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Electronic, magnetic and optical properties of cubic double perovskites Ba2CrMoO6 and Ba2CrWO6 with (d3–d1) system
Computational Materials Science 92 (2014) 298–304
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Electronic, magnetic and optical properties of cubic double perovskites Ba2CrMoO6 and Ba2CrWO6 with (d3–d1) system M. Musa Saad H.-E. ⇑ Department of Physics, College of Science, Qassim University, Buridah 51452, Saudi Arabia
a r t i c l e
i n f o
Article history: Received 22 October 2013 Received in revised form 3 April 2014 Accepted 17 May 2014
Keywords: Double perovskite Half-metallic FP-LMTO computational method LSDA
a b s t r a c t The electronic, magnetic and optical properties of cubic double perovskites Ba2CrMoO6 and Ba2CrWO6 have been studied by using full potential linear muffin-tin orbital (FP-LMTO) computational method within the local spin density approximation (LSDA) as well as taking into account the on-site Coulomb repulsive interaction in (LSDA + U) approach. The band structure results reveal a similar half-metallic (HM) ferrimagnetic (FiM) ground state in two compounds with total spin magnetic moments closed to the theoretical value (m = 2.0lB/f.u.). The LSDA and LSDA + U calculations predict an energy-gap in the spin-up bands, while a finite density of states for the spin-down bands in Ba2CrMoO6 and Ba2CrWO6. HM–FiM organizes from the ddpp super-exchange interaction (3d-t32g"–O 2pp–4d/5d-t12g;) in accordance with the Pauli Exclusion Principle and Goodenough–Kanamori rules. The real and imaginary parts of dielectric functions were calculated and the inter-band contributions to the optical properties have been analyzed. The HM–FiM nature with (SP = 1.0) implies a promising applications of Ba2CrMoO6 and Ba2CrWO6 materials in magnetoelectronic and spin-electronics technology. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction Recently, transition-metal double perovskite materials with high spin-polarization (SP) and half-metallic (HM) nature have become one of the important and hot topic in scientific research because of their remarkable properties such as crystal structure, electronic, magnetic and optical properties. Several scientists and engineers have studied and reported the crystal structure, electronic, magnetic and optical properties of transition-metal double perovskites, in particular, the renowned Cr and Fe based double perovskites such as Ba2FeMoO6 [1], A2FeReO6 (A = Ca, Sr and Ba) [2], Sr2CrMoO6 [3] and Sr2CrReO6 [4]. It is known that the ferrimagnetic half-metallic (FiM–HM) properties are predictable for the ordered double perovskites A2FeMoO6 (A = Ca, Sr and Ba) [1,5] with localized spin-up electrons of Fe3+ (3d5: t32g" e2g ") in high-spin state and itinerant spin-down electron of Mo5+ (3d1: t12g;) along with. Also, Sr2CrMoO6 is HM–FiM material with only t2g localized spinup electrons Cr3+ (3d3: t32g") [6]. The magnetic properties of A2(Fe,Cr)MoO6 are attributed to the long-range antiferromagnetic (AFM) super-exchange interaction between the two different transition-metals Cr/Fe and Mo via the intermediate O atoms Cr/Fe (3d")–O (2p)–Mo (4d;) [7,8]. Super-exchange interaction produces ⇑ Mobile: +966 509353808; fax: +966 163800911. E-mail address: [email protected] http://dx.doi.org/10.1016/j.commatsci.2014.05.030 0927-0256/Ó 2014 Elsevier B.V. All rights reserved.
FiM order between the 3d and 4d/5d ions and is frequently combined with significance HM bands [9]. In addition, HM double perovskite is metallic in one spin direction with continuous density of states (DOS > 0), while is insulator or semiconductor with energy-gap (DOS = 0) in the other spin direction. In fact, the HM property relates directly to the colossal magnetoresistance (CMR) phenomenon observed in numerous transition-metal double perovskites, such in Sr2FeMoO6 [10], Sr2FeReO6 [11], Sr2CrMO6 (M = Mo and W) [12] and Pb2FeReO6 [13]. During our research in the database and to our knowledge, no theoretical or experimental studies have been carried out on the electronic, magnetic and optical properties of chromium-based double perovskites Ba2CrMoO6 and Ba2CrWO6, except for a few experimental contributions. Dhahri et al. studied the doped Ba2CrMo1xWxO6 [14], and El-Hagary who investigated the influence of partial substitution of Cr3+ on physical properties of Ba2FeMoO6 [15]. Theoretically, Ba2CrMoO6 compound was studied in our previous study, among the 3d series of double perovskites of Ba2TMoO6 (T = V, Cr, Mn, Fe and Co) [16]. This is what motivated me to carry out a theoretical study on these two compounds, with (d3–d1) system. In the present work, two novel Cr-based double perovskites Ba2CrMoO6 and Ba2CrWO6 have been studied. The initial aim is to study the influence of 4d and 5d ions on the electronic, magnetic and optical properties. As known, molybdenum Mo (Z = 42) and tungsten W (Z = 74) elements are neighboring in
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different periods of transition-metals block in the period system. In addition, according to their similar electronic configurations, Mo: [Kr] 4d5 5s1 and W: [Xe] 4f14 5d4 6s2, Mo and W belong to the 4d and 5d transition-metals, respectively.
Table 1 Formula unit magnetic system, space group (S.G.), lattice parameter a, oxygen coordinates O (x) and volumes V, for the room temperature structure of Ba2CrMoO6 and Ba2CrWO6 compounds. The experimental values obtained (from Refs. [14,22]), and are cited between square brackets. Compound
2. Methods and computational details In this work, the electronic, magnetic and optical properties of Ba2CrMoO6 and Ba2CrWO6 double perovskites are calculated by using the local spin-density approximation (LSDA) as well as taking into account the on-site Coulomb repulsive interaction in (LSDA + U) approach within the density-functional theory (DFT). The band structure calculations are carried out using the selfconsistent (SCF) full potential linear muffin-tin orbital (FP-LMTO) method, as employed in the LMTART computational software [17]. In FP-LMTO method, the primitive cell is divided into nonoverlapped atomic spheres and interstitial regions. Within the atomic muffin-tin spheres (MTSs), the charge density q(r) is constructed from numerical solutions of radial Schrödinger’s equations (RSE) for each atom and the potential is represented via the spherical harmonic expansions. In the interstitial region, the q(r) is calculated from the Fourier transforms (FT) for the LMTOs [18] and the potential is expanded in the plane-wave (PLW) method [19]. Spin orbit coupling (SOC) is also included in the calculations. The Hartree potential is expanded in term of spherical harmonic of (lmax = 6), and an exchange–correlation potential of Von Barth– Hedin (VBH) mode is taken on. In consequence, the Ba (6s 5p 5d 4f), Cr (4s 4p 3d), Mo (5s 5p 4d), W (6s 6p 5d) and O (2s 2p) LMTOs are set as valence states, while Ba (5s), Cr (3p), Mo (4s 4p) and W (5s 5p) as semicore states. The LSDA + U scheme is considered a useful approach that provides quite satisfactory results, agreement with the experimental results, for the strong correlated systems [11,19]. The correlation parameters adopted in this study were the Coulomb U and exchange J parameters. In this work, the near maximum values were selected from the reasonable range of U (1–6.0 eV) and J (0.5–1.0 eV) [3,11,19–21]. The detailed U and J values are (U = 4.0 eV and J = 0.97 eV) for the strong localized electrons in Cr (3d) states and (U = 1.0 eV and J = 0.97 eV) for the weakly delocalized electrons in Mo/W (4d/5d) states. 3. Results and discussion
System 3+
5+
S.G.
a (Å)
x
V (Å3)
Ba2CrMoO6
Cr –Mo (3d3–4d1)
Fm3m
7.9984 [8.0098]a
0.2573 [0.2634]b
511.883 [513.884]c
Ba2CrWO6
Cr3+–W5+ (3d3–5d1)
Fm3m
7.9914 [8.0081]d
0.2575 [0.2641]e
510.673 [515.354]f
a,b,c d,e,f
Ref. [14]. Ref. [22].
