high pressure phase transitions in mg1–xcaxo: theory

Phys. Status Solidi B 248, No. 8, 1901–1907 (2011) / DOI 10.1002/pssb.201046508 p s sb

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basic solid state physics

High pressure phase transitionsin Mg1–xCaxO: Theory

Anurag Srivastava*,1, Mamta Chauhan1, R. K. Singh2, and Rishikesh Padegaonker3

1 Advanced Material Research Lab, Indian Institute of Information Technology and Management, Gwalior 474010, India2 Department of Physics, ITM University, Gurgaon 122017, India3 Indian Embassy School, P.O. Box 1154, Sana, Republic of Yemen

Received 18 September 2010, revised 8 December 2010, accepted 19 February 2011

Published online 25 March 2011

Keywords charge transfer, density functional theory, MgCaO, phase transitions, solid solutions

* Corresponding author: e-mail [email protected], Phone: þ91 751 244 9826, Fax: þ91 751 244 9813

We have analysed a B1!B2 structural phase transitions in

Mg1–xCaxO solid solutions and their ground state properties by

using first principle density functional theory and charge

transfer interaction potential (CTIP) approach. The effects of

exchange-correlation interactions are handled by the general-

ized gradient approximation with Perdew–Burke–Ernzerhof

type parameterization. CTIP approach includes the long range

modified Coulomb with charge transfer interactions and short

range part of this model includes the van der Waals as well as

Hafemeister Flygare type overlap repulsive interactions. The

study observes a linear variation of calculated transition

pressure, bulk modulus and lattice parameter of Mg1�xCaxO

as a function of Ca composition. The observed results for the

end point members are in agreement to their experimental

counterparts and the deviations have been discussed.

� 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction During the last few years the scientificcommunity has seen a rapid progress in the understanding ofthe behaviour of major mineral phases of the earth’s mantle.Recent advancements in the theory and computationaltechniques have made it possible to predict the structuralproperties of these materials through the entire pressurerange of the mantle. These kinds of studies have provided usknowledge about the fundamental changes in bondingnatures of many materials present deep inside the earth,where oxygen is the most abundant element in the earth,constituting about 43 percent by weight of the crust andmantle. While most of this mass is incorporated into silicates,the next most abundant mineral group in the planet is theoxides. In contrast to the simple oxides of monovalentcations, many oxide minerals of divalent cations are known,and several of these are of major geophysical significance. Ofthese minerals, the periclase groups with rocksalt (NaCl)structure are the most widely studied at non-ambientcondition. At least 30 monoxide minerals and compoundsare known, two-thirds of which crystallize in the cubic halite(B1) or rocksalt structure (space group Fm3m) at roomtemperature and pressure. The rocksalt structure monoxides,due to their structural simplicity provide insight into themost fundamental concepts of crystal chemistry and phase

transitions, such as size, compressibility and thermalexpansivity of cations and anions.

The D layer is a bottom layer of lower mantle andtherefore is a key to a number of problems on the dynamicsand evolution of the earth. Seismic anisotropy carriespotentially important information on the dynamics of thesolid earth which can be understood by the investigation ofthe mineral physics of anisotropy involving the elasticanisotropy of the lower mantle minerals. The relativestability of different crystallographic phases and possiblehigh-pressure phase transformations among them have beenlong interesting interest in alkaline earth oxides. However,the oxides have been discussed a lot but few importantproperties such as ultrahigh piezoelectric efficiency, colossalmagneto-resistance and high oxygen permeation throughfuel cell membranes occur only when corresponding oxidesform solid solution, most of which are distorted. In thepresent paper we consider the simplest MgO–CaO system.However, magnesiowustitle (Mg,Fe)O is believed to be amajor constituent of the Earth’s lower mantal, so that itwould be an ideal system for study, but the FeO leads to acomplex Mott insulator behaviour. The present (Mg,Ca)Osystem can be considered as a step in the direction ofunderstanding the solid solution in the minerals, as well as


