Theoretical investigation of MnFe2O4

Journal of Alloys and Compounds 580 (2013) 401–406

Contents lists available at SciVerse ScienceDirect

Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jalcom

Theoretical investigation of MnFe2O4 A. Elfalaky, S. Soliman ⇑ Department of Physics, Faculty of Science, Zagazig University, Zagazig, Egypt

a r t i c l e

i n f o

Article history: Received 4 October 2012 Received in revised form 23 May 2013 Accepted 30 May 2013 Available online 7 June 2013 Keywords: Manganese ferrite DFT Electronic structure

a b s t r a c t Generalized gradient approximation (GGA) and GGA + U (U is the Hubbard parameter) were applied to scrutinize the electronic structure and magnetic properties of the spinel ferrite MnFe2O4. Compression between GGA and GGA + U calculations were carried out. Oxygen position (z), total unit crystal spin magnetic moment, and lattice parameters were optimized. The optimized structure parameters are in good agreement with the experimental values. The experimental results described by the GGA + U calculations are better than the normal frame work. A new method for calculating U has been introduced. The calculations show that Mn ferrite has cubic structure with ordered spins. The compound should experience insulating behaviors which is experimentally observed. Such insulating behavior and spin ordering increase the ability of using Mn ferrite in high frequency applications. The moments are predominantly due to the ionic model behaviors for this compound. High spin state is favorable for the two cations Mn and Fe. From the anomalous explanation of DOS it is observed that, a ferrimagnetic spin current between Fe-3d and Mn-3d through O-2p is yielded. The flat bands, which are the more atomic-like lead to a high density of states and magnetic instability of local moment character. The effect of decreasing octahedral point group from Oh to D4h, C4v and C3v were carried out. The octahedral deformation has not been found in Mn ferrite. Total picture study for Hubbard parameter U has been carried out where accurate value for U was determined by comparing the energy gap with that obtained experimentally. Correlated behavior was proved for Manganese ferrite. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction The material science (sensulato) literature is rich in papers dealing with the structural, physical and chemical properties of phases belonging to the spinel structure type, with general chemical for2+ mula A2þ O  A3þ = Mg, Mn, Fe, Co, Ni, Cu, Zn and 2 O3 where A A3+ = Al, V, Cr, Mn, Fe, Co [1]. In spinel structures the ionic distribution may be generally represented by (A1a Ba)A[AaB2a]B O4, a is called inversion parameter where 0 6 a 6 1 stand for normal and inverse cases, respectively. The unit cell of spinel contains 8A-ions, 16B-ions and 32 oxygen ions. In the normal spinel, where a = 0, the 8A ions occupy the tetrahedral sites and the 16B ions on octahedral ones. In an inverse spinel, for a = 1, 8 of the B ions occupy the tetrahedral sites, the other 8 of B and the 8A ions occupy the octahedral sites [2]. Recently, it has been found in various magnetic spinel systems that the geometrical frustration among the B sites in the spinel structure can give rise to pronounced effects due to spin–lattice coupling [3,4]. Spinel ferrite are usually ferromagnetic due to the strong A–A ferromagnetic and A–B antiferromagnetic coupling. Curie (Nel) temperatures of these materials are spanning a broad ⇑ Corresponding author. Tel.: +20 1121740618; fax: +20 506339959. E-mail address: [email protected] (S. Soliman). 0925-8388/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jallcom.2013.05.197

