High-pressure phases of group II difluorides: polymorphism and superionicity

Joseph R. Nelson,1, ∗ Richard J. Needs,1 and Chris J. Pickard2, 3

1Theory of Condensed Matter Group, Cavendish Laboratory,J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom

2Department of Materials Science and Metallurgy, University of Cambridge,27 Charles Babbage Road, Cambridge CB3 0FS, United Kingdom

3Advanced Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba, Sendai, 980-8577, Japan(Dated: February 3, 2017)

We investigate the high-pressure behaviour of beryllium, magnesium and calcium difluorides usingab initio random structure searching and density functional theory (DFT) calculations, over thepressure range 0−70 GPa. Beryllium fluoride exhibits extensive polymorphism at low pressures,and we find two new phases for this compound − the silica moganite and CaCl2 structures − whichare stable over the wide pressure range 12−57 GPa. For magnesium fluoride, our searching resultsshow that the orthorhombic ‘O-I’ TiO2 structure (Pbca, Z = 8) is stable for this compound between40 and 44 GPa. Our searches find no new phases at the static-lattice level for calcium difluoridebetween 0 and 70 GPa; however, a phase with P62m symmetry is close to stability over this pressurerange, and our calculations predict that this phase is stabilised at high temperature. The P62mstructure exhibits an unstable phonon mode at large volumes which may signal a transition to asuperionic state at high temperatures. The Group-II difluorides are isoelectronic to a number ofother AB2-type compounds such as SiO2 and TiO2, and we discuss our results in light of thesesimilarities.

I. INTRODUCTION

The Group-II difluorides form materials with a widevariety of technological uses. BeF2, and mixtures of itwith further fluorides and difluorides, are used to createglasses for use in infrared photonics, which have excellenttransmittance in the UV. BeF2 glass itself has a largebandgap of 13.8 eV [1, 2]. BeF2 is chemically stable andis employed as a mixture component in nuclear reactormolten salts, where it is useful as a coolant, and is alsocapable of dissolving fissile materials [3]. MgF2 is bire-fringent with a wide wavelength transmission range, andis used in the manufacture of optical components suchas polarizers [4]. CaF2 also offers a high transmittanceacross a wide range of wavelengths and is used in op-tical systems, as well as an internal pressure standard[5, 6]. MgF2 and CaF2 occur naturally as the mineralssellaite and fluorite, therefore high pressure modificationsof these compounds are of interest in geophysics.

Group-II difluorides have 16 valence electrons per for-mula unit, and are isoelectronic to other AB2 compoundsof industrial or geophysical significance, such as TiO2

and SiO2. As such, these compounds share many simi-lar crystal structures, albeit stable at different pressures.Because of this structural similarity, Group-II difluorideshave been investigated as structural analogues of silicaphases [7]. BeF2 is particularly similar to silica at lowpressures [8], as in addition to being isoelectronic, thefluoride (F−) and oxide (O2−) ions have similar radii andpolarisabilities, and the Be/F and Si/O atomic radii ra-tios are similar at ≈ 0.3 [9]. MgF2 has also been exploredas a model for higher pressure silica phases [10].

∗ jn336@cam.ac.uk

We are interested in the structures and phases of Be-,Mg- and CaF2 at ambient and elevated pressures, and theimplications of such phases for other AB2 compounds.Our approach to determining stable phases in these com-pounds uses computational structure searching alongsidedensity functional theory (DFT) calculations, and weelect to explore the pressure range 0−70 GPa.

II. METHODS

The ab initio random structure searching (AIRSS)technique [11] is used to search for Group-II difluoridestructures at three pressures: 15, 30 and 60 GPa. AIRSSis a stochastic method which generates structures ran-domly with a given number of formula units. A mini-mum atom-atom separation is specified for the generatedstructures, e.g., we set the Be-Be, Be-F, and F-F mini-mum separations for the case of BeF2. These separationsare chosen based on short AIRSS runs. We can also im-pose symmetry constraints on our generated structuressuch that low symmetry structures are not considered.This strategy tends to speed up the searches because suchlow-symmetry structures are unlikely to have low ener-gies according to Pauling’s principle [12, 13], althoughwe allocate part of our searching time to check low sym-metry structures, for completeness. AIRSS has a proventrack record of predicting structures in a diverse varietyof systems that have subsequently been verified by exper-iment, such as in compressed silane, aluminium hydride,high-pressure hydrogen sulfide and xenon oxides [14–17].

We limit our searches to a maximum of 8 formulaunits (24 atoms) per cell. In addition to AIRSS, wesupplement our searches with a set of 15 known AB2-type structures taken from a variety of compounds; see

arX

iv:1

702.

0074

6v1

[co

nd-m

at.m

trl-

sci]

2 F

eb 2

017

2

the Appendix for a full list of these structures. Struc-tures generated by AIRSS or taken from known AB2

compounds are relaxed to a enthalpy minimum usinga variable-cell geometry optimization calculation. Forthis, we use density-functional theory (DFT) as imple-mented in the castep plane-wave pseudopotential code[18], with internally-generated ultrasoft pseudopotentials[19] and the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional [20]. Our DFT calculations use a800 eV plane-wave basis set cutoff, and a Brillouin zonesampling density of at least 2π × 0.04 A−1. Our cal-culations of phonon frequencies use the quasiharmonicapproximation (QHA) [21, 22] and a finite-displacementsupercell method, as implemented in castep.

As well as searches for new BeF2, MgF2 and CaF2

structures, we have also carried out variable stoichiome-try structure searching at 60 GPa, to examine the pos-sibility of other thermodynamically stable compositions.These searches predict that BeF2 and MgF2 are the onlystable stoichiometries at 60 GPa, while both CaF2 andCaF3 are stable at that pressure. Our predicted CaF3

structure is cubic with Pm3n symmetry, and the samestructure has in fact been previously predicted for high-pressure aluminium hydride (AlH3) [23]. We have notexamined the properties of high-pressure CaF3 further,and in what follows we focus only on the AB2 difluoridestoichiometry. Convex hulls for our variable stoichiome-try searches are given in the Appendix.

Semilocal density functionals such as PBE typicallyunderestimate the calculated band gap in materials. Inorder to capture the optical properties of the group II di-fluorides, we perform optical bandgap calculations withthe non-local Heyd-Scuseria-Ernzerhof (HSE06) func-tional [24]. This functional incorporates 25% screened ex-change and is expected to generally improve the accuracyof bandgap calculations carried out with DFT, thoughat a greater computational cost. We adopt the follow-ing calculation strategy: structures are first relaxed withthe PBE functional, norm-conserving pseudopotentials[25], a basis set cutoff of 1600 eV and a relatively sparseBrillouin zone sampling of 2π × 0.1 A−1. The HSE06functional is then used with PBE-relaxed geometries fora self-consistent calculation of the electronic bands of thestructure. Our stress calculations indicate that the useof the HSE06 functional with PBE geometries gives riseto small forces of . 0.1 eV/A on each atom. Electronicdensity of states calculations are then performed usingthe OptaDOS code [26–28]. Additional information onthe electronic structure calculations in this study can befound in the Appendix.

III. BERYLLIUM DIFLUORIDE

A. Low−pressure results

Beryllium fluoride has several thermodynamically ac-cessible phases which have been obtained in experimental

studies. At temperatures below its melting point (820 K)and pressures at or below atmospheric pressure, the α-quartz, β-quartz, α-cristobalite, β-cristobalite and glassphases of BeF2 have been prepared under various condi-tions [29, 30].

Our results from structure searching calculations atlow pressures (0−8 GPa) are summarised in Fig. 1(a),which shows the static-lattice enthalpies of several BeF2

structures as a function of pressure. Beryllium fluo-ride shows extensive polymorphism at low pressures; wefind at least 10 structures for BeF2 (all those shown inFig. 1(a), except Coesite-I and Coesite-II) lying withinabout 20 meV/f.u. of one another at 0 GPa. The α- andβ-quartz, and α- and β-cristobalite structures are partof this set of 10 structures. Within our current calcula-tional framework (DFT−PBE), the α-cristobalite struc-ture is lowest in enthalpy at 0 GPa, lying 8.9 meV/f.u.below α-quartz, but other low enthalpy structures arealso of interest as they can be stabilised by temperature.There is some experimental evidence for as-yet-unknownhigh-temperature low-pressure BeF2 phases [31], andmetastable structures produced in our searches provideuseful reference crystal structures that could be matchedto available experimental data at elevated temperatures.We remark here that the PBE functional may not providea completely accurate energy ordering for these struc-tures at low pressure. For example in silica, DFT−PBEalso predicts that the α-cristobalite structure has a lowerenthalpy than the α-quartz structure at 0 GPa. However,high pressure phase transition pressures in silica calcu-lated with PBE are in excellent agreement with accurateQuantum Monte−Carlo calculations [32].

