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GGA+U studies of the cubic perovskites BaMO3 (M=Pr, Th and U)
Physica B 410 (2013) 217–221
Contents lists available at SciVerse ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
GGA þU studies of the cubic perovskites BaMO3 (M ¼Pr, Th and U) Zahid Ali a,b,n, Iftikhar Ahmad a, Ali H. Reshak c a b c
Materials Modeling Center, Department of Physics, University of Malakand, Chakdara, Dir (L), Pakistan Department of Physics, Hazara University, Mansehra, Pakistan School of Complex Systems, FFPW, CENAKVA, University of South Bohemia in CB, Nove Hrady 37333, Czech Republic
a r t i c l e i n f o
abstract
Article history: Received 27 September 2012 Accepted 9 November 2012 Available online 19 November 2012
Structural and electronic properties of the cubic perovskites BaMO3 (M ¼Pr, Th and U) are theoretically investigated by using the full potential linearized augmented plane wave (FP-LAPW) method. The exchange correlation of GGA þ U is employed in the present DFT calculations to treat the f electrons properly. The structural parameters of the compounds are also evaluated by analytical techniques. The calculated structural parameters by both techniques are found in agreement with the experimental results. Furthermore, the calculated tolerance factors confirm the experimentally observed cubic structure for all the three compounds. The evaluated critical radii of the compounds provide information about the oxygen migration energy and ion conductivity and they also reveals that BaPrO3 posses the highest migration energy. The spin-polarized band structures show that BaPrO3 is a wide band gap semiconductor, BaUO3 is half-metal and BaThO3 is insulator. The magnetic studies of these compounds reveals that BaPrO3 is anti-ferromagnetic, BaThO3 is nonmagnetic/paramagnetic and BaUO3 is ferromagnetic material. & 2012 Elsevier B.V. All rights reserved.
Keywords: Perovskite oxides Cubic perovskites DFT Tolerance factor Electronic band structures
1. Introduction Perovskite is the most common structure among ternary oxides. The unique physical properties of these compounds like ferroelectricity, piezoelectricity, high temperature superconductivity, colossal magnetoresistivity and ionic conductivity make them significant in various technological applications [1–3]. They can be used as sensors, substrates, catalytic electrodes in fuel cells and are also promising candidates for optoelectronic and spintronic devices. The structural properties of the cubic perovskites BaPrO3 [2], BaThO3 [1] and BaUO3 [4] are experimentally investigated. In the cubic form of BaMO3 (M ¼Pr, Th and U), the divalent cation (Ba) is located at (0,0,0) position, O at (0.5,0.5,0), (0.5,0,0.5), (0,0.5,0.5) positions and M is located at the body centered position (0.5,0.5,0.5) of the cubic unit cell. In BaMO3, the M site ion is octahedrally coordinated with six oxygen ions (MO6). The structural phase transition in BaPrO3 has been investigated by Saines et al. [5]. BaPrO3 adsorbs atmospheric moist (H2O) and exhibits good proton conductivity. The change of oxidation states between Pr3 þ and Pr4 þ of the praseodymium ion produces high electronic conductivity [6]. The proton conductivity perovskites are potential electrolyte materials of the future generation and will probably operate at low temperatures. Barium praseodymate (BaPrO3) is a suitable compound for proton
conductivity at high temperatures [7,8]. These compounds (proton conductivity) are expected to be efficient cathode materials and will improve the electrode reaction. Kemmler-Sack et al. [2] studied that in BaPrO3, praseodymium cannot exist in the highest oxidation state Pr þ 4 in air at high temperature. Barium thorate (BaThO3), is one of the significant products of fission reactor. The structural, optoelectronic and thermodynamic properties of BaThO3 are reported in Refs. [1,9–11], while the cubic and orthorhombic structure of BaUO3 as well as the oxidation states of U in this compound are studied by Soldatov et al. [12]. In the present theoretical investigations, barium-based rareearth cubic perovskites BaPrO3, BaThO3 and BaUO3 are studied. The electronic and magnetic properties of all the three compounds are evaluated by using GGA þU. The lattice constants, Bulk moduli, ground state energies, bond lengths, electronic band structures and total and partial density of states of these compounds are calculated by using the full potential linearized augmented plane wave (FP-LAPW) method within the frame work of density functional theory (DFT). For the exchange correlation, GGA þU is employed in the calculations, so that the f electrons can be properly treated. Some of the structural properties like lattice constants, tolerance factors and critical radii are also evaluated by analytical techniques.