1902 A. Srivastava et al.: High pressure phase transitions in Mg1–xCaxOp

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important in understanding the role of Ca element. Theoptimization of the scientific knowledge about the solidsolutions is not only vital to understand the behaviour ofternary compounds but also essential for manufacturing thematerials reproducibly, effectively and economically. Theunderstanding of oxides can be useful from the perspective offerroelectric solid solutions and high temperature-supercon-ductors. A number of methods have been applied to study theMgO–CaO system. The electronic structure and total energycalculations using FPLAPW methods [1] have been used forthe constituents of this solid solution. A potential model [2]was developed to compute the phase diagram of the MgO–CaO solid solution. An ionic potential approach has alsobeen applied to study the structural phase transitions andelastic behaviour of the constituents of this solid solutionunder high compression [3, 4]. Recently, films of MgO andMgCaO ternaries were grown by Stodilka et al. [5] at lowtemperature as low temperature alternatives to SiO2 gatedielectrics for SiC MOS applications. A severe immiscibilityis observed [6] in the synthesis of CaO–MgO solid-solutionby using bulk techniques due to the large difference in ionicradius between Mg and Ca which makes difficult the use ofbulk techniques for their fabrication. Using composition-spread technique, Nishii et al. [7] have grown metastableMg1�xCaxO solid solution films on ZnO layers by pulsedlaser deposition. Miloua et al. [8] have investigated theground state properties and the stability of Ca1�xMgxOmixed oxides using full potential linearized augmented planewave (FP-LAPW) method in combination with the localdensity approximation to the exchange correlation potential.Stolbov and Cohen [9] have calculated the equilibriumlattice constants, electronic structure, and formation energyfor ordered and disordered solid solutions Mgo–CaO. Due tothe geophysical and industrial importance of these materialsand the success of CTIP model and ab initio method, wethought it pertinent to explore the extent of our computationin reference to this solid solution.

High pressure study may also be useful in the mantle ofSuper-Earth planet with �7 earth masses where pressure of�10 Mbar and temperature of �4000 K [10] is predicted. Asexperimentally it is not possible to study the behaviour ofmaterials found there at such a higher pressure andtemperature, our ab initio study may be meaningful forpredicting the behaviour of materials. The pressure inducedstructural phase transformation of MgSiO3 perovskite (withpbnm symmetry), the most abundant mineral of Earth’slower mantle, to CaIrO3-type postperovskite (with Cmcmsymmetry) at pressure–temperature conditions similar tothose expected near the core–mantle boundary of the Earth(�125 GPa and 2500 K) was discovered by Murakami et al.[11], Tsuchiya et al. [12] and Oganov and Ono [13]. Duringthe discovery of next polymorph of this compound it wasfound by Umemoto [10] that CaIrO3-type polymorph ofMgSiO3 dissociates into CsCl-type MgO and contunnite-type SiO2 at 11.2 Mbar. Similar dissociation was found byUmemoto et al. [14] at low pressure of �40 GPa in NaMgF3,a low pressure analog of MgSiO3. Two potential candidates


for post-PPV transitions in NaMgF3 were also found byUmemoto and Wentzcovitch [15] instead of dissociation.The high pressure study became interesting to understand thematerials behaviour at extreme conditions of pressure andtemperature found in Super-Earth planets [16].

MgO and CaO under normal conditions crystallize inrocksalt (B1) type structure and under compression trans-forms to CsCl (B2) type structure. Most of the investigations[2–4, 17–20] have explained their high-pressure structural,elastic and thermodynamical properties. In the present note,we have presented the structural phase transition and groundstate properties of mixed Mg1�xCaxO compounds, using aCTIP model and ab initio method. The essential theory andmethod of computation is presented in the following section.

2 Computational methodologies: A brief2.1 Ab initio approach The present computation of

structural stability and pressure induced phase transition inearth oxides has been done by using density functional theory(DFT) based SIESTA code [21] which is suitable forelectronic structure calculations and ab initio moleculardynamics simulations of molecules and solids. Underthe framework of density functional theory [22] first-principle pseudopotential approach has been used for thephase transition from original rocksalt (B1) to CsCl (B2) typephase in the host binary oxides MgO and CaO and inmixed Mg1�xCaxO compounds. The effects of exchange-correlation interactions are handled by the generalizedgradient approximation (GGA) with Perdew–Burke–Ernzerhof (PBE) [23] type parameterization in the selfconsistent run for these compounds. The pseudopotentialused in the present calculation is norm conserving and non-relativistic. The electronic wave functions are expanded in aplane wave basis set with energy cut-off of 200 Ry for MgO,Mg0.75Ca0.25O, Mg0.5Ca0.5O and Mg0.25Ca0.75O and 150 Ryfor CaO. The atomic orbital basis set employed in ourcalculations is double-z with polarization for Mg 3s and Ca4s states whereas only double-z for O 2p state. For k-pointsampling, sufficient k points are used as obtained byconvergence test for these compounds. To determine thestructural ground state properties, total energy is calculatedfor different lattice constants around the equilibrium latticeconstant for which the total energy of the system is found tobe the lowest, by the self consistent run of Kohn Shamequations [24]. The calculated total energies and theircorresponding unit cell volumes are then fitted to theMurnaghan’s equation of state [25] to get equilibriumground state properties like lattice parameter, bulk modulusand first pressure derivative of bulk modulus.