A recent study by Rakitin et al. found a C2/c struc-ture for BeF2 which is close in enthalpy to α-quartz at0 GPa [33]. Results using AIRSS produce five furtherpolymorphs for BeF2, labelled ‘Moganite’, P212121-I,P212121-II, C2/c−4×2 and P21/c in Fig. 1(a), whichhave static-lattice enthalpies of +0.9, +0.4, −8.9, −4.9and −0.3 meV/f.u. relative to the α-quartz phase at 0GPa. We discuss each of these in more detail below.

Moganite. The ‘Moganite’ polymorph, which turnedup in our searches (and was also included in our set of15 AB2 structures − see the Appendix), is BeF2 in thesilica moganite structure with Si→Be and O→F. Thishas space group C2/c and 6 formula units of BeF2 in theprimitive cell.

P212121-II. At the pressures shown in Fig. 1(a), theP212121-II polymorph has space group symmetry P43212(#96), and is actually an enantiomer of α-cristobalite. Amirror-image transformation (x, y, z) → (−y,−x, z), i.e.a reflection of atomic positions in the (110) plane, relatesthe two structures. This situation is entirely analogousto SiO2, where rigid-unit phonon modes (RUMs) dis-tort the β-cristobalite structure to one of either P41212symmetry (our α-cristobalite) or to P43212 symmetry(our P212121-II). Coh et al. [34] employ the notationα1 and α′1 respectively for these two cristobalite struc-tures. Enthalpy-pressure curves for the α-cristobalite

3

0 1 2 3 4 5 6 7 8Pressure (GPa)

-30

-20

-10

0

10

20

30

40

50E

nth

alp

y r

ela

tive

to α

-qua

rtz (

me

V/f

.u.)

α-quartz, P3121 (Z=3)

α-cristobalite, P412

12 (Z=4)

β-quartz, P6222 (Z=3)

β-cristobalite, Fd3m (Z=2)

C2/c, after Rakitin et al. (Z=6)

Coesite-I, C2/c (Z=8)

Coesite-II, P21/c (Z=32)

Moganite, C2/c (Z=6)

P212

12

1-I (Z=8)

P212

12

1-II (Z=4)

C2/c-4x2 (Z=8)

P21/c (Z=8)

(a)

5 10 15 20 25 30 35Pressure (GPa)

-100

-80

-60

-40

-20

0

20

En

tha

lpy r

ela

tive

to α

-qua

rtz (

me

V/f

.u.)

α-quartz, P3121 (Z=3)

α-cristobalite, P412

12 (Z=4)

α-PbO2, Pbcn (Z=4)

CaCl2, Pnnm (Z=2)

C2/c, after Rakitin et al. (Z=6)

Coesite-I, C2/c (Z=8)

Coesite-II, P21/c (Z=32)

Moganite, C2/c (Z=6)

P212

12

1-I (Z=8)

P212

12

1-II (Z=4) (b)

25 30 35 40 45 50 55 60 65 70Pressure (GPa)

-40

-30

-20

-10

0

10

20

30

40

En

tha

lpy r

ela

tive

to

ru

tile

(m

eV

/f.u

.)

α-PbO2, Pbcn (Z=4)

α-quartz, P3121 (Z=3)

CaCl2, Pnnm (Z=2)

Moganite, C2/c (Z=6)

P21/c (Z=8), 4x2-type

P21/c (Z=6), 3x2-type

Pbcn (Z=8), 3x3-type

Rutile, P42/mnm (Z=2)

(c)

FIG. 1. Static lattice enthalpies for BeF2 calculated with the PBE functional over the pressure ranges (a) 0−8 GPa, (b) 5−35GPa and (c) 25−70 GPa. In (a) and (b), enthalpies are shown relative to the α-quartz phase, while in (c) they are shownrelative to the rutile phase.

FIG. 2. (a) BeF2 in the C2/c moganite phase at 20 GPawith four-fold coordinated Be atoms, viewed down the b (andin this case, monoclinic) axis. The lattice a and c axes pointhorizontal and almost vertical in the page, respectively. (b)BeF2 in the Pnnm CaCl2 phase at 50 GPa, with six-foldcoordinated Be atoms. Beryllium atoms in yellow, fluorineatoms in green. Lattice parameters and atomic positions forthese structures are given in the Appendix.

and P212121-II structures lie on top of one another inFig. 1(a), and our calculations give the overall changein enthalpy going from β-cristobalite to α1 or α′1 asand −21.2 meV/BeF2; the equivalent quantity is −34.4meV/SiO2 in silica. Above 11 GPa, the P212121-IIstructure distorts to the lower space group symmetryP212121 (#19), and the enthalpy curves for it and theα-cristobalite structure start to diverge (Fig. 1(b)).

Open framework structures. The P212121-I,C2/c−4×2 and P21/c structures in Fig. 1(a) are low-density polymorphs which are previously unreported inBeF2. The notation ‘C2/c−4×2’ refers to the fact thatthis structure is a low-pressure variant of the ‘P21/c(Z=8), 4×2-type’ structure shown in Fig. 1(c). OurDFT−PBE calculations show that these polymorphs areenergetically relevant in silica, where they lie 5.5, 15.7and 5.9 meV/SiO2 in energy below α-quartz at 0 GPa,which suggests them as likely silica polymorphs as well.The monoclinic angle in the P21/c structure is close to90◦, hence we also investigate a higher symmetry versionof this structure with Pnma symmetry, whose enthalpy isfound to be 2.2 meV/SiO2 below α-quartz (this structureis not shown in Fig. 1(a)). Calculated lattice parametersand atomic positions for these structures are provided inthe Appendix.

We query the ICSD [35] and International Zeolite As-sociation (IZA) [36] databases to check if these fourstructures are already known in SiO2. Our C2/c−4×2structure matches #75654 in the ICSD, which is ‘Struc-ture 8’ in the simulated annealing structure predictionwork of Boisen et al. [37], after the latter is relaxedusing DFT. However, we find no matching SiO2 struc-tures in these databases for our P212121-I, P21/c andPnma structures. Analysis of these three structures us-ing the TOPOS code [38] shows that they are all of thesame topological type as the ABW zeolite. They haveframework densities of 18.9, 18.6 and 18.5 Si/1000A3 (cf.α-quartz, with 24.9 Si/1000A3), while the ABW silica

4

structure itself is slightly less dense at 17.6 Si/1000A3,with an enthalpy 3.9 meV/SiO2 above α-quartz. WhileBeF2 and SiO2 share many chemical similarities (as men-tioned in Sec. I), the Be−F bond is much weaker than theSi−O bond, resulting in a lower melting point and hard-ness for BeF2 [29]. Nevertheless, our results show thatBeF2 also supports open framework zeolite-like struc-tures, and highlights the utility of searching for potentialzeolite structures in model systems such as BeF2. Thelarge number of polymorphs we encounter lying close toone another in energy suggest BeF2 as a potential tetra-hedral framework material.

B. High−pressure results

Our structure searching calculations show that theapplication of pressure (0.4 GPa) favours the α-quartzphase, as seen in Fig. 1(a). Between 3.1 and 3.3 GPa,the silica ‘coesite-I’ or ‘coesite-II’ structure [39] with Z=8or 32 then becomes the lowest-enthalpy structure forBeF2. We find that over the pressure range 0-18 GPa,the coesite-I and II structures are nearly identical. Thecoesite-II structure is close to a supercell of coesite-I, butthe atomic positions in coesite-II deviate slightly fromthose expected for a perfect supercell, and the coesite-II structure lies consistently about 1 meV/BeF2 belowcoesite-I over this pressure range. The coesite-I phasehas been found in experimental studies on BeF2, at 3GPa and ≈1100 K [31]. Previous work [33] reported thata new structure of C2/c symmetry then becomes stablebetween 18 and 27 GPa; our calculations instead showthat BeF2 is most stable in the silica moganite structure,which also has C2/c symmetry, between 11.6 and 30.1GPa (see Fig. 1(b)). Above 30.1 GPa, we find that theorthorhombic CaCl2 structure with space group Pnnmbecomes stable (Fig. 1(b)), eventually giving way to thedenser α-PbO2 structure above 57.5 GPa (Fig. 1(c)). Themoganite and CaCl2 structures are depicted in Fig. 2.

