arXiv:1307.3212v1 [cond-mat.mtrl-sci] 11 Jul 2013 ihrsett l osbedsotos Second, in signature critical ( high observed relatively temperatures for with characteristics distortions. superconductors structures BCS vibrational known possible viable and stable screens all electronic dynamically typically to least one at respect or decomposition/transformation phases with to stable more respect into with thermodynamically predicted are stable that phases sure competing, structure make crystal often must metallic one two, First, of requirements. management superconductors designing careful silicon as already requires rare however of of are superconductivity pressure, superconductors structures the BCS as known of such confirmed number Experimentally predictions, large proposed. a theoretically properties being to led superconducting have based electron-phonon tools prediction structure nld iheetoi est fsae DS at (DOS) states of density ( level electronic Fermi high the a include h rsn tde.Freape n ol say could of analog one the example, of For properties superconducting MgB the studies. that recent present in shown the as properties and superconducting properties are superconductors. such BCS as which good possess considered make materials could been to therefore or have candidates mass instability low promising structural a to close with elements of ihfeunypoo modes. phonon high-frequency h eeomn fcmuainlcrystal computational of development The h iclycmsfo obnn tblt with stability combining from comes difficulty The 2 , 2 tblt n uecnutvt fC- hssa min a ambient at phases Ca-B of superconductivity and Stability 15 eateto hsc,ApidPyisadAtooy Bing Astronomy, and Physics Applied Physics, of Department CaB 1 nvriyo xod eateto aeil,PrsRoad, Parks Materials, of Department Oxford, of University h yohtclAlB hypothetical The tutr n hmclbnigpito iw h P0struc tP20 The MgB view. semiconducting of a to point relative bonding GPa chemical and structure htetabrnfrhrsaiie h rpsdB proposed the stabilizes further boron extra that tblt ftekonmtli P0paeo CaB of phase tP20 metallic known the of stability etrs sdmntae eet eusal talpressur all at unstable be to here demonstrated is features, rnfraino cP7-CaB of transformation ( temperature critical expected an with superconductor neouinr rsa tutr erh nldn stab a including search, structure high-pressur crystal candidate evolutionary of an variety isoelec remarkable system, A Ca-B the pressures. of study theory functional density fnwsnhszbesprodcigmaterials. inverse necessit superconducting synthesizable the the new and illustrates of properties study superconducting Our and stability borides. of i the stability emphasizing the analogs, alkaline-earth with comparison ihuuulbrnntok tblz teeae pressure elevated at stabilize networks boron unusual with o rtcltmeaue ( temperatures critical low td fCaB of study 2 ntesac o MgB for search the In T r da.I scmoe flight of composed is It ideal. are , .INTRODUCTION I. c E ) 6 F etrstogtt efavorable be to thought Features . ) 10 n ag otnn fcertain of softening large a and 6 1,2 Py.Rv Lett. Rev. [Phys. n ehd ocalculate to methods and 7 11,12 2 n lithium and tp tutr fCaB of structure -type OBx60,Bnhmo,NwYr 13902-6000 York New Binghamton, 6000, Box PO 2 heaShah Sheena aeil composed Materials lk hnnmdae uecnutr ehv are out carried have we superconductors phonon-mediated -like T 6 c notemr ope S6adt5 oyopsa ihpressu high at polymorphs tI56 and oS56 complex more the into ∼ ) h tblt falsrcue a enrtoaie t rationalized been has structures all of stability The K). 1 108 8,9 051(02]t erysocimtiso CaB of stoichiometries nearby to (2012)] 102501 , 13,14 Dtd ue1,2018) 11, June (Dated: 4 1 lk tutr u oceia rsuefo h ea ion. metal the from pressure chemical to due structure -like under n lke .Kolmogorov N. Aleksey and and 3–6 2 rvosysont aefvrbesuperconducting favorable have to shown previously , 4 24 n boron and a dnie ornwhg-rsuesuperconducting ( high-pressure search oI8-CaB new structure structures: four crystal identified state has ground evolutionary An acu lcrncsae tteFrilvl Calcium of level. Fermi contribution ( the high structures at a rich states compounds. have electronic superconducting all calcium stable superconductors form The to found are (0.33-0.50) calcium of concentrations medium considered, to motivation compounds. further their study us giving structures high-pressure ) mS12-CaB K), lmnsad eaiet MgB to relative and, elements tutr nedpeitdi ob superconductor a be to a it with predicted indeed structure oM-,a ohabetadhg rsue Calcium pressure. highest high the and has currently ambient pressure both under at Mg-B, to lmnsof elements Li as such proposed, u xeietlosrain ftersuperconducting unconfirmed. still their are of properties observations experimental but and four- on based boron. oI12 of layers polymorph ring eight-membered proposed our tetraboride calcium to to relative and unstable calcium thermodynamically elemental is structure the h hl opsto ag fCa of range composition whole the tteFrilvladamr rnucdvibrational hP3-CaB hypothetical pronounced the mode. more boron on in-plane study a the and of softening level phonon Fermi the at o MgB for tabetpesr sepandfo crystal a from explained is pressure ambient at ntesto aBcytlsrcue ehave we structures crystal Ca-B of set the In aiu te MgB other Various nt.Hr nepaaini ie o the for given is explanation an Here units. T atnUiest,SaeUiest fNwYork, New of University State University, hamton c ocrflyeaiebt ntedesign the in both examine carefully to y f55K ehv xeddorprevious our extended have We K. 5.5 of ) rnct gB tabetadgigapascal and ambient at Mg-B, to tronic prac ftesz ftemtlinfor ion metal the of size the of mportance rudsae aebe dnie with identified been have states ground e bv P u r on ohv very have to found are but GPa 8 above s eakln-at oooieo8CB a oI8-CaB, monoboride alkaline-earth le orlto ewe h thermodynamic the between correlation s w e ealcCaB metallic new two es; xodO13H ntdKndmand Kingdom United 3PH, OX1 Oxford T 2 c yesprodcos eew aestudied have we here superconductors, type 24,25 uei hw odsaiiea 19 at destabilize to shown is ture 9K. 29 f5 K. 50 of 2 ( > x aebe hw ofr complex form to shown been have T 22 c 2 nadto,eeetlcalcium elemental addition, In ∼ 2 16 T .0 r on ob unstable. be to found are 0.50) )adCaB and K) 1 yesprodcoshv been have superconductors type c x BC, lhuhdnmclystable, dynamically Although . ) oI12-CaB K), 5.5 = dhg pressure high nd 18 2 20,21 2 systematic a 6+ polymorphs thsahge DOS higher a has it , LiB x 2 x finding , oetn h search the extend To e The re. 19 B hrough Pasnnotation) (Pearson 6 1 . 25 − n Li and T x ( isoelectronic , c T 17 mns the amongst c 14 2 swl as well as ≈ recent A ( 2 T . K). 1.9 AlB c 22,23 ∼ 4 14 1 2

FIG. 1: (Color online) Top: the formation enthalpy of proposed structures of CaB6, CaB4, CaB2 and CaB. The plot for CaB2 also shows the unstable hP3 structure in gray. mS16-CaB cannot be seen as its enthalpy is close to oI8-CaB. Middle: the pressure ranges in which the proposed ground state structures are stable. Bottom: calculated formation enthalpies of CaxB1−x compounds at 0 (black) and 30 GPa (red) with their structures and superconducting critical temperatures (Tc).

