PNAS-Ca-2010

Exotic behavior and crystal structures of calcium under pressure Artem R. Oganova,b,1, Yanming Mac, Ying Xuc, Ion Erread,e, Aitor Bergarad,e,f, and Andriy O. Lyakhova aDepartment of Geosciences, Department of Physics and Astronomy, and New York Center for Computational Sciences, Stony Brook University, Stony Brook, NY 11794-2100; bGeology Department, Moscow State University, 119992 Moscow, Russia; cNational Lab of Superhard Materials, Jilin University, Changchun 130012, China; dMateria Kondentsatuaren Fisika Saila, Zientzia eta Teknologia Fakultatea, Euskal Herriko Unibertsitatea, 644 Postakutxatila, 48080 Bilbao, Basque Country, Spain; eDonostia International Physics Center, Paseo de Manuel Lardizabal, 20018 Donostia, Basque Country, Spain; and fCentro Fisica de Materiales, Spanish Scientific Research Council (CSIC) and the University of the Basque Country (UPV/EHU), 1072 Posta kutxatila, E-20080 Donostia, Basque Country, Spain Edited by Thomas J. Ahrens, California Institute of Technology, Pasadena, CA, and approved March 10, 2010 (received for review September 9, 2009) Experimental studies established that calcium undergoes sev- eral counterintuitive transitions under pressure: fcc→ bcc→ simple cubic→ Ca-IV→ Ca-V, and becomes a good superconductor in the simple cubic and higher-pressure phases. Here, using ab initio evolutionary simulations, we explore the behavior of Ca under pressure and find a number of new phases. Our structural sequence differs from the traditional picture for Ca, but is similar to that for Sr. The β-tin (I41∕amd) structure, rather than simple cubic, is pre- dicted to be the theoretical ground state at 0 K and 33–71 GPa. This structure can be represented as a large distortion of the simple cu- bic structure, just as the higher-pressure phases stable between 71 and 134 GPa. The structure of Ca-V, stable above 134 GPa, is a com- plex host-guest structure. According to our calculations, the pre- dicted phases are superconductors with Tc increasing under pressure and reaching approximately 20 K at 120 GPa, in good agreement with experiment. evolutionary algorithms ∣ high pressure ∣ structure prediction ∣ density functional theory ∣ superconductivity Calcium exhibits a nontrivial and somewhat mysterious beha-vior under pressure. At 19.5 GPa it transforms from the fcc to the body-centered cubic (bcc) structure, and then, at 32 GPa, to the simple cubic (sc) structure (1, 2). Such a sequence of transi- tions is exactly opposite to normal intuition, as it is accompanied by a decrease of coordination numbers (12 → 8 → 6) and sphere packing efficiency (0.74 → 0.68 → 0.52). Good metal at ambient conditions, fcc-Ca shows increasing electrical resistivity under pressure (3–5). Even more intriguingly, the resistivity of the fcc phase at and just below this maximum has negative temperature derivative, characteristic of the nonmetallic (semiconducting) state, consistent with a small band gap found in ab initio calcula- tions (6) in the same pressure range. Such demetallization under pressure is counterintuitive, because at sufficiently high pressure all materials must become free-electron metals, the expected be- havior is an increasing tendency to the free-electron limit under pressure (see ref. 7 for a discussion). Contrary to these expecta- tions, strong departure from the free-electron state under pres- sure has also been found for sodium (8, 9) and lithium (10–12) at megabar pressures. Ab initio calculations (13) confirmed the fcc → bcc → sc struc- ture sequence and yielded reasonably accurate values for the transition pressures. However, the sc phase encounters problems: It cannot be explained within the Hume–Rothery approach (Fer- mi surface–Brillouin zone interaction) unless one assumes 4 valence electrons per atom (14), and, even more seriously, lattice dynamics calculations (15, 16) showed that it is dynamically unstable, and although this dynamical instability may be lifted by anharmonic effects (16), other structures (see below) have much lower enthalpies. This seeming contradiction with experi- ments that initially showed a perfect sc structure (1, 2, 17) is lar- gely resolved by recent experimental data showing that the sc structure in reality is indeed distorted and that the observed low-temperature behavior of calcium is likely affected by metast- ability. Remarkably, Ca is a superconductor above 50 GPa (18), and its Tc rapidly (and with an increasing slope) soars with pressure, reaching 25 K at 161 GPa (19)—the highest Tc value found in any element. Whereas the sc phase is already superconducting, the most intriguing high Tc values are found in the stability fields of other phases; a recent x-ray diffraction study (2) found two further phases beyond sc—Ca-IV (stable at 113–139 GPa) and Ca-V (stable above 139 GPa). First structural models, proposed in (20), gave a structure possessing the P43212 space group and 8 atoms in the unit cell (here denoted as