The enthalpy-pressure curve for the CaCl2 structureemerges smoothly from that for the rutile structure(space group P42/mnm), which is also the case in sil-ica, where a ferroelastic phase transition occurs betweenthese two structures near 50 GPa [40]. For BeF2 at thestatic lattice level, our calculations exclude the stabil-ity of the rutile structure over the pressure range 0−70GPa, though we note that this phase lies only a fractionof a meV per BeF2 higher in enthalpy than the CaCl2phase at 30.1 GPa, as seen in Fig. 1(b). Earlier studies[33, 41] have already examined a number of the structuresdiscussed here; however, the stability of BeF2 in the mo-ganite and CaCl2 structures is new, and according to ourcalculations dominates the high-pressure phases of BeF2

over the pressure range 11.6−57.5 GPa.Fig. 1(c) shows a band of three enthalpy-pressure

curves, labelled Pbcn (Z=8), P21/c (Z=6) andP21/c (Z=8), whose energy lies in close proximityto the α-PbO2 curve. These three structures emerged

from our searches and have a similar, but slightly lowerdensity than the α-PbO2 phase for BeF2. They areclose to stability at 60 GPa, but are not predictedto be stable over the pressure range 0−70 GPa. Weidentify these phases as members of the class of silicapolymorphs introduced by Teter et al. [42], which area set of structures described as intermediaries to theCaCl2 and α-PbO2 silica phases. Our Pbcn (Z=8) andP21/c (Z=6) structures correspond to the ‘3 × 3’ and‘3× 2’ structure types, while our P21/c (Z=8) structureis not explicitly discussed in Ref. [42] and would bereferred to as ‘4 × 2’ type. We will encounter thesephases again in our results for MgF2.Summary. To briefly summarise our search results for

BeF2, we predict the following series of pressure-inducedphase transitions at the static-lattice level:

α-cristobalite (P41212)0.4 GPa−−−−−−→ α-quartz (P3121)

3.1/3.3 GPa−−−−−−−−−→ Coesite-I/II (C2/c)

11.6 GPa−−−−−−−→ Moganite (C2/c)30.1 GPa−−−−−−−→ CaCl2 (Pnnm)

57.5 GPa−−−−−−−→ α-PbO2 (Pbcn),

with the labelled arrows showing the calculated transi-tion pressures. Our searches also demonstrate numerousmetastable polymorphs for BeF2.

C. Optical bandgaps in BeF2

As mentioned in Sec. I, BeF2 has a large bandgap atambient pressure and is used in a number of optical ap-plications. We examine the optical bandgap in BeF2 asa function of pressure, with the results shown in Fig. 3.The optical gap is found to be tunable, increasing byaround 0.06 eV/GPa over the pressure range 0−70 GPa.We expect BeF2 to therefore maintain its high UV trans-mittance with increasing pressure, with potentially use-ful high pressure applications. The electronic density-of-states (DOS) of the moganite and CaCl2 phases are alsogiven in the Appendix.

IV. MAGNESIUM DIFLUORIDE

MgF2 adopts the rutile P42/mnm structure at roomtemperature and pressure. X-ray diffraction experimentsindicate a transformation to the CaCl2 structure at 9.1GPa, then to a pyrite structure with space group Pa3 andZ=4 near 14 GPa, and have also recovered a mixture ofα-PbO2 and rutile MgF2 upon decompression [10]. DFTcalculations, including those of the present study, actu-ally show that the CaCl2 structure is never stable forMgF2 and instead predict the α-PbO2 structure to havea window of stability between 10 and 15 GPa, with theCaCl2 structure slightly higher in enthalpy.

5

0 10 20 30 40 50 60 70Pressure (GPa)

6

7

8

9

10

11

12

13

14

15

16

Optical bandgap (

eV

)BeF

2

MgF2

CaF2 - 3 eV

FIG. 3. Optical bandgaps in Be-, Mg- and CaF2 as cal-culated using the HSE06 functional, over the pressure range0−70 GPa. For visibility, the bandgaps in CaF2 have beenshifted down by 3 eV. Discontinuities in the solid curves aredue to phase transitions between different structures, whilethe shaded regions serve to guide the eye. The dashed curvein CaF2 corresponds to the P62m phase, which is not stableat the static lattice level but which we predict is stabilised bytemperature.

The results from our structure searches are given inFig. 4. Based on our static-lattice results, we predict thefollowing sequence of stable structures and phase transi-tions with rising pressure:

Rutile (P42/mnm)9.4 GPa−−−−−−→ α-PbO2 (Pbcn)

15.4 GPa−−−−−−−→ Pyrite (Pa3)39.6 GPa−−−−−−−→ O-I (Pbca)

44.1 GPa−−−−−−−→ Cotunnite (Pnma).

Previous theoretical studies [43, 44] have already con-sidered the rutile, α-PbO2, pyrite and cotunnite phasesof MgF2. In the present work, we find that the PbcaO-I ‘orthorhombic-I’ structure, which has been reportedexperimentally for TiO2 near 30 GPa [45], is also sta-ble for MgF2 between 39.6 and 44.1 GPa. This struc-ture is depicted in Fig. 5. Experimental studies on MgF2

have indeed reported an unidentified ‘Phase X’ stable inthe pressure range 49−53 GPa and at 1500−2500 K be-tween the pyrite and cotunnite phases [46]. Our enthalpycalculations identify the O-I structure as the thermody-namically most likely candidate for Phase X, though theauthors of Ref. [46] note some difficulty in indexing x-raydiffraction data on Phase X to an orthorhombic struc-ture, possibly due to a mixture of phases being present.We do not find any other energetically competitive struc-tures near 50 GPa.

Silica and its stable polymorphs are of paramount im-

0 10 20 30 40 50 60 70Pressure (GPa)

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

Enth

alp

y r

ela

tive to p

yrite

(eV

/f.u

.)

α-PbO2, Pbcn (Z=4)

CaCl2, Pnnm (Z=2)

Cotunnite, Pnma (Z=4)

Fe2P-type, P62m (Z=3)

O-I, Pbca (Z=8)

P21/c (Z=8), 4x2-type

P21/c (Z=6), 3x2-type

Pbcn (Z=8), 3x3-type

Pyrite, Pa3 (Z=4)

Rutile, P42/mnm (Z=2)

8 10 12 14 16

Pressure (GPa)

-10

-5

0

5

10

Rela

tive to α

-PbO

2 (

meV

/f.u

.)

FIG. 4. Static-lattice enthalpies and results from structuresearches on MgF2 over the pressure range 0−70 GPa. Theinset plot shows the small enthalpy differences between a fewphases in the vicinity of 10 GPa. Enthalpies are shown rela-tive to the pyrite (main figure) and α-PbO2 (inset) phases.

FIG. 5. 2×1×1 slab of MgF2 in the Pbca ‘orthorhombic-I’structure (Z=8), which we predict to be stable between 39.6and 44.1 GPa. This view looks down the b axis. Magnesiumatoms in blue, fluorine atoms in green.

portance in geophysics and planetary sciences. As wellas a mineral in its own right, it is expected to be formedfrom the breakdown of post-perovskite MgSiO3 at ter-apascal pressures. SiO2 follows a very similar set ofphase transitions to MgF2 with increasing pressure [47],with the Si coordination number rising from 6 in rutileat ambient pressures to a predicted 10 in an I4/mmmstructure near 10 TPa [48]. As pointed out by pre-

6

0 10 20 30 40 50 60 70Pressure (GPa)

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

En

tha

lpy r

ela

tive

to

co

tun

nite

(e

V/f

.u.)

Rutile, P42/mnm (Z=2)

Fe2P-type, P62m (Z=3)

Cotunnite, Pnma (Z=4)

Ni2In-type, P6

3/mmc (Z=2)

Fluorite, Fm3m (Z=1)

FIG. 6. Results from structure searches on CaF2 over thepressure range 0−70 GPa: static-lattice enthalpies relative tothe Pnma (γ) CaF2 phase.

vious authors, several features of high pressure silicacan readily be modelled in MgF2, but at much lowerpressures [10, 46]. For example, the α-PbO2→pyritetransition in SiO2, which our calculations find occursat 217 GPa, takes place at a much lower pressure of15.4 GPa in MgF2. Near 690 GPa and for T & 1000K, a pyrite→ cotunnite transition is also predicted forSiO2 [49]; the analogous transition occurs at 44.1 GPa inMgF2.

