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Evidence of Coulomb correction and spin–orbit coupling in rare-earth dioxides CeO2, PrO2 and TbO2: An ab initio study
Journal of Magnetism and Magnetic Materials 324 (2012) 1397–1405
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Evidence of Coulomb correction and spin–orbit coupling in rare-earth dioxides CeO2, PrO2 and TbO2: An ab initio study Mohammed Benali Kanoun a,n, Ali H. Reshak b,c, Nawel Kanoun-Bouayed d, Souraya Goumri-Said a a
PSE, KAUST, Thuwal 23955-6900, Kingdom of Saudi Arabia School of Complex Systems, FFWP-South Bohemia University, Nove Hrady 37333, Czech Republic c School of Material Engineering, Malaysia University of Perlis, P.O Box 77, d/a Pejabat Pos Besar, 01007 Kangar, Perlis, Malaysia d PES, LPT, Universite´ de Tlemcen, Algeria b
a r t i c l e i n f o
abstract
Article history: Received 13 September 2011 Received in revised form 16 November 2011 Available online 5 December 2011
The current study investigates the structural, elastic, electronic and optical properties of CeO2, PrO2 and TbO2 using the full potential (linearized) augmented plane wave plus local orbital method within the Wu–Cohen generalized gradient approximation (GGA) with Hubbard (U) correction and spin–orbit coupling (SOC). The GGA þ U implementation lead us to describe correctly the relativistic effect on 4f electrons for CeO2. We clarify that the inclusion of the Hubbard U parameter and the spin–orbit coupling are responsible for the ferromagnetic insulating of PrO2 and TbO2. The magnetic description is achieved by the spin-density contours and magnetic moment calculations, where we show the polarization of oxygen atoms from the rare earth atoms. The mechanical stability is shown via the elastic constants calculations. The optical properties, namely the dielectric function and the reflectivity are calculated for radiation up to 12 eV, giving interesting optoelectronic properties to these dioxides. & 2011 Elsevier B.V. All rights reserved.
Keywords: Rare-earth oxides Spin-orbit coupling Coulomb correction DFT Optical properties
1. Introduction During the last 25 years, a continuous increase of the research effort on rare earths containing materials has occurred, and particularly on their oxides. It is presently acknowledged that they may have very relevant applications as catalysts [1,2], optical materials, or ionic conductors [3,4]. Some of these applications have reached the technological maturity, large scale industrial consumption of the rare earth oxides being associated with them, such is the case of the three-way catalysts [5,6], or the lighting applications of lanthanoid-containing photo-luminescent materials [3]. The application of rare earth elements has been carried out on laser cladding containing CeO2 [7] and La2O3 [8]. The wear and corrosion resistance of coatings can be remarkably increased by the addition of these oxides. On account of this obvious importance, several studies have already been conducted especially for CeO2 and little for PrO2. On the experimental side, the electronic states were experimentally determined on the basis of spectroscopic measurements [9–12] and the Brillouin zone center (BZC) phonon frequencies have been measured by several methods [13,14]. The variety of experimental results reported on the bulk modulus of CeO2 [15–18] and PrO2 [18]. The high-pressure a-PbCl2 -type phase of the CeO2 have
n
Corresponding author. E-mail address: [email protected] (M.B. Kanoun).
0304-8853/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2011.11.050
also been investigated and the transition pressures have been reported as 31 GPa. Furthermore, when Ce metal oxidizes completely to CeO2 in the presence of air, Pr occurs naturally as Pr6O11, exhibiting a slightly oxygen deficient fluorite structure. Then it stoichiometric fluorite structure exists under positive oxygen pressure. The Tb oxide occurs naturally as Tb4O7 and transforms into TbO2 under positive oxygen pressure. On the theoretical side, a little number of calculations have been published for PrO2 [19–23] in contrast to the number of studies related to CeO2. Particularly, elastic stiffness, electronic structure and dielectric functions for all rare earth dioxides have not been considered yet with an accurate approximation as well as spin–orbit coupling and strong correlation effects. As for CeO2, numerous calculations are available and are carried out by several methods such as periodic Hartree–Fock [24], self-interactioncorrected local-spin-density approximation [18,25], local-density approximation (LDA), and generalized gradient approximation (GGA) within density functional theory (DFT) [20,26–32]. The debate on the methods to investigate systems with localized (strongly correlated) f electrons still continues in the literature. Many researchers believe that conventional DFT techniques based on LDA or GGA could be unable to cope with these systems. This belief is supported by the current and recent papers on CeO2 and Ce2O3 [26,33]. The LDAþU and GGAþU methods [34,35] are applied in the study of CeO2 [36,37,39,40] and PrO2 [41] where Hartree–Fock type interactions are parameterized with Coulomb U and exchange J terms.
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Due to their technological application, it is crucial to fully understand the rare earth oxides CeO2, PrO2, and TbO2. This study is intended to provide some fundamental properties on the simplest systems related to these compounds. In this paper, we focus our intention on accurate determination of crystal parameters, electronic, and optical properties of CeO2, PrO2, and TbO2 by using DFTþ U. We discuss how these properties are affected by introducing Hubbard U and SOC, and how experimental material properties are reproducible based on the correct state. We will also investigate the magnetic stability, as well as the elastic and derived mechanical properties. The important feature of this work, unlike commonly studied nonmagnetic CeO2, is to consider the PrO2, and TbO2 in the ferromagnetic phase. The present paper is organized as follow. In the next section, we describe the computational parameters related to the fullpotential (linearized) augmented plane waves plus local orbitals (FP-(L)APWþlo) method within GGA þU and SOC. Section 3 will be devoted to the crystal optimization, magnetic stability and the mechanical properties. In Section 4, we examine the electronic structure by the calculation of the densities of states, and the charge (spin) densities contours, where we will investigate carefully, the nature of bonding. These results will be followed by the magnetic description. Optical applications are very important in dioxides, we will devote Section 5 to the optical properties, by the calculation of the imaginary and real parts of the dielectric function, and reflectivity spectra. We will then end the paper with our main findings.
2. Theoretical approaches and numerical tools The present calculations were performed using an all electron FP-(L)APWþlo method [42], based on DFT. The exchange correlation potential was computed with the Wu and Cohen generalized gradient approximation (GGA-WC) [43], while for electronic and optical properties in addition to the GGA correction the Engel and Vosko [44] scheme were also applied. The maximum value lmax for the wave functions expansion inside the atomic spheres is limited to 10. To achieve the energy eigenvalues convergence, the wave functions inside the interstitial region are expanded in plane waves with a cutoff RMT K max ¼ 9, where Kmax is the maximum
modulus for the reciprocal lattice vector. Moreover, local orbitals have been added for all atoms and valence states. The Brillouin zone integrations were performed with k meshes which were large enough to obtain well converged results. The self-consistency cycle was achieved taking 1500 points in the irreducible Brillouin zone. The spin–orbit coupling was included in the calculations based on the second variational approach [45]. The convergence of the self-consistent field calculations is attained with a total energy convergence tolerance of 0.1 mRy. To improve the description of rare earth 4f electrons we used the GGA þU which corresponds to the GGAþ U method described in Refs. [46,47] with the GGA correlation potential instead of LDA. In the GGA þU-like methods, an orbitally dependent potential is introduced for the chosen set of electron states, which in our case is 4f states of Ce, Pr and Tb. This additional potential has an atomic Hartree–Fock form but with screened Coulomb and exchange interaction parameters. In order to determine the Ueff ¼U J (setting J¼0), available experimental data such as lattice constant, bulk modulus, and band gap have been compared with the calculated values. Hence, Ueff is treated as an empirical fitting parameter. For CeO2, the lattice parameter, the bulk modulus, and band gap (Egap ¼O 2p–Ce 4f and Egap ¼O 2p–Ce 5d) values are calculated by GGAþ U, and then these results are compared with the experimental data as shown in Fig. 1(a–c). The equilibrium lattice constant (a) and the bulk modulus (B) were estimated by fitting the Murnaghan–Birch equation of states [48,49] to the resulting energy–volume data. For CeO2, the lattice parameter is overestimated by 0.13% from the purely GGA level (Ueff ¼0 eV) calculation compared to experimental data. Thus, introducing U eff 40 show that the lattice parameter is always larger than experimental value but increases as a function of Ueff (see Figs. 1(a) and 2(b)). For a Ueff value between 4 and 5 eV, the lattice parameter matches the experimental value. As indicated in Fig. 1(b), the bulk modulus value is also steadily increasing with increasing Ueff but this observable is even less affected by the latter parameter. In addition to consistent reproduction of a and B with the chosen values of Ueff for CeO2 compound, the calculated values for band gaps are reasonably good, given the well-known fact that DFT underestimates band gaps of insulators (see Fig. 1(c)). The same process were repeated for PrO2 and TbO2 where Ueff are found 5 and 6 eV, respectively.
Fig. 1. (Color online) Lattice constant (a), bulk modulus (b) and band gap (c) as a function of the Hubbard Ueff for CeO2.