We discuss the behavior of the stability with pressure (mS12), composed of a buckled sheet of boron hexagons, of calcium and other alkaline-earth metal borides in the stabilizes above 32 GPa. Finally, we propose a proposed ground state structures. stable alkaline-earth monoboride (oI8-CaB) at 20 GPa Our study of CaB4 agrees with previous DFT-ba composed of zig-zagged boron chains; the highest Tc sed calculations showing that tP20-CaB4 should be superconductor (5.5 K) found so far in the calcium stable at ambient pressure.26 However there have been borides. concerns about its stability when considering the electron count within the compound.27 Here we explain why the ThB4-type CaB4 structure should be stable at 0 GPa II. OVERVIEW but destabilizes at 19 GPa relative to a semiconducting MgB4 type structure resulting from out-of-plane chemical Figure 1 gives a summary of predicted structures pressure from the metal ion. Expanding upon our (details in Supplemental Material30), the pressures they 28 past investigation of CaB6, we show that the phase are stable over and the calculated T = 0 K formation transition to oS56 at 13 GPa is correlated to the enthalpies. This paper is split up into seven sections. length of one specific boron-boron bond length whilst Section III details the computational methodology and the transition to tI56 at 32 GPa is due to space the evolutionary crystal structure search. A summary constraints for accommodating the metal ion. Following of the stability and properties of each Ca:B composition 28 the experiments supporting the tI56 parent structure, will be given in Secs. IV-VII: IV - CaB6+x; V-CaB4; VI we find a more stable derivative, tP57-CaB6.125, created - CaB2 and VII - CaB. For proposed structures we have by the insertion of an extra boron atom into half of the rationalized their stability and examined their electronic B24 units. The stability and superconducting Tc of these (DOS and band structures) and vibrational properties. structures are shown to be highly sensitive to the boron For identified metallic structures, we have assessed the concentration. At the M B2 composition we find that strength of their electron-phonon coupling, by calculating 2 CaB2 stabilizes at 8 GPa as a low-Tc superconducting the Eliashberg spectral function, α F(ω), and estimated layered structure with four- and eight-membered boron a superconducting transition temperature (Tc) using the rings (oI12), like those observed in the composite Allen-Dynes4 equation. 29 boron-carbon layers of CaB2C2. A monoclinic structure 3

III. COMPUTATIONAL DETAILS dispersion and electron-phonon coupling calculations were performed with the Quantum ESPRESSO package41 42 We used density functional theory (DFT) calculations using ultrasoft pseudopotentials . Dense k- and 30 to assess the thermodynamic stability of over a hundred q-meshes ensured that phonon frequencies converged −1 CaxB1−x structures. Some of these structures were taken with respect to the k- and q-mesh to within 10 cm . from known metal boride and related configurations31 Energy cutoffs of 40 and 320 Ry were used for the wave listed in the Inorganic Crystal Structure Database functions and charge density respectively. (ICSD).32 The rest of the structures were generated For superconducting properties, calculation of the by performing an unconstrained structural optimization Eliashberg function determined the logarithmic average using an evolutionary algorithm within the Module phonon frequency hωlni, and the strength of the electron for Ab Initio Structure Evolution (MAISE)33. MAISE phonon coupling λ, which were both entered into 34 the Allen-Dynes equation [Eqn. 1]4 to estimate the relies on VASP to calculate the enthalpy of crystal ∗ structures and uses this information to perform a superconducting Tc. The Coulomb pseudopotential µ global optimization to find the lowest enthalpy ordered was assumed to be a constant of 0.12 as this is typically phase at a given elemental composition.35–38 It enables a good estimate for these types of superconducting 43 full structural optimization starting with random or compounds. known configurations by passing on beneficial traits from parents to offspring via crossover and mutation ωln − [1.04(1 + λ)] operations. The settings for the evolutionary algorithm Tc = exp ∗ (1) 1.2 λ − µ (1+0.62λ) search were similar to those used in our previous study.31 Table I lists the unit cells used in the evolutionary Crystal ionic radii44 were used for stability analysis search at each stoichiometry and a relative enthalpy and further details about this choice are given in 30 (H rel). H rel is defined as the ordinate distance the Supplemental Material. In addition, all crystal between the point corresponding to the most stable structure diagrams have been adapted from those structure at a given composition and the line connecting generated using VESTA45. two nearest ground states in the formation enthalpy plot (Fig. 1). Gibbs energies were calculated using 39 PHON and it was found that the vibrational entropic IV. CaB6+x (x = 0, 0.25, 0.125) contributions at elevated temperatures for structures at different compositions typically did not change by more A large number of hexaborides (lanthanides, actinides than 10-15 meV/atom. With no available experimental and alkaline-earths) adopt the same simple cubic information we have focused on the most likely to occur structure (cP7-CaB6) of B6 octahedra connected in a compositions.40 All proposed ground states have been three-dimensional network with metal ions in between. tested for thermodynamic stability relative to other Hexaborides are of interest for their valence-fluctuation 46 CaxB1−x compositions. In addition, ground states have (SmB6) and Kondo effects (CeB6), superconducting been tested for dynamic stability by calculating their properties (YB6, Tc = 7 K) and use as thermionic 46 phonon dispersions and checking for the absence of emitters (LaB6). CaB6 in particular was thought to phonon modes with imaginary frequencies. Phonon be weakly ferromagnetic47 but the observation is now attributed to impurities in the sample.48 2− Twenty electrons per boron octahedron (B6 ) are rel 49 Compound Unit cells Ca:B H (meV/atom) required for the stability of cP7- M B6. This criterion CaB12 1:12 32 may not apply for CaB6 at high pressure as X-ray CaB7 2:14 60 diffraction (XRD) measurements have indicated a cubic 50 CaB6 1:6, 2:12, 3:18, 4:24 0 to orthorhombic phase transition around 12 GPa CaB5 1:5, 2:10 81 (the observation was not confirmed in follow up experiments28,51). A theoretical high pressure study CaB4 1:14, 2:8, 3:16, 4:16 0 by Wei et al.52 compared two structures: cP7 and CaB3 2:6, 3:9, 4:12 114 53 oP14 (Pban), concluding that cP7-CaB6 does not CaB2 3:6, 4:8, 6:12 0 undergo a phase transition up to 100 GPa. Performing Ca2B3 2:3, 6:9 136 a fully unconstrained search at a particular chemical Ca5B6 5:6 153 composition enabled us to propose much lower enthalpy CaB 3:3, 4:4, 5:5, 6:6, 8:8 0 orthorhombic and tetragonal high pressure crystal 28 Ca2B 4:2, 8:4, 12:6 44 structures. Using the DFT we predicted that the cP7 structure 28 TABLE I: List of unit cells used for the evolutionary search becomes dynamically unstable at 23 GPa . At 13 across the CaxB1−x composition range and the relative GPa, cP7 is thermodynamically unstable relative to an enthalpy (as described in the text) at that composition at orthorhombic oS56 structure consisting of B6 octahedra 30 GPa. and a chain of fused octahedra which open up and now 4 share a bond (see Fig. 2). Upon applying further Frequencies of selected phonon modes at the Γ, X, pressure (32 GPa) a tetragonal tI56 structure, made M, and R high-symmetry points were calculated as a up of 24-atom units of fused boron octahedra, becomes function of pressure [see Fig. 3(d)]. thermodynamically stable.28 The existence of the parent high pressure tI56 structure has been supported with powder XRD measurements28. High pressure phase transitions have been observed in other metal hexaborides such as LaB6 and YB6. 54 The LaB6 structure was indexed as Pban but was not observed in subsequent experiments.55 Our phonon calculations also indicate no reason for the cP7 structure to distort up to 60 GPa with the proposed Pban structure relaxing back to cP7 at 15 GPa. YB6 is superconducting with a Tc of 7.1 K which has been predicted to drop with pressure. The R and M phonon modes of YB6 become imaginary at high pressures (55 and 63 GPa, 56 28 respectively) like in CaB6 . At ambient pressure, MgB6 has been suggested to stabilize as a related cubic structure which exhibits surprising magnetic and antiferroelectric properties.57 In order to generate the proposed stable structure, only one displacement from cP7 was considered. In this displacement, the magnesium ion sits closer to the face of the unit cell that had been artificially FIG. 3: (Color online) (a) High symmetry points defining forced to remain cubic in the simulation. Our the irreducible wedge in the simple cubic Brillouin zone. (b) analysis shows that the structure is unstable to Displacements of boron positions in real space viewed along M decomposition into experimentally known MgB7 and [100] corresponding to the -point phonon. (c) Calculated MgB4 by 95 meV/atom. We have also found more stable phonon dispersion for cP7-CaB6 at 30 GPa showing multiple imaginary modes. (d) Calculated variation of R and M configurations for MgB6: the proposed oS56 ground state phonon frequencies with pressure. of CaB6 and local minima around cP7-CaB6 of oP28, hR42 and tI28. The lowest energy configuration, oS56, is 37 meV/atom lower in enthalpy than the proposed magnetic cubic structure but still 59 meV/atom above To find the lower energy phase of CaB6 we constructed possible distorted structures for the 2×2×2 cP7 supercell the MgB7 ↔ MgB4 tie-line. by considering all non-equivalent combinations of the three degenerate R-point and one M-point eigenvectors