As mentioned in Sec. III B, Teter et al. [42] have intro-duced a class of SiO2 polymorphs intermediate to CaCl2and α-PbO2. At least one member of this class of poly-morphs has been synthesised in SiO2, the ‘3 × 2’ typeP21/c structure [50]. The P21/c (Z=6), P21/c (Z=8)and Pbcn (Z=8) structures of BeF2 depicted in Fig. 1(c)are members of this class, and also turn up in our MgF2

searches (Fig. 4 and its inset). Our calculations show thatthese polymorphs are closest to stability near 10 GPa inMgF2, compared to ≈100 GPa in SiO2 suggesting that,as with other features of silica, they could be studiedexperimentally at much lower pressures in MgF2.

Fig. 3 shows the optical bandgap in MgF2 as a functionof pressure. Like BeF2, the optical gap is tunable withpressure, rising by about 0.04 eV/GPa over 0−70 GPa.We also provide the electronic DOS of the O-I structurein the Appendix.

V. CALCIUM DIFLUORIDE

CaF2 crystallizes in the cubic Fm3m ‘fluorite’ struc-ture (Z=4, α-CaF2) under ambient conditions. The com-pound has a high-temperature phase above about 1400

FIG. 7. 2×2×2 slabs of (a) our predicted P62m structure,and (b) the Pnma structure (phase γ) of CaF2. Calciumatoms are in red, fluorine atoms in green. In both (a) and (b),the right-hand view is obtained from the left-hand view byrotating the structure by 90◦ about an axis running verticallyup the page.

K, known as β-CaF2, and melts near 1700 K at low pres-sures [51]. A high-pressure modification above 8−10 GPa(γ-CaF2) is also known, with CaF2 taking on the or-thorhombic Pnma cotunnite structure (Z=4) [52].

The β phase has attracted considerable interest be-cause it exhibits superionicity, with F− ions as the dif-fusing species [53]. A number of other compounds inthe same fluorite (or ‘anti-fluorite’) variants of this struc-ture, such as Li2O, are also superionic conductors [54].Such materials are of great technological interest, withapplications in solid-state battery design. A recent studyhas shown that the superionic transition temperature inCaF2 can be decreased through applied stress [55].

A. Results from structure searching

Our results from structure searching in CaF2 are shownin Fig. 6. Unlike BeF2 and MgF2, we find that the po-tential energy surface for CaF2 is relatively simple, withvery few polymorphs for this compound over the pressurerange 0−70 GPa. At the static lattice level of theory, weidentify only the sequence of stable phases and transi-tions:

Fluorite (Fm3m)7.9 GPa−−−−−−→ Cotunnite (Pnma).

7

0 10 20 30 40Pressure (GPa)

0

1000

2000

3000

4000

Tem

pera

ture

(K

)

P62m

PnmaFm3m

Liquid

(α) (γ)

Fm3m develops unstable phonons in this region (β)

P62m develops unstable phonons in this region

S/I

S/I

FIG. 8. Calculated quasiharmonic phase diagram of CaF2.We find the known Fm3m (α) and Pnma (γ) phases at lowtemperature, but our calculations indicate that a previouslyunreported phase of P62m symmetry becomes stable at hightemperature and pressure. ‘S/I’ indicates the superionic re-gion of the phase diagram, and dashed lines show boundariesdue to unstable phonons at the quasiharmonic level.

The calculated fluorite→ cotunnite transition pressurehere is in agreement with experimental results [52]. Ex-perimental studies have also shown a transition from γ-CaF2 to an Ni2In-type structure in the pressure range63−79 GPa with laser heating [56], consistent with theconvergence of the red and black-dashed curves in Fig. 6.

Our results in Fig. 6 reveal a hexagonal phase for CaF2

with P62m symmetry which is close to stability, lyingonly 6 meV/CaF2 higher in enthalpy than the γ phasenear 36 GPa. The enthalpy curves for the P62m and γphases in Fig. 6 indicate that these two structures havevery similar densities, with P62m slightly denser at pres-sures below 36 GPa and becoming less dense than γ-CaF2

at higher pressures. The P62m phase has the Fe2P struc-ture, which has also been predicted for SiO2 at very highpressures (>0.69 TPa) and low temperatures [49]. Weshow the γ and P62m structures in Fig. 7.

B. Pressure-temperature phase diagram for CaF2

The effects of nuclear zero-point motion and tempera-ture are often important and affect the relative stabilityof crystal phases, particularly in cases where there aretwo or more structures lying very close in energy [57, 58].Given the small enthalpy difference between the P62mand γ CaF2 phases, we calculate the Gibbs free energy of

these structures in the QHA as a function of pressure, aswell as that of Fm3m-CaF2. Selecting the lowest Gibbsfree energy structure at each temperature and pressuregives the phase diagram shown in Fig. 8. The solid-liquidphase boundary (dotted black line) in Fig. 8 is taken fromthe work of Cazorla et al [59]. From this, we do indeedpredict that the P62m-CaF2 structure is stabilised bytemperature, in the region P & 10 GPa and T & 1500K. We remark here that the exact phase boundaries inFig. 8 are subject to some uncertainty depending on thechoice of equation of state used for the Gibbs free energycalculation. This uncertainty is particularly noticeablealong the Pnma (γ)−P62m phase boundary.

Both the Fm3m and P62m structures develop un-stable phonon modes at sufficiently large volumes. ForFm3m-CaF2, these first occur at a static-lattice pres-sure between −7 and −6 GPa; for P62m they first occurbetween 0 and 1 GPa. Cazorla et al. also report unsta-ble phonon modes for Fm3m-CaF2 above 4.5 GPa [59],however we do not encounter any such instabilities inour calculations. We find no unstable phonon modes inγ-CaF2. In Fig. 8, black dashed lines are used to di-vide the regions of stability for the Fm3m and P62m intwo. At temperatures below the lines, these phases havevolumes corresponding to stable phonons, while abovethe lines they exhibit unstable phonon modes, and theirGibbs free energies are extrapolations of quasiharmonicresults. Dotted lines separate regions where one or bothphases have unstable phonon modes, with the exceptionof the solid-liquid boundary.

The calculated volume coefficient of thermal expan-sion, α(P, T ), can be used to assess the validity of theQHA. Applying the criteria of Karki et al. [60] andWentzcovitch et al. [61], we expect the quasiharmonicapproximation to be accurate for T (K) ≤ 28P (GPa) +453, with an uncertainty of about 100 K. The lower half ofthe α-γ phase boundary, and the γ-P62m phase bound-ary near 40 GPa, are therefore expected to be accuratewithin the QHA. At higher temperatures, the QHA isexpected to be less applicable as anharmonic effects be-come increasingly important. Further information on thecalculation of α(P, T ) and the validity of the QHA canbe found in the Appendix.

C. Superionicity in CaF2

The onset of superionicity in β-CaF2 has been dis-cussed in connection with the formation of unstablephonon modes in the fluorite CaF2 structure [62]. Indeed,this is the criterion we have used in Fig. 8, where we la-bel the region where Fm3m-CaF2 has unstable phononsas superionic (‘S/I’), or β. By this criterion, the α-βtransition is calculated to occur at ≈ 1000 K at 0 GPa,which is actually in rough agreement with the experimen-tally observed transition temperature of 1400 K, consid-ering that the QHA should be inaccurate near unstablephonon modes. Fig. 9(a) shows the phonon dispersion

8

Γ X L W X Γ0

100

200

300

400

500

ω (

cm

-1)

Γ X L W X Γ-100

0

100

200

300

ω (

cm

-1)

0 GPa

-2 GPa

-4 GPa

-6 GPa

-8 GPa

-10 GPa

(a)

(b)

FIG. 9. Softening of phonon modes in the fluorite Fm3m-CaF2 structure. (a) Phonon dispersion curves of this struc-ture at a static-lattice pressure of 0 GPa (cell volume 41.79A3/f.u.). (b) The blue-coloured mode in (a) as a function ofdecreasing static pressure, from 0 to −10 GPa. The modesoftens and first develops imaginary phonon frequencies at X(shown as negative frequencies).