range, 3K < TN,C < 1100K, depending on the A cations [5]. When A is a magnetic ion, the total magnetic moment of the eight blocks that form the unit cell in AB2O4 is due to uncompensated magnetic moments of the A and/or B magnetic sublattices. For a completely inverted structure, such as CuFe2O4, the magnetic moment per unit cell is m = 8 lB, assuming that each Cu2+ ion contributes 1 lB , and Fe+3 contributes 3 lB where lB is the Bohr magneton [6]. Manganese ferrite is well-known microwave ferrite material with a spinel crystallographic structure (space group Fd3m), in which O2 form tetrahedral and regular or deformed octahedral local symmetries as depicted before [7]. Oxygen atoms in spinel have a cubic close-packed structure, viewed along a fourfold symmetry axis which perpendicular to a face of a cubic unit cell. MnFe2O4, as most ferrites, is insulating with small energy gap varies from 0.04 to 0.06 eV [8,9]. According to neutron diffraction measurements MnFe2O4 has an intersite disorder amounts to approximately 20% Fe on the nominally Mn site in normally prepared stoichiometric samples [10,11]. This disorder and its variability have caused some confusion in analysis of both magnetic properties and band structure gap. This work aims to improve the description of the electronic structure of MnFe2O4. Additionally, solve the problem of the half metallicity appeared in the band structure calculations for this

402

A. Elfalaky, S. Soliman / Journal of Alloys and Compounds 580 (2013) 401–406

compound. This will be considered by trying to predict the structure with the available techniques like the magnetic frustration and deceasing the structure symmetry.

Table 1 Site moment optimization E0 ¼ 16021:2035Ry; DE ¼ E  E0 .

for

Structure

2. Crystal structure and calculation details It was recently reported that the local spin density approximation (LSDA) and GGA schemes are not sufficient to describe the electronic structure correctly for transition metal oxides [12]. Therefore, the GGA + U method was applied here to account for on-site correlation at the transition metal sites. The GGA + U method accounts for an orbital dependence of the Coulomb-exchange interaction was used for the present calculations. The GGA + U scheme is used as implemented in Wien2k code [13]. The energy threshold between the core and the valance states was set 87.485 eV. The muffin tin-radii ðRMT Þ were chosen to ensure nearly touching spheres and minimizing the interstitial space, where RMTMn ¼ 1:95 Å RMTFe ¼ 2:05 Å and RMTO ¼ 1:73 Å. To have convergence for the basis of the wave function, the cut of parameter RMT  KMAX ¼ 7 was used for the number of plane waves. Where RMT is the smallest atomic sphere radius in the unit cell and KMAX is the largest K vector for the basis functions in the reciprocal space, i.e. the plane wave cut-off. The expansion of the partial wave functions was set to L = 10 inside the muffin tin atomic spheres while the charge density was Fourier expanded up to G = 12. In self-consistent calculations a grid dimensions 143 form 3000 kpoints were employed in the irreducible Brillouin zone results in 104 points. It is noted that, large mesh in the volume optimization is not needed because we search about the volume. The energy and charge convergence criterion were set to 104 Ry and 103 electron respectively. Most of the more complicated structures have free internal structural parameters, which can either be taken from experiment or optimization. In optimization processes, structure parameters of an ideal crystal can be obtained by deforming the unit cell either under hydrostatic pressure (i.e. varying the volume) or by applying strain. In this process the atoms may change their positions but can be brought to equilibrium by monitoring the forces acting on them. Structural optimization have been carried out to obtain the optimized oxygen parameter u as shown in Fig. 1. Spin alignment has been studied to obtain the most stable configuration for allowed spins. Different spin alignments were studied for Mn 3d and Fe 3d electrons as shown in Table 1. From Table 1, high spin state (HS), in which Mn and Fe have 3:41lB and 3:43lB respectively, is favorable than low spin state (LS). Calculations show that MnFe2O4 prefer to be in ferrimagnetic configuration. In other

#Þ½Fe4 #Þ½Fe4 #Þ½Fe4 #Þ½Fe4

"O8 "O8 "O8 "O8

HS LS for Fe LS for Mn LS for Mn and Fe

MnFe2O4

by

GGA,

DEðRyÞ

Fe (lB)

Mn (lB)

