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ISSN 1063 7834, Physics of the Solid State, 2012, Vol. 54, No. 8, pp. 1663­1668. © Pleiades Publishing, Ltd., 2012. Original Russian Text © A.I. Lebedev, 2012, published in Fizika Tverdogo Tela, 2012, Vol. 54, No. 8, pp. 1559­1564.

PHASE TRANSITIONS

Ferroelectricity and Pressure Induced Phase Transitions in HgTiO3
A. I. Lebedev
Moscow State University, Moscow, 119899 Russia e mail: swan@scon155.phys.msu.ru
Received December 21, 2011

Abstract--Using ab initio density functional theory, the ground state of mercury titanate is determined and phase transitions occurring in it at pressures P 210 kbar are analyzed. It is shown that the R3c structure experimentally observed in HgTiO3 is metastable at P = 0. The ground state structure at T = 0 varies according to the scheme R3c R3c Pbnm with increasing pressure in agreement with available experimental data. It is shown that the appearance of ferroelectricity in HgTiO3 at P = 0 is associated with an unstable soft mode. Some properties of crystals in approximation (Eg = 2.43 eV), which in the LDA approximation (1.49 eV). of mercury titanate is possible only at DOI: 10.1134/S1063783412080185 the R 3 c phase are calculated, in particular, the band gap in the GW is in better agreement with experimental data than the value obtained An analysis of the thermodynamic stability explains why the synthesis high pressures.

There are only a few studies of mercury titanate HgTiO3 that indicate its interesting, but contradictory ferroelectric properties. Mercury titanate can be syn thesized under a pressure of 60­65 kbar [1, 2]. These crystals have a rhombohedrally distorted perovskite structure. Observation of the second harmonic gener ation in HgTiO3 at 300 K [1] enabled one to propose that the space group of the crystal is R 3 c ; however, because of a limited accuracy, the atomic coordinates were determined only for the centrosymmetric R 3 c structure. Subsequent studies of dielectric properties of mercury titanate [2, 3] did not detect pronounced dielectric anomalies: a wide strongly asymmetric peak with a maximum near 220 K and with an appreciable hysteresis in the cooling­heating cycle as well as a weak narrow peak near 515 K were observed on the temperature dependence. At 300 K, no dielectric hys teresis loops were appeared up to fields of 106 V/m [2, 3]. Scanning calorimetry detected weak anomalies in the 420­480 K temperature range [2, 3]; however, these temperatures differed from the temperatures of peaks in the dielectric constant. Furthermore, X ray studies under hydrostatic pressure [2, 3] detected non monotonic behavior of the d024 interplanar distance and the (104)­(110) doublet splitting at a pressure of ~20 kbar, which were explained by the rhombohedral to cubic phase transition. The electronic structure of the rhombohedral and cubic modifications of HgTiO3 was studied in [4]; it was shown that the rhombohedral phase is a semiconductor and the cubic phase is a metal.

To resolve the contradictions concerning the ferro electric properties of HgTiO3 and to obtain new data on this compound, in this paper, ab initio calculations of physical properties of mercury titanate were per formed. The calculations were carried out within the den sity functional theory by analogy with the previous study [5]. The pseudopotentials for Ti and O atoms used in the calculations were taken from that study, the scalar­relativistic pseudopotential for the Hg atom was constructed according to the RRKJ scheme [6] using the OPIUM program for the configuration Hg2+ (5d106s06p0) with the following parameters: rs = 1.78, rp = 2.0, rd = 1.78, qs = 7.37, qp = 7.07, qd = 7.37 a.u. (for the notation of parameters, see [5]). The plane wave energy cut off used in the calculations was 30 Ha (816 eV). Integration over the Brillouin zone was per formed on the 8 â 8 â 8 Monkhorst­Pack mesh. To test the quality of the mercury pseudopotential, the calcu lations were carried out for two polymorphs of HgO with orthorhombic and rhombohedral structures. The former one (the montroydite mineral) had a slightly lower total energy. The calculated lattice parameters of these phases (a = 3.4663 å, b = 6.6253 å, c = 5.3013 å, and a = 3.5092 å, c = 8.5417 å, respectively) were in reasonable agreement with experimental data [7] (a = 3.5215 å, b = 6.6074 å, c = 5.5254 å; a = 3.577 å, c = 8.681 å). The phonon spectra were calculated using the scheme similar to that used in [5]. The calculated phonon spectrum of HgTiO3 with a cubic perovskite structure (space group Pm3m) is shown in Fig. 1. One can see that two types of instabil