Γ K M Γ A H L Γ

0

100

200

300

400

500

ω (

cm

-1)

10 GPa

0 GPa

FIG. 10. Phonon dispersion curves for the proposed P62m-CaF2 structure. This phase has stable phonons at a staticpressure of 10 GPa (cell volume 34.45 A3/f.u.). The modecoloured blue softens and becomes unstable at the Brillouinzone K point with decreasing pressure, as shown by the bluedashed curve. Only the portion from Γ to A is shown at 0GPa.

relations in Fm3m-CaF2 at a static-lattice pressure of0 GPa, while Fig. 9(b) shows how unstable modes de-velop in this structure with increasing volume (decreas-ing static-lattice pressure). Unstable phonon modes arefirst encountered at the Brillouin zone X point. The cor-responding atomic displacements in this unstable modeleave Ca2+ ions fixed, while F− ions are displaced alongthe [100], [010] or [001] directions (referred to a conven-tional cubic cell for Fm3m-CaF2). The connection tosuperionicity is that these directions also correspond toeasy directions for F− ion diffusion in the fluorite struc-ture, and are almost barrierless at volumes correspondingto unstable X phonons [63].

Fig. 9(b) also shows that at even larger volumes, unsta-ble phonon modes develop at the Brillouin zone W pointand that the entire W−X branch becomes soft. As is thecase at X, the corresponding phonon modes involve onlyF− ion displacements, though in different directions: atW , F− ions displace along the [011] and [0-11] directions.This may explain the observed gradual onset of superi-onicity in the fluorite structure [64]: as volume increases,further low-energy diffusion pathways corresponding tounstable phonon modes are opened up in the lattice.

Fig. 10 shows the phonon dispersion relations in P62m.With increasing volume, this structure first develops un-stable phonon modes at the Brillouin zone K point, andthe atomic displacements of Ca2+ and F− ions in thecorresponding mode are depicted in Fig. 11. This modeis similar to the unstable mode found in fluorite CaF2

at X, in the sense that it involves displacements of F−

ions and a sublattice of Ca2+ ions which remain fixed.F− ions move along the [120], [210] or [1-10] directions,and all displacements are confined to the ab-plane only.The P62m structure can be visualised as layer-like: inFig. 11, all atoms that are linked by bonds belong to thesame layer, and all ‘isolated’ atoms belong to a differ-ent layer. We find that Ca2+ ions in alternating layersremain fixed in the unstable phonon mode. By analogywith fluorite CaF2, we propose that P62m-CaF2 also un-dergoes a superionic phase transition accompanying thisphonon mode, and we label the region where P62m hasunstable phonon modes in Fig. 8 as superionic (‘S/I’).As is the case for the fluorite structure, it is possiblethat other compounds in the P62m structure could alsoexhibit superionicity.

The pressure-temperature phase diagram of CaF2 hasrecently been examined by Cazorla et al. [59, 65]. Inaddition to the known α, β and γ phases, the authorspropose a high-temperature phase transition from γ toa new δ phase, which in turn is predicted to undergoa superionic transition at even higher temperatures, toa phase labelled ε-CaF2. A P21/c symmetry structurewas proposed for the δ-phase [59], however we find thatthis structure is close to Pnma symmetry, and relaxingit using DFT gives the γ-CaF2 structure. The phase di-agram of Fig. 8 is in qualitative agreement with theseresults, where we identify the δ phase with our predictedP62m structure, and the ε superionic phase with the re-

9

FIG. 11. Illustration of the unstable phonon mode in P62m-CaF2 at the Brillouin zone K point, shown here at 0 GPa.A 3×3×1 slab of the structure is depicted, viewed along thec-axis as in the left-hand panel of Fig. 7(a). Arrows indicatethe direction of movement of atoms in this mode, and arecolour-coded according to their relative amplitudes: yellow forlargest amplitude, through to dark red for smallest amplitude.Calcium atoms are in red, fluorine atoms in green.

gion where P62m has unstable phonon modes.

D. Optical bandgaps in CaF2

Our calculations show that optical bandgaps in CaF2

initially increase but then remain relatively constant overthe pressure range 0−70 GPa (Fig. 3). We also show,using a dashed line Fig. 3, the calculated bandgap forthe P62m-CaF2 phase, which begins to slowly decreaseabove 50 GPa. Semilocal DFT calculations using GGAfunctionals have also shown that this occurs for γ-CaF2

above 70 GPa [66]. The optical bandgap for γ-CaF2

lies 0.9−1.2 eV above that of P62m-CaF2 in the pres-sure range 10−70 GPa, suggesting that the formationof P62m-CaF2 might be detectable in optical measure-ments. We also give the electronic DOS of the P62mphase in the Appendix.

Low temperature CaF2 has been proposed as an in-ternal pressure standard [52]. Our calculations of thebandgap shows that it remains a wide-gap insulator upto at least 70 GPa, and likely retains its superior opticalproperties up until high pressures.

VI. CONCLUSIONS

We have explored Be-, Mg- and CaF2 at pressures upto 70 GPa through DFT calculations and computational

structure searching.BeF2 has a large number of polymorphs at ambient

pressures, and shares many of these with SiO2, such asα, β-quartz and α, β-cristobalite. Our searches show thatBeF2 has open-framework zeolite-like polymorphs, andthat framework structures predicted in BeF2 are energeti-cally relevant to SiO2, highlighting the utility of structuresearching in model systems. At higher pressures, we findthat BeF2 is stable in the moganite structure between11.6 and 30.1 GPa, and stable in the CaCl2 structurebetween 30.1 and 57.5 GPa.

In MgF2, we find that the Pbca-symmetry ‘O-I’ TiO2

structure is a stable intermediary between the pyrite andcotunnite MgF2 phases, and is the lowest enthalpy MgF2

structure between 39.6 and 44.1 GPa. A class of poly-morphs for SiO2 intermediate to the CaCl2 and α-PbO2

structures, which are relevant at Earth mantle pressuresand are close to stability near 100 GPa in SiO2, also occurin MgF2 but at much lower pressures (≈ 10 GPa).

We find that the Fe2P-type P62m-symmetry struc-ture for CaF2 lies close in enthalpy to the known γ-CaF2 phase over the pressure range 0−70 GPa. Cal-culations using the QHA show that this structure is sta-bilised at high pressure and temperature (P & 10 GPaand T & 1500 K). The P62m structure develops unstablephonon modes at high temperatures, which we proposeis associated with a superionic transition in this struc-ture. P62m-CaF2 and its region of phonon instabilityare consistent with the recently proposed δ and ε CaF2

phases.Be-, Mg- and CaF2 are wide-gap insulators. Calcula-

tions using the HSE06 functional show that the bandgapsin BeF2 and MgF2 are tunable with pressure, rising by0.06 eV/GPa and 0.04 GPa over the pressure range 0−70GPa. The optical bandgaps in CaF2 are instead rela-tively constant over this pressure range.

VII. ACKNOWLEDGEMENTS

R. J. N. acknowledges financial support from the En-gineering and Physical Sciences Research Council (EP-SRC) of the U.K. [EP/J017639/1]. C. J. P. acknowl-edges financial support from EPSRC [EP/G007489/2],a Leadership Fellowship [EP/K013688/1] and a RoyalSociety Wolfson research merit award. J. R. N., R. J.N. and C. J. P. acknowledge use of the Archer facili-ties of the U.K.’s national high-performance computingservice, for which access was obtained via the UKCPconsortium [EP/K013688/1], [EP/K014560/1]. J. R. N.acknowledges the support of the Cambridge Common-wealth Trust.

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1

Appendix for:

“High-pressure phases of group II difluorides: polymorphism and superionicity”

J. R. Nelson,1,∗ R. J. Needs,1 and C. J. Pickard2,3

1Theory of Condensed Matter Group, Cavendish Laboratory,J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom

2Department of Materials Science and Metallurgy, University of Cambridge,27 Charles Babbage Road, Cambridge CB3 0FS, United Kingdom3Advanced Institute for Materials Research, Tohoku University,

2-1-1 Katahira, Aoba, Sendai, 980-8577, Japan∗Email: jn336@cam.ac.uk

I. ELECTRONIC STRUCTURE CALCULATIONS

All calculations are carried out with version 8.0 of the Castep plane−wave pseudopotential DFT code.

Our calculations of the enthalpy vs. pressure curves in the main paper and in this appendix, lattice parameters,and supercell phonon calculations all use default ultrasoft pseudopotentials for Be, Mg, Ca, F, Si and O as gener-ated internally by the Castep code and the Perdew−Burke−Ernzerhof (PBE) exchange−correlation functional. Aplane−wave cutoff of 800 eV, Brillouin zone sampling density of at least 2π×0.04 A−1, grid scale of 2 (for representingthe density) and fine grid scale of 2.5 (grid for ultrasoft core or augmentation charges) are used in these calculations.Calculations of phonon frequencies use the finite−displacement supercell method in Castep, and the cumulant forceconstant matrix cutoff scheme [1, 2]. Details of the supercell sizes used in finite−displacement phonon calculationsare given in subsequent sections.