0.0000 0.1016 0.0987 0.2851

3.41 0.246 3.359 0.022

3.43 3.47 0.75 0.03

words, opposite spin alignment between octahedral (B-site) net spin direction and tetrahedral (A-site) net spin direction was favorable. This ferrimagnetic interaction with the existence of an inversion ratio was noticed experimentally [11,10]. Table 2 shows the dependence of both lattice parameter a and unit cell energy DE(Ry) (per formula unit shown in Table 2) on the spin alignment. Generally, it can be observed from Table 2 that the configuration ðMn # Fe #Þ½Mn " Fe3 "O8 is similar to ðFe2 #Þ½Mn2 " Fe2 "O8 from the magnetic exchange interaction A–A, B–B, and A–B point of view. Obviously, as shown in Table 2, for the three lowest energy configurations ðMn2 #Þ½Fe4 "O8 HS;ðMn # Fe #Þ½Mn " Fe3 "O8 and ðFe2 Þ # ½Mn2 " Fe2 "O8 ; the transformation of half Mn atoms from A-site to B-site or the inversion ratio below 50% reduces the lattice parameter from 8.46 Å to 8.43 Å. Additionally, more inversion above 50%, even totally inversion, has no more decrease on the lattice parameter. DOS and band structure calculations have been carried out with the configuration ðFe #Þ½Mn " Fe "O4 with lattice parameter 8.465 Å and Oxygen parameter u = 0.25515. 3. Electronic structure Previous studies utilizing GGA and LSDA, without Hubbard correlations, indicate that the normal spinel MnFe2O4 has half metallic behavior either for the ferrimagnetic or the antiferrimagnetic state [16]. DOS calculations were carried out by GGA for the lowest three energy configurations in Table 2. Half metallic behavior was observed for normal configuration ðMn #Þ½Fe2 "O4 . But for ðMn # Fe #Þ½Mn " Fe3 "O8 and ðFe #Þ½Mn " Fe "O4 metallic trends were noticed. But it is will known experimentally that Mn ferrite has semiconducting properties [9]. Therefore it is demanded investigate other techniques in order to approach the experimental behavior. 3.1. Octahedral distortion effect According to Crystal Field Theory (CFT), the interaction between a transition metal and ligands arises from the attraction between the positively charged metal cation and negative charge on the non-bonding electrons of the ligand. The theory was developed by considering energy changes of the five degenerate d-orbitals

-0.21

u = 0.25515

Unit cell energy (arb. unit)

ðMn2 ðMn2 ðMn2 ðMn2

ferro

-0.215

Table 2 The optimized lattice parameters for available cations to MnFe2O4 with high spin state (HS) except as noted, E0 ¼ 16021:3847Ry; DE ¼ E  E0 .

-0.22

-0.225

-0.23

-0.235 0.25

0.255

0.26

Oxygen parameter (u) Fig. 1. The energy dependence on the oxygen parameter (for S.G. 227 origin 1)

Structure

DERy

acal ðÅÞ

ðMn2 #Þ½Fe4 "O8 HS ðMn2 #Þ½Fe4 "O8 HS LS ðMn # Fe #Þ½Mn " Fe3 "O8 ðMn # Fe "Þ½Mn " Fe3 "O8 ðMn # Fe #Þ½Mn # Fe3 "O8 ðMn #ÞFe "Þ½Mn # Fe3 "O8 ðFe2 #Þ½Mn2 " Fe2 "O8 ðFe2 #Þ½Mn2 # Fe2 "O8 ðFe2 "Þ½Mn2 # Fe2 "O8

0.000 1.432 0.0291 0.1121 0.0337 0.0813 0.0107 0.0642 0.0548

8.465 8.376 8.43 8.456 8.44 8.48 8.43 8.456 8.44

aexp ðÅÞ In between 8.49 [14] and 8.5 [15]