P43212-8) for Ca-IV and Cmca-8 for Ca-V. The P43212-8 structure, proposed for Ca-IV, has indeed been confirmed experimentally (21). A very recent theoretical study made a different proposition—a Pnma-4 struc- ture for Ca-IV and the same Cmca-8 structure for Ca-V (22). However, using evolutionary global optimization techniques, for Ca-V we found (see below) a self-hosting structure, more stable than the Cmca-8 structure proposed in (20), and providing an excellent match to experimental diffraction data and observed stability field of Ca-V. Here we predict and examine unique crystal structures and ad- dress the counterintuitive behavior of calcium under pressure— the apparent decrease of its packing efficiency, transition into the semiconducting state at moderate pressures, link between struc- ture and superconductivity, stability of self-hosting structures un- der pressure, and unusually high Tc values observed for calcium. All these aspects make calcium one of the most anomalous and interesting elements in the periodic table. Remarks on the “Simplicity” of Calcium Before discussing the new structures predicted here, let us briefly consider two aspects of nontrivial behavior of calcium at moder- ate pressures (<50–100 GPa): (i) the apparent opening of the band gap below the fcc → bcc transition (19.5 GPa), and (ii) the anomalous fcc → bcc → sc transition sequence. With an even number (two) of valence electrons in the primi- tive cell, calcium could be an insulator—it is a metal only due to band overlap (or, in real-space language, due to the spatial extent of the wavefunctions being much larger than the shortest intera- tomic distance). Very compressible, the atomic volume of calcium shrinks by a factor of 2.4 on going from the atmospheric pressure to 50 GPa (see theoretical and experimental equations of state in ref. 6), inducing large changes in the electronic structure. Whereas in the isolated calcium atom only s- and p-orbitals Author contributions: A.R.O. designed research; A.R.O., Y.M., Y.X., I.E., A.B., and A.O.L. performed research; A.R.O. analyzed data; and A.R.O., Y.M., I.E., and A.B. wrote the paper. The authors declare no conflict of interest. This article is a PNAS Direct Submission. 1To whom correspondence should be addressed. E-mail: artem.oganov@sunysb.edu. This article contains supporting information online at www.pnas.org/cgi/content/full/ 0910335107/DCSupplemental. 7646–7651 ∣ PNAS ∣ April 27, 2010 ∣ vol. 107 ∣ no. 17 www.pnas.org/cgi/doi/10.1073/pnas.0910335107 are occupied (it is the last element in the periodic table without d-electrons), d-orbitals are low-energy and can be populated under pressure—indeed, theory predicts an s → d transition in calcium and all heavy alkali earth and alkali metals (6, 13, 23). Compression has a different effect on the energies of different orbitals (or bands), and generally higher-l orbitals that are un- occupied in the free atom become populated under pressure. In calcium’s group-I neighbor, potassium, d-electrons become dominant at the Fermi level already at approximately 10 GPa (e.g., ref. 23)—i.e., slightly earlier than in calcium. The s → d transition greatly affects reactivity of potassium and may enable it to alloy with Fe in the Earth’s core (24) (this is geochemically very important, as 40K is an important source of radiogenic heat in the Earth and the only possible radioactive element in the core); similarly large changes in reactivity may also occur in calcium. Given that the ionic radius of Ca2þ is 1.0 Å (which is an effec- tive core radius) and the 4s-orbital radius is 1.69 Å, from the equation of state of calcium we deduce that the (interatomic) core-valence overlap is large at all pressures above 30 GPa, and core–core overlap becomes significant at 300 GPa. Overlaps of valence and core orbitals of one atom with core orbitals of another atom can lead to extremely interesting physical effects, such as expulsion of valence electrons into the “empty” space of the structure leading to the formation of strong nonnuclear charge maxima and possible demetallization (8, 12, 25). The degree of localization of these electron pairs increases with pressure, which explains the observed increase of the resis- tivity and semiconducting behavior of fcc calcium under pressure. This is seen also in a very non-free-electron-like electronic den- sity of states (Fig. 1), showing a small band gap (approximately 0.1 eV) at 18 GPa. The increase of the nonnuclear density max- ima under pressure is correlated with a rapid increase of d-orbital occupancy nd: projecting wavefunctions inside atom-centered spheres of 2.0 Å radius, we find nd ¼ 0.36 at 1 atm and nd ¼ 0.83 at 18 GPa (see also refs. 6, 13, 26). Across all alkali earth metals nd > 0.92 appears to be a good phenomenological criterion for the onset of superconductivity (26). Fig. 1 shows an unusual distribution of the valence electron localization function (ELF) (27) in fcc-Ca at 1 atm: it has maxima not