Calcium with the β-tin structure at high pressure and low temperature

Abstract

Using synchrotron high-pressure X-ray diffraction at cryogenic temperatures, we have established the phase diagram for calcium up to 110 GPa and 5–300 K. We discovered the long-sought for theoretically predicted β-tin structured calcium with I4₁/amd symmetry at 35 GPa in a s mall low-temperature range below 10 K, thus resolving the enigma of absence of this lowest enthalpy phase. The stability and relations among various distorted simple-cubic phases in the Ca-III region have also been examined and clarified over a wide range of high pressures and low temperatures.

At high pressures, calcium (Ca) not only displays very rich structural changes (1–5) but also holds the record for the highest superconducting critical temperature (T_c = 26 K) among elemental materials (5–7). Ca is currently one of the most studied elements under high pressure. A series of phase transitions have been observed in previous experiments. Under ambient conditions, Ca crystallizes in a face-centered cubic (fcc) structure (Ca-I). Increasing pressure at room temperature, Ca-I transforms to Ca-II with a body-centered cubic (bcc) structure at 20 GPa, to Ca-III with a simple cubic (sc) structure at 32 GPa (1, 2, 8), to Ca-IV (P4₁2₁2) at 119 GPa (3), to Ca-V (Cmca) at 143 GPa (3), to Ca-VI (Pnma) at 158 GPa (4), and to Ca-VII at 210 GPa (5). High-precision X-ray diffraction studies revealed rhombohedral (9) and orthorhombic distortions (4) of the sc structure. However, the distortions are very small. In this paper, we refer these slightly distorted sc structure generally as Ca-III for simplicity.

Among the high-pressure phases, the Ca-III phase is particularly intriguing. The pressure-induced electronic transition from a normal metal to a superconductor occurs in the Ca-III phase region above 44 GPa (10, 11), and the T_c increases with pressure from 2 K to 10 K within the Ca-III region (6). Ca-III is also a very active test case of modern theories and experiments, where a number of paradoxes arise (12–18). Based on density-functional theory (DFT) calculations, Ca-III with sc structure has imaginary phonon frequencies, indicating dynamic instability (19–21). On the other hand, high-pressure room temperature experiments confirmed that Ca-III adopts the nearly sc structure (8). To overcome this discrepancy, Teweldeberhan et al. (16) used diffusion quantum Monte Carlo calculations, Errea et al. (18) used ab initio methods including anharmonic effects to stabilize the sc phase at room temperature, and Yao et al. (17) provided a detailed analysis of ab initio molecular dynamics simulations that could account for the stability of the sc structure at room temperature. The remaining major discrepancy is that Yao et al. (14) and Oganov et al. (13) predicted that Ca with I4₁/amd symmetry (β-tin structure) has a much lower zero-temperature enthalpy than sc, but in spite of many experimental efforts, the presumably more stable β-tin structured Ca has never been discovered. Here we report the search and discovery of the β-tin structured Ca and establish its stability range.

To study the stability of sc Ca at ground state, Mao et al. (9) performed low-temperature experiments and observed an orthorhombically distorted sc phase, Cmmm, below 30 K at 44 GPa. Nakamoto et al. confirmed the same Cmmm phase at 7 K and 47 GPa (4). Very recently, Tse et al. (22) also conducted low-temperature experiments to study the Ca sc phase; their results showed a large and anisotropic vibrations in sc Ca unit cell over a large pressure range at low temperatures. However, although the enthalpy of predicted β-tin structure Ca at 40 GPa is about 50 meV more favorable than sc structure (13, 16, 23), there have been no experimental observations of the predicted I4₁/amd phase until now. To further investigate the stability of the Ca sc phase and to search for the possible I4₁/amd tetragonal phase, we performed synchrotron X-ray diffraction (XRD) experiments on Ca through various low-temperature, high-pressure paths.

Three sets of high-pressure experiments were conducted on Ca in a diamond anvil cell (DAC). In the Run #1, Ca was compressed to 21 GPa at room temperature, and then cooled down to 8–10 K. Next, the sample was gradually compressed to approximately 45 GPa and then warmed up to room temperature. In the second experiment, the sample was compressed to 35 GPa and cooled down to 6 K. Then the pressure was gradually decreased isothermally to ambient pressure. The third experiment included a series of low-temperature scans at fixed pressure points of 55, 67, 93, and 109 GPa.