403

A. Elfalaky, S. Soliman / Journal of Alloys and Compounds 580 (2013) 401–406

in case of being surrounded by an array of point charges consisting of the ligands. As a ligand approaches the metal ion, the ligand electrons will be closer to some of the d-orbitals and further apart from others. Accordingly, the closer electrons will have high potential energy while the further apart electrons will have lower energy. Consequently splitting will inevitably occurs to the d-orbitals. The crystal spinel structure has octahedral and tetrahedral sublattices. Firstly, octahedral has six ligands form an octahedron around the metal ion. In octahedral symmetry the d-orbitals split into two sets with an energy difference, Doct (the crystal-field splitting parameter). The first state of these two states is dxy ; dxz and dyz orbitals (will be in low energy) while the second is dz2 and dx2 y2 (have high energy). This can be attributed to the fact that the former set are further apart from the ligands than the latter and therefore experiences less repulsion. The three lower-energy orbitals are collectively referred to as t2g, and the two higher-energy orbitals as eg. These labels are based on the theory of molecular symmetry. Such splitting of t2g and eg due to the CFT has been confirmed in DOS calculations Fig. 2c. Secondly, tetrahedral in which four ligands are forming a tetrahedron around the metal ion. The tetrahedral crystal field splits the dorbitals into two groups with an energy difference of Dtet . The splitting is in such a way that the dz2 and dx2y2 represent eg with lower energy whereas dx2 ; dy2 and dy2 designated by t2g with higher energy. This can be ascribed to the symmetry configuration of ligand inside the A site. Noting that in octahedral the ligands are coinciding with axis while in tetrahedral case the ligands are in between axis. This splitting of tetrahedral has also been achieved in present calculations as in Fig. 2b. In the present calculations, there is an agreement between the CFT and the splitting of the included octahedral transition metal. Consequently the octahedral site still has ideal point group symmetry Oh . In other words the deviation of the oxygen parameter u form the ideal value 0.25–0.25515 has no significant effect on the point group symmetry of the octahedral. Comparing the values of u for Zn ferrite [7] and Mn ferrite, it can be said that octahedral point group deformation has threshold value for u between 0.25515 and 0.25862. In order to assure the non-deformation of the octahedral point group; further calculations were performed as shown in Table 3. According to Table 3 the lattice parameter a and energy E have no significant change with decreasing the octahedral point group Oh to C3v through D4h and C4v respectively. Consequently the octahedral deformation has not been detected in Mn ferrite which confirms the conclusion that Mn ferrite is not deformed system as predicted for Zn ferrite [7]. This may be due to the similarity of the electronic configuration of Mn+2 and Fe+3. 3.2. Magnetic frustration

(d) DOS for O and 2p states

O 2p

DOS (states/eV)

For some crystals the Brillouin zone in the anti ferromagnetic phase is exactly the same as the Brillouin zone in the paramagnetic phase. But for some other crystals the size and shape of the Brillouin zone will change dramatically as the crystal passes from the paramagnetic phase to the antiferromagnetic phase. Such changes in the Brillouin zone are due to the magnetic ordering and not as a result of any distortion of the lattice. Sometimes distortion of Brillouin zone might occur as a result of lowering in symmetry of the antiferromagnetic phase, which is always small [17]. Volume optimization was carried out to get the lowest energy for tetragonal phase. The lattice parameters were being a = b= 8.465 Å and c = a + 0.00008 Å with corresponding energy 16021.3784Ry per formula 2MnFe2O4. Such value of energy seems to be higher than that obtained for the case of cubic structure E0 ¼ 16021:3847Ry as in Table 2. Hence, band structure calculations for the tetragonal frustrated antiferromagnetic crystal shows magnetic moment of 10lB , where Mn and Fe have 3:34lB and 3:77lB respectively. In addition a half-metallic behav-

5

0

-5

-10

-10

0

Energy (eV) Fig. 2. DOS with GGA + Ucal calculations for ðMn2 #Þ½Fe4 "O4 where UMn = 1.769 eV and UFe = 6.531 for Mn and Fe respectively.

ior was also obtained for this tetragonal structure. Hence, in the present calculations the tetragonal unit cell is characterized by

404

A. Elfalaky, S. Soliman / Journal of Alloys and Compounds 580 (2013) 401–406

Table 3 Decreasing point group symmetry of octahedral site from Oh to C3v through D4h and C4v for MnFe2O4, E0 = 16021.3847Ry, DE = E-E0. Structure ðMn2 ðMn2 ðMn2 ðMn2