only at the nuclei, but also in the octahedral voids between them (thus, ELF maxima form a NaCl-type structure). These maxima become much more pronounced on increasing pressure, and their origin can be traced to the exclusionary effect of the core electrons on the valence electrons, first predicted to occur in lithium (12, 25) and sodium (8, 9). The fcc → bcc → sc structural sequence is less anomalous than often thought. The sphere packing efficiencies (i.e., the ratio of the volume occupied by touching atomic spheres to the total volume) of the fcc, bcc, and sc structures are 74%, 68%, and 52%, respectively. If atoms were of the same size (or bonds of the same length) in all structures, fcc would be the densest. How- ever, in reality the bcc and sc structures are denser (at transition pressures) than fcc because of the shorter interatomic distances. Greater density of the bcc and sc structures has often been ex- plained by the s → d electronic transition; this is not necessary and even incorrect: indeed, magnesium (not undergoing any elec- tronic transitions) has a similar transition (hcp → bcc) at 50 GPa (28). Generally, lower-coordination structures can be denser than fcc or hcp even in the absence of electronic transitions, be- cause lower coordination corresponds to shorter bonds. Indeed, atomic sizes and bond lengths R depend on the coordination number ν (29) R ¼ R0 þ b ln ν [1] where R and R0 are expressed in Å, R0 is a bond-specific constant and b ¼ 0.37 Å. Using [1], the ratio of atomic volumes in the fcc and bcc structures at 1 atm is V bcc V fcc ¼ f bcc f fcc � Rfcc þ b lnð8∕12Þ Rfcc � 3 [2] where f are the packing efficiencies, and Rfcc is the bond length in the fcc structure. This expression does indeed show that in very many cases, without any need for electronic transitions, the bcc structure can be denser than fcc or hcp at 1 atm, and more stable under pressure. Whereas the hard-sphere model fails completely (Fig. 2), model [2] is more consistent with the computed atomic volumes, but still does not give quantitative agreement (to achieve which one might need a more complicated model, includ- ing higher coordination spheres and delocalized electrons). Fig. 2 shows that in most cases bcc and fcc structures have similar densities, with bcc being slightly denser. Thus, the fcc → bcc tran- sition is not an anomaly. On the other hand, the sc structure is usually much less dense and is denser than fcc or bcc only for a handful of elements (Li, C, Ba). Note that for calcium at 1 atm the sc structure is slightly less dense than fcc (Fig. 2) and becomes denser under pressure due to its higher compressibility. Fig. 1. Valence ELF (including semicore 3s and 3p states) and density of states of fcc-Ca at (A, C) 1 atm and 18 GPa (B, D). ELF plots (A, B): (100) sections through positions of Ca atoms and nonnuclear electron density maxima (“e”). Minimum/maximum ELF values are 0.05∕0.73 in (A) and 0.08∕0.84 in (B), with contour spacing of 0.05. Note ELF peaks in the core regions and in the interstices (valence electrons expelled from the core regions); at 18 GPa inter- stitial valence electrons are more localized than semicore electrons. In DOS plots, energies are relative to the Fermi level; note the narrow band gap at 18 GPa. Oganov et al. PNAS ∣ April 27, 2010 ∣ vol. 107 ∣ no. 17 ∣ 7647 CH EM IS TR Y Results of Evolutionary Simulations: New Phases of Calcium Evolutionary simulations at 20 GPa and 30 GPa found the bcc structure to be stable, in agreement with experiment (1, 2, 17). The lowest-enthalpy structure found at 40 GPa and 70 GPa is, however, not sc, but an I41∕amd structure (β-tin type), which has recently been reported in theoretical studies (30, 31) and can be described as a strongly distorted sc structure. Fig. 3 depicts how the β-tin structure is favored against sc and becomes the the- oretical ground state above 33 GPa, exactly the value of pressure at which very recent experiments found that fcc and bcc phases are no longer stable (17). Previous theoretical works (15, 16, 32) that found sc structure to be dynamically unstable due to strong nestings in its Fermi surface. At 40 GPa the I41∕amd phase is slightly (0.9%) less dense than the sc structure, but significantly (52 meV∕atom) more favorable. This enthalpy difference between β-tin and sc is not expected to be overcome by the zero point energy (ZPE) considering that in our calculations at 50 GPa the ZPE in the latter phase is just 5 meV∕atom lower. As shown in Fig. 4, the coordination of Ca atoms is octahedral (as in the sc phase), but distorted—with 4 nearest neighbors at 2.68 Å, and 2 neighbors at 2.77 Å, suggesting a Jahn–Teller or Peierls distortion (at this pressure, Ca atoms have an electronic configuration close to s0.5p0.5d1, which is ex- pected to show the Jahn–Teller effect). This distortion opens a pseudogap, which decreases the density of states at the Fermi level and lowers the electronic kinetic energy of the