Table 2 The average cation–anion interatomic distances (in Å) and angles h° at room temperature in Ba2CrMoO6 and Ba2CrWO6 with cubic (Fm3m) space-group, in comparison with experimental values (Exp.) taken from Refs. [14,23]. Compound
Ba2CrMoO6
Method
Theo.
Exp.
Ba2CrWO6 Theo.
Exp.
Cr (4a)–O (24e) 6 Mo/W (4b)–O (24e) 6 Ba (8c)–O (24e) 12 h hOACrAOi h hCrAOAMo/Wi
2.0585 1.9408 2.8291 90 180
2.026(3) 2.011(2) – 90 180
2.0552 1.9373 2.8265 90 180
2.112(1) 1.9353 – 90 180
Table 3 Magnetic moments of metal ions, total magnetic moments (in lB/f.u.), the number of states at EF, N(EF), the energy-gap (Eg) and the energy difference between FM and FiM configurations, DE (MeV). The spin pictures of the dynamic ions are Cr3+ (3d3; t32g" e0g" S = 3/2), Mo5+ (4d1; t12g" e0g" S = 1/2) or W5+ (5d1; t12g" e0g" S = 1/2). Compound
Ba2CrMoO6
Approach
LSDA
LSDA + U
LSDA
Ba2CrWO6 LSDA + U
Cr (3d) Mo (4d)/W (5d) Total N(EF) Eg (eV) DE(FM–FiM)
2.19 0.32 1.95 1.18; 0.45" 28.0
2.89 0.83 1.97 5.62; 1.81" 81.0
2.25 0.34 1.94 2.98; 0.46" 14.0
2.89 0.73 1.96 2.73; 1.82" 61.0
configuration, see Table 3, in a good harmony with the earlier DFT computational results for their relative strontium double perovskites Sr2CrMoO6 and Sr2CrWO6 [3,6,7,11].
3.1. Crystal structures and ground state 3.2. Electronic properties The room temperature crystal structures of Ba2CrMoO6 and Ba2CrWO6 are in face center cubic structure with space-group of (Fm3m) and lattice parameters of a = b = c = 7.9984 Å and 7.9914 Å, respectively. The lattice parameters are around the ideal value (a = b = c = 8.0 Å) and depend mainly on the ionic radii of Cr3+ (0.615 Å), Mo5+ (0.60 Å), or W5+ (0.62 Å), Ba2+ (1.49 Å) and O2 (1.32 Å). In the cubic formula unit, the atomic sites and positions are set as; two Ba (8c) atoms at (1=4 , 1=4 , 1=4 ) and (3=4 , 3=4 , 3=4 ), Cr (4a) at (½, ½, ½), Mo/W (4b) at (0, 0, 0) and six O (24e) atoms at (±x, 0, 0), (0, ±x, 0) and (0, 0, ±x), where (x = 0.2573) for Ba2CrMoO6 and (x = 0.2575) for Ba2CrWO6. The crystal structures data of Ba2CrMoO6 and Ba2CrWO6 are calculated and summarized in Tables 1 and 2, in comparison with the previous experimental data. The obtained structure information is in excellent agreement with the previous results [14,15,22]. In order to find the stable magnetic structure, the ground state, in Ba2CrMoO6 and Ba2CrWO6, the total energies of ferromagnetic (FM) and ferrimagnetic (FiM) configurations are calculated. It is found that the FiM configuration in both compounds of Ba2CrMoO6 and Ba2CrWO6 has lower total energy (ground state) than the FM
The band structures of the ground state of Ba2CrMoO6 and Ba2CrWO6 have been calculated using the LSDA and LSDA + U approaches. During the study, it has been found that FM configuration converges to FiM configuration for all the considered structure that is only FiM state is found for these compounds. Therefore, only the density of states (DOS) from cubic and FiM structure are presented. The total and partial densities of states (DOSs) are shown in Figs. 1–4, where, the upper and lower curves represent the spin-up (DOS") and spin-down (DOS;) per formula unit, respectively. The behavior of DOSs have been plotted between 10 eV to 10 eV where the main features occur, the dash lines stand for the Fermi energy (EF). From the total densities of states (TDOS) near the EF for Ba2CrMoO6 (Figs. 1(a) and 2(a)) and for Ba2CrWO6 (Figs. 3(a) and 4(a)), it is seen that at LSDA and LSDA + U calculations, two compounds exhibit half-metallic character. In LSDA scheme, the halfmetallicity can be observed with an energy-gap at the spin-up channel TDOS" of about (Eg = 0.45 eV) expands from 0.24 eV to 0.21 eV, Fig. 1(a), and (Eg = 0.47 eV) expands from 0.36 eV to
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Fig. 1. Total and partial Cr (3d), Mo (4d) and O (2p) densities of states for the formula unit of Ba2CrMoO6 versus the energy relative to the Fermi energy (E–EF), calculated using the LSDA.
0.11 eV, Fig. 3(a), for Ba2CrMoO6 and Ba2CrWO6, respectively. While, the conduction band at the spin-down channel TDOS; continues with bandwidth of (DW = 4.54 eV), extends from 1.15 eV to 3.39 eV and (DW = 1.48 eV), extends from 1.27 eV to 0.21 eV, for Ba2CrMoO6 and Ba2CrWO6, respectively. In a similar way, it seen that the LSDA + U scheme, also, gives HM behavior in two compounds of Ba2CrMoO6 and Ba2CrWO6, with an equal energygap at the spin-up channel TDOS" of about (Eg = 1.82 eV) expands from 1.60 eV to 0.22 eV. Whereas, the conduction band at the spin-down channel TDOS; with (DW = 1.37 eV) extends from 0.71 eV to 0.66 eV and (DW = 1.73 eV) extends from 1.08 eV to 0.65 eV in Ba2CrMoO6 and Ba2CrWO6, respectively. These results are attributed to the similarly electronic configuration of Mo5+ (4d1) and W5+ (5d1) ions, other than the itinerant electron in 5d level has more energy (E = 0.51 eV) than that in 4d level (E = 0.36 eV). Moreover, from the partial densities of states (PDOSs) in Figs. 1– 4 (panels b, c and d), it is clearly seen that the Mo (4d-t2g;) and W (5d-t2g;) states are responsible for the conductivity in Ba2CrMoO6 and Ba2CrWO6, respectively. In addition, the conduction bands are mainly composed from the Mo/W (4d/5d) states that hybridized with O (2p) and a little contribution of Cr (3d) states. Consequently, the interaction mechanism via the AFM long-range of Cr (3d)"–O (2p)–Mo/W (4d/5d); hybridization accountable for the HM behavior in Ba2CrMoO6 and Ba2CrWO6, resulting in a high spin-polarization (SP = 1.0) of the conduction electrons (charge carriers) in spin-down orientation. The SP ratio can be defined by means of the +DOS (spin-up) and DOS (spin-down) around the EF:
SP ¼
DOS" ðEF Þ DOS# ðEF Þ DOS" ðEF Þ þ DOS# ðEF Þ
ð1Þ
Therefore, Ba2CrMoO6 and Ba2CrWO6 are predicted to have a negative high spin-polarization (SP = 1.0) at high temperature above room temperature. The obtained perfect SP refers the HM electronic properties of those materials; this is extremely interest as promising candidates of a spin injector in spin-electronics devices.
