PHYSICAL REVIEW B 76, 064116 共2007兲

First-principles prediction of crystal structures at high temperatures using the quasiharmonic approximation Pierre Carrier and Renata Wentzcovitch Minnesota Supercomputing Institute and Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, USA

Jun Tsuchiya Geodynamics Research Center, Ehime University, Matsuyama, Japan 共Received 27 April 2007; revised manuscript received 19 July 2007; published 23 August 2007兲 We show here how first-principles quasiharmonic approximation 共QHA兲 calculations in its simplest statically constrained form can be used to predict crystal structures at high temperatures. This approximation has been extensively used to investigate thermodynamic properties of Earth forming minerals and has offered excellent results for the major mantle phases at relevant conditions. We carefully compare QHA predictions of crystal structures using the local density approximation with crystallographic data in MgSiO3 perovskite at high pressures and temperatures. Small but systematic deviations in the lattice parameters 共at most 0.3%兲 appear at high temperatures 共T ⬎ 2000 K兲 and are associated with the development of deviatoric thermal stresses. An iterative scheme is proposed to eliminate these spurious thermal stresses and further improve the quality of the predictions of this popular and successful thermodynamics method. DOI: 10.1103/PhysRevB.76.064116

PACS number共s兲: 62.20.Dc, 65.40.⫺b, 91.35.⫺x, 91.60.Gf

II. STATICALLY CONSTRAINED QUASIHARMONIC APPROXIMATION

I. INTRODUCTION

The quasiharmonic approximation1 共QHA兲 is a simple and powerful method for evaluating free energies using density functional theory.2 As opposed to molecular dynamics, its only available alternative so far, the QHA remains inexpensive computationally and valid below the Debye temperature. Here, we describe its simplest and most used form, the statically constrained QHA, indicate its conditions of validity, test its predictions for crystal structures against experimental data, and show that the quality of these predictions can be further improved by the relaxation of deviatoric thermal stresses. An ideal QHA calculation should, therefore, involve a self-consistent cycle that minimizes deviatoric thermal stresses to a predefined level, and we propose a systematic scheme to achieve this. Systematic approaches are important especially when investigating materials with a large number of structural parameters. Crystallographic measurements at combined high pressures and temperatures are also very challenging, and there are few studies to date3 that combine both high pressures and temperatures. In addition, most of these measurements register only lattice parameters, not internal ones. As an example, we chose MgSiO3 perovskite 共pv兲, the most abundant mineral in the Earth’s mantle. It has a nontrivial crystal structure with 20 atoms per unit cell and ten structural degrees of freedom, and is one of the most studied materials at high pressures and temperatures.4–9 Current crystallographic data on MgSiO3 pv vary up to 60 GPa and 2600 K, a condition pertaining to the Earth’s lower mantle. We analyze and explain why predictions of the QHA compare well with experiments and, most importantly, why they should be reliable at Earth’s lower mantle conditions. 1098-0121/2007/76共6兲/064116共5兲

The free energy according to the statically constrained QHA is given by



F共V,T兲 = U共V兲 + 兺 qj



ប␻qj共V兲 + kBT 兺 ln共1 − eប␻qj共V兲/kBT兲, 2 qj 共1兲

where U共V兲 is the static energy versus volume obtained after a full structural relaxation under isotropic pressure. ␻共V兲 is the phonon spectrum at these fully relaxed structures. The second term is the zero-point motion energy FZP, and the sum of the two terms in brackets is the energy at T = 0 K. The last term in Eq. 共1兲 is the thermal excitation energy Fth. Boltzmann’s and Planck’s constants are, respectively, kB and ប. The entropy S and pressure P are then obtained from F using standard thermodynamic relations,1

冏 冏

S=−

⳵F ⳵T

and

冏 冏

P=−

V

⳵F ⳵V

.

共2兲

T

The known quantities U, T, V, S, and P directly give the Gibbs free energy 共3兲

G = U − TS + PV.

The isothermal elastic moduli can then be evaluated from the second derivative of G with respect to the strains ⑀i and ⑀ j 共in Voigt’s notation兲,

冏 冏

cTij共P,T兲 =

⳵ 2G ⳵ ⑀ i⳵ ⑀ j

.

共4兲

P,T

At this point, it is fundamental to note that the crystal structure and phonon frequencies depend on volume alone.