2.2 Charge transfer interaction potential (CTIP)approach The Gibbs free energy describes the stability ofthe system at a given set of thermodynamic condition and canbe written as

G ¼ U þ PV�TS: (1)

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Phys. Status Solidi B 248, No. 8 (2011) 1903

Original

Paper

Here U is the internal energy which at 0 K is equal to thecohesive energy, S is the vibrational entropy at absolutetemperature T, and P is the pressure corresponding to thevolume V. The phase transition is a phenomenon where theparental structure transforms to the most stable structure atelevated pressure and/or temperature. The stability of anysystem can be explained on the basis of minimization of thetotal energy. The ionic oxides undergo structural phasetransformation from B1 to B2 structure. If GB1 and GB2 arethe Gibbs free energies of parental B1 (NaCl) and finallystable B2 (CsCl) structure then at absolute 0 K the phasetransformation can be found when

U

U

Figure 1 (online colour at: www.pss-b.com) Lattice constant as afunction of Ca composition (x) in Mg1�xCaxO.

www

GB1�GB2 ¼ 0

or UB1 þ PVB1 ¼ UB2 þ PVB2

or P ¼ UB1�UB2

VB2�VB1

:

(2)

Here VB1 and VB2 are the unit cell volumes of the respectiveB1 and B2 structures and UB1 and UB2 can be expressed as

B1 ¼ � 1:7476Ze2

r

� �½Z þ 12f ðrÞ�� C

r6�D

r8

þ 6bbþ�expðr1 þ r2�rÞ

rþ 6bbþþexp

ð2r1�krÞr

þ 6bb��expð2r2�krÞ

r;

(3)

B2 ¼ � 1:7627Ze2

r0

� �½Z þ 12f ðr0Þ�� C0

r06�D0

r08

þ 8b0bþ�expðr1 þ r2�r0Þ

r0þ 3b0bþþexp

ð2r1�kr0Þr0

þ 3b0b��expð2r2�kr0Þ

r:

(4)

Figure 2 (online colour at: www.pss-b.com) Bulk modulus as afunction of Ca composition (x) in Mg1�xCaxO.

Cohesive energies as expressed by above equationscontain range parameter bðb0Þ hardness parameter rðr0Þ andcharge transfer parameter f ðrÞðf ðr0ÞÞ and have beencomputed with the help of equilibrium conditions:d2UðrÞ=dr2 ¼ 0; the isothermal bulk modulus BT ¼ð1=ð9krÞÞ½d2UðrÞ=dr2�r¼r0

, and the knowledge of secondorder elastic constants and lattice parameter [18, 26–28]. Theexpressions for second order elastic constants are given insome other reference [29]. The values of model parametershave been computed for initial thermodynamic conditionsand then the pressure variations of the model parametershave been computed and also the different lattice energiescorresponding to the various pressures. The pressure wherethe difference in the two competitive energies comes out tobe zero the phase transition phenomenon occurred.

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3 Results and discussions In the present high-pressure study of MgO–CaO solid solution, we haveperformed first the stability analysis in its original B1(NaCl) type phase and high pressure B2 (CsCl) type phase,by using ab initio and CTIP approaches. The study alsocomputes the lattice constant and bulk modulus as the groundstate properties of Mg1�xCaxO and shown their variationas function of Ca composition (x) in Figs. 1 and 2. Thecalculated ground state properties by ab initio method andCTIP method are tabulated in Table 1 and compared with theother calculated theoretical and/or experimental values. Incase of ground state properties calculated for two end pointmembers of Mg1�xCaxO solid solution, our ab initio valuesare much closer to the experimental and theoretical results,whereas the computational findings with CTIP are somewhatdeviated from present ab initio results and other findings. InFigs. 1 and 2 present calculated ab initio values are shown bysolid line with circular dots whereas present CTIP values areshown by dotted line with square dots. Third line can bedrawn by connecting the two experimental values reportedfor the two end point members of Mg1�xCaxO or byinterpolation method applied on two host MgO and CaOcompounds shown by triangular dots.



