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ISSN 1063 7834, Physics of the Solid State, 2009, Vol. 51, No. 9, pp. 1875­1880. © Pleiades Publishing, Ltd., 2009. Original Russian Text © A.I. Lebedev, 2009, published in Fizika Tverdogo Tela, 2009, Vol. 51, No. 9, pp. 1766­1770.

MAGNETISM AND FERROELECTRICITY

Ferroelectric Phenomena in CdSnO3: A First Principles Studies
A. I. Lebedev
Moscow State University, Moscow, 119991 Russia e mail: swan@scon155.phys.msu.ru
Received December 1, 2008

Abstract--The phonon spectrum of cubic cadmium metastannate and parameters of the crystal structure of its distorted phases were calculated from first principles within the density functional theory. It is shown that the phonon spectrum and the energy spectrum of the distorted phases in CdSnO3 resemble surprisingly the corresponding characteristics of CdTiO3. The ground state of CdSnO3 is the ferroelectric Pbn21 phase, the energy gain from the phase transition to this phase from the nonpolar phase Pbnm is ~30 meV, and the spon taneous polarization is 0.25 C/m2. The analysis of the eigenvector of the ferroelectric mode in CdSnO3 and the partial densities of states indicates that the ferroelectric instability in this crystal, which does not contain transition d element atoms, is associated with the formation of a covalent bonding between Cd and O atoms. PACS numbers: 61.50.Ah, 63.20.D , 77.84.Dy DOI: 10.1134/S1063783409090182

1. INTRODUCTION Among a large family of ferroelectrics with the per ovskite structure ABO3, crystals in which A is the cad mium atom are least studied. In the CdO­SnO2 sys tem, two compounds are formed: CdSnO3 and Cd2SnO4 [1]. Cadmium metastannate CdSnO3 exists in two stable modifications: with an orthorhombically distorted perovskite structure ( modification) [1­3] and with the rhombohedral ilmenite structure ( modification) [3, 4]. As in the case of cadmium titan ate, CdSnO3 crystals with the ilmenite structure are obtained at a synthesis temperature of 800°C [5, 6] and crystals with the perovskite structure, at a temper ature of 1000­1100°C [1, 5] or at high pressures [3]. Besides, upon decomposition of CdSn(OH)6 it is pos sible to obtain CdSnO3 with a metastable spinel struc ture [7]. Cadmium orthostannate (Cd2SnO4) also exists in two crystal modifications: with the orthor hombic Sr2PbO4 structure and with the spinel struc ture [7]. All these compounds are n type semiconduc tors with a band gap of 2­3 eV [8, 9], which, due to a high concentration of native defects, are characterized by a rather high conductivity ( = 10­5­5 â 103 ­1 cm­1). The latter circumstance hampers the study of these materials by dielectric methods. Cadmium stan nates are used for fabrication of conducting thin film transparent to visible light and as gas sensors. Cadmium metastannate with the perovskite struc ture is of interest as a potential ferroelectric. Unfortu nately, there have been few studies of this material. The CdSnO3 crystal structure was studied at a tem perature of 300 K for powders [1] and single crystals [3] and identified as a structure with Pbnm space group. A refined analysis of X ray reflection intensities [10] suggested the possibility of polar state of this

phase (the proposed space group Pbn21). The same conclusion was made by the authors of [11]. The anal ysis of the optical absorption and luminescence spec tra [9] revealed, in the temperature dependence of the band gap Eg(T) for CdSnO3 single crystals, a region of rapid change in Eg (by 0.11 eV) at T 80°C and two more regions of change near 140 and 200°C, which were associated with phase transitions in the crystals. In these temperature ranges, drastic changes in the intensity of luminescence and degree of its polariza tion were also observed. If these peculiarities are really due to the ferroelectric phase transition, we deal with a rare case when the ferroelectric properties appear in perovskite crystals, which do not contain transition d elements. The objective difficulties of studying cadmium metastannate, incompleteness of experimental data in literature, and the lack of understanding of the nature of ferroelectricity, which is proposed in this com pound, make it expedient to calculate the physical properties of CdSnO3 from first principles. 2. CALCULATION TECHNIQUE The calculations were performed within the density functional theory using the pseudopotentials and plane wave expansion of wave functions as imple mented in the ABINIT code [12]. The exchange cor relation interaction was described in the local density approximation (LDA) following [13]. The pseudopo tentials used were optimized separable nonlocal pseudopotentials [14] constructed using the OPIUM software package; the local potential [15] was added to them to improve their transferability. The parameters used for constructing the pseudopotentials, the results