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©2007 The American Physical Society

PHYSICAL REVIEW B 76, 064116 共2007兲

CARRIER, WENTZCOVITCH, AND TSUCHIYA

This is because all structural parameters under pressure have been determined by static calculations and are given at Pstat共V兲 only. Note that at a certain V, Pstat indeed depends on the exchange correlation functional used, but since the structure/volume relation depends only mildly on this functional, structural predictions at high P-T’s through volumetric effects carry only a very mild dependence on the functional. Then, at T = T⬘, the pressure P⬘共V , T⬘兲, evaluated using expressions 共1兲 and 共2兲, contains contributions from zero-point and thermal energies, i.e., P⬘ = Pstat + PZP + Pth, where

,

-5

HA ali dQ e

bo

unda ry mantle adiabat 60 GPa 100 GPa pv-ppv phase boundary

0

1000

150 GPa

2000

3000

4000

Temperature [K]

FIG. 1. 共Color online兲 Thermal expansivity as a function of temperature for various pressures. The black line, labeled “QHA boundary,” is defined by the position of the inflection points of ␣共P , T兲, as described in the text. The melting curve 共Ref. 16兲, the location of the mantle adiabat 共Ref. 19兲, and the postperovskite phase boundary 共Ref. 14兲 are also shown.

the QHA, indicating that anharmonicities decrease with pressure in the absence of phonon softening. The location of the experimental melting curve of Zerr and Boehler16 is also shown. At melting, anharmonic interactions are no longer negligible.17,18 As can be seen, the valid QHA domain is clearly well below the melting curve. The QHA also remains valid along the whole mantle geotherm,19 except maybe at conditions at the top of the lower mantle, i.e., 23.5 GPa where MgSiO3 pv becomes the stable structure. We finally 4.7 4.6

Fiquet et al (1998) Utsumi et al (1995) Wang et al (1994)

4.5 4.4 4.9 4.8

Funamori et al (1996) Ross & Hazen (1989,90) theory

4.7 P⬘

In v

QHA

urv

共6兲

4.6 6.8

resulting from statically constrained calculations. Experiments show that at high temperatures, ␣ should increase linearly with T.15 However, quasiharmonic calculations display a superlinear behavior. Figure 1 shows the calculated thermal expansivity as a function of temperature for various pressures applied to MgSiO3 pv. The inflection points of ␣, as defined by 兩⳵2␣ / ⳵T2兩 P = 0, correspond to the locus where ␣ starts to vary superlinearly with increasing temperature. Therefore, the inflection curve of ␣ constitutes a realistic criterion for separating the domain of validity of the QHA from the invalid regions. This boundary is labeled “QHA boundary” in Fig. 1. In all figures, solid and dashed lines are used for predictions made within and outside the regime of validity of the QHA, respectively. We remark that the higher the pressure, the larger the temperature domain of validity of

6.7

LDA

1 ⳵V V ⳵T

30 GPa

0

No further structural relaxation is performed at T⬘. The function V共P⬘ , T⬘兲 is then obtained after inverting P⬘共V , T⬘兲. Therefore, in this statically constrained QHA calculation, if V共P⬘ , T⬘兲 = V共Pstat兲, then structural parameters and phonon frequencies at 共P⬘ , T⬘兲 and at Pstat共V兲 are the same. This fundamental consequence of the statically constrained QHA can be tested against high P-T experimental data of MgSiO3 pv. This is a direct test of the accuracy of this approximation. If the test is favorable, this procedure can be used to predict structures at high P-T’s. MgSiO3 pv has an orthorhombic structure with the symmetry group Pbnm. The unit cell contains 20 atoms. Simultaneously ten structural parameters must be relaxed: the three lattice parameters a, b, and c plus the seven internal parameters. Optimization of MgSiO3 pv is performed using the variable cell shape molecular dynamics method,10 using the quantum ESPRESSO package.11 Phonon dispersions were computed using linear response theory.12 They have been reported13 and used for elasticity calculations14 in earlier publication. Before making comparisons with experiments, one must be careful to limit this comparison to the P-T domain where quasiharmonic predictions are expected to be reliable. This P-T domain can be defined by a posteriori inspection of the thermal expansivity,14

冏 冏

α [10 /K]

. T⬘

共5兲

␣=−

gc

Invalid QHA

6.6 6.5 135

140

145

150

o

155

Volume [A3 ]

Exp. LDA+ZP



o

− ⳵Fth共V,T兲 ⳵V

ltin

Valid QHA

a [A]

T⬘



Pth =

me

Pa 0G Pa 10 G

1

o

and

3 2

b [A]