2. Theory and calculations n
Corresponding author at: Materials Modeling Center, Department of Physics, University of Malakand, Chakdara, Dir (L), Pakistan. Tel.: þ92 333 902 7401. E-mail address: [email protected] (Z. Ali). 0921-4526/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physb.2012.11.008
In the present work Kohn–Sham equations [13] are solved to calculate the structural, electronic and magnetic properties of the
218
Z. Ali et al. / Physica B 410 (2013) 217–221
cubic perovskites BaPrO3, BaThO3 and BaUO3. Details of the spinpolarized FP-LAPW method, GGA þU formulas and the wien2k software used in the present calculations can be found in Refs. [14–17]. In the full potential scheme the unit cell of the crystal is partitioned into two different regions: (1) atomic spheres and, (2) interstitial region (region outside the atomic spheres). The wave function is expanded into two different basis sets. Within each atomic sphere the wave function is expanded in atomic-like functions (radial functions times spherical harmonics) while in the interstitial region it is expanded in a plane wave basis. Inside the sphere the maximum value of l for the wave function expansion is lmax ¼ 9. In order to calculate the electronic properties of BaPrO3, BaThO3 and BaUO3 by GGA þU, it is assumed that the density matrix is diagonal and U is the same for all Coulomb interactions (Uij U) and J is also the same for all the exchange interactions (Jij J). There are several methods to incorporate the U-term [18,19], but here the self-interaction correction introduced by Anisimov et al. [20] as implemented in the wien2k package is used. In the GGAþUSIC method the total energy is E ¼ E0 þEGGA þ U SIC
EGGA þ USIC ¼
UJ 2
ð1Þ
X N n2m, s
! ð2Þ
m, s
In Eq. (2) N is the total number of electrons and nm,s is the occupation number of 9l,m, sS QUOTE state with spin s. After examining and testing several values of Hubbard potential in order to adjust the Pr-4f, Th-5f and U-5f orbitals level in the density of states an approximate corrected value of U–J for the self-interaction correction is obtained which is probably best for the strongly correlated systems and full potential scheme. The final Hubbard potential used is Ueff ¼U–J ¼9.5 eV for Pr and U and 6 eV for Th. In the full potential scheme the core electrons are treated fully relativistically and the valence electrons are treated semirelativistically. In order to ensure that no electron leakage is taking place semi-core states are included so that accurate results are achieved. Well converged results are obtained for 35 k points and RMT Kmax ¼8.00.
3. Results and discussions 3.1. Structural properties The optimum volume of each of the BaMO3 (M ¼Pr, Th and U) crystal is obtained by fitting the total energy of each unit cell against its volume using Birch–Murnaghan’s equation of state [21]. The optimization plots for all the compounds are shown in Fig. 1. The minimum energy in each plot corresponds to its optimum volume. The relaxed lattice constants, bulk moduli, and ground state energies for all the compounds are evaluated at the optimum volumes. These calculated ground state results are compared with the available experimental results in Table 1. The lattice constants of the perovskites are also calculated by the ionic radii method using the following relation [22]: a0 ¼ a þ bðr Ba þr O Þ þ gðr M þ r O Þ
ð3Þ
where a (0.06741), b (0.4905), and g (1.2921) are constants, rBa is ˚ rM is the ionic radii of Pr (1.3 A), ˚ Th the ionic radius of Ba (1.61 A), ˚ or U (0.89 A) ˚ [23], while r0 is the ionic radius of O (1.4 A). ˚ (0.94 A) It is clear from the table that our calculated lattice constants by DFT as well as ionic radii method are in close agreement (less than 1%) with the experimental results. The knowledge of bond lengths is important for the understanding of geometry and structure of a compound. The calculated bonds length between different atoms (Ba–O, Ba–M and M– O) are presented in Table 1. The calculated bonds length is also compared with the available experimental results. It is clear from the table that our results are very close to the available experimental results. Though, the experimental bonds length is not available for all the bonds but based on the available ones, we are confident to infer that our calculated results will be consistent with the experimental ones. The calculated bonds length are used in the following relation [3,24] to evaluate tolerance factors for BaPrO3, BaThO3 and BaUO3. t¼
0:707/BaOS /MOS
ð4Þ
where /BaOS and /MOS show the average bonds length between Ba and O and M (M ¼Pr, Th and U) and O respectively. The calculated tolerance factors for all the three perovskites are presented in Table 1. As for a cubic perovskite, tolerance factor lies within the range 0.93–1.02 [25], therefore the results
Fig. 1. Variation of total energy verses unit cell volume of BaMO3 (M¼ Pr, Th and U).
Z. Ali et al. / Physica B 410 (2013) 217–221
219
Table 1 Experimental and calculated values of the lattice constants (a0), ground state energies (E0), bulk moduli (B), critical radii (rC), tolerance factors (t) and bond lengths of cubic BaMO3 (M ¼ Pr, Th and U). Compound
Experimental
Present (DFT)
Present analytical
Other analytical
BaPrO3 ˚ a0 (A)
4.354a
4.3637
4.362
4.408b
EFM (Ry) EAFM (Ry) B (GPa) ˚ rC (A)
35201.349 35202.847 124.9802
t Bond lengths ˚ Ba–O (A)
0.999
0.888
3.0787 3.7706
˚ Ba–Pr (A) ˚ Pr–O (A)
2.19d
2.177
BaThO3 ˚ a0 (A)
4.48e
4.5306
Epara (Ry) B (GPa) ˚ rC (A)
69798.702 135.3302
t Bond lengths ˚ Ba–O (A)
0.999
4.513b
0.914c
3.035 3.717 2.146 4.4074f
4.38
EFM (Ry) EAFM (Ry) B (GPa) ˚ rC (A)
72891.099 72890.573 137.1814
t Bond lengths ˚ Ba–O (A)
0.999
˚ Ba–U (A) ˚ U–O (A)
4.478
0.973
˚ Ba–Th (A) ˚ Th–O (A) BaUO3 ˚ a0 (A)