A. Structural properties of CaB6 and analysis of as listed in Table II. The resulting distorted unit cells the dynamical instability were carefully relaxed (the residual forces were below 0.001 eV/atom) and found to converge to several distinct configurations judging by their space group, enthalpy, For dynamical stability analysis we calculated the 30 phonon dispersion of cP7-CaB under pressure (Fig. 3) atomic volume, simulated XRD pattern and radial 6 distribution functions (see Ref.28). For example, the and found a whole branch along the M-R direction R R R M 30 most general distortion e1 + e2 + e3 + e1 led to the becoming imaginary above 23 GPa (see Ref. ). R R M same oP28 structure as the e2 + e3 + e1 distortion. For each distinct new structure we ran a phonon dispersion calculation and established that the oP28, hR42 and tI28 30 structures of CaB6 are the local minima around the parent cP7 structure. The evolutionary search at 30 GPa revealed the proposed ground state of oS56-CaB6 which stabilizes over cP7 at 13 GPa. Applying pressure to cP7-CaB6 reduces the B-B bond between octahedra (from 1.673 A˚ at 0 GPa to 1.571 A˚ at 30 GPa). The shortened bond breaks and FIG. 2: (Color online) Structures of cP7, oS56 and tI56 half of the octahedra open up and fuse which results in CaB6. As for all structure diagrams in this article the boron the formation of zigzag strips in one direction (oS56 in atoms are in dark gray and calcium atoms in blue. The bond Fig. 2). For all alkaline-earth hexaborides at 0 - 50 GPa, between octahedra described in the text and in Fig. 4 is oS56 is found to be more stable when B-B bonds between shown in red. octahedra are longer than those in cP7 (Fig. 4). 5

number eigenvector starting final Pearson H-HcP 7 V-VcP 7 similarity dynamical combination SG# SG# symbol [meV/atom] [atom/A˚3] to oP28 stability R R R M 1 e1 + e2 + e3 + e1 14 62 oP28 -10.12 -0.107 1.0000 yes R R M 2 e2 + e3 + e1 62 62 oP28 -10.12 -0.107 0.9986 yes R R R 3 e1 + e2 + e3 167 167 hR42 -8.85 -0.093 0.9081 yes R M 4 e1 + e1 127 140 tP28 -9.96 -0.108 0.6462 yes R 5 e1 140 140 tI28 -9.96 -0.109 0.6497 yes M 6 e1 127 127 tP14 -8.49 -0.101 0.6262 no R R 7 e1 + e2 74 74 oI28 -9.07 -0.094 0.7617 no R R M 8 e1 + e2 + e1 12 12 mS56 -9.99 -0.107 0.7538 no R M 9 e2 + e1 63 63 oS56* -9.99 -0.108 0.7670 no

TABLE II: Structure and stability of nine possible distorted structures for a 2×2×2 cP7-CaB6 supercell using combinations of four eigenvectors and relaxed at 50 GPa. The table gives the starting and final space group symmetry, Pearson symbol, enthalpy gain and compression with respect to cP7, similarity to oP28 defined through radial distribution functions (Supplemental Information in Ref.28), and the structures’ dynamical stability. * Structure 9 is different from the proposed oS56 ground state.

0.3 pressure. We only tested the two most ordered configurations of boron atoms inside the units and it is 0.2

cP7 possible that others could result in lower-energy ordered

0.1 stable or disordered CaB6+x structures based on the mixture 0.0 of B24 and B25 building blocks. We find that it would

-0.1 be difficult to resolve such derivatives of the tI56-CaB6 oS56

more stable phase with powder X-ray measurements due to closeness -0.2

(eV/atom) (eV/atom) of the simulated powder XRD patterns. Out of all cP7 -0.3 alkaline-earth metal hexaborides, the calcium structure -H

Be -0.4 is stable as tP57 since it provides the smallest cuboid

Mg

oS56 30

Ca -0.5 volume in the B24 unit for a boron atom to sit. H Sr The electronic properties of known cP7 and proposed