3.3. Magnetic properties The super-exchange interaction is the most essential one for magnetism in transition-metal oxides which describes interaction between localized spin magnetic moments [7,9]. According to Goodenough–Kanamori rules (GKR), this mechanism gives rise to an antiferromagnetic (AFM) interaction between ions with same kind of d-orbital and ferromagnetic (FM) interaction between ions with more than half-filled d-orbital and less than half-filled d-orbital [24,25]. Consequently, the magnetic properties of transitionmetal double perovskites Ba2CrMoO6 and Ba2CrWO6 are governed by the 180° long-range (ddpp) super-exchange interaction and it depends mainly on the chemical states of Cr and Mo/W ions. Indeed, the result of high-spin ionic configuration in ordered (Cr3+–Mo5+) and (Cr3+–W5+) double perovskites leads to a saturation magnetization, of about (m = 2.0lB/f.u.), high SP and large ferromagnetic Curie temperature (TC); TC = 335 K for Ba2CrMoO6 [7] and TC = 145 K for Ba2CrWO6 [8]. The partial and total spin magnetic moments in two compounds have been calculated using the LSDA and LSDA + U approaches. Ba2CrMoO6 and Ba2CrWO6, with common (d3–d1) system, as the ionic-spin configurations are Cr3+ (3d3; t32g" e0g", S = 3/2), Mo5+ (4d1; t12g" e0g", S = 1/2) and W5+ (5d1; t12g" e0g", S = 1/2). The 3d and
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301
Fig. 2. Total and partial Cr (3d), Mo (4d) and O (2p) densities of states for the formula unit of Ba2CrMoO6 versus the energy relative to the Fermi energy (E–EF), calculated using the LSDA + U approach [U (Cr) = 4.0 eV, U (Mo) = 1.0 eV].
Fig. 3. Total and partial Cr (3d), W (5d) and O (2p) densities of states for the formula unit of Ba2CrWO6 versus the energy relative to the Fermi energy (E–EF), calculated using the LSDA.
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Fig. 4. Total and partial Cr (3d), W (5d) and O (2p) densities of states for the formula unit of Ba2CrWO6 versus the energy relative to the Fermi energy (E–EF), calculated using the LSDA + U approach [U (Cr) = 4.0 eV, U (W) = 1.0 eV].
4d/5d ions polarized in AFM with a theoretical spin magnetic moment of (m = 2.0lB/f.u.), consistent with the HM nature. The calculated total spin magnetic moments for Ba2CrMoO6 are (m = 1.95lB/f.u.) and (m = 1.97lB/f.u.), whereas, for Ba2CrWO6 are (m = 1.94lB/f.u.) and (m = 1.96lB/f.u.), from LSDA and LSDA + U approaches, respectively, see Table 3. Thus, FiM spins coupling with total magnetic moment of about (m = 2.0lB/f.u.) has been yielded in two compounds, in agreement with the experimental value (m = 2.0lB/f.u.) [15,23], and theoretical (m = 2.0953lB/f.u.) for Ba2CrMoO6 compound [26]. The results of partial spin magnetic moments of Cr, Mo and W, the total spin magnetic moments, the number of states at EF, [N (EF)], the energy-gaps (Eg) and the configuration energy differences (DEFM–FiM), are summarized in Table 3. The magnetic results of Ba2CrMoO6 and Ba2CrWO6 indicate ferrimagnetic (FiM) coupling between Cr3+ (3d3; t32g, S = 3/2) and Mo5+ (4d1; t12g, S = 1/2), and Cr3+ (3d3; t32g, S = 3/2) and W5+ (5d1; t12g, S = 1/ 2) ions, respectively, with HM electronic structures. The FiM coupling between Cr3+ and Mo5+, and Cr3+ and W5+ can be understood in terms of super-exchange interaction through the Cr (3d)–O (2p)–Mo (4d) and Cr (3d)–O (2p)–W (5d) bonding. As seen in Fig. 5, the spins of Cr3+–Mo5+, and Cr3+–W5+ appear to order in an anti-parallel arrangement by the super-exchange interaction and lead to FiM ordering in Ba2CrMoO6 and Ba2CrWO6. Moreover, the partial and total magnetic moment values notify that the FiM ground states of Ba2CrMoO6 and Ba2CrWO6 organize from the 180o long range ddpp super-exchange interaction [3d-t32g"–O (2pp)–4d/5d-t12g;], in conformity with the Pauli Exclusion Principle (PEP) and Goodenough–Kanamori rules (GKR). As a result, the mechanism of the FiM coupling in HM Ba2CrMoO6 and Ba2CrWO6 compounds is attributed chiefly to the strong AFM t32g–O–t12g of
Fig. 5. Schematic picture of the magnetic structure in cubic Ba2CrTO6, where T = Mo (4d), W (5d), describing the Cr3+ and Mo5+/W5+ ions in the ideal ordering of the magnetic structure with (d3–d1) system. Arrows represent the spin magnetic moments of Cr (up) and Mo/W (down), the total magnetization being in the direction of Cr ions. In AFM mechanism via the 180°-superexchange interaction Cr (t32g")–O (2pp)–Mo/W (t12g;), the 4d/5d; mobile-electron (blue) hopes between t2g states through 2p bridge states resulting in exchange ionic configuration of 3d(3"+Dq;)–4d/5d(1;Dq;), where Dq is the amount of transfer-charge and T refers to Mo (4d) or W (5d) ions. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
M.M. Saad H.-E. / Computational Materials Science 92 (2014) 298–304
(Cr–O–Mo/W) hybridization and favored by virtual charge transfer due to the half-filled and near empty t2g orbitals.