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800 CdSnO3 , cm­1 400

0
5' 3 2' 5' 5 5 3 15 25 15 25

­400 800 X

M



R

M

CdTiO3

, cm­1

400

0

5' 5 3

2' 5' 3 15 25 15 25

­400

X

M



R

M

Fig. 1. Comparison of the phonon spectrum for the cubic parent phase of CdSnO3 with the phonon spectrum of the same phase for CdTiO3 [16]. Labels near the curves indi cate the symmetry of the modes.

evident that the energy spectra of different distorted phases in CdSnO3 and CdTiO3 are very similar, which confirms the similarity of these crystals. The nonpolar phase in cadmium metastannate with the lowest energy is the orthorhombic Pbnm phase. Table 2 presents the calculated structural parame ters for the Pbnm phase of cadmium metastannate. As follows from comparison with the data in literature, the calculated lattice parameters are close to the experimentally observed ones. A slight overestimate of the calculated lattice parameters is due to the peculiar ity of the pseudopotential for a tin atom, which mani fests itself in the tests as the overestimated lattice parameters of SnO2 and gray tin. The atomic coordi nates in the unit cell agree well with the results of the structure determination of CdSnO3 [18], which was performed under the assumption that the space group of the crystal is Pbnm. Simultaneously with calculating the structure of CdSnO3, the structure of CdSnO3 (space group R 3 ) was calculated. The calculated lattice parameters of this phase (a = 5.5238 å, c = 14.6550 å) were found to be close to those obtained experimentally (a = 5.4530 å, c = 14.960 å [3]). The energy gain of the R 3 phase of cadmium metastannate was by 77 meV lower than that of the Pbn21 phase. This means that, at T = 0, the phase with the structure of distorted perovskite is metastable. It should be noted that according to our data, in CdTiO3, which also exists in ilmenite and per ovskite modifications, the energy of the R 3 phase at T = 0 is by 166 meV lower than that of the Pbnm phase. The difference in the entropy contributions to the thermodynamic potential can result in intersection of the curves for (T) for the two considered phases. This can explain why one obtains crystals with the ilmenite structure at a low synthesis temperature and crystals with the perovskite structure at a high synthe sis temperature (see Section 1). 3.2. Ferroelectric Phase in CdSnO3 In order to verify the possible existence of the polar state in CdSnO3, the frequencies of phonons at the point were calculated for the orthorhombic Pbnm phase of this crystal. The calculations revealed one unstable optical mode of symmetry B1u with a fre quency of 89i cm­1, which can be associated with the ferroelectric phase transition Pbnm Pnb21. The calculated equilibrium atomic positions and lattice parameters for the Pbn21 phase are presented in Table 2. The lattice distortion is accompanied by a noticeable rearrangement of the local environment of the Cd atom; it results in that the average distance to four nearest oxygen atoms increases by 0.017 å but another two oxygen atoms become closer by 0.23 å. Thus, the ferroelectric lattice distortion is accompa nied by an increase in the effective coordination num
Vol. 51 No. 9 2009

of testing the pseudopotentials, and other details of calculations are presented in [16]. 3. RESULTS OF THE CALCULATIONS 3.1. Comparison of the Properties of CdSnO3 and CdTiO3 Figure 1 shows the phonon spectra of CdSnO3 and CdTiO3 crystals in the cubic parent phase with the perovskite structure. The comparison of these spectra reveals their surprising resemblance. This enables one to suppose that other physical properties of these crys tals are similar as well. Table 1 presents the energies of different low sym metry phases formed in distortion of the cubic parent phase of cadmium metastannate according to the eigenvectors of unstable phonons that are present in its phonon spectrum. For comparison, the results of pre vious calculations of the energy of distorted phases in cadmium titanate [16, 17] are included in the table. The existence of one more unstable M5 mode in CdSnO3 corresponding to a distortion of the structure with rotation of octahedra leads to two more low sym metry phases with space groups Cmma and Pmna. It is