-0.6 Ba structures of CaB6 (oS56, tI56) and tP57-CaB6.125 -0.7 were calculated within the GGA (Fig. 5). These -0.02 0.00 0.02 0.04 0.06 tests were intended only for preliminary analysis of

oS56-cP7 B-B (Å)

oct the electronic states because standard DFT incorrectly 61 predicts cP7-CaB6 as being metallic . Since there is FIG. 4: (Color online) Difference between boron-boron no significant change in the electronic density of states bond length between octahedra (B-Boct) in oS56 and cP7 or band structure for the cP7 structure under pressure, against the difference in their formation enthalpies for all the driving force for the phase transition at 13 GPa is alkaline-earth metals at increasing pressures from left to right thought to be structural (Sec. IV A). of 0, 10, 20, 30, 40 and 50 GPa. The B-B bond is shown in red in Fig. 2. The kink at 50 GPa for the Be point is thought The decomposed band structure plots in Fig. 5 show orbital contributions to specific bands close to the to be due to collapse of the highly unstable BeB6. Fermi level. The valence bands close to and below the Fermi level are dominated by boron p states in all three structures. The conduction band for the cP7 structure At 32 GPa all octahedra open up, rearrange and across Γ-X-M is composed of a hybrid between B(p) fuse with three others to form a 24-atom boron unit and Ca(d 2 ) and Ca(d 2− 2 ). In the oS56 structure, it in a tetragonal tI56 structure. The evolutionary search z x y is the nearly free-electron Ca(dyz) states that hybridize revealed a metastable state where one boron atom had with the B-p states near the Fermi level and produce a migrated to the center of the B24 unit. As there is nearly constant DOS in the pseudogap region from -2 to enough space in the middle for an extra boron atom 0 eV. The Fermi level lies on the slope of a DOS peak we constructed unit cells with additional boron atoms. further suggesting that this is a suboptimal configuration Filling all of the B24 units led to a tI58-CaB6.25 phase and a more stable related structure might be found by metastable by 6 meV/atom with respect to γ-B and considering larger unit cells. In the tI56 structure, just tI56-CaB6 at 50 GPa. However, filling just half of the as in cP7, the Ca (d) states close to the Fermi level B24 units resulted in a stable tP57-CaB6.125 phase and are from Ca (dz2 ) and Ca (dx2−y2 ). The bottom of the made the empty tI56-CaB6 metastable by 5 meV/atom Ca d bands in both oS56 and tI56 structures now lies with respect to tP57-CaB6.125 and oP20-CaB4 at this 6

band crossing Γ around -1 eV in tI56 is moved down by 0.7 eV in tP57 due to the filling of hybrid B(p) and Ca(dz2 ) band.

B. Superconducting properties of CaB6

We predict that the metastable tI58-CaB6.25 would be superconducting at 50 GPa with a Tc of 1.9 K (λ = −1 0.41, ωln = 653.9 cm ). tI56-CaB6 is predicted to be non-superconducting. Due to the large structure size, we could not complete the calculation of the Eliashberg function for the more stable tP57-CaB6.125. However, judging by the DOS at EF one could expect the structure 30 to be superconducting with Tc of 1K .

V. CaB4

Twenty metal tetraborides are known to have a common structure, the tetragonal ThB4-type tP20 structure, first determined by Zalkin and Templeton62 in 1952. Tetraborides are of interest for their electronic 63 (e.g. LaB4) and magnetic properties (e.g. M B4 where M = Nd, Sm, Er and Yb).64 Calcium tetraboride is the only alkaline-earth metal thought to form this structure and although it was first synthesized in 1961,65 the compound has regained some interest recently.27,64,66,67 Previous DFT-based calculations have found tP20-CaB4 to be stable at ambient pressure.26 Liu et al.67 recently reported the experimental synthesis of CaB4 under 2 GPa of pressure, however from computational analysis they concluded that the formation of CaB4 is not favorable under ambient pressure in comparison to CaB6. FIG. 5: (Color online) Calculated band structure and density of states plots of known (cP7-CaB6) and predicted (oS56-CaB6, tI56-CaB6 and tP57-CaB6.125 ) structures relaxed at 30 GPa. The size of the colored circles indicates the orbital contributions to the bands. The decomposition was done within the default PAW radius. The k-paths were generated using Refs.58–60. The conventional unit cell was used for tI56 to illustrate the differences in the tI56 and tP57 band structures.