To search for the predicted I4₁/amd phase at the ground state, we cooled the sample to 8–10 K and monitored the XRD patterns while increasing the pressure at constant temperature. Selected XRD patterns are displayed in Fig. 1. At 21.5 GPa and 299 K, Ca shows a mixture of the fcc and bcc phases. After cooling down to 8 K with the pressure increased to 25.7 GPa, the dominant phase in the mixture changed from the fcc to the bcc phase. Between 25.7 and 37.5 GPa, Ca became stable in the bcc phase. Above 37.5 GPa, Ca again appeared as a mixture of the orthorhombic (Cmmm) phase and a previously undescribed phase, as shown in Fig. 1. When we increased the temperature from 10 K to 50 K, the peaks from the previously undescribed phase disappeared, but the peaks from the Cmmm phase remained. Finally, when the sample was warmed up to 299 K, the broad orthorhombic peaks gradually became narrower, indicating a transition from Cmmm to the sc phase, as described in ref. 9.

Fig. 1.

In situ XRD patterns of Ca at different temperatures during compression. The dotted lines, which serve as guides to the eye, show the peak shifts corresponding to increasing pressure. Peaks are observed that cannot be indexed by fcc, bcc, sc, or orthorhombic (O) with Cmmm symmetry. X-ray wavelength, 0.413281 Å.

To better characterize the previously undescribed phase, we conducted Run #2 experiment, aiming at obtaining a pure phase from the mixture of phases. Fig. 2 presents the results from this experiment. In this experiment, we cooled the Ca-III sc phase at 35 GPa 298 K to 6 K. During cooling, the 100 peak (at 2θ = 8.7°) of the sc phase broadens, whereas the 110 peak splits into three peaks at 101 K. These changes indicate that the Ca-III sc phase had an orthorhombic distortion at low temperatures. After cooling down to 6 K, we held the temperature constant and started to decompress the sample. With decreasing pressure, the peaks from the orthorhombic (Cmmm) phase gradually lost intensity, and peaks from a previously undescribed phase began to appear. At 32.8 GPa, the Cmmm peaks vanished completely, and only peaks from the previously undescribed phase remained. The peaks marked with asterisks are from an unknown impurity; they are present in all of the XRD patterns and do not interfere with the peaks from the previously undescribed phase. Under further decompression at 6 K, Ca transformed from the previously undescribed phase to the bcc phase at 31 GPa and to the fcc phase at approximately 18 GPa.

Fig. 2.

In situ XRD patterns of Ca at different temperatures during decompression.The diffraction peaks marked with asterisks are from impurities. Different phases are represented as fcc, bcc, sc, orthorhombic (O) with Cmmm symmetry and Re (rhenium gasket). X-ray wavelength, 0.407204 Å.

We confirmed that the XRD patterns from the previously undescribed phase obtained at approximately 35 GPa and 6 K can be best fitted with the structural model that is based on the theoretically predicated I4₁/amd tetragonal phase. Results from the full profile refinement are shown in Fig. 3A for XRD pattern at 32.8 GPa and 6 K during decompression and in Fig. S1 for XRD pattern at 40 GPa and 10 K during compression. The lattice parameters obtained from the refinement for I4₁/amd tetragonal Ca at 32.8 GPa-6 K are a = 5.322(2) Å, c = 2.831(1) Å. As shown in Fig. 3A and Table S1, the XRD pattern can be fitted with a β-tin structure (I4₁/amd) very well. Meanwhile the mixture of phases during the phase transition from the bcc phase can be well modeled with the Cmmm and I4₁/amd phases (see Fig. S1). Crystal structures of I4₁/amd (6 K) and sc (room temperature) at 32.8 GPa are shown in Fig. 3 B and C for comparison; the β-tin structure I4₁/amd phase can be viewed as a distortion of the sc structure. In the sc phase, Ca atom has six nearest neighbors at 2.728 Å occupying the octahedral site (24), while in the β-tin structure phase, the octahedron was distorted with four nearest neighbors at 2.754 Å and two next nearest neighbors at 2.831 Å.