4

o

− ⳵FZP共V,T兲 ⳵V

5

c [A]



PZP =

6

160

165

FIG. 2. 共Color online兲 Comparison between lattice parameters predicted by the QHA and experimental data as a function of volume. The LDA and LDA plus zero-point motion equilibrium volumes are also compared to the experimental equilibrium volume at 0 GPa. Experimental data are from Refs. 4–9. Temperatures vary from 295 to 1024 K for Wang et al. 共Ref. 9兲, from 293 to 2668 K for Fiquet et al. 共Ref. 4兲, from 298 to 1173 K for Utsumi et al. 共Ref. 8兲, from 293 to 2000 K for Funamori et al. 共Ref. 5兲, and from 77 to 400 K for Ross and Hazen 共Ref. 6 and 7兲. Solid lines are from theory.

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PHYSICAL REVIEW B 76, 064116 共2007兲

FIRST-PRINCIPLES PREDICTION OF CRYSTAL…

Mg y

0.55 0.5

3 2 1 0 -1 -2

δσ1 [GPa]

0.6

Mgx O z2 Oy1

0.45

3 2 1 0 -1 -2

O y2

δσ2 [GPa]

0.2

Ox2 Ross & Hazen (1989,90)

0.15 Ox1 0.1 140

145

150

o

155

Volume [A3 ]

160

165

FIG. 3. 共Color online兲 Comparison between internal parameters predicted by the QHA 共solid lines兲 and experiments 共data points兲 as a function of volume. Experimental data are from Refs. 6 and 7.

notice that the postperovskite 共ppv兲 transition phase boundary20 falls within the domain of validity of the QHA for pv and ppv phases. Figure 2 shows the variation of lattice parameters a, b, and c as functions of volume obtained by several high P-T experiments 共data points兲 as compared with the static local density approximation 共LDA兲 prediction 共full line兲. Figure 3 shows the same variation now for the internal parameters, as compared to available experimental data points. Experiments were performed under various P-T’s, for temperatures varying from 70 共Refs. 6 and 7兲 to above ⬃2600 K 关e.g., in Funamori et al., T’s vary from 293 to 2000 K 共Ref. 5兲; in Fiquet et al., T’s vary from 293 to 2668 K 共Ref. 4兲兴 and pressures varying between 0 and 60 GPa. Clearly, from Fig. 2, all experimental lattice parameters obtained at various P-T’s lie on or very close to the theoretical lines. Additional experimental data should test 共likely confirm兲 the trend for the internal parameters shown in Fig. 3. These results indicate that, indeed as predicted, the statically constrained QHA is a good approximation in the P-T range of these experiments. We also indicate in Fig. 2 the LDA equilibrium volume at 0 GPa, including 共not including兲 zero-point 共ZP兲 motion energy, LDA+ ZP, and compare both with the experimental value.6,13 The inclusion of ZP motion increases the equilibrium volume at zero pressure. It becomes larger than, but in better agreement with, the experimental zero pressure volume. A similar trend has been found for all other mantle silicates and oxides investigated by this method so far. A closer examination of Fig. 2 at small volumes 共i.e., higher pressures兲 shows that although all experimental data lie close to the predicted theoretical straight lines, there exist still some small but systematic deviations. Some of these data points include very high temperature data. For instance, both data of Fiquet et al. and of Funamori et al. for the lattice parameter a are slightly larger than the QHA prediction. The discrepancy is of the order of 0.3% or 0.01 Å. The opposite is observed for the lattice parameter b. The discrepancy in this case is of the order of −0.1% or −0.005 Å. No clear

1000 K 2000 K 3000 K

3000 K

4000 K

4000 K 2000 K

1000 K

3 2 3000 K 4000 K 200 1 1000 0KK 0 300 K -1 0 K -2 -3 0 20 40 60

300 K

0K

δσ 3 [GPa]

135

0K 300 K

80

P [GPa]

100

120

140

FIG. 4. 共Color online兲 Deviatoric stresses 关see Eq. 共7兲兴 along a, b, and c axes.

discrepancies are noticeable in the lattice parameter c. Although minor, these discrepancies remain systematic and could originate either in the LDA or in the QHA. In the following, we address this problem. III. DEVIATORIC THERMAL STRESS RELAXATION

In the statically constrained QHA, energies are computed according to Eq. 共1兲 and pressures according to Eq. 共2兲. This procedure implicitly assumes that pressure remains isotropic at all temperatures, but this is only true for static calculations. The zero-point motion and the thermal pressure contributions to P⬘ are not necessarily isotropic. This effect is explicitly quantified by the deviatoric stresses defined as the difference between the nominal pressure and the diagonal stress components,

冋冏

␦␴i = P⬘ − −

1 ⳵G共P,T兲 V ⳵⑀i

冏 册 P⬘,T⬘

.