0.951c
4.413
4.455b
0.951 0.934c
3.0899 3.7843 2.19f,
g
2.1849 Fig. 2. Spin dependent (a) band structures and (b) total and partial densities of states for BaPrO3.
a
[2]. [23]. c [9]. d [26]. e [3]. f [4]. g [12]. b
presented in the table reveals that BaPrO3, BaThO3 and BaUO3 perovskites are cubic in nature. Critical radius plays an important role in the activation energy of oxygen (O) migration, ion conductivity and it also provides a guideline for doping selection. The critical radii for all the three compounds are calculated by using the following mathematical formula [25]: rc ¼
1:414r M r O 3:414r Ba þ5:828ðr Ba r O Þ2 2r Ba þ 0:828r M þ 2:828r O
ð5Þ
It is clear from the table that the critical radius of BaThO3 is greater than the other two compounds. Hence it shows smaller migration energy. 3.2. Electronic properties The spin-polarized electronic band structures for BaMO3 (M¼Pr, Th and U) perovskites are calculated using GGA with the insertion of coulomb interaction U (GGAþU). The calculated band structures of all the three compounds are presented in Fig. 2–Fig. 4(a), respectively. It is clear from Fig. 2(a) that BaPrO3 is a wide bandgap semiconductor for spin up (m) as well as for spin down (k) states. Interestingly, the band structure in Fig. 3(a) confirms
the insulating behavior of BaThO3 for both spin states (majority as well as minority spin channels), which is in agreement with the simple GGA results [1]. It is evident from Fig. 4(a), that BaUO3 is metallic for spin up state (majority spin state) and wide band gap semiconductor for spin down state. Hence, our calculations show that BaPrO3 is a wide band gap semiconductor for both spin states, BaThO3 is insulator and BaUO3 is half-metal. In order to visualize the electronic origin of band structures, total and partial densities of states for all the three compounds are calculated. The densities of states for BaPrO3, BaThO3 and BaUO3 are presented in Figs. 2–Fig. 4(b). It is clear from Fig. 2(b) that in the spin-up state; Ba-4p, Ba-6s and Pr-5p make the core of BaPrO3 and Pr-4f state is localized at 5.5 eV, while O-2p state makes the valence band from 3.5 to the Fermi level. It is evident from the figure that for spin-up state, BaPrO3 exhibits semiconducting behavior with a bandgap of 3.2 eV at G–R symmetry point. Interestingly, in the spin-down state, Pr-4f is pulled by the conduction band and though the compound is still a semiconductor but the bandgap is increased to 3.8 eV. In conduction band the mix contribution of Ba-4d, Pr-5d and Ba-4 f states takes place. The spin-polarized band structures as well as density of states predict spin dependent variable semiconductor nature of BaPrO3 (narrow gap semiconductor in one spin- state and wide in the other spin state). Fig. 3(b) shows, total and partial densities of states for BaThO3, where the energy is scaled with respect to the Fermi level. It is clear from the figure that Ba-6s, Ba-4p and Th-6p form the core, while O-2p state forms the valence band of BaThO3
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Z. Ali et al. / Physica B 410 (2013) 217–221
Fig. 3. Spin dependent (a) band structures and (b) total and partial densities of states for BaThO3.
and occurs in the range 4 eV to the Fermi level. The conduction band of this compound is constituted of Th-5f, Ba-5d and Th-6d. There is a large band gap between the conduction band and valence band, for both majority and minority spin-channels. This confirms the insulating behavior of the compound. It is also clear from Fig. 3(a and b) that there is no difference in the shape and symmetry of the majority and minority spin-states. Hence BaThO3 is insulator. Spin-polarized total and partial densities of states for BaUO3 are given in Fig. 4(b). It is clear from the figure that the bottom of the top of the core state is occupied by Ba-6s at 30 eV and U-6p state, which occurs in the range 23.7 to 17 eV. The peak at 15 eV is due to Ba-4p state and O-2p state covers the valance band maxima from 8 to 4.4 eV. A prominent role in the overall physical properties of the compound is played by U-5f state. For spin-up state, the f- state is localized at the Fermi level, makes the compound metallic. Hence BaUO3 exhibits metallic character for spin-up state, while for spin-down state, U-5f state is pulled by the conduction band generating a band gap of 3.9 eV between the conduction band and valence band at G symmetry point, which is also shown in Fig. 4(a). Hence BaUO3 is a spin dependent half-metal. 3.3. Magnetic properties Magnetism in perovskite oxides arises due to indirect exchange interaction between rare-earths–rare-earths via nonmagnetic
Fig. 4. Spin dependent (a) band structures and (b) total and partial densities of states for BaUO3.