Fig. 3.

Typical full profile refinement of an in situ XRD pattern at 32.8 GPa, 6 K during decompression (A) and (B, C) comparison between crystal structures of β-tin-type tetragonal with I4₁/amd symmetry at 32.8 GPa 6 K and sc at 32.8 GPa and room temperature (24). Rietveld refinements are performed using GSAS. Experimental data, red crosses; full-profile refinements, green curves; difference patterns, pink curves; tick marks, peak positions calculated from the refined lattice parameters of different phases: β-tin type tetragonal (Ca_T), rhenium (Re). The diffraction peak marked with asterisk is from impurity.

Our results indicate that I4₁/amd is indeed the most stable low-temperature Ca phase between 30 and 40 GPa. At low temperature, however, the kinetic barrier dominates and requires a special path to form this phase. The most favorable path is through the Cmmm phase at low pressure. Yao et al. (14) calculated the transition energy barrier to be the lowest at the lower pressure range. In our first experiment of increasing pressure at low temperature, the bcc phase remains metastable and only transforms into a mixture of Cmmm and I4₁/amd phases above 37.5 GPa. In our second experiment, we cooled Cmmm phase down to the I4₁/amd region, but the barrier was still too high at 35 GPa. By decompression to 32 GPa at low temperature, finally, we were able to lower the barrier sufficiently for the complete conversion of Cmmm to the pure I4₁/amd phase.

To further explore the stability field of the sc phase in the pressure–temperature (P–T) phase diagram, we performed more temperature scanning measurements in Run #3 at several fixed pressures, i.e., 38, 55, 67, 93, and 109 GPa. We focused on the 110 peak of the sc phase because it is sensitive to changes in the structure of the sc phase. The results are displayed in Fig. 4. During cooling at a pressure of 38 GPa, as shown in Fig. 4A, the orthorhombic distortion gradually became apparent because the 110 peak of the sc phase split into three separated peaks, which is in agreement with the results of Mao (9) and Nakamoto (4) at 44 GPa and 30 K, and 47 GPa and 7 K, respectively. However, at other pressures [see Fig. 4 B–E], there was not a detectable splitting of the 110 peak of the sc phase, as was the case at 38 GPa. Fig. 4F illustrates how the full width at half maximum (FWHM) of the 110 peak of the sc phase changes during cooling at 55, 67, 93, and 109 GPa. The FWHM of all of the 110 peaks of the sc phase at different pressures show a broadening trend during cooling. Therefore, at higher pressures (55 GPa ≤ P ≤ 109 GPa) and low temperatures down to 6 K, there are no apparent distortions of the sc structure associated with peak splitting within the capability of X-ray powder diffraction measurements, but the distortion becomes larger during cooling.

Fig. 4.

Temperature effects on the changes of the 110 peak of the sc phase (sc-110) corresponding to the stability of the Ca sc phase at a few pressure points. (A) Approximately 38 GPa, (B) approximately 55 GPa, (C) approximately 67 GPa, (D) approximately 93 GPa, and (E) approximately 109 GPa; (F) FWHM changes during cooling for four pressure points.

According to the results from our experiments, we propose the P–T phase diagram for Ca (see Fig. 5). The axes of pressure and temperature are rescaled to present the details of the I4₁/amd and Cmmm phases. The solid colored symbols show the P–T points that were examined in this study. The dotted lines link the phase transition points at room temperature, and the low-temperature points show the proposed phase boundary. The yellow area at approximately 35 GPa and 6 K shows that tetragonal I4₁/amd Ca exists. The thick dot-dashed line that connects one data point in this study and ones from refs. 4 and 9 indicates the proposed area where Ca will take a severe distortion of the sc with peak splitting under low temperature with a Cmmm structure.

Fig. 5.

Proposed phase diagram of Ca at high pressure and low temperature. The solid symbols show some of the P–T points covered in this study; the triangle and diamond solid symbols are data from refs. 4 and 9, respectively. The dotted lines show the proposed phase boundary; the dot-dashed line shows the boundary of the Cmmm orthorhombic phase (severely distorted sc structure) and the less distorted sc phase.