Bandgap calculations (Fig. 3 of the main paper) are instead carried out with norm−conserving pseudopotentials,which are required in order to use hybrid density functionals in Castep. Generation strings for these potentialsare taken from the version 16.0 Castep Norm-Conserving Pseudopotential (‘NCP’) library, and used in version 8.0of the code. Our bandgap calculations use the ‘HSE06’ variant of the Heyd-Scuseria-Ernzerhof hybrid functional, aplane−wave cutoff of 1600 eV, a Brillouin zone sampling density of at least 2π×0.1 A−1, and grid and fine grid scalesof 2 and 2.5 respectively. The structures for bandgap calculations were obtained by geometry optimisation using thesame plane−wave cutoff, Brillouin zone sampling and grid scales, but with the PBE functional instead of HSE06.

Optical bandgaps for BeF2 are calculated for the α-quartz, coesite-I, moganite, CaCl2 and α-PbO2 structures andshown in Fig. 3 of the main paper, over the pressure ranges for which these structures are calculated to be stable atthe static-lattice level. We therefore exclude the small stability range of α-cristobalite BeF2 (0−0.4 GPa), and use thecoesite-I structure instead of coesite-II on the grounds that these two structures are almost identical at low pressures.Optical bandgaps for MgF2 are calculated for the rutile, α-PbO2, pyrite, O-I and cotunnite structures and likewiseshown in Fig. 3 of the main paper. Finally for CaF2, we show optical bandgaps for the fluorite, cotunnite and P62mstructures.

2

II. STRUCTURE DATA

A. Beryllium fluoride structure data

Moganite, space group C2/c (#15)The table below gives the DFT-relaxed BeF2 moganite structure at 20 GPa. A conventional cell (36 atoms) with fullHermann-Mauguin (HM) symbol C12/c1 (standard setting for this space group) is given; the primitive cell has 18atoms.Fig. 2(a) of the main paper illustrates this structure, but uses different lattice vectors, obtained from those belowthrough the relations a′ = a + c, b′ = −b, c′ = −c. This has full HM symbol I12/c1 and a smaller monoclinic angle(β = 90.446◦).

Lattice parameters Atomic coordinates Wyckoff(A, deg.) Atom x y z site

a=12.008 b=4.191 c=7.103 Be 0.0000 0.9572 0.2500 4eα=90.000 β=125.818 γ=90.000 Be 0.1658 0.3268 0.1989 8f

F 0.2909 0.1705 0.2289 8fF 0.1237 0.1811 0.3455 8fF 0.0421 0.2694 0.9492 8f

CaCl2 structure, space group Pnnm (#58)The unit cell is primitive and has Z=2 (6 atoms). The positions below correspond to 50 GPa, and the structure isdepicted in Fig. 2(b) of the main paper.

Lattice parameters Atomic coordinates Wyckoff(A, deg.) Atom x y z site

a=3.796 b=3.959 c=2.445 Be 0.0000 0.0000 0.0000 2aα=90.000 β=90.000 γ=90.000 F 0.2736 0.3233 0.0000 4g

B. Magnesium fluoride structure data

Orthorhombic-I structure, space group Pbca (#61)The unit cell is primitive and has Z=8 (24 atoms). The positions below correspond to 42 GPa, and the structure isdepicted in Fig. 5 of the main paper.

Lattice parameters Atomic coordinates Wyckoff(A, deg.) Atom x y z site

a=4.782 b=4.556 c=8.863 Mg 0.9586 0.2259 0.8870 8cα=90.000 β=90.000 γ=90.000 F 0.1099 0.1721 0.2008 8c

F 0.2339 0.9977 0.4653 8c

C. Calcium fluoride structure data

Fe2P-type structure, space group P62m (#189)The unit cell is primitive and has Z=3 (9 atoms). The positions below correspond to 30 GPa, and the structure isdepicted in Fig. 7(a) of the main paper.

Lattice parameters Atomic coordinates Wyckoff(A, deg.) Atom x y z site

a=5.697 b=5.697 c=3.255 Ca 0.0000 0.0000 0.5000 1bα=90.000 β=90.000 γ=120.000 Ca 0.3333 0.6666 0.0000 2c

F 0.7433 0.0000 0.0000 3fF 0.4126 0.0000 0.5000 3g

3

D. Low−pressure SiO2 polymorphs

Three low−enthalpy structures for BeF2, labelled P212121-I, C2/c − 4 × 2, and P21/c in Fig. 1(a) of the mainpaper, are examined as possible low−enthalpy polymorphs for SiO2. The monoclinic angle in P21/c is close to 90◦

(=89.958◦), and so we also examine a higher symmetry version of this with α=β=γ=90◦, with the space groupPnma. Structure data for these polymorphs are shown below, alongside their enthalpies relative to the α-quartzphase. All data is given at 0 GPa.

C2/c− 4× 2 structure, space group #15 (−15.7 meV/f.u.)A conventional unit cell has Z=8 (24 atoms), and is given below. This has full HM symbol C12/c1.

Lattice parameters Atomic coordinates Wyckoff(A, deg.) Atom x y z site

a=8.940 b=5.031 c=8.977 Si 0.1877 0.1694 0.3129 8fα=90.000 β=111.563 γ=90.000 O 0.2055 0.5806 0.7058 8f

O 0.0000 0.0798 0.2500 4eO 0.2500 0.2500 0.5000 4d

P21/c structure, space group #14 (−5.9 meV/f.u.)The unit cell is primitive and has Z=8 (24 atoms). The cell given below has full HM symbol P121/n1 (ITA uniqueaxis b, cell choice 2).

Lattice parameters Atomic coordinates Wyckoff(A, deg.) Atom x y z site

a=8.583 b=9.626 c=5.207 Si 0.0618 0.1781 0.2632 4eα=90.000 β=89.958 γ=90.000 Si 0.3654 0.3576 0.2649 4e

O 0.1448 0.0263 0.2455 4eO 0.1914 0.2978 0.2191 4eO 0.4278 0.3066 0.5443 4eO 0.4830 0.3001 0.0441 4e

P212121-I structure, space group #19 (−5.5 meV/f.u.)The unit cell is primitive and has Z=8 (24 atoms).

Lattice parameters Atomic coordinates Wyckoff(A, deg.) Atom x y z site

a=5.195 b=8.519 c=9.560 Si 0.2933 0.6088 0.8607 4aα=90.000 β=90.000 γ=90.000 Si 0.7068 0.1942 0.3192 4a

O 0.7237 0.1042 0.4693 4aO 0.9249 0.3308 0.3101 4aO 0.5746 0.7739 0.2020 4aO 0.2499 0.5688 0.3056 4a

Pnma structure, space group #62 (−2.2 meV/f.u.)The unit cell is primitive and has Z=8 (24 atoms).

Lattice parameters Atomic coordinates Wyckoff(A, deg.) Atom x y z site

a=9.415 b=5.285 c=8.683 Si 0.6850 0.2500 0.5529 4cα=90.000 β=90.000 γ=90.000 Si 0.3577 0.2500 0.6389 4c

O 0.5303 0.2500 0.6363 4cO 0.3087 0.2500 0.8176 4cO 0.2041 0.4997 0.0538 8d

4

III. CONVEX HULLS FOR BE, MG AND CA−F AT 60 GPA

In addition to our BeF2, MgF2 and CaF2 searches, we carry out variable stoichiometry structure searches for theBe-F, Mg-F and Ca-F systems at 60 GPa. Convex hulls depicting the results of these searches are shown below.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x in Be

1-xF

x

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

Form

ation E

nth

alp

y (

eV

/ato

m)

BeF2 (α-PbO

2)

Be (hcp) F (Cmce)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x in Mg

1-xF

x

-5

-4

-3

-2

-1

0

Form

ation E

nth

alp

y (

eV

/ato

m)

Mg (hcp) F (Cmce)

MgF2 (Cotunnite)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x in Ca

1-xF

x

-5

-4

-3

-2

-1

0

Form

ation E

nth

alp

y (

eV

/ato

m)

Ca (I41/amd) F (Cmce)

CaF2 (Cotunnite)

CaF3

(Pm3n)

FIG. 12. Convex hulls showing the results of structure searching at the static-lattice level of theory at 60 GPa in the Be-F(left), Mg-F (middle) and Ca-F (right) systems.