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Table 1 Equilibrium theoretical lattice parameter (aÞ, bulk modulus (BÞ, its pressure derivative (B0 ), and total energy difference (DEðAFMFMÞ Þ between antiferromagnetic (AFM) and ferromagnetic (FM) phases of CeO2, PrO2 and TbO2 compared to other ˚ B in GPa, B0 dimensionless and DE in theoretical and experimental data. (a is in A, meV). (FP LMTO, full potential linear muffin tin orbital; PW PP, plane-wave pseudopotential method; PAW, projector-augmented wave method; SIC, selfinteraction corrected; LSD, local spin density approximation; HGH, Hartwigsen– Goedecker–Hutter; TB LMTO, Tight binding linear muffin tin orbital). Compounds
Approximations
a
B
B0
CeO2
This work Experiment [15–18,48] SIC-LSD [18] FPLMTO [26] PW-PP [28] PAW GGAþ U [29] PAW GGAþ U [30] PAW GGA [31] PAW PBE0 [32] PAW PBE [32] L/APWþ LO PBE [32] PAW LDAþ U [36] HGH PP [37] PAW LDAþ U [38] FPLMTO-SIC-LSD [25]
5.426 5.410 5.384 5.48 5.48 5.48 5.38 5.45 5.39 5.47 5.47 5.40 5.40 5.366 3.408
207.1 204–236 176.9 187.7 178.0 187.0 202.4 194
4.60
PrO2
This work Experiment [18,48] SIC-LSD [26] TB-LMTO [37] FPLMTO-SIC-LSD [25]
5.380 5.394 5.364 5.392 3.397
189.38 187 176.8 378
4.01
0.56
TbO2
This work Experiment [48] FPLMTO-SIC-LSD [25]
5.272 5.222 3.317
192.03
4.76
37.41
Fig. 2. (Color online) The total and partial density of states at GGA þU þ SOC for (a) CeO2. Since the spin-up and spin-down channels are identical, only the spin-up channel is shown. The vertical solid line denotes the Fermi level.
3. Structural and elastic properties The ground-state structures of CeO2, PrO2, and TbO2 were obtained by performing DFT/GGA (add U) calculations with spinpolarization effects. The equilibrium lattice constant (a), bulk modulus (B) and its pressure derivative (B0 ) are listed in Table 1 in comparison with experimental data [15–18,50] and recent theoretical calculations [20,25,26,28–32,36–38] for comparison. As a general remark, the deviations for lattice constants are estimated to be about 0.07–0.7% for CeO2, PrO2 and TbO2 compared to available experimental data. However, our theoretical approach seems to considerably underestimate the bulk modulus, the calculated values being 191 GPa and 189.4 GPa for CeO2 and PrO2, respectively, which are 6–19% and 3% lower than the experimental data [15–18,50]. It is observed that the experimental values for the bulk modulus of CeO2 are confined to a 15% range 204–236 GPa, whereas there is large scatter of the calculated values. Therefore, it is worth noticing that all GGA calculations tend to underestimate the bulk modulus. From Table 1, it is clear that the predicted bulk modulus value for CeO2 is larger than PrO2 and TbO2. In order to compute the ground state magnetic phase between antiferromagnetic (AFM) and ferromagnetic (FM) phases, we calculated the total energy difference DEðAFMFMÞ between them. Our results are also summarized in Table 1. We must recall here that the FM phase is stabilized over the AFM phase when DE 40, whereas the AFM is stabilized over the FM phases when (DE o 0). From Table 1, we found that the ground state is the FM phase for TbO2. For PrO2, the AFM phase is more stable but DEðAFMFMÞ are small. This result indicates that this system is weak AFM. The elastic stiffness determines the response of the crystal to an externally applied strain (or stress) and provides information about the bonding characteristics, mechanical and structural stability. The elastic constants Cij were calculated within the total-energy method, where the unit cell is subjected to a number of finite-size strains along several strain directions. Cubic lattices have three independent elastic constants [51], namely, C11, C12, and C44. The obtained elastic stiffness constants C ij are gathered in Table 2. Furthermore, the mechanically stable phases or macroscopic stability is dependent on the positive definiteness of stiffness matrix. For a stable cubic structure, the independent elastic constants should satisfy the well-known Born–Huang criterion [52], given by: C44 40,
172 170 217 211.11 210.1
DEðAFMFMÞ –
4.417 4.4
Table 2 Calculated elastic constants C ij (GPa), Bulk moduli BH (GPa), shear moduli G (GPa), Young’s modulus Y (GPa), and Poisson’s ratio n for CeO2, PrO2 and TbO2. Compounds
C11
C12
C44
BH
CeO2
390.86 390a 386b 354.79c 403d
128.5 130 124b 139.27c 105d
57.0 82 73b 51.19c 60d
215.95
PrO2 TbO2
344.33 300.2
126.5 108.63
181.1 130.0
199.11 184.94
Y
n
80.14
213.96
0.334
147.68 119.33
355.23 294.63
0.202 0.234
G
a
Ref. [38]. Ref. [37]. Ref. [36]. d Ref. [60] experimental data. b c
C11 49C12 9 and C11 þ2C12 4 0. One may see that the rare earth dioxides are mechanically stable because all elastic constants are positive and satisfy the Born mechanical stability restrictions. Considering the experimental data set due to Nakajima et al. [17], the C11, C12 and C44 for CeO2 values are underestimated by 3%, 22% and 5%, respectively, whereas the agreement is good for all elastic constants. Note also that our results are in close agreement with the available theoretical works [37,38]. In order to measure the stiffness of the solid, Voight–Reuss–Hill [53] estimates BH, G, Y and n of polycrystalline bulk modulus, shear modulus, Young’s modulus, and Poisson’s ratio computed from Cij are presented in Table 2 (following the procedure detailed in Ref. [54]). From this table, the bulk modulus agrees with that from the Birch–Murnaghan equation of state. It can be remark that the shear and Young’s modulus of the PrO2 is higher than CeO2 and TbO2. The larger shear modulus is mainly due to its larger C44.
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4. Electronic properties 4.1. CeO2 The calculated total and partial density of states (DOS) curves at the predicted equilibrium lattice constants for CeO2 are shown in Fig. 2 within the framework of GGA approach with and without U, where the vertical line is the Fermi level (EF). A common trend can be proposed from total DOS graphs that the influence of Ueff on the electronic structure is basically restricted to the unoccupied f state and this does not drastically change the electronic distribution especially occupied states. Hence, the gap between the valence O (2p) and conduction Ce (5d) bands is 5.5 eV, however this calculated value is slightly smaller than the measured one at 6 eV [55] and agrees with a previous theoretical study (5–5.5) eV [26,29,40]. Moreover the distance between the valence band and the Ce 4f states is 2.5 eV, to be compared with 3 eV in experiments [55]. So, the width of the O 2p is 4.38 eV, which agrees with experimental data and theoretical calculation. Another consequence of higher values of Ueff is a slightly reduced width value of the Ce 4f peak and increases the intensity of this peak, which reflects stronger localization. We notice an important feature in DOS that they are the O 2p–Ce 5d band gap and the empty Ce 4f states in the gap above the O 2p band. To gain a more detailed insight into the bonding characters of nonmagnetic ceria, we calculate the charge density distribution. The charge-density contours for a cut in the (110) plane of CeO2 are displayed in Fig. 3(a). It is clearly seen that CeO2 is
Fig. 3. (Color online) Charge density distributions (a) and electron-localization function (b) (e=A 3 ) in (110) plane of CeO2. ˚
characterized by a nearly spherical charge-density distribution around the Ce and O atoms and a low charge density in the interstitial region. It can be also seen that the ionic bonding is the determinant of the chemical bonding in CeO2, even though the covalency is strong in the sense of orbital participation in forming the valence state. We have also calculated the electronic localized function (ELF) [56] using the VASP code [57,58]. In fact, the ELF has proven to be a useful companion to the density in the task of providing us with intuition on the electronic structure. Then, when ELF ¼1, it corresponds to a perfect electron localization. ELF is a dimensionless localization index restricted to the range of [0,1]. A high ELF value stands for a low probability of finding a second electron with the same spin in the neighboring region of the reference electron, i.e., the reference electron is highly localized. Based on its definition, one can simply interpret high ELF values as covalent bonds, lone pairs, or inert cores. In Fig. 3(b), we plot the ELF 2D-contours of CeO2. Undoubtedly, the interaction between Ce and O ions is principally ionic. It is also clearly seen that nonzero ELF values ( 0:5) are present between the Ce and O atoms. These values are not large enough to define strong covalent bonds, but we can certainly attribute them to the weak hybridization of O 2p states with partially occupied Ce 4f or 5d states as shown in figures ELF and DOS. So, our ELF analysis obtained by GGAþU calculation show that some covalent components contribute to the Ce 4f(5d)–O 2p interaction. 4.2. PrO2 Fig. 4(a) provides the total and partial DOS calculated by GGAþU without and with SOC for PrO2 in the ferromagnetic phase. So, Pr þ 4 is surrounded by O 2-cube and its point symmetry group is Oh. In this case, Pr þ 4 has one electron on 4f states. The standard GGA and also GGA þU calculations without SOC make f state bands cross the Fermi level without any gap opening (see Fig. 4(a)). Thus, we apply the GGAþU with SOC to open a narrow gap between occupied and empty 4f states principally in majority spin part. Moreover, we can see that the highest occupied Pr 4f state for spin up are situated around 4 eV below the Fermi energy and concentrates them into a sharp peak. Note also that the peak from the O 2p DOS at 4 eV comes from separate bands without 4f contributions and thus these 4f states are localized and not hybridized with the O 2p states. Moreover, the Pr 4f–O 2p hybridization can be observed from some similarities in the positions of the peaks at 0.8 and 0.3 eV between the Pr 4f and O 2p curves. Noteworthy is that the spin-up and spin-down Pr 4f DOSs above 3 eV are very similar. Above the Fermi level we find the unoccupied f-levels. The fundamental band gap separates the valence band maximum and unoccupied Pr 4f is about 1.8 eV. Furthermore, the measured onset of the optical conductivity of PrO2d is about 2 eV [60], which is well reproduced by our calculation. Then, at about 5 eV above the Fermi energy, states predominantly of Pr 5d character begin to appear. Derived from experimental data, Van der Kolk and Dorenbos [61] proposed a model to predict the insulating or metallic behavior and chemical stability of lanthanide materials. From their model, it was predicted that for PrO2, the 4f empty states are situated at about 2 eV above the top of the valence band, a feature which is quite well obtained by our calculated. Spin-polarized calculations give a total magnetic moment of 0:999mB per cell, principally induced by Pr. A relative amount of the moment is found in the interstitial region about 0:06mB , while oxygen shows weak antiparallel spin moment close to 0:101mB . To further examine the origin and distribution of this moment, we have calculated self-consistently the spin density distribution for PrO2, see Fig. 5(a,b). The iso-surface plot of the spin density gives a clear scheme about the strongly localized spin polarization near
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Fig. 4. (Color online) Calculated total and partial density of states at GGAþ U and GGA þ Uþ SOC for PrO2 for majority (upper part) and minority (lower part) spin states.