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Table 1 Lattice constant (a), bulk modulus (B0), pressure derivative (B0’) and phase transition pressure (PT) of Mg1�xCaxO using abinitio method and CTIP method.

Mg1�xCaxO a (A) B0 (GPa) B00 Pr (GPa)

B1 B2 B1 B2 B1 B2

x¼ 0 Present 4.165a, 2.67a 176.01a, 171.95a 3.25a 3.19a 495.00a,4.44b 222.0b – – – 288.50b,

Experimental 4.213c, 4.20d, 4.212e – 160.0c, – 4.15c, – >100w

156.0f, 4.7f,Others 4.17g,4.165h, 4.213i, 4.259j, 2.604h, 173.15g, 163.0h, 4.29h, 3.396h, 3.54m, 515l,

4.25k, 4.167l, 4.162m, 2.6m, 171.0h, 170.0m, 4.01j, 2.94n 451k,4.20n, 4.214o 2.572n, 167.6i, 193.0n 4.26k, 1050s,

2.604i 160.0j, 4.09l, 220n,159.7k, 3.40m, 251t

172.0l, 3.53n

185.9m,186.0n

x¼ 0.25 Present 4.48a, 2.73a 134.53a, 130.14a 3.77a 4.17a 265.00a,4.58b 200.27b 244.64b

Experimental – – – – – – –Others 4.35g – 157.12g – – – –

x¼ 0.50 Present 4.65a, 2.81a 131.15a, 126.07a 3.54a 4.07a 165.00a,4.7b 178.70b 200.00b

Experimental – – – – – – –Others 4.49g – 144.41g – – – –

x¼ 0.75 Present 4.812a, 2.907a 118.76a, 109.01a 3.64a 4.11a 125a,4.82b 156.78b 156.68b

Experimental – – – – – – –Others 4.61g 135.84g – – – –

x¼ 1 Present 4.953a, 2.969a 106.47a, 104.66a 3.62a 4.34 86.00a,4.920b, 135.00b, 113.00b,

Experimental 4.81p – 110.0p, – 4.26p – 70� 10v,116.1q, 63x,

Others 4.71g, 2.855h 127.0g, 132.8h 4.11h, 4.37h 55.0t,4.72h, 4.810o, 128.0h, 4.47l, 4.0j, 54.2l,4.714l, 4.84j, 129.0l 4.41r 121u

4.838r

aPresent ab initio result, bPresent CTIP result, cRef. [27], dRef. [40], eRef. [41], fRef. [42], gRef. [8], hRef. [43], iRef. [44], jRef. [45], kRef. [38], lRef. [1],mRef. [46], nRef. [35], oRef. [39], pRef. [47], qRef. [28], rRef. [48], sRef. [37], tRef. [36], uRef. [49], vRef. [33], wRef. [34], xRef. [50].

The comparison of trends as shown in graphs representsthat the present CTIP values of lattice constant follows theVegard’s law, however, the values are comparatively higher,whereas the ab initio results are slightly deviated fromVigard’s law but closer to the experimental trend. In case ofbulk modulus, the predicted CTIP values follow the lineardependence on composition x and are again showing a cleardeviation from their experimental counterparts, wheras theab initio results are much closer to the experimental trendand only slightly deviated from linear dependence oncomposition x.

Ab initio values of lattice constant follow an increasingquadratic function of composition x as given in the followingequation:

� 20

aðxÞ ¼ 0:661x3�1:410x2 þ 1:537xþ 4:166: (5)

11 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

The deviation of CTIP results with our ab initio andothers may be due to the use of Vigard’s law in CTIP model
for generating the input parameters for intermediatecompounds whereas in ab initio the calculation have beenmade for different compositions by replacing the percentageof atoms out of the total atoms taken into consideration forthe host material. Another reason of deviation might be thetype of interactions taken into consideration in both theapproaches, like van der Waals and charge transferinteractions that becomes very dominant under highcompression. In case of doping, it is not necessary thatintermediate values should always follow the Vigard’s law,which has recently been supported by Murphy and Chroneosin their article [30]. They have discussed in detail thepossible reasons of deviation from Vigard’s law in case ofsolid solution of AlAs and GaAs as the deviation from anylinear interpolation depends on (i) the relative size of the
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Figure 5 Enthalpy as a function of pressure for Mg0.5Ca0. 5O.

constituents, (ii) the relative volume per valence electron,(iii) Brillouin-zone effects and (iv) electrochemical differ-ences between the elements. However, a similar kind ofexplanation was reported by Axon and Hume-rothery [31].In another article, Chizmeshya et al. [32] discussed theexperimental and theoretical study of deviation fromVigard’s law in their article on a different solid solutionSnxGe1�x.