1663


1664 1000 800 600
1

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, cm-

400 200 0
5' 5' 5 3 3' 3 15 25 25

-200 -400

X

M



R

M

Fig. 1. Phonon spectrum of HgTiO3 in the cubic Pm3m phase. Labels near the curves indicate the symmetry of unstable modes. The absence of LO­TO splitting at the point is associated with metallic conductivity of the phase.

ity simultaneously appear in mercury titanate: the stronger one associated with deformation and rotation of octahedra, which is not accompanied with the appearance of dipole moments (the 25­X3­M3­25­ R25­M3 branch), and the ferroelectric (antiferroelec tric at the Brillouin zone boundary) one (the 15­ X 5' ­ M 3' ­15 branch). The absence of LO­TO mode split
Table 1. Energy and volume per formula unit in different distorted phases of HgTiO3 at P = 0 Unstable mode ­ X3 15 15 25 15, 25 X5 X5 25 R25 M3 R25 + M R25 R25 A
2u

Space group Pm3m P42/mmc R3m R4mm P 4 m2 Amm2 Pmma Cmcm R32 I4/mcm P4/mbm Pbnm Imma R 3c R3c R3

Energy, meV 0 ­88 ­94 ­122 ­139 ­151 ­202 ­306 ­467 ­778 ­809 ­936 ­940 ­974 ­982 ­1059

Cell volume, å3 57.573 57.275 58.923 59.444 57.088 59.749 57.916 57.867 56.956 56.188 56.195 55.853 56.099 56.336 56.632 60.140

3

ting at the point is explained by the metallic charac ter of the band structure of cubic HgTiO3. To determine the ground state structure, energies of different distorted phases which arise from the cubic perovskite structure during condensation of the above determined unstable modes were calculated taking into account their degeneracy (Table 1). Among these phases, the R3c one has the lowest energy. It is formed from the cubic structure by antiphase rotations of octahedra about all three fourfold axes as a result of condensation of the triply degenerate R25 mode at the Brillouin zone boundary (the a­a­a­ Glazer tilt sys tem). The energy of this phase is even lower than that of the Pbnm phase, in contrast to other titanates of Group II elements [5]. We note that as the structure becomes more distorted, the overlap of the energy bands disappears, and all phases with energies below 300 meV are semiconductors. The ferroelectric instability of the parent cubic structure of HgTiO3 is also retained in the R 3 c phase. Calculations show that unstable modes of symmetry A2u and Eu with frequencies 135i and 21i cm­1 are observed in the phonon spectrum of this phase at the point. Among the corresponding ferroelectrically dis torted phases, the R3c phase has the lowest energy. The fact that this phase corresponds to the ground state is proved by that all phonon frequencies at the Brillouin zone center and at high symmetry points A, D, and Z at its boundary are positive, and the elastic moduli matrix is positive definite (see below). The calculated lattice parameters and atomic coordinates in the R 3 c and R3c structures are given in Table 2. It is seen that they are in good agreement with experimental data [1]. The calculated interatomic dis tances for the R 3 c phase and average calculated dis tances for the R3c phase are also in good agreement with interatomic distances obtained from X ray dif fraction studies (Table 3). In addition to the phases derivative of the cubic perovskite structure, the possible formation of other phases should be considered, in particular, the ilmenite structure characteristic of titanates of Group II elements, i.e., MgTiO3, ZnTiO3, and CdTiO3 [7]. Calculations showed that the ilmenite structure (space group R 3 ) has the lowest energy at normal pressure (P = 0) among the considered phases (Table 1). The fact that the R 3 c or R3c phases are observed in X ray experiment enables to suppose that these phases are metastable. Their metastability is obviously associated with the significant difference between R3c ( R 3 c ) and R 3 structures in both the lattice parameter and the rhombohedral angle (see Table 2). Therefore, the phase transition between them is the first order transi tion for which a wide metastability region is character

­

Note: The energy of the cubic phase is taken as the energy reference. The phase with the lowest specific energy and the phase with the lowest specific volume are denoted by bold font.