303
3.4. Optical properties
The data of the real and imaginary parts of dielectric function allows calculating an important optical function such as the refractive index nðxÞ. The nðxÞ can be calculated by means of e1 ðxÞ and e2 ðxÞ according to the following expression [27]:
The optical properties of double perovskites can be expressed effectively by frequency-dependent dielectric function that takes the complex form:
2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 31=2 e21 ðxÞ þ e22 ðxÞ þ e1 ðxÞ 5 nðxÞ ¼ 4 2
eðxÞ ¼ e1 ðxÞ þ ie2 ðxÞ
ð2Þ
Direct calculation of the imaginary part e2 ðxÞ of the dielectric functions are available in LMTART code by evaluating the matrix elements of the electric dipole operator between the occupied and unoccupied electronic states in the valence band (VB), including the deeply located bands as well, and in the conduction band (CB), respectively. The real and imaginary parts of the dielectric function e1 ðxÞ and e2 ðxÞ; respectively, are then obtained from the Kramers–Kronig transform (KKT); where the e2 ðxÞ part describes dissipation of electromagnetic energy (EME) and e1 ðxÞ part describes the dispersion of EME. The integral relations between e1 ðxÞ and e2 ðxÞ parts can be derived using the KKT method, as follows:
e1 ðxÞ ¼ 1 þ e2 ðxÞ ¼
2
p
2
p
Z
1 0
Z 0
1
x0 e2 ðx0 Þ 0 dx x2 x02
xe1 ðx0 Þ 0 dx x2 x02
Fig. 6. The calculated imaginary part approaches.
ð3Þ
ð4Þ
ð5Þ
The calculated real and imaginary parts of the dielectric function for Ba2CrMo6 and Ba2CrWO6, by the LSDA and LSDA + U approaches, are shown in Fig. 6. Taking the square root of their real parts in the limit of infinite wavelengths, it can estimate the values of the refractive index that turns out to be 2.0. Unfortunately, no experimental or theoretical values of the refractive index were found in the literature for Ba2CrMo6 and Ba2CrWO6 to predict and evaluate the present calculated results. The imaginary part e2 ðxÞ of dielectric function eðxÞ determines the absorption properties of the material. In Fig. 6(a) and (b), there are five similar intensive peaks of the absorption, centered at energies of about (0.66, 1.36, 3.60, 4.93 and 7.14 eV) due to the VB–CB transitions. These five peaks appear since the CBs have sub-structures composed of five sub-bands; compare to DOS graphs in Figs. 1–4. The optical results are perfect consistent with the band structure calculation of Ba2CrMo6 and Ba2CrWO6. Consideration of the DOSs diagrams in Figs. 1–4, aid to assign the peaks positions and transitions, the transitions from the occupied to the unoccupied states, can be estimated by the band structures. It has been suggested that the
e2 ðxÞ and real part e1 ðxÞ of the dielectric function eðxÞ for Ba2CrMoO6 and Ba2CrWO6, calculated by means of the LSDA and LSDA + U
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absorption edges due to the charge-transfer transitions from the highest filled O (2p) orbitals to the lowest empty Cr (3d) and Mo/W (4d/5d) orbitals. Based on the band structures, the origin of five peaks (0.66, 1.36, 3.60, 4.93 and 7.14 eV) in Fig. 6(a) and (b) can be assigned as the O (2p) ? Mo/W (4d/5d-t2g), O (2p) ? Cr (3d-t2g), O (2p) ? Mo/W (4d/5d-eg), p–d transitions, respectively. These peaks have attributes of dipole-allowed p–d transitions have large strengths. The other two small peaks can be assigned as the Cr (3d-t2g) ? Mo/W (4d/5d-t2g) and Cr (3d-eg) ? Mo/W (4d/5d-eg), d–d transitions. The strength of these two peaks is very small; because the d–d transitions between the Cr (3d) and Mo/W (4d/5d) bands. The energy differences are 5.5 eV between peaks (at 1.36 eV) and (at 4.93 eV), and 1.02 eV between peaks (at 1.36 eV) and (at 3.6 eV) represent the crystal field splitting energies of Mo/W (4d/5d) and Cr (3d), respectively. The increase of the absorption of two compounds above 5.0 eV can probably be attributed to the chargetransfer transition between the O (2p) spin-up and spin-down bands at 1.0 eV (in LSDA–DOSs) and 2.0 eV (in LSDA + U–DOSs) below the EF into Cr/Mo–O or Cr/W–O bands the at +1.0 eV above EF. The absorption shoulder around 1.0 eV coincides roughly with the energy-gaps at EF in spin-up bands and are, therefore, probably caused by the transition from Cr (t2g") states below the EF into Mo/ W–O bands around +1.0 eV above EF. 4. Conclusion Using the FP-LMTO computational method, the electronic, magnetic and optical properties of ordered double perovskites Ba2CrMoO6 and Ba2CrWO6 were studied. The calculations were performed by the local spin density approximation (LSDA) as well as taking into account the on-site Coulomb repulsive interaction in (LSDA + U) approach within the density functional theory (DFT). At room temperature, Ba2CrMoO6 and Ba2CrWO6 crystallize in face space group) with almost center cubic crystal structure (Fm3m equal lattice parameters (a = 7.9984 Å) and (a = 7.9914 Å), respectively. The LSAD and LSDA + U calculation approaches depicted well the proper half-metallic ferrimagnetic (HM–FiM) groundstates with total spin magnetic moment of (m = 2.0lB/f.u.) in Ba2CrMoO6 and Ba2CrWO6. HM–FiM nature organizes from the ddpp super-exchange interaction (3d-t32g"–O-2pp–4d/5d-t12g;), in conformity with the Pauli Exclusion Principle and Goodenough– Kanamori rules. The real and imaginary parts of the dielectric function, and the optical constant were calculated by using the
LSAD and LSDA + U approaches and the inter-band contributions to the optical properties were analyzed. It found that the optical results are consistent with the band structure (DOSs) results. The obtained electronic and magnetic results demonstrate a promising applications of negative high spin-polarization (SP = 1.0), halfmetallic and ferrimagnetic Ba2CrMoO6 and Ba2CrWO6 double perovskite materials in magnetoelectronic and spin-electronics technology. References [1] J. Kim, J.G. Sung, H.M. Yang, B.W. Lee, J. Magn. Magn. Mater. 290–291 (2005) 1009–1011. [2] N. Auth, G. Jakob, W. Westerburg, C. Ritter, I. Bonn, C. Felser, W. Tremel, J. Magn. Magn. Mater. 272–276 (2004) e607–608. [3] Q.F. Li, X.F. Zhu, L.F. Chen, J. Phys.: Condens. Matter 20 (255230) (2008) 1–8. [4] C.Q. Tang, Y. Zhang, J. Dai, Solid State Commun. 133 (2005) 219–222. [5] Z. Szotek, W.M. Temmerman, A. Svane, L. Petit, G.M. Stocks, H. Winter, J. Magn. Magn. Mater. 272–276 (2004) 1816–1817. [6] C.M. Bonilla, D.A. Landínez, J. Arbey Rodríguez, E. Vera López, J. Roa-Rojas, Phys. B: Condens. Matter 398 (2007) 208–211. [7] D. Serrate, J.M. De Teresa, M.R. Ibarra, Phys.: Condens. Matter 19 (023201) (2007) 1–86. [8] J.B. Philipp, P. Majewski, L. Alff, A. Erb, R. Gross, T. Graf, M.S. Brandt, J. Simon, T. Walther, W. Mader, D. Topwal, D.D. Sarma, Phys. Rev. B 68 (144431) (2003) 1– 13. [9] D.D. Sarma, Priya Mahadevan, T. Saha-Dasgupta, Sugata Ray, Ashwani Kumar, Phys. Rev. Lett. 85 (2000) 2549–2552. [10] K.