共7兲

Figure 4 shows these deviatoric stresses versus P along the crystalline directions 关100兴, 关010兴, and 关001兴 at various T’s. With increasing T, ␦␴1 becomes more negative. This means that the statically constrained system becomes overcompressed along 关100兴 at high T’s. The opposite appears along 关010兴, and minor changes affect the 关001兴 direction. This result is consistent with Fig. 2, where the experimental data lie above the static QHA for a, below for b, and essentially on top for c. These deviatoric stresses can be relaxed to first order if one knows the compliance tensor ␬ij共P⬘ , T⬘兲 = c−1 ij 共P⬘ , T⬘兲,

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PHYSICAL REVIEW B 76, 064116 共2007兲

CARRIER, WENTZCOVITCH, AND TSUCHIYA

o

a [A]

4.7 4.65 4.6

K 4000000 K K 3 000 K 2 1000 K 300

o

b [A]

4.8

o

c [A]

0K

4.75 4.7 6.75 6.7 6.65

K K 300 1000 0 K 200 3000 K 000 K 4

IV. SELF CONSISTENT QUASIHARMONIC APPROXIMATION

0 K 00 K 3 00 K 10 000 K K 2 3000

6.6 6.55 140

400 - 1500 K 1501-2668 K

0K

4.55 4.85

mal stresses, especially in the range of temperatures of the data of Fiquet et al., up to 2668 K,4 and of Funamori et al., up to 2000 K,5 we conclude that these deviations are caused by these stresses. After the relaxation of deviatoric thermal stresses according to Eq. 共8兲, the agreement with experiments improves considerably, especially in the high temperature ranges of the data of Fiquet et al.’s, between 1501 and 2668 K,4 and of Funamori et al., between 400 and 1500 K.5 Corrections of elastic constants to first order are also possible and should be done, but these will be discussed somewhere else.

4000

142

144

146

K

148

o3

150

152

154

Volume [A ] FIG. 5. 共Color online兲 Corrected lattice parameters 关a − ⑀1a兴, 关b − ⑀2b兴, and 关c − ⑀3c兴 where ⑀i’s are solutions of Eq. 共8兲, compared with the data of Funamori et al. 共Ref. 5兲 data between 400 and 1500 K 共orange dots兲 and Fiquet et al.4 data above 1500 K 共green triangles兲. Solid lines are from theory. As indicated by the data of Fiquet et al.’s data experimental errors increase with temperature.

where cij共P⬘ , T⬘兲 is the elastic constant tensor given in Eq. 共4兲. cij共P⬘ , T⬘兲 for MgSiO3 pv has already been determined 共Ref. 14兲. The correction is then carried out by evaluating the strains ⑀i involved in the relaxation of these deviatoric thermal stresses,

⑀i共P⬘,T⬘兲 = 兺 ␬ij共P⬘,T⬘兲␦␴ j .

共8兲

j

Figure 5 shows the resulting corrections on the lattice parameters as a function of volume at various T’s, combined with the experimental temperature dependent data of Fiquet et al.4 and Funamori et al.5 These high temperature data were collected nonsystematically therefore, the trends of deviations with temperature are not systematic either. Besides, uncertainties in pressure scales at high temperatures are also large, which further obscures these trends. It appears clearly that the magnitude of the errors in the lattice parameters increase with T. This error could be caused by anharmonic effects, but it could also be caused by thermally induced deviatoric stresses. Because the agreement with experiments improves considerably after the relaxation of deviatoric ther-

These results indicate that QHA predictions of thermodynamics and structural properties, which are already in excellent agreement with experiments within the domain of validity of this approximation, can be further improved, particularly the crystal structure, if computations include the relaxation of deviatoric thermal stresses. This can be accomplished by an iterative scheme. After thermal stresses are completely relaxed, static energies, phonons, and the free energy,