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Table 1. Comparison of the relative energies of different low symmetry phases of CdTiO3 and CdSnO3 (the energy of the cubic phase was taken as the energy origin) CdTiO3 [16, 17] unstable mode R15 X3 25 X5 15 X5 15 15, 25 25 R25 M3 R25 R15 R25 + M B2u B1u space group I4/mmm P42/mmc P 4 m2 Pmma R3m Cmcm P4mm Amm2 R32 I4/mcm P4/mbm R 3c C2/m Pbnm Pb21m Pbn21 energy, meV ­24 ­45 ­134 ­160 ­245 ­282 ­340 ­412 ­486 ­912 ­920 ­1197 ­1202 ­1283 ­1285 ­1290 unstable mode M5 M5 X3 X5 R15 25 15 X5 15 15, 25 25 M3 R25 R15 R25 R25 + M B1u CdSnO3 space group Cmma Pmna P42/mmc Pmma I4/mmm P 4 m2 R3m Cmcm P4mm Amm2 R32 P4/mbm I4/mcm C2/m R 3c Pbnm Pbn21 energy, meV ­141 ­262 ­404 ­474 ­536 ­540 ­753 ­886 ­1079 ­1259 ­1450 ­1617 ­1659 ­2455 ­2460 ­2575 ­2605

3

3

Note: The bold number is the energy of the ground state.

Table 2. Lattice parameters a, b, and c (in å) and atomic coordinates in CdSnO3 crystals with space groups Pbnm and Pbn21 This work Parameter Pbnm a b c Cdx Cdy Cdz Snx Sny Snz O1x O1y O1z O2ax O2ay O2az O2bx O2by O2bz 5.5024 5.5982 7.9770 ­0.00861 +0.04816 +0.25000 +0.00000 +0.50000 +0.00000 +0.12498 +0.42809 +0.25000 +0.68331 +0.31304 +0.06830 +0.31669 +0.68696 ­0.06830 Pbn21 5.5284 5.5972 7.9584 ­0.00553 +0.03818 +0.26021 +0.00036 +0.51455 ­0.00209 +0.12340 +0.42802 +0.24122 +0.66193 +0.35381 +0.04275 +0.29516 +0.72657 ­0.09020 [1] 5.547 5.577 7.867 [3] 5.4578 5.5773 7.8741 [11] 5.4593 5.5804 7.8771 [18]* 5.4588 5.5752 7.8711 ­0.0092 +0.0423 +0.2500 +0.0000 +0.5000 +0.0000 +0.114 +0.455 +0.2500 +0.695 +0.301 +0.058 +0.305 +0.699 ­0.058 Experiment

* In the structure determination, the space group was assumed to be Pbnm. PHYSICS OF THE SOLID STATE Vol. 51 No. 9 2009


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Table 3. Eigenvectors of the ferroelectric mode B1u and effective atomic charges Z* in CdSnO3 and CdTiO3 crystals with space group Pbnm and in BaTiO3 with space group Pm3m Atom Cd Sn O1 O2 Cd Ti O1 O2 Ba Ti O1 O2
x



y



z

Z* xx +2.462 +4.379 ­2.052 ­2.395 +2.570 +7.363 ­2.194 ­3.869

Z* yy +2.506 +4.407 ­1.894 ­2.509 +2.500 +7.693 ­1.957 ­4.118

Z* zz +2.440 +4.338 ­2.905 ­1.937 +2.592 +7.260 ­5.650 ­2.101 +2.738 +7.761 ­6.128 ­2.186