Fig. 5. (a) Isosurface spin-density plot (at an isovalue of 0.06 electron/A˚ 3) for PrO2. The blue balls (very light gray) are Pr atoms and the red (dark gray) are O atoms. (b) Two-dimensional distribution of spin density (electron/A˚ 3) for the (1 1 2 0) plane passing Pr and O. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 6. (Color online) Calculated total and partial density of states at GGAþ U and GGAþ Uþ SOC for TbO2.
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4.3. TbO2
the Pr atoms. This localized spin polarization reflects the local magnetic moments in the system. In Fig. 5(b), we show the twodimensional spin density distribution in the (1 1 0) plane passing Pr, and O. The distribution shows that polarized components mainly are located at Pr atoms.
We have plotted in Fig. 6 the total and local DOS of TbO2. This figure provide the local DOSs 4f states with the effects of spin– orbit interaction and on-site Coulomb repulsion. In the DOS
Fig. 7. (Color online) (a) Spin-density plot and (b) isosurface at an isovalue of 0.6 electron/A˚ 3) for TbO2.
10
10
CeO2
CeO2 5
6
1 ()
2 ()
8
4
0
2 -5
0 8
PrO2
PrO2 5
1 ()
2 ()
6 4
0
2 0
-5
10
10
TbO2
TbO2
5
6
1 ()
2 ()
8
4
0
2 -5
0 0
4
8 Energy (eV)
12
0
4
8
12
Energy (eV)
Fig. 8. The calculated (a) imaginary (e2 ðoÞ) and (b) real (e1 ðoÞ) parts of dielectric functions of CeO2, PrO2 and TbO2.
M.B. Kanoun et al. / Journal of Magnetism and Magnetic Materials 324 (2012) 1397–1405
0.8
R ()
0.4
0.2
0 0.8
PrO2 0.6
0.4
0.2
5. Optical properties
0 0.6
TbO2
0.4
R ()
From our electronic structure calculations, the imaginary part of the dielectric function e2 ðoÞ has been derived by summing transitions from occupied to unoccupied states for energies much larger than those of the phonons. The real part of the diagonal dielectric functions is computed using the Kramers–Kronig transformation [62]. The knowledge of both the real and imaginary parts of the dielectric tensor allows one to calculate other important optical spectra. We also computed and analyzed the reflectivity RðoÞ. In order to assess convergence of the optical properties, we calculated e2 ðoÞ with increasingly finer meshes for the discretization of the BZ: 1500 k points with the spin–orbit interaction, and we apply a broadening equal to 0.1 (this value is typical of the experimental accuracy) in order to bring out all the structures. The analysis of e2 ðoÞ curves of Fig. 8(a) shows that the only CeO2, PrO2 and TbO2 compounds have an optical gap of about 3.8, 1.8 and 0.5 eV. The threshold energy (first critical point) of the dielectric function for these three compounds occurs at 3.8, 1.8 and 0.5 eV. This point gives the threshold for direct optical transitions between the highest valence and the lowest conduction band. This is known as the fundamental absorption edge. The origin of these peaks is attributed to the inter-band transitions form the occupied states to the unoccupied states. Beyond these threshold energies (first critical points), the curve increases rapidly. This is due to the fact that the number of points contributing toward e2 ðoÞ increases abruptly. Note that we do not include phonon contributions to the dielectric screening. The main peak of e2 ðoÞ for CeO2, PrO2 and TbO2 is situated around 2.9, 3.5, and 2.0 eV. The density of states of CeO2 (PrO2) suggests that the first peak in e2 ðoÞ at about 2.9 eV (3.5 eV) is due to the transition from O p to Ce (Pr) f states, while the second peak corresponds to transitions of the O p to the Tb f states. Moreover, the fine structure of this peak in e2 ðoÞ is similar to the fine structure of the DOS in the energy interval 4 to Fermi level. The
CeO2
0.6
R ()
calculated by GGAþU, we observe only one intense peak whereas with the introduction of the spin orbit coupling, the f-states are represented by several peaks, which correspond to the electronic nature of these states. One may notice that the SOC effect gives rise to the broadened 4f state. We can also see that the spin up Tb 4f are centered around 10 eV below Fermi level and for spin down bands, they are empty and situated between the valence band maximum and conduction band minimum with a more localized peak about 2 eV. Thus, the dominance of Tb 4f to majority spin in the valence band and of the minority spin in the forbidden band. Like to PrO2, the total moment TbO2 comes principally from Tb elements which is about 7:016mB per cell. Then, the magnetic moments of all atoms have the same direction, which indicates that there is ferromagnetic coupling between the Tb and O atoms. To complete the magnetic structure description, we display in Fig. 7, the two- and three-dimensional spin densities of TbO2. One may observe from two-dimensional spin density distribution in the (1 1 0) plane that interactions appeared between heavy Tb atoms show a pronounced effect on the spin densities maps. The spin density distribution around the oxygens has a circular appearance but slightly modified due to the polarization induced by the presence of Tb atom. The iso-surface plot of the spin density shows that the shape of the spin densities distribution around the Tb atom is circular. This distribution depends on the filling of orbitals, mainly the f one. We are able to conclude that the hybridization between different orbitals affect the spin density contours [63,64].
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0.2
0 0
4
8
12
Energy (eV) Fig. 9. Reflectivity spectra of CeO2, PrO2 and TbO2.
most realistic explanation for the above observation is that the final state for the optical transitions, i.e., the unoccupied 4f states of Ce (Pr), is a localized atomic-like state. The same effect is obtained for TbO2, where the first peak in e2 ðoÞ is due to transitions of O p to the Tb f states (spin up plus spin down), while the second peak corresponds to transitions of the O p to the Tb d states. The real part of the electronic dielectric function spectrum for the all rare earth dioxides is shown in Fig. 8(b). The CeO2, PrO2 and TbO2 compounds show almost the same main features. The estimated values of static dielectric constant e1 ð0Þ of these compounds are given by 4.84, 5.13 and 6.52, respectively. Our theoretical e1 ð0Þ values of CeO2 are compared to three sets of
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experimental results 4.7 [9], 5.31 [65] and 6 [66], the differences are smaller and range between an overestimation by 3% and an underestimation by 8% and 19%. The reflectivity spectra RðoÞ for CeO2, PrO2 and TbO2 are presented in Fig. 9. The figures show a small reflectivity up to 8.0 eV then increasing abruptly at higher energies. We should emphasize that the reflectivity minimum between 4.0 and 8.0 eV for the CeO2, PrO2 and TbO2. The depth of the plasmon minimum is determined by the imaginary part of the dielectric function at the plasma resonance and is representative of the degree of overlap between the inter-band absorption regions.
6. Summary To conclude this paper, we have performed a detailed investigations on the structural, elastic, electronic, magnetic and optical properties of CeO2, PrO2, and TbO2 within the DFT framework based on full potential approaches. The application of GGAþ U and the spin–orbit coupling, on the electronic, magnetic, and optical properties was detailed and demonstrated to be relevant of the correctness of our results. Elastic constants calculation proves the mechanical stability of CeO2, PrO2 and TbO2. The electronic structure description has shown that these structures are stable in ferromagnetic phases and good agreement with experiment can be obtained for the magnetic moment. Moreover, we found that the GGAþ U calculation with SOC reproduces very well the observed energy gap and other electronic features compared to any other schemes. This fact emphasizes the role of both the Hubbard U and SOC in calculating the electronic states. The magnetic contribution of the rare earth atom in each dioxide was achieved by analyzing the spin-density contours, where the polarization on the first neighbor oxygen is observed.