F共V, P⬘,T⬘兲 = U关V共P⬘,T⬘兲兴 + 兺 qj

ប␻qj关V共P⬘,T⬘兲兴 2

+ kBT⬘ 兺 ln共1 − eប␻qj关V共P⬘,T⬘兲兴/kBT⬘兲,



共9兲

qj

should be recomputed, and so should Eqs. 共2兲–共4兲, along with Eqs. 共7兲 and 共8兲, until ␦␴i’s in Eq. 共7兲 are negligible. The small magnitude of ␦␴i’s 共Fig. 4兲 suggests that one extra cycle in this iterative procedure may suffice. This iterative scheme apparently involves an excessive number of calculations. However, this systematic procedure minimizes the number of calculations for structures with a larger number of degrees of freedom, as in the present case. Finally, we have focused here on the relaxation of lattice degrees of freedom only, but internal parameters should be relaxed also. The self-consistent scheme should relax forces that develop at high temperatures. In this proposed iterative scheme, forces are only relaxed along with the lattice degrees of freedom, and this may suffice as well, at least for the purpose of computing structural and thermodynamic properties. This work was supported by NSF Grants No.EAR0230319, No. EAR-0635990, and No. ITR-0428774. We especially thank Shuxia Zhang from the Minnesota Supercomputing Institute for her assistance with optimizing the PWSCF code performance on the BladeCenter Linux Cluster.

064116-4

PHYSICAL REVIEW B 76, 064116 共2007兲

FIRST-PRINCIPLES PREDICTION OF CRYSTAL… C. Wallace, Thermodynamics of Crystals 共Dover, Mineola, 1972兲. 2 P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 共1964兲; W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 共1965兲. 3 Y. Kuwayama, K. Hirose, N. Sata, and Y. Ohishi, Science 309, 923 共2005兲. 4 G. Fiquet, D. Andrault, A. Dewaele, T. Charpin, M. Kunz, and D. Hausermann, Phys. Earth Planet. Inter. 105, 21 共1998兲. 5 N. Funamori, T. Yagi, W. Utsumi, T. Kondo, T. Ushida, and M. Funamori, J. Geophys. Res. 101, 8257 共1996兲. 6 N. Ross and R. Hazen, Phys. Chem. Miner. 16, 415 共1989兲. 7 N. Ross and R. Hazen, Phys. Chem. Miner. 17, 228 共1990兲. 8 W. Utsumi, N. Funamori, T. Yagi, E. Ito, T. Kikegawa, and O. Shimomura, Geophys. Res. Lett. 22, 1005 共1995兲. 9 Y. Wang, D. Weidner, R. Liebermann, and Y. Zhao, Phys. Earth Planet. Inter. 83, 13 共1994兲. 10 R. M. Wentzcovitch, Phys. Rev. B 44, 2358 共1991兲. 11 S. Baroni, A. Dal Corso, S. de Gironcoli, P. Giannozzi, C. Cavazzoni, G. Ballabio, S. Scandolo, G. Chiarotti, P. Focher, A. Pas1 D.

quarello, K. Laasonen, A. Trave, R. Car, N. Marzari, and A. Kokalj, http://www.pwscf.org/ 12 S. Baroni, S. de Gironcoli, A. Dal Corso, and P. Giannozzi, Rev. Mod. Phys. 73, 515 共2001兲. 13 B. B. Karki, R. M. Wentzcovitch, S. de Gironcoli, and S. Baroni, Phys. Rev. B 62, 14750 共2000兲. 14 R. M. Wentzcovitch, B. B. Karki, M. Cococcioni, and S. de Gironcoli, Phys. Rev. Lett. 92, 018501 共2004兲. 15 O. L. Anderson, Equations of State of Solids for Geophysics and Ceramic Science 共Oxford University Press, New York, 1995兲. 16 A. Zerr and R. Boehler, Science 262, 553 共1993兲. 17 L. D. Landau and E. M. Lifshitz, Statistical Physics 共AddisonWesley, Reading, MA, 1958兲. 18 P. F. Choquard, The Anharmonic Crystal 共Benjamin, New York, 1967兲. 19 R. Boehler, Rev. Geophys. 38, 221 共2000兲. 20 T. Tsuchiya, J. Tsuchiya, K. Umemoto, and R. M. Wentzcovitch, Earth Planet. Sci. Lett. 224, 241 共2004兲.

064116-5

First-principles prediction of crystal structures at high ...

mental data, and show that the quality of these predictions ..... 101, 8257 1996. 6 N. Ross and R. Hazen, Phys. Chem. Miner. 16, 415 1989. 7 N. Ross and R.

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