+0.00000 +0.00257 +0.00000 ­0.13111 +0.00000 +0.01863 +0.00000 ­0.07161

+0.00000 +0.12515 +0.00000 +0.20564 +0.00000 ­0.19067 +0.00000 +0.18255

+0.18913 ­0.01935 ­0.05940 ­0.19455 +0.10628 +0.15463 ­0.16286 ­0.19322 +0.02988 +0.67340 ­0.54043 ­0.35607

ber of the Cd atom. The energy gain from the phase transition Pbnm Pbn21 is E = 30.5 meV (Table 1). The sufficiently large energy gain allows one to expect that the structure will be ferroelectric at room temper

300 CdSnO3 200 d p s Cd states 0~ 50 ~ 0~ ~ 50 0~ ~ 150 100 50 0 ­0.3 ­0.2 ­0.1 0 Energy, Ha 0.1 0.2 ~ ~ ~ ~

ature in agreement with the data of [10, 11]. Indeed, the phase transition temperature (E/k ~ 350 K) esti mated from the energy gain from the transition to the ferroelectric phase is found to be close to the temper ature of 80°C, at which the most drastic changes in the absorption spectra were observed [9]. The calculation of the spontaneous polarization in CdSnO3 with Pbn21 m structure using the Berry phase method [19] gives an unexpectedly high value of Ps = 0.25C/m2, which is close to spontaneous polarization in barium titanate. 4. DISCUSSION OF THE RESULTS The results of our calculations show that the ferro electricity can appear in perovskite crystals that do not contain the atoms of transition d elements. Let us try to understand the nature of such phase transitions. The appearance of the ferroelectric phase transi tion in CdSnO3 cannot be associated with an off center position of cadmium atoms. Although, in the cubic parent phase of CdSnO3, the diagonal element of the on site force constant matrix for Cd atoms is ­ 0.0111 Ha/Bohr2, which indicates its off center posi tion, in the Pbnm phase, after the unit cell volume decreased by 8.4%, the minimum value of the diagonal element increases to +0.0714 Ha/Bohr2 and the potential well for the Cd atom becomes on center.1 Now, we consider the characteristics of the ferro electric soft mode. Table 3 presents the eigenvectors of the dynamic matrix for the B1u mode in CdSnO3 and CdTiO3 crystals and, for comparison, the 15 mode in BaTiO3. Here, O1 denotes oxygen atoms forming, with B atoms, chains that go in the direction of the polar axis of the ferroelectric phase and O2
1

PDOS, (Ha · cell)­1

100

Sn states O1 states

~ ~ O2 states

Fig. 2. Partial contributions of the s, p, and d states of the Cd, Sn, O1, and O2 atoms to the density of states in the orthorhombic phase Pbnm of CdSnO3. The energy origin is taken at the top of the valence band.

In the present work, the values of the force constant matrix and the partial densities of states are given in the Hartree system of atomic units. Vol. 51 No. 9 2009