#Þ½Fe4 #Þ½Fe4 #Þ½Fe4 #Þ½Fe4

"O8 "O8 "O8 "O8

DEðRyÞ

aÅ

point group

0.0000 0.0026 0.0026 0.0027

8.46 8.46 8.46 8.46

Oh D4h C4v C3v

having higher energy compared with cubic unit cell besides halfmetallic behaviors. In accord such results confirm that Mn ferrite has cubic structure with ordered spins. 3.3. Effect of on-site Coulomb repulsion term U on the band gap with GGA + U The d-orbital of the transition metals have small radius, 0.4 Å, and inside the 4s levels and to some-extend core-like [18]. The small size of the d-orbital enhances interaction effects due to squeezing several electrons into a small space. Thus tends to decrease overlap with other orbitals and to reduces the kinetic energy. This is why the correlations are strong and adding the Hubbard parameter U becomes reasonable in some compounds. A new method to calculate U for correlated systems is introduced. The Hubbard parameter U is the Coulomb energy costs to place two electrons at the same site which is, for d metals, the energy costs for moving a d electron between two atoms both of which initially have nd electrons: nþ1

U ¼ Eðd

n1

Þ þ Eðd

n

Þ  2Eðd Þ

ð1Þ n

The procedure is as follows: The ground state energy E(d ) of a transition metal (d) impurity hybridizes with the anion valance band is firstly calculated. This involves n-electrons with taking into account the d–d Coulomb and exchange interactions. In this way the charge neutrality of the system is assumed without considering the actual number of electrons and holes in the ground state. Then the procedure was repeated for the states with one electron removed and obtain again the lowest energy ionized state E(dn  1) of (n  1) electron system and also for electron affinity states containing (n + 1) electrons of which lowest energy is E(dn + 1). Accordingly it is required to perform two sets of calculations in order to find the energies for nd  1=2 electron at the atom for which U has been calculated. For example in MnFe2O4, F2+ usually has 6d electrons and thus one need the energies for 6.5 and 5.5d electrons. The first calculation for ðnd þ 1=2Þ ¼ 6:5 electrons is performed with 3.5", 3;and the second calculation ðnd  1=2Þ ¼ 5:5 electrons is performed with 3.5", 2;. The same was carried out with Mn, where for Mn the calculations were performed with 4.5 and 5.5 electrons. In the present work the Coulomb potential exerted on one d electron for Fe and Mn was found to be 6.531 eV and 1.769 eV respectively in MnFe2O4 compound. Such calculated values of Hubbard parameters Ucal for Fe and Mn (6.531 eV and 1.769 eV) were applied for electronic structure investigations of MnFe2O4. Regarding the density of states (DOS) Fig. 2a–d for normal ferrimagnetic compound MnFe2O4 calculated by GGA + Ucal. In Fig. (2a–d) the density of states (DOS) of the normal ferrimagnetic compound MnFe2O4 applying GGA + Ucal is plotted. The upper panel displays the majority spin densities and the lower one is for the minority spin densities. Fig. 2b represents the DOS of Mn 3d which are just below Fermi level. Fig. 2c introduces DOS of Fe 3d which are accumulated at about 8 eV far away from Fermi level. On the other hand the width (W) of the 3d states of Mn is about 2 eV and that of Fe 3d is 1.7 eV. The Fe 3d electrons are strongly correlated since UFe/WFe  3.8 > 1, but for Mn 3d electrons UMn/ WMn is about 0.9 < 1. Therefore the large difference in the calcu-