-I. Kobayashi, T. Kimura, H. Sawada, K. Terakura, Y. Tokura, Nature 395 (1998) 677–680. [11] Horng-Tay Jeng, G.Y. Guo, Phys. Rev. B 67 (094438) (2003) 1–7. [12] T.S. Chan, R.S. Liu, G.Y. Guo, S.F. Hu, J.G. Lin, J.-F. Lee, L.-Y. Jang, C.-R. Chang, C.Y. Huang, Solid State Commun. 131 (2004) 531–535. [13] Yan Zhang, Vincent Ji, Ke-Wei Xu, Phys. B: Condens. Matter 407 (2012) 2617– 2621. [14] A. Dhahri, J. Dhahri, S. Zemni, M. Oumezzine, H. Vincent, J. Alloy. Compd. 420 (2006) 15–19. [15] M. El-Hagary, J. Alloy. Compd. 502 (2010) 376–381. [16] H.-E. Mohamed Musa Saad, Phys. B: Condens. Matter 407 (13) (2012) 2512– 2518. [17] S.Yu. Savrasov, D.Y. Savrasov, Phys. Rev. B 46 (1992) 12181–12195. [18] S.Y. Savrasov, Phys. Rev. B 54 (1996) 16470–16486. [19] I.V. Solovyev, Phys. Rev. B 65 (144446) (2002) 1–17. [20] J.-S. Kang, H. Han, B.W. Lee, C.G. Olson, S.W. Han, K.H. Kim, J.I. Jeong, J.H. Park, B.I. Min, Phys. Rev. B 64 (024429) (2001) 1–6. [21] Hua Wu, Phys. Rev. B 64 (125126) (2001) 1–7. [22] Abdul Majid, Asifullah Khan, Tae-Sun Choi, Comput. Mater. Sci. 50 (2011) 1879–1888. [23] A. Dhahri, J. Dhahri, M. Oumezzine, Mater. Lett. 63 (2009) 121–123. [24] J.B. Goodenough, J. Phys. Chem. Solids 6 (1958) 287–297. [25] J. Kanamori, J. Phys. Chem. Solids 10 (1959) 87–98. [26] M. Musa Saad H.-E., J. Alloy. Compd. 587 (2014) 652–658. [27] Alo Dutta, T.P. Sinha, Solid State Commun. 150 (2010) 1173–1177.
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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Electronic, magnetic and optical properties of cubic double perovskites Ba2CrMoO6 and Ba2CrWO6 with (d3–d1) system M. Musa Saad H.-E. ⇑ Department of Physics, College of Science, Qassim University, Buridah 51452, Saudi Arabia
a r t i c l e
i n f o
Article history: Received 22 October 2013 Received in revised form 3 April 2014 Accepted 17 May 2014
Keywords: Double perovskite Half-metallic FP-LMTO computational method LSDA
a b s t r a c t The electronic, magnetic and optical properties of cubic double perovskites Ba2CrMoO6 and Ba2CrWO6 have been studied by using full potential linear muffin-tin orbital (FP-LMTO) computational method within the local spin density approximation (LSDA) as well as taking into account the on-site Coulomb repulsive interaction in (LSDA + U) approach. The band structure results reveal a similar half-metallic (HM) ferrimagnetic (FiM) ground state in two compounds with total spin magnetic moments closed to the theoretical value (m = 2.0lB/f.u.). The LSDA and LSDA + U calculations predict an energy-gap in the spin-up bands, while a finite density of states for the spin-down bands in Ba2CrMoO6 and Ba2CrWO6. HM–FiM organizes from the ddpp super-exchange interaction (3d-t32g"–O 2pp–4d/5d-t12g;) in accordance with the Pauli Exclusion Principle and Goodenough–Kanamori rules. The real and imaginary parts of dielectric functions were calculated and the inter-band contributions to the optical properties have been analyzed. The HM–FiM nature with (SP = 1.0) implies a promising applications of Ba2CrMoO6 and Ba2CrWO6 materials in magnetoelectronic and spin-electronics technology. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction Recently, transition-metal double perovskite materials with high spin-polarization (SP) and half-metallic (HM) nature have become one of the important and hot topic in scientific research because of their remarkable properties such as crystal structure, electronic, magnetic and optical properties. Several scientists and engineers have studied and reported the crystal structure, electronic, magnetic and optical properties of transition-metal double perovskites, in particular, the renowned Cr and Fe based double perovskites such as Ba2FeMoO6 [1], A2FeReO6 (A = Ca, Sr and Ba) [2], Sr2CrMoO6 [3] and Sr2CrReO6 [4]. It is known that the ferrimagnetic half-metallic (FiM–HM) properties are predictable for the ordered double perovskites A2FeMoO6 (A = Ca, Sr and Ba) [1,5] with localized spin-up electrons of Fe3+ (3d5: t32g" e2g ") in high-spin state and itinerant spin-down electron of Mo5+ (3d1: t12g;) along with. Also, Sr2CrMoO6 is HM–FiM material with only t2g localized spinup electrons Cr3+ (3d3: t32g") [6]. The magnetic properties of A2(Fe,Cr)MoO6 are attributed to the long-range antiferromagnetic (AFM) super-exchange interaction between the two different transition-metals Cr/Fe and Mo via the intermediate O atoms Cr/Fe (3d")–O (2p)–Mo (4d;) [7,8]. Super-exchange interaction produces ⇑ Mobile: +966 509353808; fax: +966 163800911. E-mail address: [email protected] http://dx.doi.org/10.1016/j.commatsci.2014.05.030 0927-0256/Ó 2014 Elsevier B.V. All rights reserved.
FiM order between the 3d and 4d/5d ions and is frequently combined with significance HM bands [9]. In addition, HM double perovskite is metallic in one spin direction with continuous density of states (DOS > 0), while is insulator or semiconductor with energy-gap (DOS = 0) in the other spin direction. In fact, the HM property relates directly to the colossal magnetoresistance (CMR) phenomenon observed in numerous transition-metal double perovskites, such in Sr2FeMoO6 [10], Sr2FeReO6 [11], Sr2CrMO6 (M = Mo and W) [12] and Pb2FeReO6 [13]. During our research in the database and to our knowledge, no theoretical or experimental studies have been carried out on the electronic, magnetic and optical properties of chromium-based double perovskites Ba2CrMoO6 and Ba2CrWO6, except for a few experimental contributions. Dhahri et al. studied the doped Ba2CrMo1xWxO6 [14], and El-Hagary who investigated the influence of partial substitution of Cr3+ on physical properties of Ba2FeMoO6 [15]. Theoretically, Ba2CrMoO6 compound was studied in our previous study, among the 3d series of double perovskites of Ba2TMoO6 (T = V, Cr, Mn, Fe and Co) [16]. This is what motivated me to carry out a theoretical study on these two compounds, with (d3–d1) system. In the present work, two novel Cr-based double perovskites Ba2CrMoO6 and Ba2CrWO6 have been studied. The initial aim is to study the influence of 4d and 5d ions on the electronic, magnetic and optical properties. As known, molybdenum Mo (Z = 42) and tungsten W (Z = 74) elements are neighboring in
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different periods of transition-metals block in the period system. In addition, according to their similar electronic configurations, Mo: [Kr] 4d5 5s1 and W: [Xe] 4f14 5d4 6s2, Mo and W belong to the 4d and 5d transition-metals, respectively.
