significant change in the superconductivity temperature under ∼5 GPa.³⁶ The Hall coefficient, related to the inverse of the effective mass, is the second derivative of the band dispersion of the electronic bands near the Fermi surface, and a very sensitive symbol for the subtle change in electronic band structure. Hence, the observed c/a anomaly in CaB₄ is likely to cause subtle changes in Hall coefficient and electrical resistivity. Based on this judgment, we measured the Hall coefficient and electrical resistivity of CaB₄ under pressure in order to extract more information of pressure effect on the sample.

The pressure dependent resistivity (ρ) and Hall coefficient (R_H) measurements of CaB₄ up to 25 GPa at room temperature are mapped in Fig. 6. According to Fig. 6, with an initial value of 1.3 × 10⁻⁴ Ω cm the ρ shows a positive correlation with pressure within 25 GPa while reveals an inflexion at 11.7 GPa. The R_H, the initial value of which is −4.8 × 10⁻³ cm³ C⁻¹, shows a negative correlation with pressure and shares a similar inflexion position with resistivity at 11.7 GPa. As mentioned above, Hall coefficient is the second derivative of the band dispersion of the electronic bands near the Fermi surface. So the discontinuity occurred at 11.7 GPa, indicating a change in the electronic band dispersion near the Fermi surface.

Fig. 6 Pressure dependence of resistivity (ρ) and Hall coefficient (R_H) of CaB₄ at room temperature.

In order to further explore the abnormal change of the electrical transport parameters, we performed the carrier concentration (n) and mobility (μ) measurements. As shown in Fig. 7, under ambient pressure, the initial values of n and μ are 1.3 × 10²¹ cm⁻³ and 35 cm² V^-1 s⁻¹, respectively. In the whole pressure range, n increases with pressure, indicating that the excited carriers don't saturate. From ambient pressure to ∼11 GPa, μ increases with pressure. Above 11 GPa, μ remains nearly constant up to 25 GPa. Meanwhile, ρ and R_H arise some subtle changes, the trend has become gentler than before, indicating that the variation of ρ and R_H is mainly caused by μ. And the transition point accords well with the observed anomaly in the c/a ratio. This result confirms that the anomaly of c/a ratio in CaB₄ under compression is caused by the subtle change of the electronic structure within the Brillouin zone.

Fig. 7 Pressure dependence of (a) carrier concentration (n) and (b) mobility (μ) of CaB₄ at room temperature.

To further account for the observed anomaly of c/a ratio, we carried out the temperature dependent electrical resistivity (ρ) measurements under different pressures³⁷ (Fig. 8). The electrical resistivity shows a positive relationship with the temperature, indicating that CaB₄ presents metallic conduction in the whole pressure range. And the energy band structure (Fig. 4) and phonon dispersion curves (Fig. 5) also confirm this conclusion, which is consistent with the results of Yahia et al.¹⁴ For metallic conductive material, the variation of resistivity with temperature can be written as:

ρ = ρ₀(1 + αT)

where ρ is the resistivity (Ω cm), T is the temperature (K), ρ₀ is the resistivity at 100 K, and α is the temperature coefficient (K⁻¹). According to the linear fitting of above-mentioned relation in Fig. 8, we can obtain the variation of temperature coefficient versus pressure (Fig. 9). According to Fig. 9, the temperature coefficient shows a negative correlation with pressure, indicating that the lattice scattering effect on carrier decreases with increasing pressure. Above 9.5 GPa, the temperature coefficient shows a positive correlation with pressure, indicating that the lattice scattering effect on carrier becomes stronger with increasing pressure, and then prevents further increase of carrier mobility. It should be note that the anomaly of temperature coefficient occurs at about 10 GPa earlier than that of c/a ratio (11.7 GPa). We attribute this pressure difference partly to the non-hydrostatic effect.^38,39 Otherwise, during temperature dependent resistivity measurement the temperature effect should be take account in. It is reasonable that the pressure difference mentioned above is partly from the temperature effect.

Fig. 8 Normalized electrical resistivity of CaB₄ versus temperature under different pressures.

Fig. 9 Variation of temperature coefficient versus pressure.

According to above analysis, we believe, all changes in physical parameters within 9.5–11.7 GPa, can be attribute to the pressure induced isostructural phase transition in CaB₄. The electrical transport measurement can provide another routine to probe isostructural phase transition.

Conclusions

In summary, a pressure-induced isostructural phase transition in CaB₄ has been found by the combination of synchrotron X-ray diffraction and electrical transport measurements. An abnormal change of c/a ratio and some unusual changes in electrical parameters can be attribute to this transition. The theoretical analysis has also found the anomaly of c/a ratio. The first principle calculations indicate that no significant changes in electron band structure and phonon mode softening happen during this isostructural transition. An ETT driven mechanism for the anomaly of c/a ratio can be ruled out. The electronic origin of the c/a anomaly is due to the small changes in the curvature of the electronic band dispersion near the Fermi surface.

Acknowledgements

This work was supported by the National Basic Research Program of China (Grant no. 2011CB808204), the National Natural Science Foundation of China (Grant no. 91014004, 11074094, and 11374121), China Postdoctoral Science Foundation (Grant no. 2013M540243), and the Fundamental Research Funds for Jilin University, China (Grant no. 450060491500).

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c4ra04102j

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