Acknowledgments A part of calculations with Wien2k and VASP codes have been performed in ISCF computers of FUNDP University (Belgium) during the research stays of S.G.-S. and M.B.K. VASP calculations were carried out partially in the Beowulf class heterogeneous computer cluster at KAUST. S.G.-S. and M.B.K. wish to thank Alberto Otero de la Roza from Universidad de Oviedo (Spain) for providing us with RUNWIEN code. For the author A.H.R., his work was supported from the program RDI of the Czech Republic, the project CENAKVA (No. CZ.1.05/2.1.00/01.0024), the Grant no. 152/ 2010/Z of the Grant Agency of the University of South Bohemia. School of Material Engineering, Malaysia University of Perlis, P.O. Box 77, d/a Pejabat Pos Besar, 01007 Kangar, Perlis, Malaysia. References [1] G. Adachi, Z.C. Kang, in: G. Adachi, N. Imanaka, Z.C. Kang (Eds.), Binary Rare Earth Oxides, Kluwer Academic Publishers, 2004. [2] A. Trovarelli, Structural properties and nonstoichiometric behavior of CeO2, in: A. Trovarelli (Ed.), Catalysis by Ceria and Related Materials, Imperial College Press, London, 2002, p. 15; M.W. Shafer, J.B. Torrance, T.J. Perry, Journal of Physics and Chemistry of Solids 33 (1972) 2251. [3] K.A. Gschneidner, Jr. (Ed.), Industrial Applications of Rare Earth Elements, ACS Symposium Series 164, American Chemical Society, Washington, DC, 1981. [4] G.R. Choppin, Chemical properties of the rare earth elements, in: J.-C.G. Bunzli, G.R. Choppin (Eds.), Lanthanide Probes in Life, Chemical and Earth Sciences, Elsevier, Amsterdam, 1989, p. 1. [5] L. Eyring, The binary rare earth oxides: synthesis and identification, in: G. Meyer, L.R. Morss (Eds.), Synthesis of Lanthanide and Actinide Compounds, Kluwer Academic Publishers, Dordrecht, 1990, p. 87. [6] L. Eyring, The higher oxides of the rare earths; a paradigm for solid state chemistry, in: R. Saez Puche, P.A. Caro (Eds.), Rare Earths, Editorial Complutense, Madrid, 1998, p. 119.
[7] L. Mukin, M. Chen, S. Dechun, Journal of Rare Earths 9 (1991) 294. [8] X.H. Wang, Z.D. Zou, S.L. Song, S.Y. Qu, Journal of Materials Science Letters 22 (2003) 713. [9] M. Marabelli, P. Wachter, Physical Review B 36 (1987) 1238. [10] A. Fujimori, Physical Review B 27 (1983) 3992 28 (1983) 2281. [11] R.C. Karnatak, J.-M. Esteva, H. Dexpert, M. Gasgnier, P.E. Caro, L. Albert, Physical Review B 36 (1987) 1745. [12] H. Ogasawara, A. Kotani, K. Okada, B.T. Thole, Physical Review B 43 (1991) 854. [13] J.E. Spanier, R.D. Robinson, F. Zhang, S.-W. Chan, I.P. Herman, Physical Review B 64 (2001) 245407. [14] S. Wang, W. Wang, J. Zuo, Y. Qian, Materials Chemistry and Physics 68 (2001) 246. [15] S.J. Duclos, Y.K. Vohra, A.L. Ruoff, A. Jayaraman, G.P. Espinosa, Physical Review B 38 (1988) 7755. [16] L. Gerward, J.S. Olsen, Powder Diffraction 8 (1993) 127. [17] A. Nakajima, A. Yoshihara, M. Ishigama, Physical Review B 50 (1994) 13297. [18] L. Gerward, J. Staun Olsen, L. Petit, G. Vaitheeswaran, V. Kanchana, A. Svanee, Journal of Alloys and Compounds 400 (2005) 56. [19] J. Dabrowski, V. Zavodinsky, A. Fleszar, Microelectronics Reliability 41 (2001) 1093. [20] S. Mehrotra, A.K. Bandyopadhyay, Journal of Alloys and Compounds 436 (2007) 56. [21] M. Diviˇs, J. Rusz, V. Nekvasil, Czechoslovak Journal of Physics 54 (2004) D291. [22] M. Diviˇs, J. Rusz, Journal of Magnetism and Magnetic Materials 290–291 (2005) 1015. [23] P. Nova´k, M. Diviˇs, Physica Status Solidi B 244 (2007) 3168. [24] S. Gennard, F. Cora , C.R.A. Catlow, Journal of Physical Chemistry B 103 (1999) 10158. [25] L. Petit, A. Svane, Z. Szotek, W.M. Temmerman, Physical Review B 72 (2005) 205118. [26] N.V. Skorodumova, R. Ahuja, S.I. Simak, I.A. Abrikosov, B. Johansson, B.I. Lundqvist, Physical Review B 64 (2001) 115108. [27] D.D. Koelling, A.M. Boring, J.H. Wood, Solid State Communications 47 (1983) 227. [28] S. Fabris, S. de Gironcoli, S. Baroni, G. Vicario, G. Balducci, Physical Review B 71 (2005) 041102. [29] C. Loschen, J. Carrasco, K.M. Neyman, F. Illas, Physical Review B 75 (2007) 035115. [30] Y. Jiang, J.B. Adams, M. van Schilfgaarde, Journal of Chemical Physics 123 (2005) 064701. [31] Z. Yang, T.K. Woo, M. Baudin, K. Hermansson, Journal of Chemical Physics 120 (2004) 7741. [32] J.L.F. Da Silva, M.V. Ganduglia-Pirovano, J. Sauer, V. Bayer, G. Kresse, Physical Review B 75 (2007) 045121. [33] N.V. Skorodumova, M. Baudin, K. Hermansson, Physical Review B 69 (2004) 075401. [34] V.I. Anisimov, J. Zaanen, O.K. Andersen, Physical Review B 44 (1991) 943. [35] V.I. Anisimov, F. Aryasetiawan, A.I. Lichtenstein, Journal of Physics: Condensed Matter 9 (1997) 767. [36] S. Shi, Y. Tang, C. Ouyang, L. Cui, X. Xin, P. Li, W. Zhou, H. Zhang, M. Lei, L. Chen, Journal of Physics and Chemistry of Solids 71 (2010) 788. ¨ [37] T. Gurel, R. Eryigit, Physical Review B 74 (2006) 014302. [38] C. Sevik, T. C - agin, Physical Review B 80 (2009) 014108. [39] M. Nolan, S. Grigoleit, D.C. Sayle, S.C. Parker, G.W. Watson, Surface Science 576 (2005) 217. [40] D.A. Andersson, S.I. Simak, B. Johansson, I.A. Abrikosov, N.V. Skorodumova, Physical Review 75 (2007) 035109. [41] F. Tran, J. Schweifer, P. Blaha, K. Schwarz, Physical Review B 77 (2008) 085123. [42] P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, J. Luitz, Wien2k, An Augmented-Plane-Wave þLocal Orbitals Program for Calculating Crystal Properties, TU Wien, Austria, 2001. [43] Z. Wu, R.E. Cohen, Physical Review B 73 (2006) 235116. [44] E. Engel, S.H. Vosko, Physical Review B 47 (1993) 13164. [45] J. Kuneˇs, P. Nova´k, M. Diviˇs, P.M. Oppeneer, Physical Review B 63 (2001) 205111. [46] V.I. Anisimov, I.V. Solovyev, M.A. Korotin, M.T. Czyzyk, G.A. Sawatzky, Physical Review B 48 (1993) 16929. [47] A.I. Liechtenstein, V.I. Anisimov, J. Zaanen, Physical Review B 52 (1995) R5467. [48] F.D. Murnaghan, Proceedings of the National Academy of Sciences of the United States of America 30 (1944) 244. [49] F. Birch, Physical Review B 71 (1947) 809; F. Birch, Journal of Geophysical Research 86 (1978) 1257. [50] P. Villars, L.D. Calvert, Pearson’s Handbook of Crystallographic Data for Intermetallic Phases, 2nd ed., ASM International, Ohio, 1991. [51] J.F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, Oxford University Press, Oxford, 1985. [52] M. Born, K. Hang, Dynamical Theory and Experiments I, Springer Verlag Publishers, Berlin, 1982. [53] R. Hill, Proceedings of the Physical Society, London, Section A 65 (1952) 349. [54] N. Kanoun-Bouayed, M.B. Kanoun, S. Goumri-Said, Central European Journal of Physics 9 (2011) 205. [55] E. Wuilloud, B. Delley, W.-D. Schneider, Y. Baer, Physical Review Letters 53 (1984) 202. [56] B. Silvi, A. Savin, Nature (London) 371 (1994) 683.
M.B. Kanoun et al. / Journal of Magnetism and Magnetic Materials 324 (2012) 1397–1405
¨ [57] G. Kresse, J. Furthmuller, Computational Materials Science 6 (1996) 15. [58] The calculations were performed within DFT using the projector augmented wave (PAW) method as implemented in the VASP package [57]. The exchange and correlations were treated within GGA-PBE [59] including Hubbard correction. An energy cutoff of 500 eV for the plane-wave expansion of the PAWs is used. The Brillouin zone is sampled on a 11 11 11 k-point grid. [59] J.P. Perdew, K. Burke, M. Ernzerhof, Physical Review Letters 77 (1996) 3865. [60] S. Kimura, F. Arai, M. Ikezawa, Journal of Electron Spectroscopy and Related Phenomena 78 (1996) 135.
1405
[61] E. van der Kolk, P. Dorenbos, Chemistry of Materials 18 (2006) 3458. [62] H.Z. Tributsch, Naturforsch A 32A (1977) 972. [63] N. Kanoun-Bouayed, S. Goumri-Said, A.E. Merad, M.B. Kanoun, Materials Science Forum 609 (2009) 167. [64] S. Goumri-Said, M.B. Kanoun, Journal of Physics D: Applied Physics 41 (2008) 035004. [65] S. Mochizuki, Physica Status Solidi B 114 (1982) 189. [66] N.I. Santha, M.T. Sebastian, P. Mohanan, N.M. Alford, K. Sarma, R.C. Pullar, S. Kamba, A. Pashkin, P. Samukhina, J. Petzelt, Journal of the American Ceramic Society 87 (2004) 1233.