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denotes the remaining oxygen atoms. The analysis of the eigenvector of the B1u mode in CdSnO3 shows that the main contribution to the ferroelectric mode is made by Cd and O2 atoms. At the same time, the Sn atom substantially shifts only in the y direction per pendicular to the polar axis and its small shift along the polar axis takes place in the opposite phase with cad mium atoms and in phase with oxygen atoms. This means that the main role in the appearance of ferro electricity in CdSnO3 is played by cadmium atoms. The comparison of eigenvectors of ferroelectric modes in these three crystals reveals that cadmium metastan nate and barium titanate are the limiting cases, in which the appearance of ferroelectricity is caused by A and B atoms in the perovskite structure ABO3, and cadmium titanate is an intermediate case. The effective atomic charges Z* in the Pbnm phase of cadmium metastannate (Table 3) are close to formal valences of atoms. In the cubic parent phase, the charge for Sn atoms is slightly smaller than that in the orthorhombic phase (Z* = 4.17) and, for Cd atoms, it is slightly greater (Z* = 3.26). For better understanding of the nature of ferroelec tric instability and of the character of chemical bond ing in CdSnO3, the partial densities of states (con tributions of s, p, and d orbitals of Cd, Sn, O1, and O2 atoms to the total density of states) were calculated. The results of these calculations are presented in Fig. 2. As follows from the figure, the overlap of Cd 4d states and O 2p states plays an important role in the formation of chemical bonding in the crystal; compa rable contributions from these states give evidence for a noticeable covalent component in this bonding. The contribution of Sn 5s and Sn 5p states to the valence band is much smaller than that of Cd atoms, which suggests a predominantly ionic character of the Sn­O bonds. The Sn 5s states overlapping with O 2p states make the main contribution to the conduction band (E > 0.037 Ha in Fig. 2). It should be noted that, although the partial densities of states for CdSnO3 were already calculated in [18], that work did not ana lyze the role of Cd 4d states, whose contribution to the density of states, according to our calculations, is sev eral times greater than the contribution of Sn 5s states. The conclusion of an important role of covalent inter action between Cd and O atoms, which follows from the analysis of partial densities of states, agrees with the results given by the analysis of the eigenvector of the ferroelectric mode and suggests that the rearrange ment of these bonds is the cause of the ferroelectric instability in CdSnO3. Table 4 presents calculated frequencies of the modes active in Raman and infrared (IR) spectra for crystals of CdSnO3 with space groups Pbnm and Pbn21. In the crystal with the Pbnm structure, there are 24 Raman active modes (modes with Ag, B1g, B2g, and B3g symmetry) and 25 IR active modes (modes with B1u, B2u, and B3u symmetry). When the symmetry of crystals is lowered to Pbn21, all 57 optical modes
PHYSICS OF THE SOLID STATE Vol. 51 No. 9

Table 4. Calculated frequencies of the IR and Raman active optical modes i for CdSnO3 crystals with space groups Pbnm and Pbn21 Space Mode group Pbnm Ag B1g B2g B3g B1u B2u B3u A1 A
2

i, cm­1 79, 130, 196, 290, 413, 452, 522 108, 150, 232, 347, 440, 500, 671 87, 207, 446, 474, 668 107, 142, 375, 500, 590 89i, 118, 163, 237, 390, 551, 594 62, 150, 175, 226, 264, 302, 420, 512, 605 90, 131, 186, 216, 267, 362, 420, 476, 640 78, 117, 138, 144, 201, 213, 241, 271, 375, 413, 436, 504, 562, 574 93, 105, 121, 130, 149, 202, 226, 263, 351, 396, 432, 486, 539, 599, 655 90, 120, 152, 206, 236, 251, 273, 351, 396, 443, 466, 483, 610, 658 88, 131, 151, 193, 208, 238, 261, 298, 364, 421, 462, 515, 579, 603

Pbn21

B B

1

2

become active in the Raman spectra and 42 modes of A1, B1, and B2 symmetry become active in the IR spec tra. Unfortunately, the IR absorption spectra in the range 250­800 cm­1 for CdSnO3 [20] available in literature consist of five very wide bands and their identification failed. As for the prospects of further studies of ferroelec tric properties of CdSnO3, we should note the fol lowing. As was noted in Section 1, these crystals are n type semiconductors and their temperature depen dence of conductivity has a thermoactive character with an activation energy of 0.3 eV [3]. Therefore, if the hopping conductivity is not high, studies at low temperatures are possible. Donor levels supplying electrons to the conduction band are usually associ ated with the oxygen vacancies. However, a rather small effect of variations in the partial pressure of O2 during annealing on the conductivity of CdSnO3 [8] makes this explanation very unlikely. In [21], in the discussion of the properties of Cd2SnO4, the energy position of various defects was calculated and it was shown that the most important defect in this crystal is the antisite defect CdSn. Perhaps, the same defects can determine the electrical conductivity of CdSnO3. We hope that improvements in the technology of the crystal growth and doping of them with acceptors will make it possible in the future to obtain high resistivity crystals of CdSnO3, which will enable one to per form direct dielectric measurements. 5. CONCLUSIONS The first principles calculations confirm the exist ence of the stable ferroelectric phase Pbn21 in cad