Table 1 Formula unit magnetic system, space group (S.G.), lattice parameter a, oxygen coordinates O (x) and volumes V, for the room temperature structure of Ba2CrMoO6 and Ba2CrWO6 compounds. The experimental values obtained (from Refs. [14,22]), and are cited between square brackets. Compound
2. Methods and computational details In this work, the electronic, magnetic and optical properties of Ba2CrMoO6 and Ba2CrWO6 double perovskites are calculated by using the local spin-density approximation (LSDA) as well as taking into account the on-site Coulomb repulsive interaction in (LSDA + U) approach within the density-functional theory (DFT). The band structure calculations are carried out using the selfconsistent (SCF) full potential linear muffin-tin orbital (FP-LMTO) method, as employed in the LMTART computational software [17]. In FP-LMTO method, the primitive cell is divided into nonoverlapped atomic spheres and interstitial regions. Within the atomic muffin-tin spheres (MTSs), the charge density q(r) is constructed from numerical solutions of radial Schrödinger’s equations (RSE) for each atom and the potential is represented via the spherical harmonic expansions. In the interstitial region, the q(r) is calculated from the Fourier transforms (FT) for the LMTOs [18] and the potential is expanded in the plane-wave (PLW) method [19]. Spin orbit coupling (SOC) is also included in the calculations. The Hartree potential is expanded in term of spherical harmonic of (lmax = 6), and an exchange–correlation potential of Von Barth– Hedin (VBH) mode is taken on. In consequence, the Ba (6s 5p 5d 4f), Cr (4s 4p 3d), Mo (5s 5p 4d), W (6s 6p 5d) and O (2s 2p) LMTOs are set as valence states, while Ba (5s), Cr (3p), Mo (4s 4p) and W (5s 5p) as semicore states. The LSDA + U scheme is considered a useful approach that provides quite satisfactory results, agreement with the experimental results, for the strong correlated systems [11,19]. The correlation parameters adopted in this study were the Coulomb U and exchange J parameters. In this work, the near maximum values were selected from the reasonable range of U (1–6.0 eV) and J (0.5–1.0 eV) [3,11,19–21]. The detailed U and J values are (U = 4.0 eV and J = 0.97 eV) for the strong localized electrons in Cr (3d) states and (U = 1.0 eV and J = 0.97 eV) for the weakly delocalized electrons in Mo/W (4d/5d) states. 3. Results and discussion
System 3+
5+
S.G.
a (Å)
x
V (Å3)
Ba2CrMoO6
Cr –Mo (3d3–4d1)
Fm3m
7.9984 [8.0098]a
0.2573 [0.2634]b
511.883 [513.884]c
Ba2CrWO6
Cr3+–W5+ (3d3–5d1)
Fm3m
7.9914 [8.0081]d
0.2575 [0.2641]e
510.673 [515.354]f
a,b,c d,e,f
Ref. [14]. Ref. [22].
Table 2 The average cation–anion interatomic distances (in Å) and angles h° at room temperature in Ba2CrMoO6 and Ba2CrWO6 with cubic (Fm3m) space-group, in comparison with experimental values (Exp.) taken from Refs. [14,23]. Compound
Ba2CrMoO6
Method
Theo.
Exp.
Ba2CrWO6 Theo.
Exp.
Cr (4a)–O (24e) 6 Mo/W (4b)–O (24e) 6 Ba (8c)–O (24e) 12 h hOACrAOi h hCrAOAMo/Wi
2.0585 1.9408 2.8291 90 180
2.026(3) 2.011(2) – 90 180
2.0552 1.9373 2.8265 90 180
2.112(1) 1.9353 – 90 180
Table 3 Magnetic moments of metal ions, total magnetic moments (in lB/f.u.), the number of states at EF, N(EF), the energy-gap (Eg) and the energy difference between FM and FiM configurations, DE (MeV). The spin pictures of the dynamic ions are Cr3+ (3d3; t32g" e0g" S = 3/2), Mo5+ (4d1; t12g" e0g" S = 1/2) or W5+ (5d1; t12g" e0g" S = 1/2). Compound
Ba2CrMoO6
Approach
LSDA
LSDA + U
LSDA
Ba2CrWO6 LSDA + U
Cr (3d) Mo (4d)/W (5d) Total N(EF) Eg (eV) DE(FM–FiM)
2.19 0.32 1.95 1.18; 0.45" 28.0
2.89 0.83 1.97 5.62; 1.81" 81.0
2.25 0.34 1.94 2.98; 0.46" 14.0
2.89 0.73 1.96 2.73; 1.82" 61.0
configuration, see Table 3, in a good harmony with the earlier DFT computational results for their relative strontium double perovskites Sr2CrMoO6 and Sr2CrWO6 [3,6,7,11].
3.1. Crystal structures and ground state 3.2. Electronic properties The room temperature crystal structures of Ba2CrMoO6 and Ba2CrWO6 are in face center cubic structure with space-group of (Fm3m) and lattice parameters of a = b = c = 7.9984 Å and 7.9914 Å, respectively. The lattice parameters are around the ideal value (a = b = c = 8.0 Å) and depend mainly on the ionic radii of Cr3+ (0.615 Å), Mo5+ (0.60 Å), or W5+ (0.62 Å), Ba2+ (1.49 Å) and O2 (1.32 Å). In the cubic formula unit, the atomic sites and positions are set as; two Ba (8c) atoms at (1=4 , 1=4 , 1=4 ) and (3=4 , 3=4 , 3=4 ), Cr (4a) at (½, ½, ½), Mo/W (4b) at (0, 0, 0) and six O (24e) atoms at (±x, 0, 0), (0, ±x, 0) and (0, 0, ±x), where (x = 0.2573) for Ba2CrMoO6 and (x = 0.2575) for Ba2CrWO6. The crystal structures data of Ba2CrMoO6 and Ba2CrWO6 are calculated and summarized in Tables 1 and 2, in comparison with the previous experimental data. The obtained structure information is in excellent agreement with the previous results [14,15,22]. In order to find the stable magnetic structure, the ground state, in Ba2CrMoO6 and Ba2CrWO6, the total energies of ferromagnetic (FM) and ferrimagnetic (FiM) configurations are calculated. It is found that the FiM configuration in both compounds of Ba2CrMoO6 and Ba2CrWO6 has lower total energy (ground state) than the FM
The band structures of the ground state of Ba2CrMoO6 and Ba2CrWO6 have been calculated using the LSDA and LSDA + U approaches. During the study, it has been found that FM configuration converges to FiM configuration for all the considered structure that is only FiM state is found for these compounds. Therefore, only the density of states (DOS) from cubic and FiM structure are presented. The total and partial densities of states (DOSs) are shown in Figs. 1–4, where, the upper and lower curves represent the spin-up (DOS") and spin-down (DOS;) per formula unit, respectively. The behavior of DOSs have been plotted between 10 eV to 10 eV where the main features occur, the dash lines stand for the Fermi energy (EF). From the total densities of states (TDOS) near the EF for Ba2CrMoO6 (Figs. 1(a) and 2(a)) and for Ba2CrWO6 (Figs. 3(a) and 4(a)), it is seen that at LSDA and LSDA + U calculations, two compounds exhibit half-metallic character. In LSDA scheme, the halfmetallicity can be observed with an energy-gap at the spin-up channel TDOS" of about (Eg = 0.45 eV) expands from 0.24 eV to 0.21 eV, Fig. 1(a), and (Eg = 0.47 eV) expands from 0.36 eV to
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Fig. 1. Total and partial Cr (3d), Mo (4d) and O (2p) densities of states for the formula unit of Ba2CrMoO6 versus the energy relative to the Fermi energy (E–EF), calculated using the LSDA.