FIG. 6: (Color online) A top view of one layer of the crystal well below the Fermi level (by ∼1 eV) while the top structure of the tP20-CaB4 ground state at 0 GPa and a side of the predominantly B p bands lies above the Fermi view of the oP20 MgB4-type structure of CaB4, a proposed ground state at 30 GPa. level (by ∼0.5 eV). These observations suggest that the predicted polymorphs should be metallic with multiple Fermi surfaces of mixed Ca-d/B-p or predominantly B-p The ThB4 type structure (Fig. 6) consists of a boron 68 character but more accurate methods, such as GW or layer with B6 octahedra, as seen in M B6, joined by B2 68 time-dependent DFT, should be used to determine the dimers as in each unit cell of M B2 . Metal atoms sit exact position of these electronic states and whether the above and below the boron layer and B-B bonds join the band gap in cP7 closes at high pressures. layers of octahedra forming a three-dimensional covalent The insertion of an extra boron atom into half of boron network. It has been suggested that insufficient tI56-CaB6 boron units results in mostly boron states electrons are provided by the calcium ion to the boron 27,66 at EF . These states are mainly B(py) which lie in the network. For CaB4 to be stable, the dimer must 69 2+ direction of the filled boron unit layers. The Ca(dx2−y2 ) be a double bond taking two electrons from a Ca ion. The rarity of a boron-boron double bond has led to 7 the conclusion that three electrons are required from the metal atom for the stability of a single bonded dimer.27,66 However, it has been shown that semi-filled bonding bands are sufficient for the stability of lanthanide and actinide tetraborides.64 It is therefore clear that some questions about the stability and electronic structure of CaB4 still remain unanswered. In addition, to our knowledge there is no evidence of a pressure-induced phase transition from ThB4-type calcium tetraboride. Transition metal tetraboride structures (e.g. CrB4, MoB4 and WB4) are not relevant for this study as their alkaline-earth analogs are unstable. It is however interesting to note that even for other tetraborides there have been many open questions at ambient pressure. For example, we predicted that the ground state crystal structure of the known CrB4 compound has lower symmetry70 than thought originally71 and our proposed structure type has been subsequently confirmed with electron diffraction experiments.72 Here we explain why ThB4-type CaB4 is stable at FIG. 7: (Color online) The space filled by the metal ion in the ambient pressure and give a general rule for the stability c axis plotted against the ionic radius, for the tP20 structure of all ThB4-type tetraborides with pressure. Stability of M B4. The space filled is the ratio between the smaller is shown to be a balance between the size restrictions and larger red arrows on the crystal structure diagram. posed by the metal ion within the boron lattice and the The gray shaded region (49-57%) shows the space filling corresponding to a stable tP20 structure. Heavy elements distribution of charge between B6 and B2 units. Using an evolutionary search we found that at 19 GPa the oP20 are these lanthanides and actinides: Th, Lu, Ce, Nd, Y, Gd, Sm, Tb, Er, Pr, Dy, La, Pu, Ho, U and Tm. Alkaline-earth MgB -type structure becomes energetically favorable 4 metals are assumed to be divalent. Lanthanides are assumed relative to the low-pressure tP20 ThB4 type (Fig. to be trivalent as known in the literature.64 6). oP20-CaB4 is composed of a chain of pentagonal pyramids with metal atoms sitting in channels running in one direction. At 30 GPa, the tP20 structure is 39 meV/atom higher in enthalpy than the proposed range which is the reason for the tP20 structure being oP20 ground state (Fig. 1). Two higher pressure 105 and 293 meV/atom higher in enthalpy than the (50 GPa) semiconducting metastable states (mP20 and ground state at these two pressures. The experimentally hR10) were also revealed in the search as detailed in the observed tetraborides of the heavy elements, lanthanides Supplemental Material.30 and actinides, are plotted here and all fall within this range. CaB4 is stable in the tP20 structure up to 19 GPa before becoming thermodynamically unstable A. Structural stability of CaB4 relative to the oP20 structure. tP20-CaB4 at 20 GPa lies outside the proposed allowed filling range. At high Metal ions sit directly on top of each other in space filling, direct metal-metal repulsion makes this tP20-CaB4 and since these ions are closer together than tP20 configuration thermodynamically unfavorable. those in the plane, one can expect this interplanar At ambient pressure, tetraborides with metallic ions separation to have an effect on stability. Figure 7 shows of radii 0.99-1.18 A˚ are predicted to form the tP20 how the percentage space filled up by the divalent metal structure. This prediction can be used to explain the ion perpendicular to the boron plane varies with ionic formation of tetraborides for all metals. Muetterties68 radius and pressure for the alkaline-earth metal borides. defined an atomic radius range over which tetraborides Going down the group, the interplanar separation (lattice were stable (1.97 - 1.68 A,˚ center - 1.82 A)˚ but we vector c) does not increase greatly to accommodate have defined it in terms of an ionic radius. This is the larger ionic radius so the space filled up by the because: (i) charge transfer from the metal to the boron ion increases almost linearly. As expected, applying network is known to occur and (ii) the ion space filling pressure reduces the space available to the ion but only is more strongly correlated to the compounds’ stability slightly as the rigidity of the boron lattice in three than atomic space filling.30 dimensions is maintained by the B-B bond between the Monovalent sodium is the only alkali metal that falls layers. All known metal tetraborides with the tP20 within the range; however, as two electrons are not structure fit neatly over a small range where 49-57% available to be transferred to the boron network, sodium of the interplanar distance can be filled with the metal does not form a stable structure, lying 120 meV/atom ion. At both 0 and 30 GPa, MgB4 falls below this above the Na3B20-Na tie-line. RbB4 has been suggested 8 to stabilize as tP2026 but our calculations indicate that it is unstable with a positive formation enthalpy of 253 meV/atom. Across the row of 3d transition metals, only one ion (1+, 2+, 3+, 4+ or 5+), Ti2+, fits within the range. Since titanium prefers to be in the 3+ or 4+ state, tP20-TiB4 does not form. Among 4d transition metals, yttrium with an ionic radius of 1.04 A˚ falls within the window and forms a stable tP20 tetraboride. Divalent palladium, silver and cadmium have ionic radii of 1.00, 1.08 and 1.09 A,˚ respectively, falling within the stability window. The tP20 structure is not stable for these borides as the metals are thought to prefer to be in other valence states (for example silver prefers to be monovalent). No cadmium borides are known and palladium only forms metal rich borides. FIG. 8: (Color online) The electronic band structure of Praseodymium and europium tetraborides are the only tP20-CaB4 at 0 GPa. Blue and red circles show the B2 and lanthanide tetraborides that are not stable at ambient B6 bands respectively. pressure.64 Trivalent praseodymium has an ionic radius of 1.13 A,˚ so theoretically we would expect this to form a tP20 structure; however, we are unlikely to see on the octahedron does not change much with pressure. any experimental studies to prove this due to the low Low charge around the dimer could be thought to lead abundance of the element. Trivalent europium has an to instability of the B unit and thus the whole tP20 ionic radius of 1.09 A;˚ however, it is thought that the 2 framework. Using this boundary one could also predict element prefers to be in a 2+ state which would make destabilization of YB and LaB above 50 GPa.30 the ion too large (1.31 A)˚ to form the tP20 tetraboride 4 4 The electronic band structure shows that tP20-CaB structure. All the known actinide tetraborides fall within 4 is metallic (Fig. 8). Unlike the other Ca-B structures the window (Th, U, Np, Pu, Am). The trivalent ions of discussed in this study, the DOS at the Fermi level is Pa, Cm, Bk and Cf are of an appropriate size to fit into dominated by boron p states. The semi-filled bands the window however the rarity of these elements makes around the Fermi level are mainly from the B2 dimer. it difficult to form a sound conclusion as to whether or The dimer bands have also been found to be semi-filled not they will form a tetraboride. in stable lanthanide and actinide tetraborides.64 Our The phase transition from tP20 to oP20 makes CaB4 GGA calculations indicate that the high pressure oP20 more compact: at 30 GPa the volume per ion for oP20 is structure is semiconducting. 8.45 A,˚ while for tP20 it is 8.97 A.˚ Naslain et al.73 studied the structure of magnesium tetraboride and explained that larger ions prefer to be in the tP20 structure while VI. CaB2 the smaller magnesium ion prefers the oP20 structure. We extend this idea to include the effect of externally applied pressure and to explain why the oP20 structure The diborides have become the most studied boride composition after the discovery of superconductivity destabilizes at high pressures for all alkaline-earths. We in MgB ,76 though among the alkaline-earths, only find that there is an ideal volume for the oP20 structure 2 ˚3 30 magnesium diboride is stable at ambient pressure. A to be stable, 7.78 - 9.1 A per atom. 77,78 stable phase of BeB2 has previously been proposed. However, the experiments were not able to fully characterize the structure. Furthermore, our calculations B. Electronic properties of CaB4 indicate that the theoretically determined F4¯3m structure proposed77 is thermodynamically unstable Examination of the charge on the B dimer in various 2 (by 84 meV/atom) relative to at least one structure tP20-type metal tetraborides could give insights into found in our preliminary evolutionary search of buckled their stability. Unfortunately, there is no universal hexagonal sheets of boron with a double layer of metal definition of an atomic charge in the DFT. Bader 30 74 ions. A CaB2 structure of linear boron chains was charge decomposition has revealed interesting charge 79 72,75 recently predicted to stabilize at ambient pressure. This transfer trends in several systems. We have used structure was previously proposed (δ-CaB ) and found to the highest accuracy settings (dense k-meshes and high 2 be highly unstable against phase separation into CaB6 cutoff energies) to calculate the charge distribution in and fcc-Ca by 144 meV/atom due to the large ionic size tP20-CaB as a function of pressure. We observed that 4 of calcium.17 at pressures above 20 GPa, the excess Bader charge on At high pressure, MgB is known to be stable in the the dimer (relative to three electrons per boron) falls 2 hP3-AlB (hP3) configuration up to at least 57 GPa.80 below 0.4 electrons.30 In comparison, the excess charge 2 As can be seen in Fig. 9, the hP3 structure consists 9 of hexagonal layers of boron sandwiching closed-packed calcium atom. Boron states are mainly from py and layers of metal atoms. It has been suggested that pz orbitals which lie perpendicular and parallel to the 79 CaB2 stabilizes as hP3 above 8 GPa. Our calculations boron plane. Calcium states are mainly from dxy and indicate that at 30 GPa, hP3-CaB2 lies 56 meV/atom dxz orbitals. above the CaB4-Ca tie-line (see Fig. 1). We propose

a different superconducting oI12 structure above 8 GPa, Total B Ca consisting of ABA stacking of B-Ca layers with four and hP3