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Evidence of Coulomb correction and spin–orbit coupling in rare-earth dioxides CeO2, PrO2 and TbO2: An ab initio study Mohammed Benali Kanoun a,n, Ali H. Reshak b,c, Nawel Kanoun-Bouayed d, Souraya Goumri-Said a a
PSE, KAUST, Thuwal 23955-6900, Kingdom of Saudi Arabia School of Complex Systems, FFWP-South Bohemia University, Nove Hrady 37333, Czech Republic c School of Material Engineering, Malaysia University of Perlis, P.O Box 77, d/a Pejabat Pos Besar, 01007 Kangar, Perlis, Malaysia d PES, LPT, Universite´ de Tlemcen, Algeria b
a r t i c l e i n f o
abstract
Article history: Received 13 September 2011 Received in revised form 16 November 2011 Available online 5 December 2011
The current study investigates the structural, elastic, electronic and optical properties of CeO2, PrO2 and TbO2 using the full potential (linearized) augmented plane wave plus local orbital method within the Wu–Cohen generalized gradient approximation (GGA) with Hubbard (U) correction and spin–orbit coupling (SOC). The GGA þ U implementation lead us to describe correctly the relativistic effect on 4f electrons for CeO2. We clarify that the inclusion of the Hubbard U parameter and the spin–orbit coupling are responsible for the ferromagnetic insulating of PrO2 and TbO2. The magnetic description is achieved by the spin-density contours and magnetic moment calculations, where we show the polarization of oxygen atoms from the rare earth atoms. The mechanical stability is shown via the elastic constants calculations. The optical properties, namely the dielectric function and the reflectivity are calculated for radiation up to 12 eV, giving interesting optoelectronic properties to these dioxides. & 2011 Elsevier B.V. All rights reserved.
Keywords: Rare-earth oxides Spin-orbit coupling Coulomb correction DFT Optical properties
1. Introduction During the last 25 years, a continuous increase of the research effort on rare earths containing materials has occurred, and particularly on their oxides. It is presently acknowledged that they may have very relevant applications as catalysts [1,2], optical materials, or ionic conductors [3,4]. Some of these applications have reached the technological maturity, large scale industrial consumption of the rare earth oxides being associated with them, such is the case of the three-way catalysts [5,6], or the lighting applications of lanthanoid-containing photo-luminescent materials [3]. The application of rare earth elements has been carried out on laser cladding containing CeO2 [7] and La2O3 [8]. The wear and corrosion resistance of coatings can be remarkably increased by the addition of these oxides. On account of this obvious importance, several studies have already been conducted especially for CeO2 and little for PrO2. On the experimental side, the electronic states were experimentally determined on the basis of spectroscopic measurements [9–12] and the Brillouin zone center (BZC) phonon frequencies have been measured by several methods [13,14]. The variety of experimental results reported on the bulk modulus of CeO2 [15–18] and PrO2 [18]. The high-pressure a-PbCl2 -type phase of the CeO2 have
n
Corresponding author. E-mail address: [email protected] (M.B. Kanoun).
0304-8853/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2011.11.050
also been investigated and the transition pressures have been reported as 31 GPa. Furthermore, when Ce metal oxidizes completely to CeO2 in the presence of air, Pr occurs naturally as Pr6O11, exhibiting a slightly oxygen deficient fluorite structure. Then it stoichiometric fluorite structure exists under positive oxygen pressure. The Tb oxide occurs naturally as Tb4O7 and transforms into TbO2 under positive oxygen pressure. On the theoretical side, a little number of calculations have been published for PrO2 [19–23] in contrast to the number of studies related to CeO2. Particularly, elastic stiffness, electronic structure and dielectric functions for all rare earth dioxides have not been considered yet with an accurate approximation as well as spin–orbit coupling and strong correlation effects. As for CeO2, numerous calculations are available and are carried out by several methods such as periodic Hartree–Fock [24], self-interactioncorrected local-spin-density approximation [18,25], local-density approximation (LDA), and generalized gradient approximation (GGA) within density functional theory (DFT) [20,26–32]. The debate on the methods to investigate systems with localized (strongly correlated) f electrons still continues in the literature. Many researchers believe that conventional DFT techniques based on LDA or GGA could be unable to cope with these systems. This belief is supported by the current and recent papers on CeO2 and Ce2O3 [26,33]. The LDAþU and GGAþU methods [34,35] are applied in the study of CeO2 [36,37,39,40] and PrO2 [41] where Hartree–Fock type interactions are parameterized with Coulomb U and exchange J terms.
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Due to their technological application, it is crucial to fully understand the rare earth oxides CeO2, PrO2, and TbO2. This study is intended to provide some fundamental properties on the simplest systems related to these compounds. In this paper, we focus our intention on accurate determination of crystal parameters, electronic, and optical properties of CeO2, PrO2, and TbO2 by using DFTþ U. We discuss how these properties are affected by introducing Hubbard U and SOC, and how experimental material properties are reproducible based on the correct state. We will also investigate the magnetic stability, as well as the elastic and derived mechanical properties. The important feature of this work, unlike commonly studied nonmagnetic CeO2, is to consider the PrO2, and TbO2 in the ferromagnetic phase. The present paper is organized as follow. In the next section, we describe the computational parameters related to the fullpotential (linearized) augmented plane waves plus local orbitals (FP-(L)APWþlo) method within GGA þU and SOC. Section 3 will be devoted to the crystal optimization, magnetic stability and the mechanical properties. In Section 4, we examine the electronic structure by the calculation of the densities of states, and the charge (spin) densities contours, where we will investigate carefully, the nature of bonding. These results will be followed by the magnetic description. Optical applications are very important in dioxides, we will devote Section 5 to the optical properties, by the calculation of the imaginary and real parts of the dielectric function, and reflectivity spectra. We will then end the paper with our main findings.
2. Theoretical approaches and numerical tools The present calculations were performed using an all electron FP-(L)APWþlo method [42], based on DFT. The exchange correlation potential was computed with the Wu and Cohen generalized gradient approximation (GGA-WC) [43], while for electronic and optical properties in addition to the GGA correction the Engel and Vosko [44] scheme were also applied. The maximum value lmax for the wave functions expansion inside the atomic spheres is limited to 10. To achieve the energy eigenvalues convergence, the wave functions inside the interstitial region are expanded in plane waves with a cutoff RMT K max ¼ 9, where Kmax is the maximum
modulus for the reciprocal lattice vector. Moreover, local orbitals have been added for all atoms and valence states. The Brillouin zone integrations were performed with k meshes which were large enough to obtain well converged results. The self-consistency cycle was achieved taking 1500 points in the irreducible Brillouin zone. The spin–orbit coupling was included in the calculations based on the second variational approach [45]. The convergence of the self-consistent field calculations is attained with a total energy convergence tolerance of 0.1 mRy. To improve the description of rare earth 4f electrons we used the GGA þU which corresponds to the GGAþ U method described in Refs. [46,47] with the GGA correlation potential instead of LDA. In the GGA þU-like methods, an orbitally dependent potential is introduced for the chosen set of electron states, which in our case is 4f states of Ce, Pr and Tb. This additional potential has an atomic Hartree–Fock form but with screened Coulomb and exchange interaction parameters. In order to determine the Ueff ¼U J (setting J¼0), available experimental data such as lattice constant, bulk modulus, and band gap have been compared with the calculated values. Hence, Ueff is treated as an empirical fitting parameter. For CeO2, the lattice parameter, the bulk modulus, and band gap (Egap ¼O 2p–Ce 4f and Egap ¼O 2p–Ce 5d) values are calculated by GGAþ U, and then these results are compared with the experimental data as shown in Fig. 1(a–c). The equilibrium lattice constant (a) and the bulk modulus (B) were estimated by fitting the Murnaghan–Birch equation of states [48,49] to the resulting energy–volume data. For CeO2, the lattice parameter is overestimated by 0.13% from the purely GGA level (Ueff ¼0 eV) calculation compared to experimental data. Thus, introducing U eff 40 show that the lattice parameter is always larger than experimental value but increases as a function of Ueff (see Figs. 1(a) and 2(b)). For a Ueff value between 4 and 5 eV, the lattice parameter matches the experimental value. As indicated in Fig. 1(b), the bulk modulus value is also steadily increasing with increasing Ueff but this observable is even less affected by the latter parameter. In addition to consistent reproduction of a and B with the chosen values of Ueff for CeO2 compound, the calculated values for band gaps are reasonably good, given the well-known fact that DFT underestimates band gaps of insulators (see Fig. 1(c)). The same process were repeated for PrO2 and TbO2 where Ueff are found 5 and 6 eV, respectively.
Fig. 1. (Color online) Lattice constant (a), bulk modulus (b) and band gap (c) as a function of the Hubbard Ueff for CeO2.