Table 2 Calculated total, interstitial and local magnetic moments in the units of mB per unit cell for the cubic perovskites BaMO3 (M ¼ Pr, Th and U). Magnetic moment T
M Mint MBa MPr/Th/U MO
BaPrO3
BaThO3
BaUO3
0.9989 0.0372 0.0011 1.0508 0.0300
0.0000 0.00120 0.00004 0.00012 0.00045
2.00 0.509 0.058 1.647 0.070
anion O (M–O–M). There are two possible exchanges, one is double-exchange and the other one is super exchange [27]. In most cases double exchange is ferromagnetic and super exchange is anti-ferromagnetic. While in some cases the situation is reverse. Zener [28] studied that ferromagnetic interactions are favored when magnetic atoms are well separated and conduction electrons are present in a crystal. He introduced the concept of immediate transfer of electron from rare-earths metal (M) to the oxygen and from the oxygen to the neighboring rare-earths. The stable magnetic state of all the three compounds is calculated on the basis of optimization; in which system lowers its ground state energy. For anti-ferromagnetic calculations a subcell is selected for each of the three compounds, in which the spin of one magnetic element is being up while the other is down like in our previous work [3].
Z. Ali et al. / Physica B 410 (2013) 217–221
221
compound is 2 mB. The integer value of the total magnetic moment is one of the significance of the half metallic nature of a material. Hence BaUO3 is half metallic ferromagnetic material.
4. Conclusions
Fig. 5. AFM DOS for the double cell of the BaPrO3.
In the present calculations the stable magnetic state for BaPrO3 is anti-ferromagnetic in which the system lowers its ground state energy as compared to ferromagnetic and paramagnetic states. The calculated energies of different magnetic states are presented in the Table 1. Our anti-ferromagnetic result for BaPrO3 is in agreement with the experimental work reported in Ref. [29], which shows that BaPrO3 is an anti-ferromagnetic material. The calculated magnetic moments per unit cell are presented in Table 2. A self consistent field AFM calculations has been performed for the double cell of the BaPrO3 in which the spin of one Pr atom is sets up (m) while the other is down (k). The calculated spin dependent total density of states is shown in Fig. 5. It is clear from the figure that the density is symmetric for both spin states. The magnetic moments of the constituent atoms cancel each other result and hence zero net magnetic moment is observed. This is one of the significance of the anti-ferromagnetic nature of a compound. Hence BaPrO3 is anti-ferromagnetic material. The stable magnetic state of BaThO3 is nonmagnetic/paramagnetic insulator. The calculated magnetic moments of interstitial, local and total magnetic moments per unit cell are presented in Table 2. It is clear from the table that the constituent magnetic moments for all the adjacent atoms as well as interstitial magnetic moments are negligible and also cancel the effect of each other, resulting zero magnetic moment per unit cell. These results reveal that BaThO3 is a nonmagnetic/paramagnetic material. Similarly the stable magnetic state of the BaUO3 compound is ferromagnetic in which the ground state energy of the material is minimum as compared to the other magnetic states. The calculated magnetic moments per unit cell for interstitial, local and total magnetic moments are presented in Table 2. It is clear from the table that the effective magnetic moment for U is 1.647 mB, while the literature reported [30] magnetic moment of U in BaUO3 x is 2 mB, which shows that our results are closer to the experimental ones. In this compound double-exchange interaction between uranium atoms via nonmagnetic oxygen anion (U þ 4–O–U þ 5) takes place. The total magnetic moment of the