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LEBEDEV 9. E. N. Myasnikov, R. I. Spinko, E. A. Shalaeva, and T. P. Myasnikova, Ferroelectrics 214, 177 (1998). 10. V. N. Lebedev, R. V. Kolesova, and E. G. Fesenko, in Crystallization and Properties of Crystals (Novocherkassk Polytechnical Institute, Novocherkassk, 1977), Issue 4, p. 96 [in Russian]. 11. N. V. Prutsakova and Yu. V. Kabirov, Issled. Ross. (èle ktron. Zh.) 7, 2402 (2004); http://zhurnal.ape.relarn.ru/ articles/2004/226.pdf. 12. X. Gonze, J. M. Beuken, R. Caracas, F. Detraux, M. Fuchs, G. M. Rignanese, L. Sindic, M. Verstraete, G. Zerah, F. Jollet, M. Torrent, A. Roy, M. Mikami, Ph. Ghosez, J. Y. Raty, and D. C. Allan, Comput. Mater. Sci. 25, 478 (2002). 13. J. P. Perdew and A. Zunger, Phys. Rev. B: Condens. Matter 23, 5048 (1981). 14. A. M. Rappe, K. M. Rabe, E. Kaxiras, and J. D. Joan nopoulos, Phys. Rev. B: Condens. Matter 41, 1227 (1990). 15. N. J. Ramer and A. M. Rappe, Phys. Rev. B: Condens. Matter 59, 12 471 (1999). 16. A. I. Lebedev, Fiz. Tverd. Tela (St. Petersburg) 51 (2), 341 (2009) [Phys. Solid State 51 (2), 362 (2009)]. 17. A. I. Lebedev, Fiz. Tverd. Tela (St. Petersburg) 51 (4), 757 (2009) [Phys. Solid State 51 (4), 802 (2009)]. 18. H. Mizoguchi, H. W. Eng, and P. M. Woodward, Inorg. Chem. 43, 1667 (2004). 19. N. Sai, K. M. Rabe, and D. Vanderbilt, Phys. Rev. B: Condens. Matter 66, 104 108 (2002). 20. I. L. Botto and E. J. Baran, Z. Anorg. Allg. Chem. 465, 186 (1980). 21. S. B. Zhang and S. H. Wei, Appl. Phys. Lett. 80, 1376 (2002).

mium metastannate, the energy of which is 30.5 meV lower than the energy of the nonpolar phase Pbnm, and the spontaneous polarization is 0.25 C/m2. The analysis of the eigenvector of the ferroelectric mode in CdSnO3 and the partial densities of states shows that the ferroelectric instability in this crystal, which does not contain transition d element atoms, is asso ciated with the formation of the covalent bonding between Cd and O atoms. ACKNOWLEDGMENTS This work was supported by the Russian Founda tion for Basic Research (project no. 08 02 01436). REFERENCES
1. A. J. Smith, Acta Crystallogr. 13, 749 (1960). 2. I. NÀray SzabÑ, Naturwiss. 31, 202 (1943). 3. R. D. Shannon, J. L. Gillson, and R. J. Bouchard, J. Phys. Chem. Solids 38, 877 (1977). 4. I. Morgenstern Badarau, P. Poix, and A. Michel, C. R. Acad. Sci., ser. IV 258, 3036 (1964). 5. Z. Tianshu, S. Yusheng, D. Qiang, and F. Huajun, J. Mater. Sci. Lett. 13, 1647 (1994). 6. Y. L. Liu, Y. Xing, H. F. Yang, Z. M. Liu, Y. Yang, C. L. Shen, and R. Q. Yu, Anal. Chim. Acta 527, 21 (2004). 7. Landolt­BÆrnstein: Numerical Data and Functional Relationships in Science and Technology. New Series. Group III (Springer, Berlin, 1986), Vol. 7d1g, p. 178. 8. F. Golestani Fard, C. A. Hogarth, and D. N. Waters, J. Mater. Sci. Lett. 2, 505 (1983).

Translated by E. Chernokozhin

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