0.50

eight membered rings of boron. The 4-8 arrangement of 0.00

oI12 the covalent boron network makes room for the large Ca

0.25 ion to sit over the eight-membered ring. At 32 GPa a 0.00 monoclinic structure (mS12) of buckled layers of boron mS12

0.25

hexagons becomes thermodynamically more stable. f.u)] spin [states/(eV PDOS

0.00

-12 -10 -8 -6 -4 -2 0 2 4 6 -1 0 1

E-E (eV) E-E (eV)

F F

FIG. 10: (Color online) Projected density of states of three structures of CaB2 relaxed to 30 GPa. PDOS is stated in units of states per eV per spin per formula unit (f.u.) against the energy relative to the Fermi energy (EF ). On the right there is a close up of the states around the Fermi level. The scale for hP3 is double that for oI12 and mS12. FIG. 9: (Color online) A top view of the planes that form the crystal structures of three considered CaB2 ground states at 30 GPa: the AlB2-type hP3 structure and proposed mS12 and oI12 structures. The dark blue in oI12 represents the calcium atom in the next layer which is shifted relative to the C. Superconducting properties of CaB2 top layer.

oI12-CaB2 at 30 GPa and mS12-CaB2 at 50 GPa are expected to be poor superconductors with a Tc of 0.8 and 1.0 K respectively. Compared to CaB, which will A. Structural properties and stability of potential be discussed next, oI12-CaB2 has a higher hωlni but its ground states of CaB2 DOS at EF is under half of that in CaB resulting in a lower λ. From a structure search of Sr4B8 we find

hP3-CaB2 is unstable at ambient pressure as metal ions oI12-SrB2 to be stable at 30 GPa, and predict it to be are too close to each other in the plane. We have found superconducting with a Tc of 0.7 K. Despite Sr having ω more energetically favorable configurations of CaB2 at a larger mass than Ca, therefore lowering its h lni, the 0 GPa. The lowest energy structure is the proposed structure has a similar predicted Tc. high-pressure phase of oI12-CaB2. At 30 GPa, the formation enthalpy of CaB2 structures is generally found to be dependent on the shortest metal-boron distance VII. CaB (M-B) within the structure.30 The M-B in oI12 is 2.61 A,˚ which was the largest of all the structures considered. Monoboride structures are unknown among the The eight-membered boron ring in oI12, restricts the alkaline-earth metals at both ambient and high pressure metal ion to sit above the center of such a large ring, as the number of electrons in the compound is not resulting in a large M-B. optimal. Using the results of an evolutionary search we find that CaB stabilizes at 20 GPa as zigzagged boron chains (oI8) and a metastable state of snake like boron B. Electronic properties of CaB2 chains becomes thermodynamically more stable above 35 GPa. The two structures are also stable for strontium The electronic DOS for the unstable hP3 structure and boride. two proposed ground state structures oI12 and mS12 are shown in Fig. 10. All three structures are metallic with the DOS at the Fermi level for the hP3 structure being A. Structural properties of CaB around double that for the ground states, illustrating the basic problem of combining desired superconductivity With only one boron atom per metal atom there are and stability features in one structure. The hP3 and limited configurations that a covalent boron network 81 oI12 structures have equal contributions from both boron can form. All involve individual chains since a slightly and calcium atoms at the Fermi level, while the mS12 higher concentration of boron is required for double structure has a slightly higher contribution from the chains (as in Cr3B4) and a slightly lower concentration 10

81 is required for branched chains (as in Ru11B8). The C. Superconducting properties of CaB evolutionary search revealed three low-energy types of boron chains that we have compared: oP8, linear; mS16, Figure 13 shows the Eliashberg spectral function and snake-like and oI8 - zigzag (Fig. 11). oI8-CaB is an ABA the electron-phonon coupling strength λ for oI8-CaB. layered structure with calcium atoms sitting in the bends We predict that oI8-CaB will be superconducting at of the chains. Linear chained oP8 is 53 meV/atom higher 30 GPa with a Tc of 5.5 K, total λ of 0.58 and −1 in enthalpy than oI8. Boron chains fit themselves around hωlni of 330.2 cm . 68% of the total λ results from group II metal ions, therefore, the larger the ion, the modes below 360 cm−1, which are mainly displacements larger the bond angle in the zigzagged structure (oI8). of calcium atoms. The decomposition of the phonon Comparing to the linear chains (oP8) we find that the density of states into contributions from the atoms (Fig. smallest ions (Be and Mg) relax away from linear chains 13) shows that the in-plane movement of boron atoms and back to zigzag chains (oI8). Calcium boride relaxes (around 500 cm−1) has a negligible contribution to the as linear chains only when 20 GPa of pressure is applied. electron-phonon coupling in this compound. Similarly, a calcium phonon mode is thought to play a large role in the linear increase of the Tc of CaC6 with pressure (11.5 K at 0 GPa up to a maximum of 15.1 K at 7.5 GPa).82 It would also be interesting to compare the superconducting properties of the CaB and the related isoelectronic oS8-LiC compounds. The latter has been recently predicted to form under pressure.83 FIG. 11: (Color online) The crystal structures of the three ) -1 lowest enthalpy phases of CaB found using the evolutionary 0.15

Total search. The oI8 ground state consisting of zig-zag boron B Ca layers, mS16 metastable state consisting of bent boron chains 0.10 and the linear chained oP8 structure that is the highest in 0.05 enthalpy out of the three structures. 0.00

0 200 400 600 800 1000 1200 PHDOS (states/cm

0.60 ) 2

F( ) (

0.40

),

0.20 F( B. Electronic properties of CaB 2

0.00

0 200 400 600 800 1000 1200

-1 The electronic density of states (Fig. 12) shows the (cm ) metallic nature of all the considered CaB structures. The DOS at Fermi level of oI8 and oP8 are 0.45 and 0.65 FIG. 13: (Color online) Top: total and projected phonon states/(eV spin f.u.), respectively, coming mainly from density of states (PHDOS), bottom: Eliashberg function α2 ω λ the calcium atom, in particular Ca(dx2−y2 ) states which F( ) and of oI8-CaB at 30 GPa. lie along the line of the boron chains and between the layers. Boron py and pz orbitals which lie in the zig-zag plane contribute most to boron states at the Fermi level. The mS16 structure has a much lower DOS at EF [0.20 states/(eV spin f.u.)], possibly resulting from a higher VIII. CONCLUSIONS contribution (50%) to the total DOS from the boron. We have used an evolutionary structure search to

oI8 1.00 identify high-pressure stable phases of CaxB1−x. At

30 GPa three new structures of CaB6, CaB2 and CaB 0.00

mS16

1.00 stabilize, while CaB4 stabilizes as the known MgB4

structure type. For all compositions we have found that 0.00

oP8 the stability of a particular structure is highly dependent 1.00

on the size of the metal ion, shaping the boron framework PDOS [states/(eV spin f.u)] spin [states/(eV PDOS

0.00

-12 -10 -8 -6 -4 -2 0 2 4 6 -1 0 1 around it.