In summary, the structural and electronic properties of the cubic perovskites BaPrO3, BaThO3 and BaUO3 are calculated using FP-LAPW method with GGA þU. The structural properties are also evaluated by analytical techniques. The calculated spin-polarized band structures reveal that, band gap exists around the Fermi level for the spin down states of all the three compounds, while for spin up states BaPrO3 is semiconductor, BaThO3 is insulator and BaUO3 is metallic in nature. Furthermore, the bandgap for the spin up state of BaPrO3 is larger than the spin down state. Hence, it is concluded that BaPrO3 is spin dependent semiconductor, BaUO3 is a half-metal and BaThO3 is insulator. The magnetic properties of these compounds reveal that BaPrO3 is anti-ferromagnetic, BaThO3 is nonmagnetic/paramagnetic and BaUO3 is a ferromagnetic compound. References [1] G. Murtaza, I. Ahmad, B. Amin, A. Afaq, M. Maqbool, J. Maqssod, I. Khan, M. Zahid, Opt. Mater. 33 (2011) 553. [2] S. Kemmler-Sack, I. Hofelich, Z. Naturforsch 26b (1971) 539. [3] Z. Ali, I. Ahmad, I. Khan, B. Amin, http://dx.doi.org/10.1016/j.intermet.2012. 08.001. [4] Y. Hinatsu, J. Solid State Chem. 102 (1993) 566. [5] P.J. Saines, B.J. Kennedy, R.I. Smith, Mater. Res. Bull. 44 (2009) 874. [6] C.Y. Jones, J. Wu, L.P. Li, S.M. Haile, J. Appl. Phys. 97 (2005) 114908. [7] V.P. Gorelov, B.L. Kuzin, V.B. Balakireva, N.V. Sharova, G.K. Vdovin, S.M. Beresnev, Y.N. Kleshchev, V.P. Brusentsov, Russ. J. Electrochem. 37 (2001) 505. [8] A. Mitsui, M. Miyayama, H. Yanagida, Solid State Ionics 22 (1987) 213. [9] R.L. Moreira, A. Dias, J. Phys. Chem. Solids 68 (2007) 1617. [10] R.V. Krishnan, K. Nagarajan, P.R.V. Rao, J. Nucl. Mater. 99 (2001) 28. [11] R. Mishra, M.A. Basu, S.R. Bharadwaj, A.S. Kerkar, D. Das, S.R. Dharwadkar, J. Alloy Compd. 290 (1999) 97. [12] A.V. Soldatov, D. Lamoen, M.J. Konstantinovic, S. Van den Berghe, A.C. Scheinost, M. Verwerft, J. Solid State Chem. 180 (2007) 54. [13] W. Kohn, L.S. Sham, Phys. Rev. A 140 (1965) 1133. [14] K. Schwarz, J. Solid State Chem. 176 (2003) 319. [15] O.K. Andersen, Phys. Rev. B 12 (1975) 3060. [16] P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, J. Luitz, WIEN2K: AnAugmented Plane Wave þLocal Orbital Program for Calculating Crystal Properties, Techn Universitat, Wien, Austria, 2001. [17] B. Amin, I. Ahmad, J. App. Phys. 109 (2011) 023109. [18] A.G. Petukhov, I.I. Mazin, Phys. Rev. B 67 (2003) 153106. [19] P. Novak, J. Kunes, L. Chaput, W.E. Pickett, Phys. Status Solidi B 243 (2006) 563. [20] V.I. Anisimov, I.V. Solovyev, M.A. Korotin, M.T. Czyzyk, G.A. Sawatzky, Phys. Rev. B 48 (1993) 16929. [21] F. Birch, Phys. Rev. 71 (1947) 809. [22] R. Ubic, J. Am. Ceram. Soc. 90 (2007) 3326. [23] A.S. Verma, A. Kumar, S.R. Bhardwaj, Phys. Status Solidi B 245 (2008) 1520. [24] J.B. Goodenough, Rep. Prog. Phys. 67 (2004) 1915. [25] N. Xu, H. Zhao, X. Zhou, W. Wei, X. Lu, W. Ding, F. Li, J. Hydrogen Energy 35 (2010) 7295. [26] M. Marezio, J.P. Remeika, P.D. Dernier, Acta Cryst. B 26 (1970) 2008. [27] S. Blundell, Magnetism in Condensed Matter, Oxford University Press, New York. 2001. [28] C. Zener, Phys. Rev. 82 (1951) 403. [29] P.N. Lisboa-Filho, J.P. Rodrigues, W.A. Ortiz, J. Mater. Sci. Lett. 22 (2003) 623. [30] S. Van den Berghe, A. Leenaers, C. Ritter, J. Solid State Chem. 177 (2004) 2231.
Contents lists available at SciVerse ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
GGA þU studies of the cubic perovskites BaMO3 (M ¼Pr, Th and U) Zahid Ali a,b,n, Iftikhar Ahmad a, Ali H. Reshak c a b c
Materials Modeling Center, Department of Physics, University of Malakand, Chakdara, Dir (L), Pakistan Department of Physics, Hazara University, Mansehra, Pakistan School of Complex Systems, FFPW, CENAKVA, University of South Bohemia in CB, Nove Hrady 37333, Czech Republic
a r t i c l e i n f o
abstract
Article history: Received 27 September 2012 Accepted 9 November 2012 Available online 19 November 2012
Structural and electronic properties of the cubic perovskites BaMO3 (M ¼Pr, Th and U) are theoretically investigated by using the full potential linearized augmented plane wave (FP-LAPW) method. The exchange correlation of GGA þ U is employed in the present DFT calculations to treat the f electrons properly. The structural parameters of the compounds are also evaluated by analytical techniques. The calculated structural parameters by both techniques are found in agreement with the experimental results. Furthermore, the calculated tolerance factors confirm the experimentally observed cubic structure for all the three compounds. The evaluated critical radii of the compounds provide information about the oxygen migration energy and ion conductivity and they also reveals that BaPrO3 posses the highest migration energy. The spin-polarized band structures show that BaPrO3 is a wide band gap semiconductor, BaUO3 is half-metal and BaThO3 is insulator. The magnetic studies of these compounds reveals that BaPrO3 is anti-ferromagnetic, BaThO3 is nonmagnetic/paramagnetic and BaUO3 is ferromagnetic material. & 2012 Elsevier B.V. All rights reserved.