0.11 eV, Fig. 3(a), for Ba2CrMoO6 and Ba2CrWO6, respectively. While, the conduction band at the spin-down channel TDOS; continues with bandwidth of (DW = 4.54 eV), extends from 1.15 eV to 3.39 eV and (DW = 1.48 eV), extends from 1.27 eV to 0.21 eV, for Ba2CrMoO6 and Ba2CrWO6, respectively. In a similar way, it seen that the LSDA + U scheme, also, gives HM behavior in two compounds of Ba2CrMoO6 and Ba2CrWO6, with an equal energygap at the spin-up channel TDOS" of about (Eg = 1.82 eV) expands from 1.60 eV to 0.22 eV. Whereas, the conduction band at the spin-down channel TDOS; with (DW = 1.37 eV) extends from 0.71 eV to 0.66 eV and (DW = 1.73 eV) extends from 1.08 eV to 0.65 eV in Ba2CrMoO6 and Ba2CrWO6, respectively. These results are attributed to the similarly electronic configuration of Mo5+ (4d1) and W5+ (5d1) ions, other than the itinerant electron in 5d level has more energy (E = 0.51 eV) than that in 4d level (E = 0.36 eV). Moreover, from the partial densities of states (PDOSs) in Figs. 1– 4 (panels b, c and d), it is clearly seen that the Mo (4d-t2g;) and W (5d-t2g;) states are responsible for the conductivity in Ba2CrMoO6 and Ba2CrWO6, respectively. In addition, the conduction bands are mainly composed from the Mo/W (4d/5d) states that hybridized with O (2p) and a little contribution of Cr (3d) states. Consequently, the interaction mechanism via the AFM long-range of Cr (3d)"–O (2p)–Mo/W (4d/5d); hybridization accountable for the HM behavior in Ba2CrMoO6 and Ba2CrWO6, resulting in a high spin-polarization (SP = 1.0) of the conduction electrons (charge carriers) in spin-down orientation. The SP ratio can be defined by means of the +DOS (spin-up) and DOS (spin-down) around the EF:
SP ¼
DOS" ðEF Þ DOS# ðEF Þ DOS" ðEF Þ þ DOS# ðEF Þ
ð1Þ
Therefore, Ba2CrMoO6 and Ba2CrWO6 are predicted to have a negative high spin-polarization (SP = 1.0) at high temperature above room temperature. The obtained perfect SP refers the HM electronic properties of those materials; this is extremely interest as promising candidates of a spin injector in spin-electronics devices.
3.3. Magnetic properties The super-exchange interaction is the most essential one for magnetism in transition-metal oxides which describes interaction between localized spin magnetic moments [7,9]. According to Goodenough–Kanamori rules (GKR), this mechanism gives rise to an antiferromagnetic (AFM) interaction between ions with same kind of d-orbital and ferromagnetic (FM) interaction between ions with more than half-filled d-orbital and less than half-filled d-orbital [24,25]. Consequently, the magnetic properties of transitionmetal double perovskites Ba2CrMoO6 and Ba2CrWO6 are governed by the 180° long-range (ddpp) super-exchange interaction and it depends mainly on the chemical states of Cr and Mo/W ions. Indeed, the result of high-spin ionic configuration in ordered (Cr3+–Mo5+) and (Cr3+–W5+) double perovskites leads to a saturation magnetization, of about (m = 2.0lB/f.u.), high SP and large ferromagnetic Curie temperature (TC); TC = 335 K for Ba2CrMoO6 [7] and TC = 145 K for Ba2CrWO6 [8]. The partial and total spin magnetic moments in two compounds have been calculated using the LSDA and LSDA + U approaches. Ba2CrMoO6 and Ba2CrWO6, with common (d3–d1) system, as the ionic-spin configurations are Cr3+ (3d3; t32g" e0g", S = 3/2), Mo5+ (4d1; t12g" e0g", S = 1/2) and W5+ (5d1; t12g" e0g", S = 1/2). The 3d and
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301
Fig. 2. Total and partial Cr (3d), Mo (4d) and O (2p) densities of states for the formula unit of Ba2CrMoO6 versus the energy relative to the Fermi energy (E–EF), calculated using the LSDA + U approach [U (Cr) = 4.0 eV, U (Mo) = 1.0 eV].
Fig. 3. Total and partial Cr (3d), W (5d) and O (2p) densities of states for the formula unit of Ba2CrWO6 versus the energy relative to the Fermi energy (E–EF), calculated using the LSDA.
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Fig. 4. Total and partial Cr (3d), W (5d) and O (2p) densities of states for the formula unit of Ba2CrWO6 versus the energy relative to the Fermi energy (E–EF), calculated using the LSDA + U approach [U (Cr) = 4.0 eV, U (W) = 1.0 eV].
4d/5d ions polarized in AFM with a theoretical spin magnetic moment of (m = 2.0lB/f.u.), consistent with the HM nature. The calculated total spin magnetic moments for Ba2CrMoO6 are (m = 1.95lB/f.u.) and (m = 1.97lB/f.u.), whereas, for Ba2CrWO6 are (m = 1.94lB/f.u.) and (m = 1.96lB/f.u.), from LSDA and LSDA + U approaches, respectively, see Table 3. Thus, FiM spins coupling with total magnetic moment of about (m = 2.0lB/f.u.) has been yielded in two compounds, in agreement with the experimental value (m = 2.0lB/f.u.) [15,23], and theoretical (m = 2.0953lB/f.u.) for Ba2CrMoO6 compound [26]. The results of partial spin magnetic moments of Cr, Mo and W, the total spin magnetic moments, the number of states at EF, [N (EF)], the energy-gaps (Eg) and the configuration energy differences (DEFM–FiM), are summarized in Table 3. The magnetic results of Ba2CrMoO6 and Ba2CrWO6 indicate ferrimagnetic (FiM) coupling between Cr3+ (3d3; t32g, S = 3/2) and Mo5+ (4d1; t12g, S = 1/2), and Cr3+ (3d3; t32g, S = 3/2) and W5+ (5d1; t12g, S = 1/ 2) ions, respectively, with HM electronic structures. The FiM coupling between Cr3+ and Mo5+, and Cr3+ and W5+ can be understood in terms of super-exchange interaction through the Cr (3d)–O (2p)–Mo (4d) and Cr (3d)–O (2p)–W (5d) bonding. As seen in Fig. 5, the spins of Cr3+–Mo5+, and Cr3+–W5+ appear to order in an anti-parallel arrangement by the super-exchange interaction and lead to FiM ordering in Ba2CrMoO6 and Ba2CrWO6. Moreover, the partial and total magnetic moment values notify that the FiM ground states of Ba2CrMoO6 and Ba2CrWO6 organize from the 180o long range ddpp super-exchange interaction [3d-t32g"–O (2pp)–4d/5d-t12g;], in conformity with the Pauli Exclusion Principle (PEP) and Goodenough–Kanamori rules (GKR). As a result, the mechanism of the FiM coupling in HM Ba2CrMoO6 and Ba2CrWO6 compounds is attributed chiefly to the strong AFM t32g–O–t12g of
Fig. 5. Schematic picture of the magnetic structure in cubic Ba2CrTO6, where T = Mo (4d), W (5d), describing the Cr3+ and Mo5+/W5+ ions in the ideal ordering of the magnetic structure with (d3–d1) system. Arrows represent the spin magnetic moments of Cr (up) and Mo/W (down), the total magnetization being in the direction of Cr ions. In AFM mechanism via the 180°-superexchange interaction Cr (t32g")–O (2pp)–Mo/W (t12g;), the 4d/5d; mobile-electron (blue) hopes between t2g states through 2p bridge states resulting in exchange ionic configuration of 3d(3"+Dq;)–4d/5d(1;Dq;), where Dq is the amount of transfer-charge and T refers to Mo (4d) or W (5d) ions. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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(Cr–O–Mo/W) hybridization and favored by virtual charge transfer due to the half-filled and near empty t2g orbitals.