In each case, we search over 29 stoichiometries, including the elements (Be or Mg or Ca, and F). The convex hullsdepict the lowest enthalpy structure at each stoichiometry, relaxed to the same standard as the results in the mainpaper (800 eV basis set cutoff, Brillouin zone sampling of 2π×0.04 A−1, PBE exchange).

At 60 GPa, beryllium has the α-Be hexagonal closed-packed (hcp) structure, space group P63/mmc [3], which wealso verify directly through structure searches. We find a molecular crystal comprised of F2 molecules with spacegroup Cmce to be the lowest enthalpy fluorine structure at 60 GPa; this also applies to the Mg-F and Ca-F systems.Of the stoichiometries searched, only BeF2 is stable at this pressure in the Be-F system.

Magnesium undergoes a phase transition from hcp to bcc between about 45 and 60 GPa; within this pressure range,the coexistence of the hcp and bcc phases is observed at low temperatures [4]. Our PBE−DFT calculations find thatthe hcp−Mg phase is very slightly lower in enthalpy than bcc−Mg at 60 GPa, so we use the hcp−Mg phase in ourhulls above. As is the case for BeF2, we find that MgF2 is the only stable stoichiometry in the Mg-F system at 60GPa.

DFT and structure searching calculations predict that the β-tin I41/amd-symmetry structure is the lowest enthalpyphase for calcium at 60 GPa [5], and we also confirm this in the present study. The β-tin phase is well known to beslightly at odds with experimental results, which instead observe simple cubic calcium (or distorted versions of it)at 60 GPa [6], although the β-tin phase has been synthesised near 35 GPa. We carry out our calculations with thelowest enthalpy DFT phase for calcium (β-tin) in the Ca−F hulls above. Our searches find that both CaF2 and CaF3

are stable stoichiometries at 60 GPa.

5

IV. ELECTRONIC AND PHONON DOS OF SELECTED STRUCTURES

In this section we show the electronic and phonon density−of−states (DOS) for some of the newly predictedstructures discussed in the main paper. Specifically, these are the moganite−BeF2 and CaCl2−BeF2 structures,stable in the pressure ranges 11.6−30.1 GPa and 30.1−57.5 GPa respectively (Fig. 13), and the O-I−MgF2 andP62m-CaF2 structures (Fig. 14). The O-I−MgF2 structure is predicted to be stable in the pressure range 39.6−44.1GPa, while the P62m-CaF2 phase is calculated to become stable at high temperature.

The electronic DOS ((a) in the figures below) is calculated with the Optados code using the same DFT parametersas our bandgap calculations (see Sec. I above) and a Gaussian smearing of 0.25 eV. The valence band maximum (VBM)is set to 0 eV. The majority contributor (species and orbital) to each feature in the DOS is labelled in Figs. 13 and14 below. In all cases, we find that the valence bands consist of fluorine 2p orbitals, while the conduction bands comefrom unoccupied p or d orbitals in beryllium, magnesium or calcium.

The dynamic stability of the predicted structures is indicated by the phonon DOS ((b) in the figures below) forthese structures having no protrusion into negative (imaginary) frequencies (shown as grey regions).

-30 -20 -10 0 10 20 300

10

20

Pa

rtia

l D

OS

0 200 400 600 800 10000.0

0.1

0.2

Ph

on

on

DO

S

(eV)

F 2s F 2p

Be 2p

(cm-1

)

(a)

(b)

Band gap

-30 -20 -10 0 10 20 300

1

2

3

4

5

6

Pa

rtia

l D

OS

0 200 400 600 800 1000 12000.00

0.02

0.04

0.06

Ph

on

on

DO

S

(eV)

F 2s F 2p

Be 2p

(cm-1

)

(a)

(b)

Band gap

FIG. 13. Electronic (a) and phonon (b) density of states for the moganite−BeF2 phase at 20 GPa (left panel), and for theCaCl2−BeF2 phase at 50 GPa (right panel). A 4×4×3 (864 atom) and 4×4×6 (576 atom) supercell is used for the moganiteand CaCl2 phonon calculations, respectively.

-30 -20 -10 0 10 20 300

10

20

30

Pa

rtia

l D

OS

0 200 400 600 8000.0

0.1

0.2

Ph

on

on

DO

S

(eV)

F 2s F 2p

Mg 3p

(cm-1

)

(a)

(b)

Bandgap

-40 -30 -20 -10 0 10 200

5

10

15

Pa

rtia

l D

OS

0 200 400 6000.00

0.05

0.10

0.15

Ph

on

on

DO

S

(eV)

Ca

(cm-1

)

(a)

(b)

Bandgap

F 2s

F 2p

Ca 3s

Ca 3p3d

FIG. 14. Electronic (a) and phonon (b) density of states for the O-I−MgF2 phase at 42 GPa (left panel), and for the P62m-CaF2 phase at 30 GPa (right panel). A 4×3×2 (576 atom) and 3×3×6 (486 atom) supercell is used for the O-I and P62mphonon calculations, respectively.

6

V. KNOWN AB2 STRUCTURES USED

A list of AB2-type structures that were considered in addition to our AIRSS searches is given below. The atomsare substituted appropriately, for example Si→Be, O→F, and the structure relaxed. Structures are taken from theInorganic Crystal Structure Database (ICSD) [7] or from literature. We give the 5-digit ICSD reference number forthe former.

In the table below, Z is the number of formula units in the primitive unit cell, and so 3Z is the total number ofatoms in the primitive cell. Our AIRSS searches were restricted to Z ≤ 8.

Compound SG (HM) SG (#) Lattice Z Name(s) ReferenceHfO2 P21/c 14 monoclinic 4 Baddeleyite 60903SiO2 P21/c 14 monoclinic 32 Coesite-II [8]SiO2 C2/c 15 monoclinic 8 Coesite-I [8]SiO2 C2/c 15 monoclinic 6 Moganite 67669BeF2 C2/c 15 monoclinic 6 − [9]SiO2 Pnnm 58 orthorhombic 2 CaCl2-type 26158SiO2 Pbcn 60 orthorhombic 4 α-PbO2

(a) [10]TiO2 Pbca 61 orthorhombic 8 Orthorhombic-I, OI [11]ZrO2 Pnma 62 orthorhombic 4 Cotunnite, Ortho-II (PbCl2) [12]SiO2 P41212 92 tetragonal 4 α-cristobalite 75300SiO2 P42/mnm 136 tetragonal 2 Rutile, β-PbO2, Plattnerite 9160SiO2 P3121 152 trigonal 3 α-quartz 27745SiO2 P6222 180 hexagonal 3 β-quartz 31088

SiO2 Pm3m 221 cubic 46 Melanophlogite 159046(b)

SiO2 Fd3m 227 cubic 2 β-cristobalite 77458(a) Has a number of aliases: Seifertite (SiO2), Scrutinyite (α-PbO2), Columbite (TiO2)(b) Interstitial carbon atoms (representing methane) were removed from this structure before relaxing

For the orthorhombic space groups, a number of different settings are possible with different HM symbols. We listsome of the orthorhombic ones below [13]:

SG (#) Standard setting Alternative settings58 Pnnm Pmnn, Pnmn60 Pbcn Pcan, Pnca, Pnab, Pbna, Pcnb61 Pbca Pcab62 Pnma Pmnb, Pbnm, Pcmn, Pmcn, Pnam

7

VI. QUASIHARMONIC PHASE DIAGRAM FOR CAF2

A. Computational details

To construct the phase diagram shown in Fig. 8 of the main paper, we perform finite-displacement phonon calcu-lations at several different volumes for the Fm3m, P62m and Pnma CaF2 structures. We use 5×5×5 (375 atom),3×3×6 (486 atom), and 5×3×3 (540 atom) supercells for the Fm3m, P62m and Pnma structures, respectively.Volumes corresponding to the following static-lattice pressures are used for these calculations:

Fm3m: −6, −4, −2, 0, 4, 8, 12, 16, 20 GPaP62m: 10, 20, 30, 40, 50, 60, 70 GPaPnma: 0, 2, 6, 10, 20, 30, 40, 50, 60, 70 GPa

This gives us the phonon free energies F (V, T ) at each volume for these structures.We fit the resulting energies with a polynomial in V . For the Fm3m and P62m structures, the calculated F (V, T )

values vary reasonably slowly with volume and are well reproduced by a quadratic in V . For the Pnma structure, wefind that the calculated F (V ) values tend to initially increase rapidly with decreasing volume, but then increase lessrapidly at lower volumes (see Fig. 15). A cubic polynomial in V is therefore used to capture this trend. Some insightinto this behaviour can be found by looking at the dependence of the lattice parameters of these structures on V ; forFm3m and P62m, the lattice parameters steadily decrease on compression. The Pnma structure, however, showsslightly unusual behaviour on compression in that lattice parameter a initially decreases, but then starts increasingfor static-lattice pressures above 56 GPa, while lattice parameter c decreases more rapidly at that pressure as well(Fig. 16).