Keywords: Perovskite oxides Cubic perovskites DFT Tolerance factor Electronic band structures
1. Introduction Perovskite is the most common structure among ternary oxides. The unique physical properties of these compounds like ferroelectricity, piezoelectricity, high temperature superconductivity, colossal magnetoresistivity and ionic conductivity make them significant in various technological applications [1–3]. They can be used as sensors, substrates, catalytic electrodes in fuel cells and are also promising candidates for optoelectronic and spintronic devices. The structural properties of the cubic perovskites BaPrO3 [2], BaThO3 [1] and BaUO3 [4] are experimentally investigated. In the cubic form of BaMO3 (M ¼Pr, Th and U), the divalent cation (Ba) is located at (0,0,0) position, O at (0.5,0.5,0), (0.5,0,0.5), (0,0.5,0.5) positions and M is located at the body centered position (0.5,0.5,0.5) of the cubic unit cell. In BaMO3, the M site ion is octahedrally coordinated with six oxygen ions (MO6). The structural phase transition in BaPrO3 has been investigated by Saines et al. [5]. BaPrO3 adsorbs atmospheric moist (H2O) and exhibits good proton conductivity. The change of oxidation states between Pr3 þ and Pr4 þ of the praseodymium ion produces high electronic conductivity [6]. The proton conductivity perovskites are potential electrolyte materials of the future generation and will probably operate at low temperatures. Barium praseodymate (BaPrO3) is a suitable compound for proton
conductivity at high temperatures [7,8]. These compounds (proton conductivity) are expected to be efficient cathode materials and will improve the electrode reaction. Kemmler-Sack et al. [2] studied that in BaPrO3, praseodymium cannot exist in the highest oxidation state Pr þ 4 in air at high temperature. Barium thorate (BaThO3), is one of the significant products of fission reactor. The structural, optoelectronic and thermodynamic properties of BaThO3 are reported in Refs. [1,9–11], while the cubic and orthorhombic structure of BaUO3 as well as the oxidation states of U in this compound are studied by Soldatov et al. [12]. In the present theoretical investigations, barium-based rareearth cubic perovskites BaPrO3, BaThO3 and BaUO3 are studied. The electronic and magnetic properties of all the three compounds are evaluated by using GGA þU. The lattice constants, Bulk moduli, ground state energies, bond lengths, electronic band structures and total and partial density of states of these compounds are calculated by using the full potential linearized augmented plane wave (FP-LAPW) method within the frame work of density functional theory (DFT). For the exchange correlation, GGA þU is employed in the calculations, so that the f electrons can be properly treated. Some of the structural properties like lattice constants, tolerance factors and critical radii are also evaluated by analytical techniques.
2. Theory and calculations n
Corresponding author at: Materials Modeling Center, Department of Physics, University of Malakand, Chakdara, Dir (L), Pakistan. Tel.: þ92 333 902 7401. E-mail address: [email protected] (Z. Ali). 0921-4526/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physb.2012.11.008
In the present work Kohn–Sham equations [13] are solved to calculate the structural, electronic and magnetic properties of the
218
Z. Ali et al. / Physica B 410 (2013) 217–221
cubic perovskites BaPrO3, BaThO3 and BaUO3. Details of the spinpolarized FP-LAPW method, GGA þU formulas and the wien2k software used in the present calculations can be found in Refs. [14–17]. In the full potential scheme the unit cell of the crystal is partitioned into two different regions: (1) atomic spheres and, (2) interstitial region (region outside the atomic spheres). The wave function is expanded into two different basis sets. Within each atomic sphere the wave function is expanded in atomic-like functions (radial functions times spherical harmonics) while in the interstitial region it is expanded in a plane wave basis. Inside the sphere the maximum value of l for the wave function expansion is lmax ¼ 9. In order to calculate the electronic properties of BaPrO3, BaThO3 and BaUO3 by GGA þU, it is assumed that the density matrix is diagonal and U is the same for all Coulomb interactions (Uij U) and J is also the same for all the exchange interactions (Jij J). There are several methods to incorporate the U-term [18,19], but here the self-interaction correction introduced by Anisimov et al. [20] as implemented in the wien2k package is used. In the GGAþUSIC method the total energy is E ¼ E0 þEGGA þ U SIC
EGGA þ USIC ¼
UJ 2
ð1Þ
X N n2m, s
! ð2Þ
m, s
In Eq. (2) N is the total number of electrons and nm,s is the occupation number of 9l,m, sS QUOTE state with spin s. After examining and testing several values of Hubbard potential in order to adjust the Pr-4f, Th-5f and U-5f orbitals level in the density of states an approximate corrected value of U–J for the self-interaction correction is obtained which is probably best for the strongly correlated systems and full potential scheme. The final Hubbard potential used is Ueff ¼U–J ¼9.5 eV for Pr and U and 6 eV for Th. In the full potential scheme the core electrons are treated fully relativistically and the valence electrons are treated semirelativistically. In order to ensure that no electron leakage is taking place semi-core states are included so that accurate results are achieved. Well converged results are obtained for 35 k points and RMT Kmax ¼8.00.
3. Results and discussions 3.1. Structural properties The optimum volume of each of the BaMO3 (M ¼Pr, Th and U) crystal is obtained by fitting the total energy of each unit cell against its volume using Birch–Murnaghan’s equation of state [21]. The optimization plots for all the compounds are shown in Fig. 1. The minimum energy in each plot corresponds to its optimum volume. The relaxed lattice constants, bulk moduli, and ground state energies for all the compounds are evaluated at the optimum volumes. These calculated ground state results are compared with the available experimental results in Table 1. The lattice constants of the perovskites are also calculated by the ionic radii method using the following relation [22]: a0 ¼ a þ bðr Ba þr O Þ þ gðr M þ r O Þ
ð3Þ
where a (0.06741), b (0.4905), and g (1.2921) are constants, rBa is ˚ rM is the ionic radii of Pr (1.3 A), ˚ Th the ionic radius of Ba (1.61 A), ˚ or U (0.89 A) ˚ [23], while r0 is the ionic radius of O (1.4 A). ˚ (0.94 A) It is clear from the table that our calculated lattice constants by DFT as well as ionic radii method are in close agreement (less than 1%) with the experimental results. The knowledge of bond lengths is important for the understanding of geometry and structure of a compound. The calculated bonds length between different atoms (Ba–O, Ba–M and M– O) are presented in Table 1. The calculated bonds length is also compared with the available experimental results. It is clear from the table that our results are very close to the available experimental results. Though, the experimental bonds length is not available for all the bonds but based on the available ones, we are confident to infer that our calculated results will be consistent with the experimental ones. The calculated bonds length are used in the following relation [3,24] to evaluate tolerance factors for BaPrO3, BaThO3 and BaUO3. t¼
0:707/BaOS /MOS
ð4Þ
where /BaOS and /MOS show the average bonds length between Ba and O and M (M ¼Pr, Th and U) and O respectively. The calculated tolerance factors for all the three perovskites are presented in Table 1. As for a cubic perovskite, tolerance factor lies within the range 0.93–1.02 [25], therefore the results
Fig. 1. Variation of total energy verses unit cell volume of BaMO3 (M¼ Pr, Th and U).