303
3.4. Optical properties
The data of the real and imaginary parts of dielectric function allows calculating an important optical function such as the refractive index nðxÞ. The nðxÞ can be calculated by means of e1 ðxÞ and e2 ðxÞ according to the following expression [27]:
The optical properties of double perovskites can be expressed effectively by frequency-dependent dielectric function that takes the complex form:
2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 31=2 e21 ðxÞ þ e22 ðxÞ þ e1 ðxÞ 5 nðxÞ ¼ 4 2
eðxÞ ¼ e1 ðxÞ þ ie2 ðxÞ
ð2Þ
Direct calculation of the imaginary part e2 ðxÞ of the dielectric functions are available in LMTART code by evaluating the matrix elements of the electric dipole operator between the occupied and unoccupied electronic states in the valence band (VB), including the deeply located bands as well, and in the conduction band (CB), respectively. The real and imaginary parts of the dielectric function e1 ðxÞ and e2 ðxÞ; respectively, are then obtained from the Kramers–Kronig transform (KKT); where the e2 ðxÞ part describes dissipation of electromagnetic energy (EME) and e1 ðxÞ part describes the dispersion of EME. The integral relations between e1 ðxÞ and e2 ðxÞ parts can be derived using the KKT method, as follows:
e1 ðxÞ ¼ 1 þ e2 ðxÞ ¼
2
p
2
p
Z
1 0
Z 0
1
x0 e2 ðx0 Þ 0 dx x2 x02
xe1 ðx0 Þ 0 dx x2 x02
Fig. 6. The calculated imaginary part approaches.
ð3Þ
ð4Þ
ð5Þ
The calculated real and imaginary parts of the dielectric function for Ba2CrMo6 and Ba2CrWO6, by the LSDA and LSDA + U approaches, are shown in Fig. 6. Taking the square root of their real parts in the limit of infinite wavelengths, it can estimate the values of the refractive index that turns out to be 2.0. Unfortunately, no experimental or theoretical values of the refractive index were found in the literature for Ba2CrMo6 and Ba2CrWO6 to predict and evaluate the present calculated results. The imaginary part e2 ðxÞ of dielectric function eðxÞ determines the absorption properties of the material. In Fig. 6(a) and (b), there are five similar intensive peaks of the absorption, centered at energies of about (0.66, 1.36, 3.60, 4.93 and 7.14 eV) due to the VB–CB transitions. These five peaks appear since the CBs have sub-structures composed of five sub-bands; compare to DOS graphs in Figs. 1–4. The optical results are perfect consistent with the band structure calculation of Ba2CrMo6 and Ba2CrWO6. Consideration of the DOSs diagrams in Figs. 1–4, aid to assign the peaks positions and transitions, the transitions from the occupied to the unoccupied states, can be estimated by the band structures. It has been suggested that the
e2 ðxÞ and real part e1 ðxÞ of the dielectric function eðxÞ for Ba2CrMoO6 and Ba2CrWO6, calculated by means of the LSDA and LSDA + U
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absorption edges due to the charge-transfer transitions from the highest filled O (2p) orbitals to the lowest empty Cr (3d) and Mo/W (4d/5d) orbitals. Based on the band structures, the origin of five peaks (0.66, 1.36, 3.60, 4.93 and 7.14 eV) in Fig. 6(a) and (b) can be assigned as the O (2p) ? Mo/W (4d/5d-t2g), O (2p) ? Cr (3d-t2g), O (2p) ? Mo/W (4d/5d-eg), p–d transitions, respectively. These peaks have attributes of dipole-allowed p–d transitions have large strengths. The other two small peaks can be assigned as the Cr (3d-t2g) ? Mo/W (4d/5d-t2g) and Cr (3d-eg) ? Mo/W (4d/5d-eg), d–d transitions. The strength of these two peaks is very small; because the d–d transitions between the Cr (3d) and Mo/W (4d/5d) bands. The energy differences are 5.5 eV between peaks (at 1.36 eV) and (at 4.93 eV), and 1.02 eV between peaks (at 1.36 eV) and (at 3.6 eV) represent the crystal field splitting energies of Mo/W (4d/5d) and Cr (3d), respectively. The increase of the absorption of two compounds above 5.0 eV can probably be attributed to the chargetransfer transition between the O (2p) spin-up and spin-down bands at 1.0 eV (in LSDA–DOSs) and 2.0 eV (in LSDA + U–DOSs) below the EF into Cr/Mo–O or Cr/W–O bands the at +1.0 eV above EF. The absorption shoulder around 1.0 eV coincides roughly with the energy-gaps at EF in spin-up bands and are, therefore, probably caused by the transition from Cr (t2g") states below the EF into Mo/ W–O bands around +1.0 eV above EF. 4. Conclusion Using the FP-LMTO computational method, the electronic, magnetic and optical properties of ordered double perovskites Ba2CrMoO6 and Ba2CrWO6 were studied. The calculations were performed by the local spin density approximation (LSDA) as well as taking into account the on-site Coulomb repulsive interaction in (LSDA + U) approach within the density functional theory (DFT). At room temperature, Ba2CrMoO6 and Ba2CrWO6 crystallize in face space group) with almost center cubic crystal structure (Fm3m equal lattice parameters (a = 7.9984 Å) and (a = 7.9914 Å), respectively. The LSAD and LSDA + U calculation approaches depicted well the proper half-metallic ferrimagnetic (HM–FiM) groundstates with total spin magnetic moment of (m = 2.0lB/f.u.) in Ba2CrMoO6 and Ba2CrWO6. HM–FiM nature organizes from the ddpp super-exchange interaction (3d-t32g"–O-2pp–4d/5d-t12g;), in conformity with the Pauli Exclusion Principle and Goodenough– Kanamori rules. The real and imaginary parts of the dielectric function, and the optical constant were calculated by using the
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