M.B. Kanoun et al. / Journal of Magnetism and Magnetic Materials 324 (2012) 1397–1405
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Table 1 Equilibrium theoretical lattice parameter (aÞ, bulk modulus (BÞ, its pressure derivative (B0 ), and total energy difference (DEðAFMFMÞ Þ between antiferromagnetic (AFM) and ferromagnetic (FM) phases of CeO2, PrO2 and TbO2 compared to other ˚ B in GPa, B0 dimensionless and DE in theoretical and experimental data. (a is in A, meV). (FP LMTO, full potential linear muffin tin orbital; PW PP, plane-wave pseudopotential method; PAW, projector-augmented wave method; SIC, selfinteraction corrected; LSD, local spin density approximation; HGH, Hartwigsen– Goedecker–Hutter; TB LMTO, Tight binding linear muffin tin orbital). Compounds
Approximations
a
B
B0
CeO2
This work Experiment [15–18,48] SIC-LSD [18] FPLMTO [26] PW-PP [28] PAW GGAþ U [29] PAW GGAþ U [30] PAW GGA [31] PAW PBE0 [32] PAW PBE [32] L/APWþ LO PBE [32] PAW LDAþ U [36] HGH PP [37] PAW LDAþ U [38] FPLMTO-SIC-LSD [25]
5.426 5.410 5.384 5.48 5.48 5.48 5.38 5.45 5.39 5.47 5.47 5.40 5.40 5.366 3.408
207.1 204–236 176.9 187.7 178.0 187.0 202.4 194
4.60
PrO2
This work Experiment [18,48] SIC-LSD [26] TB-LMTO [37] FPLMTO-SIC-LSD [25]
5.380 5.394 5.364 5.392 3.397
189.38 187 176.8 378
4.01
0.56
TbO2
This work Experiment [48] FPLMTO-SIC-LSD [25]
5.272 5.222 3.317
192.03
4.76
37.41
Fig. 2. (Color online) The total and partial density of states at GGA þU þ SOC for (a) CeO2. Since the spin-up and spin-down channels are identical, only the spin-up channel is shown. The vertical solid line denotes the Fermi level.
3. Structural and elastic properties The ground-state structures of CeO2, PrO2, and TbO2 were obtained by performing DFT/GGA (add U) calculations with spinpolarization effects. The equilibrium lattice constant (a), bulk modulus (B) and its pressure derivative (B0 ) are listed in Table 1 in comparison with experimental data [15–18,50] and recent theoretical calculations [20,25,26,28–32,36–38] for comparison. As a general remark, the deviations for lattice constants are estimated to be about 0.07–0.7% for CeO2, PrO2 and TbO2 compared to available experimental data. However, our theoretical approach seems to considerably underestimate the bulk modulus, the calculated values being 191 GPa and 189.4 GPa for CeO2 and PrO2, respectively, which are 6–19% and 3% lower than the experimental data [15–18,50]. It is observed that the experimental values for the bulk modulus of CeO2 are confined to a 15% range 204–236 GPa, whereas there is large scatter of the calculated values. Therefore, it is worth noticing that all GGA calculations tend to underestimate the bulk modulus. From Table 1, it is clear that the predicted bulk modulus value for CeO2 is larger than PrO2 and TbO2. In order to compute the ground state magnetic phase between antiferromagnetic (AFM) and ferromagnetic (FM) phases, we calculated the total energy difference DEðAFMFMÞ between them. Our results are also summarized in Table 1. We must recall here that the FM phase is stabilized over the AFM phase when DE 40, whereas the AFM is stabilized over the FM phases when (DE o 0). From Table 1, we found that the ground state is the FM phase for TbO2. For PrO2, the AFM phase is more stable but DEðAFMFMÞ are small. This result indicates that this system is weak AFM. The elastic stiffness determines the response of the crystal to an externally applied strain (or stress) and provides information about the bonding characteristics, mechanical and structural stability. The elastic constants Cij were calculated within the total-energy method, where the unit cell is subjected to a number of finite-size strains along several strain directions. Cubic lattices have three independent elastic constants [51], namely, C11, C12, and C44. The obtained elastic stiffness constants C ij are gathered in Table 2. Furthermore, the mechanically stable phases or macroscopic stability is dependent on the positive definiteness of stiffness matrix. For a stable cubic structure, the independent elastic constants should satisfy the well-known Born–Huang criterion [52], given by: C44 40,
172 170 217 211.11 210.1
DEðAFMFMÞ –
4.417 4.4
Table 2 Calculated elastic constants C ij (GPa), Bulk moduli BH (GPa), shear moduli G (GPa), Young’s modulus Y (GPa), and Poisson’s ratio n for CeO2, PrO2 and TbO2. Compounds
C11
C12
C44
BH
CeO2
390.86 390a 386b 354.79c 403d
128.5 130 124b 139.27c 105d
57.0 82 73b 51.19c 60d
215.95
PrO2 TbO2
344.33 300.2
126.5 108.63
181.1 130.0
199.11 184.94
Y
n
80.14
213.96
0.334
147.68 119.33
355.23 294.63
0.202 0.234
G
a
Ref. [38]. Ref. [37]. Ref. [36]. d Ref. [60] experimental data. b c
C11 49C12 9 and C11 þ2C12 4 0. One may see that the rare earth dioxides are mechanically stable because all elastic constants are positive and satisfy the Born mechanical stability restrictions. Considering the experimental data set due to Nakajima et al. [17], the C11, C12 and C44 for CeO2 values are underestimated by 3%, 22% and 5%, respectively, whereas the agreement is good for all elastic constants. Note also that our results are in close agreement with the available theoretical works [37,38]. In order to measure the stiffness of the solid, Voight–Reuss–Hill [53] estimates BH, G, Y and n of polycrystalline bulk modulus, shear modulus, Young’s modulus, and Poisson’s ratio computed from Cij are presented in Table 2 (following the procedure detailed in Ref. [54]). From this table, the bulk modulus agrees with that from the Birch–Murnaghan equation of state. It can be remark that the shear and Young’s modulus of the PrO2 is higher than CeO2 and TbO2. The larger shear modulus is mainly due to its larger C44.
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4. Electronic properties 4.1. CeO2 The calculated total and partial density of states (DOS) curves at the predicted equilibrium lattice constants for CeO2 are shown in Fig. 2 within the framework of GGA approach with and without U, where the vertical line is the Fermi level (EF). A common trend can be proposed from total DOS graphs that the influence of Ueff on the electronic structure is basically restricted to the unoccupied f state and this does not drastically change the electronic distribution especially occupied states. Hence, the gap between the valence O (2p) and conduction Ce (5d) bands is 5.5 eV, however this calculated value is slightly smaller than the measured one at 6 eV [55] and agrees with a previous theoretical study (5–5.5) eV [26,29,40]. Moreover the distance between the valence band and the Ce 4f states is 2.5 eV, to be compared with 3 eV in experiments [55]. So, the width of the O 2p is 4.38 eV, which agrees with experimental data and theoretical calculation. Another consequence of higher values of Ueff is a slightly reduced width value of the Ce 4f peak and increases the intensity of this peak, which reflects stronger localization. We notice an important feature in DOS that they are the O 2p–Ce 5d band gap and the empty Ce 4f states in the gap above the O 2p band. To gain a more detailed insight into the bonding characters of nonmagnetic ceria, we calculate the charge density distribution. The charge-density contours for a cut in the (110) plane of CeO2 are displayed in Fig. 3(a). It is clearly seen that CeO2 is
Fig. 3. (Color online) Charge density distributions (a) and electron-localization function (b) (e=A 3 ) in (110) plane of CeO2. ˚
characterized by a nearly spherical charge-density distribution around the Ce and O atoms and a low charge density in the interstitial region. It can be also seen that the ionic bonding is the determinant of the chemical bonding in CeO2, even though the covalency is strong in the sense of orbital participation in forming the valence state. We have also calculated the electronic localized function (ELF) [56] using the VASP code [57,58]. In fact, the ELF has proven to be a useful companion to the density in the task of providing us with intuition on the electronic structure. Then, when ELF ¼1, it corresponds to a perfect electron localization. ELF is a dimensionless localization index restricted to the range of [0,1]. A high ELF value stands for a low probability of finding a second electron with the same spin in the neighboring region of the reference electron, i.e., the reference electron is highly localized. Based on its definition, one can simply interpret high ELF values as covalent bonds, lone pairs, or inert cores. In Fig. 3(b), we plot the ELF 2D-contours of CeO2. Undoubtedly, the interaction between Ce and O ions is principally ionic. It is also clearly seen that nonzero ELF values ( 0:5) are present between the Ce and O atoms. These values are not large enough to define strong covalent bonds, but we can certainly attribute them to the weak hybridization of O 2p states with partially occupied Ce 4f or 5d states as shown in figures ELF and DOS. So, our ELF analysis obtained by GGAþU calculation show that some covalent components contribute to the Ce 4f(5d)–O 2p interaction. 4.2. PrO2 Fig. 4(a) provides the total and partial DOS calculated by GGAþU without and with SOC for PrO2 in the ferromagnetic phase. So, Pr þ 4 is surrounded by O 2-cube and its point symmetry group is Oh. In this case, Pr þ 4 has one electron on 4f states. The standard GGA and also GGA þU calculations without SOC make f state bands cross the Fermi level without any gap opening (see Fig. 4(a)). Thus, we apply the GGAþU with SOC to open a narrow gap between occupied and empty 4f states principally in majority spin part. Moreover, we can see that the highest occupied Pr 4f state for spin up are situated around 4 eV below the Fermi energy and concentrates them into a sharp peak. Note also that the peak from the O 2p DOS at 4 eV comes from separate bands without 4f contributions and thus these 4f states are localized and not hybridized with the O 2p states. Moreover, the Pr 4f–O 2p hybridization can be observed from some similarities in the positions of the peaks at 0.8 and 0.3 eV between the Pr 4f and O 2p curves. Noteworthy is that the spin-up and spin-down Pr 4f DOSs above 3 eV are very similar. Above the Fermi level we find the unoccupied f-levels. The fundamental band gap separates the valence band maximum and unoccupied Pr 4f is about 1.8 eV. Furthermore, the measured onset of the optical conductivity of PrO2d is about 2 eV [60], which is well reproduced by our calculation. Then, at about 5 eV above the Fermi energy, states predominantly of Pr 5d character begin to appear. Derived from experimental data, Van der Kolk and Dorenbos [61] proposed a model to predict the insulating or metallic behavior and chemical stability of lanthanide materials. From their model, it was predicted that for PrO2, the 4f empty states are situated at about 2 eV above the top of the valence band, a feature which is quite well obtained by our calculated. Spin-polarized calculations give a total magnetic moment of 0:999mB per cell, principally induced by Pr. A relative amount of the moment is found in the interstitial region about 0:06mB , while oxygen shows weak antiparallel spin moment close to 0:101mB . To further examine the origin and distribution of this moment, we have calculated self-consistently the spin density distribution for PrO2, see Fig. 5(a,b). The iso-surface plot of the spin density gives a clear scheme about the strongly localized spin polarization near
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Fig. 4. (Color online) Calculated total and partial density of states at GGAþ U and GGA þ Uþ SOC for PrO2 for majority (upper part) and minority (lower part) spin states.