For the determination of phase transition pressure wehave plotted enthalpy (H¼UþPV) as a function of pressureat zero temperature and the pressure at which enthalpies ofboth the B1 and B2 type phases are found to be equal has beenconsidered as the phase transition pressure and shown inFigs. 3–7. The findings observed that our computedtransition pressures of Mg1�xCaxO solid solutions by theab initio and CTIP method are in good match with each otherexcept for host MgO. Here, in case of host CaO, transitionpressure calculated by ab initio method is much closer to theexperimental value [33] as compared to that calculated byCTIP method, whereas in case of host MgO no accurateexperimental value of phase transition is reported yet andone conclusion [34] has been drawn that this pressure mustbe greater than 100 GPa. However, some theoretical findingspredicted the lower transition pressure for MgO at around200 GPa [35, 36] and few reported its higher value of1050 GPa [37] and best estimated value of phase transitionpressure is found to be 500 GPa [38, 39]. In view of the best

Figure 6 Enthalpy as a function of pressure for Mg0.25Ca0.75O.

Figure 7 Enthalpy as a function of pressure for CaO.

Figure 3 Enthalpy as a function of pressure for MgO.

Figure 4 Enthalpy as a function of pressure for Mg0.75Ca0.25O.

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estimates for the host binary oxides of Mg1�xCaxO, our abinitio results are very close to its experimental and theoreticalcounterparts, whereas in case of solid solutionsMg0.75Ca0.25O, Mg0.5Ca0.5O and Mg0.25Ca0.75O no exper-imental or theoretical evidences are available to our knowl-edge for the phase transition.

In Fig. 8 we have plotted phase transition pressure(PT) as a function of Ca composition for mixed oxides,computed by ab initio and CTIP approaches, where it isobserved that the CTIP values are following the linear



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Figure 8 (onlinecolourat:www.pss-b.com)Transitionpressureasa function of Ca composition (x).

dependence on composition whereas our ab initio resultsillustrates a quadratic trend. Contrast in the two findings maybe due to the reasons discussed earlier in our article and thetype of interactions involved in two approaches andhence needs more discussion on the behaviour of shortrange interactions specifically the van der Waals and chargetransfer under high-pressure conditions, as these interactionshave exclusively been taken into consideration in CTIPmodel. In another observation, the non-zero values of chargetransfer parameter f(r) very well supports the Cauchydiscrepancy (C11 6¼C12) present in these materials anddefends the importance of non-central forces. The systematicchange seems to be related to the differences in bondingnature of cations. These results suggest that the degree ofnon-central force in alkaline earth oxides increases withdecreasing volume by replacing the large cation with thesmall one or by compression.

4 Conclusion In the present comparative study of twoapproaches in prediction of stability of B1 and B2 phases,structural phase transition and ground state properties ofMg1�xCaxO, the structures of host binary as well as solidsolutions are found to be stable in its low pressure B1 phaseand high pressure B2 phase. The B1 to B2 phase transitionhas been observed in the solid solutions for the whole rangeof Ca composition. The variation of lattice constant, bulkmodulus and transition pressure as a function of Cacomposition has also been discussed and the deviations inthe results of two approaches used in this study have beenanalysed in terms of short range forces, specially the van derWaals and charge transfer. Hence, in view of the aboveobservations and discussions we conclude that the findings ofthis paper will certainly stimulate the work on the mixedsystems and may serve as input to other theoretical andexperimental workers.

Acknowledgements The authors are thankful to Dr. S.L.Chaplet, BARC for his valuable discussions. RP and MC, the co-authors are thankful to ABV-IIITM, Gwalior, for providing thenecessary facility during our stay at IIITM, Gwalior.


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high pressure phase transitions in mg1–xcaxo: theory

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