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Table 2. Lattice parameters and atomic coordinates in the mercury titanate phases with space groups R3c, R 3 c, and R 3 at P = 0 and the space group Pbnm at 141 kbar Phase R3c a, å 5.4984 , deg 58.4093 Atom Hg Ti O Hg Ti O Hg Ti O Hg Ti O Hg Ti O O Position 2a 2a 6b 2a 2b 6e 2a 2b 6e 2c 2c 6f 4c 4b 4c 4d x 0.24904 ­0.00333 0.66598 0.25000 0.00000 0.65983 0.25 0.0 0.665 0.36869 0.84974 0.55966 ­0.00445 0.50000 0.08502 0.69594 y 0.224904 ­0.00333 ­0.15240 0.25000 0.00000 ­0.15983 0.25 0.0 ­0.165 0.36869 0.84974 ­0.03220 0.03190 0.00000 0.47264 0.30216 z 0.24904 ­0.00333 0.25846 0.25000 0.00000 0.25000 0.25 0.0 0.25 0.36869 0.84974 0.19275 0.25000 0.00000 0.25000 0.04431

R 3c

5.4881

58.4252

R 3c (exp.)* [1] R3

5.4959

58.59

5.8304

53.9320

Pbnm

5.2678(a) 5.2983(b) 7.5501(c)

­

* Coordinates were recalculated for the rhombohedral setting.

istic. The fact that namely the metastable R3c ( R 3 c ) phase appears during the synthesis can be caused by that HgTiO3 is synthesized at a pressure of 60­65 kbar at which (as shown below) the R 3 c phase is the most stable. The energy of two more possible hexagonal HgTiO3 phases with the two layer BaNiO3 structure and the six layer hexagonal BaTiO3 structure (the space group of both is P63/mmc) is higher than the energy of the Pm3m phase by 269 and 73 meV, respec tively. We discuss now the ferroelectric properties and the nature of the ferroelectric phase transition in HgTiO3. Since the change in the Hg­O bond lengths during phase transition to the ferroelectric phase does not exceed 0.1 å, and the energy difference of the R3c and R 3 c phases is only 8.1 meV, it is unlikely that the Curie temperature in HgTiO3 will exceed 300 K. Therefore, it is in better agreement with a temperature of 220 K at which the dielectric constant maximum was experimentally observed. The absence of the dielectric hysteresis loops at 300 K also supports this interpretation. Therefore, in what follows, being ori ented on experiments performed at 300 K, we will focus on the properties of the R 3 c phase. The fact that the authors of [1] observed a signal of the second har monic at room temperature can be associated with the sample defectness which is discussed in what follows.
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An analysis of the eigenvector of the A2u ferroelec tric mode in the R 3 c phase shows that the displace ments of Hg atoms in this mode is smaller than that of Ti atoms by a factor of 22. This means that collective displacements of the titanium atoms with respect to the oxygen ones are responsible for the ferroelectric phase transition, rather than the mercury atom jumps between wells of the two well potential. The effective Born charges of the mercury atoms also indicate their * * weak ferroelectric activity: they are Z xx = Z yy = 3.20 * and Z zz = 2.42 and differ slightly from the nominal ionic charge. The calculated static dielectric constant at 0 K in the R3c phase is almost isotropic (xx = 97, zz = 101);
Table 3. Interatomic distances in the HgTiO3 phases with space groups R3c and R 3 c Distance, å Atom pair Hg­O Hg­O Hg­O Ti­O this work Ref. [1] R3c 2.198 2.698, 2.888 3.172 1.906, 2.064 R 3c 2.195 2.786 3.162 1.977 2.20(4) 2.77(4) ­ 1.96(4) 3 3+3 3 3+3 Number of bonds