Fig. 5. (a) Isosurface spin-density plot (at an isovalue of 0.06 electron/A˚ 3) for PrO2. The blue balls (very light gray) are Pr atoms and the red (dark gray) are O atoms. (b) Two-dimensional distribution of spin density (electron/A˚ 3) for the (1 1 2 0) plane passing Pr and O. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 6. (Color online) Calculated total and partial density of states at GGAþ U and GGAþ Uþ SOC for TbO2.
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4.3. TbO2
the Pr atoms. This localized spin polarization reflects the local magnetic moments in the system. In Fig. 5(b), we show the twodimensional spin density distribution in the (1 1 0) plane passing Pr, and O. The distribution shows that polarized components mainly are located at Pr atoms.
We have plotted in Fig. 6 the total and local DOS of TbO2. This figure provide the local DOSs 4f states with the effects of spin– orbit interaction and on-site Coulomb repulsion. In the DOS
Fig. 7. (Color online) (a) Spin-density plot and (b) isosurface at an isovalue of 0.6 electron/A˚ 3) for TbO2.
10
10
CeO2
CeO2 5
6
1 ()
2 ()
8
4
0
2 -5
0 8
PrO2
PrO2 5
1 ()
2 ()
6 4
0
2 0
-5
10
10
TbO2
TbO2
5
6
1 ()
2 ()
8
4
0
2 -5
0 0
4
8 Energy (eV)
12
0
4
8
12
Energy (eV)
Fig. 8. The calculated (a) imaginary (e2 ðoÞ) and (b) real (e1 ðoÞ) parts of dielectric functions of CeO2, PrO2 and TbO2.
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0.8
R ()
0.4
0.2
0 0.8
PrO2 0.6
0.4
0.2
5. Optical properties
0 0.6
TbO2
0.4
R ()
From our electronic structure calculations, the imaginary part of the dielectric function e2 ðoÞ has been derived by summing transitions from occupied to unoccupied states for energies much larger than those of the phonons. The real part of the diagonal dielectric functions is computed using the Kramers–Kronig transformation [62]. The knowledge of both the real and imaginary parts of the dielectric tensor allows one to calculate other important optical spectra. We also computed and analyzed the reflectivity RðoÞ. In order to assess convergence of the optical properties, we calculated e2 ðoÞ with increasingly finer meshes for the discretization of the BZ: 1500 k points with the spin–orbit interaction, and we apply a broadening equal to 0.1 (this value is typical of the experimental accuracy) in order to bring out all the structures. The analysis of e2 ðoÞ curves of Fig. 8(a) shows that the only CeO2, PrO2 and TbO2 compounds have an optical gap of about 3.8, 1.8 and 0.5 eV. The threshold energy (first critical point) of the dielectric function for these three compounds occurs at 3.8, 1.8 and 0.5 eV. This point gives the threshold for direct optical transitions between the highest valence and the lowest conduction band. This is known as the fundamental absorption edge. The origin of these peaks is attributed to the inter-band transitions form the occupied states to the unoccupied states. Beyond these threshold energies (first critical points), the curve increases rapidly. This is due to the fact that the number of points contributing toward e2 ðoÞ increases abruptly. Note that we do not include phonon contributions to the dielectric screening. The main peak of e2 ðoÞ for CeO2, PrO2 and TbO2 is situated around 2.9, 3.5, and 2.0 eV. The density of states of CeO2 (PrO2) suggests that the first peak in e2 ðoÞ at about 2.9 eV (3.5 eV) is due to the transition from O p to Ce (Pr) f states, while the second peak corresponds to transitions of the O p to the Tb f states. Moreover, the fine structure of this peak in e2 ðoÞ is similar to the fine structure of the DOS in the energy interval 4 to Fermi level. The
CeO2
0.6
R ()
calculated by GGAþU, we observe only one intense peak whereas with the introduction of the spin orbit coupling, the f-states are represented by several peaks, which correspond to the electronic nature of these states. One may notice that the SOC effect gives rise to the broadened 4f state. We can also see that the spin up Tb 4f are centered around 10 eV below Fermi level and for spin down bands, they are empty and situated between the valence band maximum and conduction band minimum with a more localized peak about 2 eV. Thus, the dominance of Tb 4f to majority spin in the valence band and of the minority spin in the forbidden band. Like to PrO2, the total moment TbO2 comes principally from Tb elements which is about 7:016mB per cell. Then, the magnetic moments of all atoms have the same direction, which indicates that there is ferromagnetic coupling between the Tb and O atoms. To complete the magnetic structure description, we display in Fig. 7, the two- and three-dimensional spin densities of TbO2. One may observe from two-dimensional spin density distribution in the (1 1 0) plane that interactions appeared between heavy Tb atoms show a pronounced effect on the spin densities maps. The spin density distribution around the oxygens has a circular appearance but slightly modified due to the polarization induced by the presence of Tb atom. The iso-surface plot of the spin density shows that the shape of the spin densities distribution around the Tb atom is circular. This distribution depends on the filling of orbitals, mainly the f one. We are able to conclude that the hybridization between different orbitals affect the spin density contours [63,64].
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0.2
0 0
4
8
12
Energy (eV) Fig. 9. Reflectivity spectra of CeO2, PrO2 and TbO2.
most realistic explanation for the above observation is that the final state for the optical transitions, i.e., the unoccupied 4f states of Ce (Pr), is a localized atomic-like state. The same effect is obtained for TbO2, where the first peak in e2 ðoÞ is due to transitions of O p to the Tb f states (spin up plus spin down), while the second peak corresponds to transitions of the O p to the Tb d states. The real part of the electronic dielectric function spectrum for the all rare earth dioxides is shown in Fig. 8(b). The CeO2, PrO2 and TbO2 compounds show almost the same main features. The estimated values of static dielectric constant e1 ð0Þ of these compounds are given by 4.84, 5.13 and 6.52, respectively. Our theoretical e1 ð0Þ values of CeO2 are compared to three sets of
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experimental results 4.7 [9], 5.31 [65] and 6 [66], the differences are smaller and range between an overestimation by 3% and an underestimation by 8% and 19%. The reflectivity spectra RðoÞ for CeO2, PrO2 and TbO2 are presented in Fig. 9. The figures show a small reflectivity up to 8.0 eV then increasing abruptly at higher energies. We should emphasize that the reflectivity minimum between 4.0 and 8.0 eV for the CeO2, PrO2 and TbO2. The depth of the plasmon minimum is determined by the imaginary part of the dielectric function at the plasma resonance and is representative of the degree of overlap between the inter-band absorption regions.
6. Summary To conclude this paper, we have performed a detailed investigations on the structural, elastic, electronic, magnetic and optical properties of CeO2, PrO2, and TbO2 within the DFT framework based on full potential approaches. The application of GGAþ U and the spin–orbit coupling, on the electronic, magnetic, and optical properties was detailed and demonstrated to be relevant of the correctness of our results. Elastic constants calculation proves the mechanical stability of CeO2, PrO2 and TbO2. The electronic structure description has shown that these structures are stable in ferromagnetic phases and good agreement with experiment can be obtained for the magnetic moment. Moreover, we found that the GGAþ U calculation with SOC reproduces very well the observed energy gap and other electronic features compared to any other schemes. This fact emphasizes the role of both the Hubbard U and SOC in calculating the electronic states. The magnetic contribution of the rare earth atom in each dioxide was achieved by analyzing the spin-density contours, where the polarization on the first neighbor oxygen is observed.