E-E (eV) E-E (eV) F F A variety of stable superconducting calcium boride compounds are predicted but have relatively low FIG. 12: (Color online) The projected density of states of critical temperatures, exemplifying the inverse the three lowest enthalpy structures (oI8, mS16 and oP8) of CaB found using the evolutionary search. See Fig. 10 for the correlation between stability and superconducting graph labels. Tc. Stable superconductors are only found for medium concentrations of calcium where x = 0.33-0.5. Their superconducting nature is thought to be due to the high 11 contribution of calcium states at the Fermi level. The size of the metal ion (Figure 7). According to the rule, main results for each Ca-B composition are summarized at ambient pressure, metals with ionic radii of 0.99-1.18 below. A˚ should form tP20-M B4. tP20-CaB4 destabilizes CaB6: The known semiconducting cubic structure at 19 GPa relative to the MgB4-type semiconducting (cP7) at ambient pressure becomes dynamically unstable structure. around 25 GPa. Three derived dynamically stable CaB2: No stable polymorphs have been found at structures have been identified via a systematic analysis ambient pressure but a four- and eight-membered boron of multiple imaginary-frequency phonon modes. The ring layered metallic structure (oI12) is expected to form stability of a metallic orthorhombic structure (oS56), under 8 GPa of pressure. oI12-CaB2 and its SrB2 proposed to be the CaB6 ground state in the 13-32 GPa analog are superconducting at ∼1 K. Above 32 GPa, a pressure range,28 is shown to correlate with the length monoclinic structure (mS12) composed of bucked sheets of a boron bond in all alkaline-earth hexaborides (Fig. of hexagons, which is poorly superconducting, becomes 4). The parent tI56-CaB6 structure, predicted to be thermodynamically more stable. stable above 32 GPa and supported by XRD data in our CaB: No alkaline-earth monoborides are known at previous study,28 is shown here to stabilize further with ambient pressure. Above 20 GPa, a superconducting the addition of a small amount of boron. The more stable zigzag chained structure of oI8-CaB stabilizes with a tP57-CaB6.125 structure derived from tI56-CaB6 may be predicted Tc of 5.5 K. The boron chains are zigzagged a low-Tc superconductor. as they fit themselves around the metal ion. CaB4: The ThB4-type structure (tP20) is demonstrated to be thermodynamically stable at We acknowledge the support of the EPSRC CAF ambient pressure. A general rule has been found, for all EP/G004072/1. pressures, relating the stability of tP20-M B4 with the

1 S. Curtarolo, G. L. W. Hart, M. B. Nardelli, N. Mingo, review letters 88, 127001 (2002). S. Sanvito, and O. Levy, Nat Mater 12, 191 (2013). 19 M. Calandra, Physical review. C, Nuclear physics 75, 2 G. Hautier, A. Jain, and S. P. Ong, Journal of Materials 144506 (2007), 0556-2813. Science 47, 7317 (2012). 20 A. Lazicki, R. J. Hemley, W. E. Pickett, and C.-S. Yoo, 3 G. Eliashberg, Soviet Physics - JETP 11, 696 (1960). Phys. Rev. B 82, 180102 (2010). 4 P. B. Allen and R. C. Dynes, Physical Review B 12, 905 21 A. Bharathi, S. Jemima Balaselvi, M. Premila, T. N. (1975). Sairam, G. L. N. Reddy, C. S. Sundar, and Y. Hariharan, 5 E. R. Margine and F. Giustino, Physical Review B 87, Solid State Communications 124, 423 (2002). 024505 (2013). 22 H. Fujihisa, Y. Nakamoto, M. Sakata, K. Shimizu, 6 M. Cohen, in BCS: 50 Years, edited by L. Cooper and T. Matsuoka, Y. Ohishi, H. Yamawaki, S. Takeya, and D. Feldman (World Scientific, Singapore, 2011). Y. Gotoh, Phys. Rev. Lett. 110, 235501 (2013). 7 K. J. Chang and M. L. Cohen, Phys. Rev. B 30, 5376 23 A. R. Oganov, Y. Ma, Y. Xu, I. Errea, A. Bergara, and (1984). A. O. Lyakhov, Proceedings of the National Academy of 8 J. B. Neaton and N. W. Ashcroft, Nature 400, 141 (1999). Sciences 107, 7646 (2010). 9 K. Shimizu, H. Ishikawa, D. Takao, T. Yagi, and 24 A. R. Oganov, J. Chen, C. Gatti, Y. Ma, Y. Ma, C. W. K. Amaya, Nature 419, 597 (2002). Glass, Z. Liu, T. Yu, O. O. Kurakevych, and V. L. 10 W. E. Pickett, Physica C: Superconductivity 468, 126 Solozhenko, Nature 457, 863 (2009). (2008). 25 E. Y. Zarechnaya, L. Dubrovinsky, N. Dubrovinskaia, 11 C. Buzea and T. Yamashita, Superconductor Science and N. Miyajima, Y. Filinchuk, D. Chernyshov, and Technology 18 (2005). V. Dmitriev, Science and Technology of Advanced 12 R. Lortz, Y. Wang, U. Tutsch, S. Abe, C. Meingast, Materials 9, 044209 (2008). P. Popovich, W. Knafo, N. Shitsevalova, Y. B. Paderno, 26 R. V. Chepulskii and S. Curtarolo, Phys. Rev. B 79, and A. Junod, Phys. Rev. B 73, 024512 (2006). 134203 (2009). 13 J. A. Flores-Livas, M. Amsler, T. J. Lenosky, 27 M. B. Yahia, O. Reckeweg, R. Gautier, J. Bauer, L. Lehtovaara, S. Botti, M. A. L. Marques, and T. Schleid, J.-F. Halet, and J.-Y. Saillard, Inorganic S. Goedecker, Phys. Rev. Lett. 108, 117004 (2012). Chemistry 47, 6137 (2008). 14 A. N. Kolmogorov, M. Calandra, and S. Curtarolo, 28 A. N. Kolmogorov, S. Shah, E. R. Margine, A. K. Kleppe, Physical Review B 78, 094520 (2008). and A. P. Jephcoat, Phys. Rev. Lett. 109, 075501 (2012). 15 J. M. An and W. E. Pickett, Phys. Rev. Lett. 86, 4366 29 B. Albert and K. Schmitt, Inorganic Chemistry 38, 6159 (2001). (1999). 16 H. J. Choi, S. G. Louie, and M. L. Cohen, Physical Review 30 S. Shah and A. Kolmogorov, See supplemental information B 80, 064503 (2009). for this article with the publication in the physical review 17 A. N. Kolmogorov and S. Curtarolo, Phys. Rev. B 74, b. 224507 (2006). 31 A. N. Kolmogorov, S. Shah, E. R. Margine, A. F. Bialon, 18 H. Rosner, A. Kitaigorodsky, and W. E. Pickett, Physical T. Hammerschmidt, and R. Drautz, Physical review letters 12