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Table 1 Experimental and calculated values of the lattice constants (a0), ground state energies (E0), bulk moduli (B), critical radii (rC), tolerance factors (t) and bond lengths of cubic BaMO3 (M ¼ Pr, Th and U). Compound
Experimental
Present (DFT)
Present analytical
Other analytical
BaPrO3 ˚ a0 (A)
4.354a
4.3637
4.362
4.408b
EFM (Ry) EAFM (Ry) B (GPa) ˚ rC (A)
35201.349 35202.847 124.9802
t Bond lengths ˚ Ba–O (A)
0.999
0.888
3.0787 3.7706
˚ Ba–Pr (A) ˚ Pr–O (A)
2.19d
2.177
BaThO3 ˚ a0 (A)
4.48e
4.5306
Epara (Ry) B (GPa) ˚ rC (A)
69798.702 135.3302
t Bond lengths ˚ Ba–O (A)
0.999
4.513b
0.914c
3.035 3.717 2.146 4.4074f
4.38
EFM (Ry) EAFM (Ry) B (GPa) ˚ rC (A)
72891.099 72890.573 137.1814
t Bond lengths ˚ Ba–O (A)
0.999
˚ Ba–U (A) ˚ U–O (A)
4.478
0.973
˚ Ba–Th (A) ˚ Th–O (A) BaUO3 ˚ a0 (A)
0.951c
4.413
4.455b
0.951 0.934c
3.0899 3.7843 2.19f,
g
2.1849 Fig. 2. Spin dependent (a) band structures and (b) total and partial densities of states for BaPrO3.
a
[2]. [23]. c [9]. d [26]. e [3]. f [4]. g [12]. b
presented in the table reveals that BaPrO3, BaThO3 and BaUO3 perovskites are cubic in nature. Critical radius plays an important role in the activation energy of oxygen (O) migration, ion conductivity and it also provides a guideline for doping selection. The critical radii for all the three compounds are calculated by using the following mathematical formula [25]: rc ¼
1:414r M r O 3:414r Ba þ5:828ðr Ba r O Þ2 2r Ba þ 0:828r M þ 2:828r O
ð5Þ
It is clear from the table that the critical radius of BaThO3 is greater than the other two compounds. Hence it shows smaller migration energy. 3.2. Electronic properties The spin-polarized electronic band structures for BaMO3 (M¼Pr, Th and U) perovskites are calculated using GGA with the insertion of coulomb interaction U (GGAþU). The calculated band structures of all the three compounds are presented in Fig. 2–Fig. 4(a), respectively. It is clear from Fig. 2(a) that BaPrO3 is a wide bandgap semiconductor for spin up (m) as well as for spin down (k) states. Interestingly, the band structure in Fig. 3(a) confirms
the insulating behavior of BaThO3 for both spin states (majority as well as minority spin channels), which is in agreement with the simple GGA results [1]. It is evident from Fig. 4(a), that BaUO3 is metallic for spin up state (majority spin state) and wide band gap semiconductor for spin down state. Hence, our calculations show that BaPrO3 is a wide band gap semiconductor for both spin states, BaThO3 is insulator and BaUO3 is half-metal. In order to visualize the electronic origin of band structures, total and partial densities of states for all the three compounds are calculated. The densities of states for BaPrO3, BaThO3 and BaUO3 are presented in Figs. 2–Fig. 4(b). It is clear from Fig. 2(b) that in the spin-up state; Ba-4p, Ba-6s and Pr-5p make the core of BaPrO3 and Pr-4f state is localized at 5.5 eV, while O-2p state makes the valence band from 3.5 to the Fermi level. It is evident from the figure that for spin-up state, BaPrO3 exhibits semiconducting behavior with a bandgap of 3.2 eV at G–R symmetry point. Interestingly, in the spin-down state, Pr-4f is pulled by the conduction band and though the compound is still a semiconductor but the bandgap is increased to 3.8 eV. In conduction band the mix contribution of Ba-4d, Pr-5d and Ba-4 f states takes place. The spin-polarized band structures as well as density of states predict spin dependent variable semiconductor nature of BaPrO3 (narrow gap semiconductor in one spin- state and wide in the other spin state). Fig. 3(b) shows, total and partial densities of states for BaThO3, where the energy is scaled with respect to the Fermi level. It is clear from the figure that Ba-6s, Ba-4p and Th-6p form the core, while O-2p state forms the valence band of BaThO3
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Fig. 3. Spin dependent (a) band structures and (b) total and partial densities of states for BaThO3.