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Table 4. Calculated frequencies of optical phonons at the point of the Brillouin zone for HgTiO3 with R3c, R 3 , and Pbnm structures (the latter is at P = 147 kbar) Structure R3c mode A
1

Structure Pbnm v, cm 81 121 139 165 274 312 443 495 515 94 189 311 442 565 148 256 371 452
­1

v, cm 78 181 379 476 62 347 355 417 753 73 204 306 440 651 133 348 477 647

­1

mode E

mode A
g

v, cm 68 113 144 277 417 462 559 65 74 108 141 303 375 498 539 80 103 139 350

­1

mode B
1g

v, cm­1 439 502 782 104 266 450 542 819 118 226 353 539 732 38 84 134 243 387 475

mode B B
1u 2u

v, cm­1 530 38 86 141 190 345 356 431 496 524 58 114 165 246 301 380 403 448 547

B

2g

A

2

A

u

B

3g

Structure R 3 A
g

E

g

B

3u

B

1u

A

u

E

u

B

1g

in the R 3 phase, the dielectric constant is notably lower (xx = 28, zz = 27). For comparison, the maxi mum dielectric constant experimentally observed at 220 K is ~800 [2, 3]. The calculated spontaneous polarization in the R3c phase appears unexpectedly high, Ps = 0.37 C/m2. This is probably a result of the large effective charge of the A2u mode in the paraelec * tric phase ( Z eff = 12.66). We consider now some other physical properties of HgTiO3 at P = 0. The calculations of the electronic structure performed in this work confirmed the data of [4] that the R 3 c phase is a direct gap semiconductor with the band extrema located at the point. The band LDA gap of mercury titanate at P = 0 is E g = 1.49 eV, its pressure coefficient is dE g / dP = +0.44 meV/kbar. The value of Eg obtained in the LDA approximation agrees with the value of 1.6 eV found in the GGA approximation using the FP LAPW method [4]. However, both results disagree with the experimental fact that HgTiO3 crystals are light yellow [1]. It is well known that the density functional theory always underestimates the band gap because of its limitations with respect to the calculation of the excited states energy. One of the approaches that yield the Eg values
LDA