105, 217003 (2010). 58 M. I. Aroyo, J. M. Perez-Mato, C. Capillas, 32 G.Bergerhoff and I.D.Brown, Crystallographic Databases E. Kroumova, S. Ivantchev, G. Madariaga, A. Kirov, (International Union of Crystallography, 1987). and H. Wondratschek, Zeitschrift fr Kristallographie 221, 33 A. Kolmogorov, URL www.maise-guide.org. 15 (2006). 34 G. Kresse and J. Furthm¨uller, Phys. Rev. B 54, 11169 59 M. I. Aroyo, A. Kirov, C. Capillas, J. M. Perez-Mato, and (1996). H. Wondratschek, Acta Crystallographica Section A 62, 35 P. E. Bl¨ochl, Physical Review B 50, 17953 (1994). 115 (2006). 36 H. J. Monkhorst and J. D. Pack, Physical Review B 13, 60 W. Setyawan, R. M. Gaume, S. Lam, R. S. Feigelson, and 5188 (1976). S. Curtarolo, ACS Combinatorial Science 13, 382 (2011). 37 J. D. Pack and H. J. Monkhorst, Physical Review B 16, 61 B. Lee and L.-W. Wang, Applied Physics Letters 87, 1748 (1977). 262509 (2005). 38 J. P. Perdew, K. Burke, and M. Ernzerhof, Physical Review 62 A. Zalkin and D. H. Templeton, Acta Crystallographica 6, Letters 77, 3865 (1996). 269 (1953). 39 D. Alf`e, Computer Physics Communications 180, 2622 63 J. M. Lafferty, Journal of Applied Physics 22, 299 (1951). (2009). 64 Z. P. Yin and W. E. Pickett, Phys. Rev. B 77, 035135 40 J. R. Rodgers and P. Villars, Journal of Alloys and (2008). Compounds 197, 167 (1993). 65 R. W. Johnson and A. H. Daane, The Journal of Physical 41 G. Paolo and et al., Journal of Physics: Condensed Matter Chemistry 65, 909 (1961). 21, 395502 (2009), 0953-8984. 66 R. Schmitt, B. Blaschkowski, K. Eichele, and H. J. Meyer, 42 We used the pseudopotentials Ca.pbe-nsp-van.UPF and Inorganic Chemistry 45, 3067 (2006). B.pbe-n-van.UPF from www.quantum-espresso.org. 67 Z. Liu, Appl. Phys. Lett. 96, 031903 (2010). 43 S. Y. Savrasov and D. Y. Savrasov, Phys. Rev. B 54, 16487 68 E. L. Mutterities, The Chemistry of Boron and its (1996). Compounds (John Wiley & Sons, Inc., 1967). 44 R. Shannon, Acta Crystallographica Section A 32, 751 69 Burdett and Canadell, ChemInform 22, 7207 (1991). (1976). 70 A. F. Bialon, T. Hammerschmidt, R. Drautz, S. Shah, 45 K. Momma and F. Izumi, Journal of Applied E. R. Margine, and A. N. Kolmogorov, Applied Physics Crystallography 44, 1272 (2011). Letters 98, 081901 (pages 3) (2011). 46 X. H. Ji, Q. Y. Zhang, J. Q. Xu, and Y. M. Zhao, Progress 71 S. Andersson and T. Lundstrom, Acta Chem Scand. 22, in Solid State Chemistry 39, 51 (2011). 3103 (1968). 47 D. P. Young, D. Hall, M. E. Torelli, Z. Fisk, J. L. Sarrao, 72 H. Niu, J. Wang, X.-Q. Chen, D. Li, Y. Li, P. Lazar, J. D. Thompson, H. R. Ott, S. B. Oseroff, R. G. Goodrich, R. Podloucky, and A. N. Kolmogorov, Phys. Rev. B 85, and R. Zysler, Nature 397, 412 (1999). 144116 (2012). 48 J.-S. Rhyee and B. K. Cho, Journal of Applied Physics 95, 73 M. R. Naslain, A. Guette, and M. Barret, Journal of Solid 6675 (2004). State Chemistry 8, 68 (1973), ISSN 0022-4596. 49 H. C. Longuet-Higgins and M. De V. Roberts, Proceedings 74 W. Tang, E. Sanville, and G. Henkelman, Journal of of the Royal Society of London. Series A. Mathematical Physics: Condensed Matter 21, 084204 (2009). and Physical Sciences 224, 336 (1954). 75 R. F. Zhang, D. Legut, Z. J. Lin, Y. S. Zhao, H. K. Mao, 50 M. Li, W. Yang, L. Li, H. Wang, S. Liang, and C. Gao, and S. Veprek, Phys. Rev. Lett. 108, 255502 (2012). Physica B: Condensed Matter 406, 59 (2011). 76 C. Buzea and T. Yamashita, Superconductor Science and 51 Y. Li, J. Yang, X. Cui, T. Hu, C. Liu, Y. Tian, H. Liu, Technology 14, R115 (2001). Y. Han, and C. Gao, Physica Status Solidi (b) 248, 1162 77 A. Hermann, N. W. Ashcroft, and R. Hoffmann, Inorganic (2011). Chemistry 51, 9066 (2012). 52 Y.-K. Wei, J.-X. Yu, Z.-G. Li, Y. Cheng, and G.-F. Ji, 78 D. E. Sands, C. F. Cline, A. Zalkin, and C. L. Hoenig, Physica B: Condensed Matter 406, 4476 (2011). Acta Crystallographica 14, 309 (1961). 53 This structure was proposed as a high pressure phase of 79 P. Vajeeston, P. Ravindran, and H. Fjellvag, RSC Adv. 2, 54 LaB6 by Teredesai et. al. . 11687 (2012). 54 P. Teredesai, D. V. S. Muthu, N. Chandrabhas, 80 A. F. Goncharov and V. V. Struzhkin, Physica C: S. Meenakshi, V. Vijayakumar, P. Modak, R. S. Rao, Superconductivity 385, 117 (2003). B. K. Godwal, S. K. Sikka, and A. K. Sood, Solid State 81 J. Etourneau and P. Hagenmuller, Philosophical Magazine Communications 129, 791 (2004). Part B 52, 589 (1985). 55 B. K. Godwal, E. A. Petruska, S. Speziale, J. Yan, S. M. 82 A. Gauzzi, S. Takashima, N. Takeshita, C. Terakura, Clark, M. B. Kruger, and R. Jeanloz, Phys. Rev. B 80, H. Takagi, N. Emery, C. H´erold, P. Lagrange, and 172104 (2009). G. Loupias, Phys. Rev. Lett. 98, 067002 (2007). 56 Y. Xu, L. Zhang, T. Cui, Y. Li, Y. Xie, W. Yu, Y. Ma, 83 C. Xing-Qiu, C. L. Fu, and C. Franchini, Journal of and G. Zou, Phys. Rev. B 76, 214103 (2007). Physics: Condensed Matter 22, 292201 (2010). 57 I. Popov, N. Baadji, and S. Sanvito, Phys. Rev. Lett. 108, 107205 (2012).