The structures of the Cmmm (orthorhombic) and I4₁/amd (tetragonal) phases can both be viewed as a distortion of the sc structure (9, 13, 14), and they were all observed by exploring the low-temperature areas. Our results indicate that the I4₁/amd and Cmmm structures of Ca only exist in a very small pressure range just above the transition pressure from bcc to sc phase at room temperature. At low temperatures and low pressure ranges from approximately 32 GPa to approximately 47 GPa, the Cmmm phase (severely distorted sc phase) or even the I4₁/amd phase are preferred. However, at higher pressures (55 GPa ≤ P ≤ 109 GPa), the distortions of the sc structure become smaller, where less distorted sc structures exist. This result indicates that the I4₁/amd and Cmmm phases are the results from competitions between density, temperature, and enthalpy factors, and up to this point, the sc structure Ca-III is considered to be a metastable phase at room temperature, whose stability is most likely facilitated by temperature-induced anhomonicity and density effects. Furthermore, we believe that the complex behavior of Ca could be closely related to the s-d electron transfer process (13, 14).

In conclusion, we have observed the theoretically predicted lowest enthalpy phase of Ca that has a β-tin structure (I4₁/amd) using in situ XRD at approximately 35 GPa and 6 K. However, this structure only exists in a small area in the phase diagram. This study indicates that this more stable phase of tetragonal I4₁/amd Ca does exist and could be obtained by experiments at low temperatures. Temperature effects are one of the most important aspects to understanding the complexity of the Ca-III phase, and they are also indispensable for theoretical calculations. At higher pressure ranges from approximately 55 GPa to approximately 109 GPa, the temperature effects are not sufficient for overcoming the energy barrier that results in a phase transition from sc to Cmmm, I4₁/amd, or other structural models such as the C2/c (Sr-IV structure) (13, 25) but will only result in a slightly distorted sc structure. This Ca study may have relevance for other alkaline earth metals (26, 27) and other materials that exhibit s-d transitions under high pressure.

Methods

A Mao-type diamond anvil cell (DAC) and a plate DAC (28) were used to generate high pressures. The culets of the diamond anvils were either approximately 300 μm or approximately 100 μm for the low and high-pressure experiments, respectively. Rhenium and tungsten with cubic boron nitride powder inserts were used as the gaskets. Sample chambers were drilled into the center of preindented gaskets. Polycrystalline Ca samples (99.98%) were loaded into the sample chamber without a pressure-transmitting medium. The samples were loaded into the DACs inside of a glovebox with an Ar atmosphere to avoid oxidization. Pressures were determined using the ruby fluorescence method or by measuring an X-ray powder diffraction of the sample and calculating the pressure using its known equation of state (8, 24), especially for the high-pressure range (> 40 GPa). The uncertainty of the pressure measurements was estimated to be ΔP =  ± 0.4 GPa and ΔP =  ± 1 GPa for the ruby pressure scale and the sample equation of state, respectively. In situ XRD patterns were recorded at the various pressure-temperature conditions on beamlines 16-IDB and 16-BMD of the HPCAT sector, Advanced Photon Source, Argonne National Laboratory. The low-temperature experiments were conducted in a liquid-flow helium cryostat. All of the XRD patterns were integrated using the Fit2D software and analyzed using GSAS (29).

Capability of controlling the P–T paths at will in order to approach the I4₁/amd stability region through different directions is the key for discovering this phase. Changing pressure in a cryostat is often challenging. Pressure can be changed simply at ambient temperature, and the sample can be cooled down to the desired temperature, then warmed up to ambient temperature for further pressure change. This is the procedure used in our Run #3 as well as in many previous studies (e.g., ref. 9) that failed to reach I4₁/amd. For Run #1, we used a helium gas-driven membrane device (30) to increase pressure of the symmetric DAC in the cryostat at low temperature. The device worked well in increasing pressure but had difficulties in controlling pressure release. For Run #2, we used a mechanical gear box linkage to change the load and counteracted by springs to push the symmetric DAC apart for pressure release. Finally we were able to vary pressures smoothly up and down at 6 K to synthesize the pure I4₁/amd Ca phase as reported above.