in good agreement with experiment is based on the consideration of many body effects (electron correla tions, dynamic screening, local field effects) within the GW approximation [8]. The calculations per formed in this work within this approximation yielded GW E g = 2.43 eV in the R 3 c phase, which agrees much better with the sample color reported in [1] than the band gap in the LDA approximation. The elastic moduli tensor in the R 3 c phase is pre sented by seven independent components: C11 = C22 = 348.0 GPa, C33 = 260.3 GPa, C12 = 178.5 GPa, C13 = C23 = 149.9 GPa, C44 = C55 = 76.3 GPa, C66 = 84.8 GPa, and C14 = ­C24 = C56 = 18.3 GPa. The bulk elastic modulus calculated from them is B = 205.8 GPa; it slightly differs from the value of 178 GPa obtained in [4] ignoring the relaxation of internal degrees of freedom. To interpret future experiments on infrared (IR) reflection and Raman scattering, the phonon frequen cies calculated at the point for R3c, R 3 phases at normal pressure and the Pbnm phase at 147 kbar, given in Table 4, can be useful. In the low temperature R3c phase, the A1 and E modes are active in both IR and Raman spectra. In the R 3 phase, the Au and Eu modes
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are IR active; the Ag and Eg modes are Raman active. In the high pressure Pbnm phase, the B1u, B2u, and B3u modes are IR active; the Ag, B1g, B2g, and B3g modes are active in Raman spectra. To discuss the experimental data [2, 3] on the effect of the hydrostatic pressure on the HgTiO3 structure, we calculated the properties of mercury titanate under pressure. At a nonzero pressure, the phase which is characterized by the lowest enthalpy H = Etot + PV, rather than by the lowest total energy Etot, is thermo dynamically stable at T = 0. To compare phases with different number of molecules in the unit cell, we use the specific total energy and the specific unit cell vol ume defined per one formula unit. Calculations show that as the pressure changes, the relative contribution of the PV term to the change in H is ~95% in the crys tals under study. Therefore, the phases with the lowest specific volume should become the most stable as the pressure increases. As follows from Table 1, the Pbnm phase has the smallest volume at P = 0 among the con sidered phases; then, in order of increasing the specific volume, the phases are arranged as follows: Imma, I4/mcm, P4/mbm, R 3 c , R3c. The largest unit cell vol ume is characteristic of the ilmenite R 3 phase, whose total energy at P = 0 is the lowest. This enables to expect that the sequence of stable phases will vary with increasing pressure as follows: R 3 R3c Pbnm. In addition, the suppression of ferroelectricity (the R3c R 3 c phase transition) should be observed with increasing pressure. The differences between enthalpies of the phases under consideration and the enthalpy of the R 3 c phase as a function of pressure are shown in Fig. 2. It R3c and the R 3 c Pbnm is seen that the R 3 phase transitions occur in HgTiO3 at P = 38 and 141 kbar, respectively. Since the specific volume of the unit cell changes stepwise (by 6 and 0.71%, respectively) at both phase transitions, they should be of the first order. The Imma phase, whose enthalpy at P = 0 is lower than that of the Pbnm phase, becomes energetically less favorable with increasing pressure, and can be excluded from the consideration. The similar behav ior, when the stable R 3 phase transformed to the pres sure stabilized Pbnm phase under high temperatures and pressures and then relaxed from the latter phase to the metastable R 3 c phase as pressure was released, was also observed in MnTiO3 [9], FeTiO3 [10], and ZnGeO3 [11]. Ferroelectric LiTaO3 also exhibits the R3c Pbnm phase transition at high pressures [12]. The present calculations enable to propose a new interpretation of the phase transition observed in X ray experiments at hydrostatic pressure [2, 3]. The energy of the cubic Pm3m phase, which was attributed to the high pressure phase in [2, 3], is higher than that
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250 200 150 H, meV 100 50 0 -50 -100 R3c Pbnm Imma R3

0

50

100 P, kbar

150

200

Fig. 2. Difference between the enthalpies of different phases and the enthalpy of the R 3 c phase in HgTiO3 as a function of hydrostatic pressure.

of the R 3 c phase by almost 1 eV, and the specific cell volume of this phase exceeds that of the R 3 c phase (Table 1). This means that the sufficiently large differ ence between the enthalpies of these phases will only increase with increasing pressure. Therefore, the Pm3m phase cannot be considered as a high pressure phase. According to our calculations, the rhombohe dron angle in the R 3 c phase increases with a rate of 0.0054°/kbar with increasing pressure; therefore, at P = 20 kbar the structure should remain strongly dis torted, and the relative decrease in the interplane dis tance should be P/3B 0.32%, which is several times smaller than that observed at the phase transition. However, if we admit that the pressures in [2, 3] were determined with an error (according to our estimates, they are underestimated by a factor of 5­7) and take the relative change in the interplane distance d0241 as a measure of pressure, the agreement between the present calculations and the experimental data becomes satisfactory. For example, at the R 3 c Pbnm transition point (141 kbar), the calculated decrease in the d024 interplane distance in comparison with the case of P = 0 is 2.0%, whereas the experimen tal value is 2.3%, and the calculated jump of the aver age interplane distance2 at the phase transition point (0.054%) is close to the experimental value of ~0.05%.
1

The d024 value at P = 0, shown in Fig. 3 of [2], does not corre spond to the lattice parameters given in the paper for this pres sure (disagreement is ~5%). During the R 3 c Pbnm phase transition, the peak (012) splits into two components with indices (110) and (002).