preferred I41∕amd phase over the sc. The same structure type is known for other elements—Sn, and high-pressure forms of Ge, Si. Most interestingly, it is adopted by the phase III of strontium in the pressure range 26–35 GPa (33, 34). At 71 GPa the β-tin structure distorts into a C2∕c-12 (Sr-IV) structure with 12 atoms in the conventional unit cell, and this structure remains stable up to 89 GPa. C2∕c-12 (Sr-IV) structure is a helical distortion of the β-tin structure with a triplication of the unit cell size. The same helical distortion of the β-tin structure was reported in Sr-IV phase (ref. 34, where the space group, how- ever, was misdetermined as Ia, whereas a closer inspection shows it to be C2∕c). Up to this point, the observed (33) sequence of phase transitions in strontium and the predicted sequence in cal- cium are identical: fcc → bcc → I41∕amd → C2∕c-12 (Sr-IV). The next higher-pressure phase of strontium, Sr-V, is an incom- mensurate host–guest structure. We do find such a structure also for calcium, but preceded by two other phases. On increasing pressure to approximately 100 GPa, we find extremely distorted versions of the simple cubic structure. One of these is a metastable (but very competitive) Cmca-16 structure, which can be described as a frustrated structure intermediate be- tween sc and hexagonal close-packed structures. Fig. 3 shows that in the pressure range 100–130 GPa there are several energetically extremely close structures, and the most stable structure at 89–116 GPa is the P43212 − 8 structure proposed by Ishikawa et al. (20), this structure also can be derived from sc by a large distortion and has recently been experimentally confirmed for Ca-IV phase (25). The Pnma-4 structure is the stable phase at higher pressures, 116–134 GPa, and is the last stable structure related to sc. The structure proposed in (24) for Ca-V (also be- longing to the family of sc-derived structures) is never stable at T ¼ 0 K. The structure we found for Ca-V is a host–guest structure, si- milar to Sr-Vand Ba-IV (35) (Fig. 5). This result is similar to that recently reported by Arapan et al. (36), who found this structure by educated guess based on a possible analogy with Sr. Such Fig. 2. Atomic volumes in the bcc (Gray Squares) and sc (Black Circles) phases, relative to those in the fcc structure. In the horizontal axis the ele- ments are arranged by the metallic radius in the fcc phase. All results are based on present GGA calculations. Whereas bcc phases have similar densities to fcc (and for most elements are even slightly denser), hypothetical sc phases are usually less dense, except C, Li and Ba. It is clear that predictions of the hard-sphere model are inconsistent with ab initio calculations. Fig. 3. (A) Enthalpies of the bcc, sc, and I41∕amd (β-tin phase) structures (relative to fcc). (B) Enthalpies of several competitive phases (relative to the β-tin phase). A few other structures [C2∕c-32 (host–guest), C2∕c-24, I4∕mcm-32 (host–guest)], very nearly degenerate with the ground states, are not shown for clarity. Fig. 4. Structures of (A) simple cubic, (B) β-tin-type tetragonal distortion (I41∕amd) of the simple cubic structure for Ca at 40 GPa. The I41∕amd struc- ture is significantly more favorable. Bond lengths are indicated. 7648 ∣ www.pnas.org/cgi/doi/10.1073/pnas.0910335107 Oganov et al. phases for Sr and Ba are incommensurate—since incommensu- rate structures cannot be predicted within strictly imposed periodic boundary conditions, we can only produce their com- mensurate approximants. In fact, our evolutionary simulations yielded several energetically nearly degenerate and geometrically very similar structures, such as C2∕m-32 (host–guest) (the most stable structure) and I4∕mcm-32 (host–guest) and C2∕c-32 (host–guest) structures. Specific details on these structures are available in SI Text. This near degeneracy suggests that the system is frustrated by competing interactions—a common reason be- hind incommensurate phases. In experiments (2), Ca-V was seen coexisting with Ca-IV above 139 GPa. This metastable coexis- tence implies a large energy barrier for the phase transition, likely due to large structural differences between Ca-IVand Ca-V. This is consistent with our results, whereas in (20) both proposed structures are rather similar derivatives of the sc structure. We also note that because Ca-IV is related to the sc structure, finding it with a neighborhood search method (metadynamics, starting from the sc structure, was used in ref. 20) is very efficient, whereas Ca-V would be more challenging to neighborhood algorithms and a fully global search based on evolutionary algorithms should be preferred. Interestingly, all distortions of the simple cubic structure (β-tin, C2∕c-12 (Sr-IV), P43212-8 and Pnma-4), predicted to be stable below 134 GPa, have a highly coupled soft mode associated to the distortion as a major contributor to the observed supercon- ductivity (SI Text). For ins