and occurs in the range 4 eV to the Fermi level. The conduction band of this compound is constituted of Th-5f, Ba-5d and Th-6d. There is a large band gap between the conduction band and valence band, for both majority and minority spin-channels. This confirms the insulating behavior of the compound. It is also clear from Fig. 3(a and b) that there is no difference in the shape and symmetry of the majority and minority spin-states. Hence BaThO3 is insulator. Spin-polarized total and partial densities of states for BaUO3 are given in Fig. 4(b). It is clear from the figure that the bottom of the top of the core state is occupied by Ba-6s at 30 eV and U-6p state, which occurs in the range 23.7 to 17 eV. The peak at 15 eV is due to Ba-4p state and O-2p state covers the valance band maxima from 8 to 4.4 eV. A prominent role in the overall physical properties of the compound is played by U-5f state. For spin-up state, the f- state is localized at the Fermi level, makes the compound metallic. Hence BaUO3 exhibits metallic character for spin-up state, while for spin-down state, U-5f state is pulled by the conduction band generating a band gap of 3.9 eV between the conduction band and valence band at G symmetry point, which is also shown in Fig. 4(a). Hence BaUO3 is a spin dependent half-metal. 3.3. Magnetic properties Magnetism in perovskite oxides arises due to indirect exchange interaction between rare-earths–rare-earths via nonmagnetic
Fig. 4. Spin dependent (a) band structures and (b) total and partial densities of states for BaUO3.
Table 2 Calculated total, interstitial and local magnetic moments in the units of mB per unit cell for the cubic perovskites BaMO3 (M ¼ Pr, Th and U). Magnetic moment T
M Mint MBa MPr/Th/U MO
BaPrO3
BaThO3
BaUO3
0.9989 0.0372 0.0011 1.0508 0.0300
0.0000 0.00120 0.00004 0.00012 0.00045
2.00 0.509 0.058 1.647 0.070
anion O (M–O–M). There are two possible exchanges, one is double-exchange and the other one is super exchange [27]. In most cases double exchange is ferromagnetic and super exchange is anti-ferromagnetic. While in some cases the situation is reverse. Zener [28] studied that ferromagnetic interactions are favored when magnetic atoms are well separated and conduction electrons are present in a crystal. He introduced the concept of immediate transfer of electron from rare-earths metal (M) to the oxygen and from the oxygen to the neighboring rare-earths. The stable magnetic state of all the three compounds is calculated on the basis of optimization; in which system lowers its ground state energy. For anti-ferromagnetic calculations a subcell is selected for each of the three compounds, in which the spin of one magnetic element is being up while the other is down like in our previous work [3].
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compound is 2 mB. The integer value of the total magnetic moment is one of the significance of the half metallic nature of a material. Hence BaUO3 is half metallic ferromagnetic material.
4. Conclusions
Fig. 5. AFM DOS for the double cell of the BaPrO3.
In the present calculations the stable magnetic state for BaPrO3 is anti-ferromagnetic in which the system lowers its ground state energy as compared to ferromagnetic and paramagnetic states. The calculated energies of different magnetic states are presented in the Table 1. Our anti-ferromagnetic result for BaPrO3 is in agreement with the experimental work reported in Ref. [29], which shows that BaPrO3 is an anti-ferromagnetic material. The calculated magnetic moments per unit cell are presented in Table 2. A self consistent field AFM calculations has been performed for the double cell of the BaPrO3 in which the spin of one Pr atom is sets up (m) while the other is down (k). The calculated spin dependent total density of states is shown in Fig. 5. It is clear from the figure that the density is symmetric for both spin states. The magnetic moments of the constituent atoms cancel each other result and hence zero net magnetic moment is observed. This is one of the significance of the anti-ferromagnetic nature of a compound. Hence BaPrO3 is anti-ferromagnetic material. The stable magnetic state of BaThO3 is nonmagnetic/paramagnetic insulator. The calculated magnetic moments of interstitial, local and total magnetic moments per unit cell are presented in Table 2. It is clear from the table that the constituent magnetic moments for all the adjacent atoms as well as interstitial magnetic moments are negligible and also cancel the effect of each other, resulting zero magnetic moment per unit cell. These results reveal that BaThO3 is a nonmagnetic/paramagnetic material. Similarly the stable magnetic state of the BaUO3 compound is ferromagnetic in which the ground state energy of the material is minimum as compared to the other magnetic states. The calculated magnetic moments per unit cell for interstitial, local and total magnetic moments are presented in Table 2. It is clear from the table that the effective magnetic moment for U is 1.647 mB, while the literature reported [30] magnetic moment of U in BaUO3 x is 2 mB, which shows that our results are closer to the experimental ones. In this compound double-exchange interaction between uranium atoms via nonmagnetic oxygen anion (U þ 4–O–U þ 5) takes place. The total magnetic moment of the
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