2

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1668 (104) (012) (110) R3c Intensity, arb. units (024) (202)

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(112, 020, 200) (002, 110) Pbnm (022, 202) (111) 20 30 (021) 40 2, deg 50 60

generation signal can be caused by the existence of defects. Indeed, according to the data of [1], the sam ples rapidly darkened upon exposure to light. The absence of a sharp peak in the dielectric constant can also be caused by defects. This propensity of HgTiO3 for the defect formation is confirmed by simple ab ini tio calculations of its thermodynamics: at P = 0, the enthalpy of the R 3 c phase of mercury titanate is higher than the sum of enthalpies of orthorhombic HgO and rutile TiO2 by 150 meV (per formula unit). This means that mercury titanate is thermodynami cally unstable with respect to its decomposition into starting components. However, since the specific vol ume of the HgTiO3 unit cell is noticeably smaller than the sum of specific volumes of HgO and TiO2, the sta bility of HgTiO3 increases with increasing pressure. For example, at a pressure of 58.8 kbar, the HgTiO3 enthalpy appears lower than the sum of enthalpies of orthorhombic HgO and rutile TiO2 by 75 meV. This explains why the synthesis of mercury titanate is pos sible only under high pressure conditions. The calculations presented in this paper were per formed on the laboratory computer cluster (16 cores). REFERENCES
1. A. W. Sleight and C. T. Prewitt, J. Solid State Chem. 6, 509 (1973). 2. Y. J. Shan, Y. Inaguma, T. Nakamura, and L. J. Gauck ler, Ferroelectrics 326, 117 (2005). 3. Y. J. Shan, Y. Inaguma, H. Tetsuka, T. Nakamura, and L. J. Gauckler, Ferroelectrics 337, 71 (2006). 4. H. S. Nabi, R. Pentcheva, and R. Ranjan, J. Phys.: Condens. Matter 22, 045504 (2010). 5. A. I. Lebedev, Phys. Solid State 51 (2), 362 (2009). 6. A. M. Rappe, K. M. Rabe, E. Kaxiras, and L. D. Joan nopoulos, Phys. Rev. B: Condens. Matter 41, 1227 (1990). 7. Springer Materials: The Landolt BÆrnstein Database; URL http://www.springermaterials.com/navigator/. 8. G. Onida, L. Reining, and A. Rubio, Rev. Mod. Phys. 74, 601 (2002). 9. N. L. Ross, J. Ko, and C. T. Prewitt, Phys. Chem. Miner. 16, 621 (1989). 10. K. Leinenweber, W. Utsumi, Y. Tsuchida, T. Yagi, and K. Kurita, Phys. Chem. Miner. 18, 244 (1991). 11. H. Yusa, M. Akaogi, N. Sata, H. Kojitani, R. Yama moto, and Y. Ohishi, Phys. Chem. Miner. 33, 217 (2006). 12. J. Li, X. Zhou, W. Zhu, J. Li, and F. Jing, J. Appl. Phys. 102, 083503 (2007).

Fig. 3. Calculated X ray diffraction patterns of R 3 c and Pbnm phases of mercury titanate at P = 141 kbar (for CuK radiation).

The lattice parameters and atomic coordinates in the Pbnm structure at 141 kbar are given in Table 2. The calculated X ray diffraction patterns of R 3 c and Pbnm phases at 141 kbar are shown in Fig. 3. The cal culated X ray diffraction pattern of the Pbnm phase is indeed similar to the experimental X ray diffraction pattern of the high pressure phase [2, 3]. During the transition to the orthorhombic phase, the (012) line broadens, since a pair of closely spaced reflections (002) and (110) appears in the orthorhombic phase. The new line (021) appearing in the orthorhombic phase is easily seen in the X ray diffraction patterns recorded as the pressure is released. A notable dis agreement between the experimental and calculated X ray diffraction patterns is the absence of the (111) line in the X ray diffraction pattern of the high pres sure phase. It is possible that this disagreement results from an incompleteness of the structural transforma tion. Therefore, to confirm the interpretation pro posed in this paper, additional studies of HgTiO3 under pressure are required. As noted above, the contradiction between the absence of dielectric hysteresis loops in HgTiO3 at 300 K and the observation of the second harmonic

Translated by A. Kazantsev

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