Static-lattice calculations provide relaxed geometries for our calculations, and give us sets of enthalpies H, volumesV and static-lattice pressures (or electron pressures) Pst. Within the QHA, at a given volume V , the correspondingGibbs free energy is the sum of the electronic enthalpy and phonon free energy: G(V, T ) = H(V ) + F (V, T ). Thepressure P at that volume is P = Pst + Pph, where Pph ≡ −∂F (V, T )/∂V is the phonon pressure.

34 36 38 40 42 44 46

Volume (Å3/f.u.)

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

Fre

e en

ergy

(eV

/f.u.

)

0 K

500 K

1000 K

1500 K

2000 K

Fm3m26 28 30 32 34 36

Volume (Å3/f.u.)

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

Fre

e en

ergy

(eV

/f.u.

)

0 K

500 K

1000 K

1500 K

2000 K

P62m24 26 28 30 32 34 36 38 40

Volume (Å3/f.u.)

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5F

ree

ener

gy (

eV/f.

u.)

2000 K

0 K

500 K

1000 K

1500 K

Pnma

FIG. 15. Phonon free energies F (V ) for the Fm3m (left), P62m (middle) and Pnma (right) structures of CaF2, as a functionof volume and of temperature. The ‘+’ marks indicate values from supercell calculations, while the solid curves show theresulting polynomial fits to F (V ).

B. Validity of the QHA

Here we calculate the volume coefficient of thermal expansion:

α(P, T ) =1

V

(∂V

∂T

)P

at a variety of pressures for the Fm3m, P62m and Pnma calcium fluoride structures. The results are shown inFig. 17.

8

34 36 38 40 42 44 46

Volume (Å3/f.u.)

3.5

3.6

3.7

3.8

3.9

4.0

4.1

Latti

ce p

aram

eter

(s)

(Å)

a (=b=c)

Fm3m26 28 30 32 34 36

Volume (Å3/f.u.)

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

Latti

ce p

aram

eter

s (Å

)

a (=b)

c

P62m24 26 28 30 32 34 36 38 40

Volume (Å3/f.u.)

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

7.5

Latti

ce p

aram

eter

s (Å

)

a

b

c

Pnma

FIG. 16. Lattice parameters of the Fm3m (left), P62m (middle) and Pnma (right) structures of CaF2 as a function ofvolume, calculated at the static lattice level. The lattice parameter for Fm3m is given for the primitive cell (Z=1, a=b=c,α=β=γ=60◦); the lattice parameter for the corresponding conventional cell (Z=4) is

√2 times larger. Our simulation cell for

the Pnma structure has a < c < b, and is actually set in the alternative Pmcn setting for that space group (#62).

The calculated thermal expansion coefficient can be used to assess the validity of the QHA [14, 15]. We can considerthe QHA valid at temperatures lower than the higher temperature inflection point in α(P, T ), i.e. where[

∂2α(P, T )

∂T 2

]P

changes sign. Beyond this point, we generally find that[∂2α(P,T )∂T 2

]P≥ 0 and the QHA is expected to be less

applicable with further increases in temperature, as anharmonic effects become more important.

For the Fm3m and P62m structures, Fig. 17 shows this inflection point in α(P, T ) using open circles at each pressure.Fitting the resulting (P, T ) coordinates of these inflection points gives us the validity rule T (K) ≤ 28P (GPa) + 453given in the main paper. We note that locating the inflection point is subject to some numerical uncertainty, becauseα(P, T ) changes quite slowly in the vicinity of this point (Fig. 17). An uncertainty of about 100 K is therefore quotedin the main paper.

This inflection point is harder to locate in the α(P, T ) curves for the Pnma structure. The thermal expansioncoefficients are slightly unusual in that they initially increase, but then decline at higher temperatures. This is partlydue to the fact that the phonon pressure in Pnma declines at higher volumes, as can be seen in the slope of theF (V, T ) curves for Pnma in the right−hand panel of Fig. 15.

0 200 400 600 800 1000 1200 1400Temperature (K)

0

20

40

60

80

100

120

140

α (

x10

-6 K

-1)

0 GPa 5 GPa10 GPa

Fm3m0 1000 2000 3000 4000

Temperature (K)

0

10

20

30

40

50

60

70

α (

x10

-6 K

-1)

20 GPa30 GPa40 GPa

P62m0 500 1000 1500 2000 2500 3000

Temperature (K)

0

10

20

30

40

50

α (

x10

-6 K

-1)

10 GPa20 GPa30 GPa40 GPa

Pnma

FIG. 17. Calculated volume thermal expansion coefficients α(P, T ) at a variety of pressures as a function of temperature forthe Fm3m (left), P62m (middle) and Pnma (right) CaF2 structures. The open circles in the left and middle panels showinflection points, where d2α/dT 2 changes sign.

9

VII. UNSTABLE MODES IN Fm3m AND P62m CAF2

In Figs. 9 and 10 of the main paper, the Fm3m and P62m structures develop unstable phonon frequencies withdecreasing pressure / increasing volume. For the Fm3m structure, imaginary (‘negative’) phonon frequencies firstdevelop at the Brillouin zone X point, and then at larger volumes at the W point. Fig. 18 shows the smallest modefrequency at X and W as a function of volume for this structure. The solid blue points in Fig. 18 were obtainedusing a 2×4×4 supercell (96 atoms) for Fm3m, as the X and W points are both commensurate with a supercellof this size. For comparison, red crosses in Fig. 18 show the results obtained using a 5×5×5 supercell (375 atoms),the size used in our phase diagram calculations. The X and W points are not commensurate with this supercell, sothe frequencies shown with red crosses have been obtained by interpolation. These results differ from the exactlycommensurate 2×4×4 supercell results by <4 cm−1. The thin solid and dotted black lines connecting the blue pointsare shape−preserving interpolants fitted to the data and are intended to guide the eye.

38 40 42 44 46 48 50 52

Volume (Å3/f.u.)

-100

-50

0

50

100

150

200

Fre

quen

cy a

t X o

r W

(cm

-1)

X W

Fm3m

32 33 34 35 36 37 38

Volume (Å3/f.u.)

-40

-20

0

20

40

60

80

100

120

Fre

quen

cy a

t K (

cm-1

)

P62m

K

FIG. 18. (Left) Lowest mode frequency at the Brillouin zone X (left−hand curve) and W (right−hand curve) points forFm3m-CaF2, as a function of volume. (Right) Mode frequencies at the Brillouin zone K point for P62m-CaF2 as a functionof volume. Imaginary frequencies are shown as negative values.

For the P62m structure, imaginary phonons develop at the Brillouin zone K point, and the right−hand panel ofFig. 18 also shows the lowest mode frequency at this point as a function of volume. The solid blue points are obtainedusing a 3×3×1 supercell (81 atoms) commensurate with K, while the red crosses show results using a 3×3×6 supercell(486 atoms), as used in phase diagram calculations. As with the Fm3m structure, the red crosses are interpolations;however, we note that K is also commensurate with this larger supercell, and the interpolated results are almostidentical to the exactly commensurate ones in this case. The thin solid and dotted black lines are again guides to theeye.

The volume at which these structures first develop imaginary phonon frequencies directly affects the phase diagramof Fig. 8 of the main paper. In particular, this volume determines the location of the dashed lines between thered/light−red and blue/light−blue regions of the phase diagram. We have taken this volume to be simply halfwaybetween the first blue point in Fig. 18 with a frequency >0 cm−1, and the first ‘negative-frequency’ blue point. ForFm3m, this corresponds to V = 46.47 A3/f.u., and is at a static−lattice pressure between −7 and −6 GPa. ForP62m, we have V = 37.70 A3/f.u., occurring between 0 and 1 GPa.

10

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