Acknowledgments A part of calculations with Wien2k and VASP codes have been performed in ISCF computers of FUNDP University (Belgium) during the research stays of S.G.-S. and M.B.K. VASP calculations were carried out partially in the Beowulf class heterogeneous computer cluster at KAUST. S.G.-S. and M.B.K. wish to thank Alberto Otero de la Roza from Universidad de Oviedo (Spain) for providing us with RUNWIEN code. For the author A.H.R., his work was supported from the program RDI of the Czech Republic, the project CENAKVA (No. CZ.1.05/2.1.00/01.0024), the Grant no. 152/ 2010/Z of the Grant Agency of the University of South Bohemia. School of Material Engineering, Malaysia University of Perlis, P.O. Box 77, d/a Pejabat Pos Besar, 01007 Kangar, Perlis, Malaysia. References [1] G. Adachi, Z.C. Kang, in: G. Adachi, N. Imanaka, Z.C. Kang (Eds.), Binary Rare Earth Oxides, Kluwer Academic Publishers, 2004. [2] A. Trovarelli, Structural properties and nonstoichiometric behavior of CeO2, in: A. Trovarelli (Ed.), Catalysis by Ceria and Related Materials, Imperial College Press, London, 2002, p. 15; M.W. Shafer, J.B. Torrance, T.J. Perry, Journal of Physics and Chemistry of Solids 33 (1972) 2251. [3] K.A. Gschneidner, Jr. (Ed.), Industrial Applications of Rare Earth Elements, ACS Symposium Series 164, American Chemical Society, Washington, DC, 1981. [4] G.R. Choppin, Chemical properties of the rare earth elements, in: J.-C.G. Bunzli, G.R. Choppin (Eds.), Lanthanide Probes in Life, Chemical and Earth Sciences, Elsevier, Amsterdam, 1989, p. 1. [5] L. Eyring, The binary rare earth oxides: synthesis and identification, in: G. Meyer, L.R. Morss (Eds.), Synthesis of Lanthanide and Actinide Compounds, Kluwer Academic Publishers, Dordrecht, 1990, p. 87. [6] L. Eyring, The higher oxides of the rare earths; a paradigm for solid state chemistry, in: R. Saez Puche, P.A. Caro (Eds.), Rare Earths, Editorial Complutense, Madrid, 1998, p. 119.
[7] L. Mukin, M. Chen, S. Dechun, Journal of Rare Earths 9 (1991) 294. [8] X.H. Wang, Z.D. Zou, S.L. Song, S.Y. Qu, Journal of Materials Science Letters 22 (2003) 713. [9] M. Marabelli, P. Wachter, Physical Review B 36 (1987) 1238. [10] A. Fujimori, Physical Review B 27 (1983) 3992 28 (1983) 2281. [11] R.C. Karnatak, J.-M. Esteva, H. Dexpert, M. Gasgnier, P.E. Caro, L. Albert, Physical Review B 36 (1987) 1745. [12] H. Ogasawara, A. Kotani, K. Okada, B.T. Thole, Physical Review B 43 (1991) 854. [13] J.E. Spanier, R.D. Robinson, F. Zhang, S.-W. Chan, I.P. Herman, Physical Review B 64 (2001) 245407. [14] S. Wang, W. Wang, J. Zuo, Y. Qian, Materials Chemistry and Physics 68 (2001) 246. [15] S.J. Duclos, Y.K. Vohra, A.L. Ruoff, A. Jayaraman, G.P. Espinosa, Physical Review B 38 (1988) 7755. [16] L. Gerward, J.S. Olsen, Powder Diffraction 8 (1993) 127. [17] A. Nakajima, A. Yoshihara, M. Ishigama, Physical Review B 50 (1994) 13297. [18] L. Gerward, J. Staun Olsen, L. Petit, G. Vaitheeswaran, V. Kanchana, A. Svanee, Journal of Alloys and Compounds 400 (2005) 56. [19] J. Dabrowski, V. Zavodinsky, A. Fleszar, Microelectronics Reliability 41 (2001) 1093. [20] S. Mehrotra, A.K. Bandyopadhyay, Journal of Alloys and Compounds 436 (2007) 56. [21] M. Diviˇs, J. Rusz, V. Nekvasil, Czechoslovak Journal of Physics 54 (2004) D291. [22] M. Diviˇs, J. Rusz, Journal of Magnetism and Magnetic Materials 290–291 (2005) 1015. [23] P. Nova´k, M. Diviˇs, Physica Status Solidi B 244 (2007) 3168. [24] S. Gennard, F. Cora , C.R.A. Catlow, Journal of Physical Chemistry B 103 (1999) 10158. [25] L. Petit, A. Svane, Z. Szotek, W.M. Temmerman, Physical Review B 72 (2005) 205118. [26] N.V. Skorodumova, R. Ahuja, S.I. Simak, I.A. Abrikosov, B. Johansson, B.I. Lundqvist, Physical Review B 64 (2001) 115108. [27] D.D. Koelling, A.M. Boring, J.H. Wood, Solid State Communications 47 (1983) 227. [28] S. Fabris, S. de Gironcoli, S. Baroni, G. Vicario, G. Balducci, Physical Review B 71 (2005) 041102. [29] C. Loschen, J. Carrasco, K.M. Neyman, F. Illas, Physical Review B 75 (2007) 035115. [30] Y. Jiang, J.B. Adams, M. van Schilfgaarde, Journal of Chemical Physics 123 (2005) 064701. [31] Z. Yang, T.K. Woo, M. Baudin, K. Hermansson, Journal of Chemical Physics 120 (2004) 7741. [32] J.L.F. Da Silva, M.V. Ganduglia-Pirovano, J. Sauer, V. Bayer, G. Kresse, Physical Review B 75 (2007) 045121. [33] N.V. Skorodumova, M. Baudin, K. Hermansson, Physical Review B 69 (2004) 075401. [34] V.I. Anisimov, J. Zaanen, O.K. Andersen, Physical Review B 44 (1991) 943. [35] V.I. Anisimov, F. Aryasetiawan, A.I. Lichtenstein, Journal of Physics: Condensed Matter 9 (1997) 767. [36] S. Shi, Y. Tang, C. Ouyang, L. Cui, X. Xin, P. Li, W. Zhou, H. Zhang, M. Lei, L. Chen, Journal of Physics and Chemistry of Solids 71 (2010) 788. ¨ [37] T. Gurel, R. Eryigit, Physical Review B 74 (2006) 014302. [38] C. Sevik, T. C - agin, Physical Review B 80 (2009) 014108. [39] M. Nolan, S. Grigoleit, D.C. Sayle, S.C. Parker, G.W. Watson, Surface Science 576 (2005) 217. [40] D.A. Andersson, S.I. Simak, B. Johansson, I.A. Abrikosov, N.V. Skorodumova, Physical Review 75 (2007) 035109. [41] F. Tran, J. Schweifer, P. Blaha, K. Schwarz, Physical Review B 77 (2008) 085123. [42] P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, J. Luitz, Wien2k, An Augmented-Plane-Wave þLocal Orbitals Program for Calculating Crystal Properties, TU Wien, Austria, 2001. [43] Z. Wu, R.E. Cohen, Physical Review B 73 (2006) 235116. [44] E. Engel, S.H. Vosko, Physical Review B 47 (1993) 13164. [45] J. Kuneˇs, P. Nova´k, M. Diviˇs, P.M. Oppeneer, Physical Review B 63 (2001) 205111. [46] V.I. Anisimov, I.V. Solovyev, M.A. Korotin, M.T. Czyzyk, G.A. Sawatzky, Physical Review B 48 (1993) 16929. [47] A.I. Liechtenstein, V.I. Anisimov, J. Zaanen, Physical Review B 52 (1995) R5467. [48] F.D. Murnaghan, Proceedings of the National Academy of Sciences of the United States of America 30 (1944) 244. [49] F. Birch, Physical Review B 71 (1947) 809; F. Birch, Journal of Geophysical Research 86 (1978) 1257. [50] P. Villars, L.D. Calvert, Pearson’s Handbook of Crystallographic Data for Intermetallic Phases, 2nd ed., ASM International, Ohio, 1991. [51] J.F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, Oxford University Press, Oxford, 1985. [52] M. Born, K. Hang, Dynamical Theory and Experiments I, Springer Verlag Publishers, Berlin, 1982. [53] R. Hill, Proceedings of the Physical Society, London, Section A 65 (1952) 349. [54] N. Kanoun-Bouayed, M.B. Kanoun, S. Goumri-Said, Central European Journal of Physics 9 (2011) 205. [55] E. Wuilloud, B. Delley, W.-D. Schneider, Y. Baer, Physical Review Letters 53 (1984) 202. [56] B. Silvi, A. Savin, Nature (London) 371 (1994) 683.
M.B. Kanoun et al. / Journal of Magnetism and Magnetic Materials 324 (2012) 1397–1405
¨ [57] G. Kresse, J. Furthmuller, Computational Materials Science 6 (1996) 15. [58] The calculations were performed within DFT using the projector augmented wave (PAW) method as implemented in the VASP package [57]. The exchange and correlations were treated within GGA-PBE [59] including Hubbard correction. An energy cutoff of 500 eV for the plane-wave expansion of the PAWs is used. The Brillouin zone is sampled on a 11 11 11 k-point grid. [59] J.P. Perdew, K. Burke, M. Ernzerhof, Physical Review Letters 77 (1996) 3865. [60] S. Kimura, F. Arai, M. Ikezawa, Journal of Electron Spectroscopy and Related Phenomena 78 (1996) 135.
1405
[61] E. van der Kolk, P. Dorenbos, Chemistry of Materials 18 (2006) 3458. [62] H.Z. Tributsch, Naturforsch A 32A (1977) 972. [63] N. Kanoun-Bouayed, S. Goumri-Said, A.E. Merad, M.B. Kanoun, Materials Science Forum 609 (2009) 167. [64] S. Goumri-Said, M.B. Kanoun, Journal of Physics D: Applied Physics 41 (2008) 035004. [65] S. Mochizuki, Physica Status Solidi B 114 (1982) 189. [66] N.I. Santha, M.T. Sebastian, P. Mohanan, N.M. Alford, K. Sarma, R.C. Pullar, S. Kamba, A. Pashkin, P. Samukhina, J. Petzelt, Journal of the American Ceramic Society 87 (2004) 1233.