lated Hubbard parameter value UMn and UFe can be attributed to three aspects; (i) the correlation of Fe 3d electrons is higher than that for Mn 3d electrons (ii) the difference of the 3d states energy position which is about 8 eV away from Fermi level for Fe and just down Fermi level for Mn as shown in Fig. 2, accordingly Fe 3d electrons are more localized and near to each other (WFe  1.7 eV) (iii) Fe-3d electrons occupying the crowded B-site while Mn-3d occupy the less electrons crowded A-site. Now, from this anomalous behavior for the Fe 3d energy states position, one can say that these Fe 3d electrons are not able to share the conduction mechanism (8 eV away from Fermi level). Conduction mechanism in spinel principally depends on the supper exchange between A and B sites i.e. between Mn and Fe cations through the anion (Oxygen atom). Accordingly, the Fe-3d electrons site for this cycle of conduction (Fe, O, and Mn) is missing. Hence the cycle is opened and the compound should experience insulating behavior which is experimentally observed. Such insulating behavior and spin ordering increase the ability of using Mn ferrite in high frequency applications. From Fig. 2a no contribution from minority states until about 2 eV away from Fermi level was observed. In Fig. 2c, the Fe 3d width WFe is about 1.7 eV which means that 3d states (eg and t2g) nearly have close energies. With this small difference in m-resolved 3d state energies, electrons will prefer the unpaired distribution than the paring distribution in which electrons should exerts the pairing energy. Such unpaired distribution is a good evidence for having high spin state (4:2 lB ) for Fe+3 in this compound. Regarding Fig. 2c and d overlapping states between Fe-3d bands and some states splitting from O-2p bands occurs in the down channel. From Fig. 2b and d, the Mn-3d majority states just below Fermi level were overlapping with some majority splitting states splitting from O-2p down Fermi level. From the above anomalous explanation of DOS figures, it can be observed that, a ferrimagnetic spin current between Fe-3d and Mn-3d through O-2p is yielded according to: Spin open loop 1) Fe–O spin exchange interaction in minority chaneel between 10 and 8 eV.

Fe  3d # þO  2p ! Fe  3d þ O  2p #

ð2Þ

2) O–O spin exchange interaction between minority and majority.

O  2p # ðminorityÞ þ O  2p ðmajorityÞ ! O  2p "

ð3Þ

3) O–Mn spin exchange interaction in majority chaneel between 2 and 0 eV.

O  2p " ; Mn  3d ! O  2p; Mn  3d "

ð4Þ

Spin closed loop Such p–d overlapping is the reason for ferrimagnetic supper exchange interaction between tetra and octahedral sites in ferrite. Besides, ferrites are characterized by high hardness which confirms such p–d overlapping for oxygen and TM is too much strong. The DOS of the minority states are absent at the Fermi level. Whereas the majority states have two peaks one for Mn-eg from about 1.8 to 1 eV and the second for Mn-t2g is just down Fermi level. This indicates that there is a small energy difference between Mn-t2g and Mn-eg which forces the electrons to be unpaired instead of exerting the paring energy. Therefore, these spin polarized electronic structure calculations indicate that, the moments are predominantly due to the ionic model behaviors. Such high spin state is favorable for the two cations Mn and Fe as proved in Table 1. For arrangement ðMn #ÞFe # ½Mn " Fe3 "O8 which shows low energy in Table 2, the GGA + Ucal calculations showed that it still

405

A. Elfalaky, S. Soliman / Journal of Alloys and Compounds 580 (2013) 401–406

1.5

spin up direction

Energy gap (eV)

1

0.5

0

-0.5

2

3

4

5

6

Hubbard parameter U (eV) Fig. 4. Energy gap variation with the Hubbard parameter for up channel with U = 0, 1, . . . 6 eV.

energy gap is demanded. The effect of Hubbard parameter (U) on the energy gap is shown in Fig. 4 for U = 0, 1 . . . 6 eV for upper channel which responsible for the half metallic behavior for this compound. As shown in Fig. 4, half metallic behavior was destroyed between U = 2 and 3 eV. A direct gap of about 0.1 eV for majority carriers is obtained at U = 3 eV. Accordingly for U = 3 eV the value of lattice parameter is reduced to about 8.565 Å without considering the effect of inversion ratio. The calculations show that, the down channel energy gap of about 2.4 eV for U = 0 has been changed to 3.6 eV for U = 3. A set of calculations for each value of UMn from 1, 1.5, . . ., 3 eV were performed for each value of UFe from 3, 2.5, . . ., 1 eV. It has been found a decrease in lattice constant is obtained while the half metallic trend destroyed at U P 3 eV for both Mn and Fe. Fig. 3. Band structure diagram for the two spin channel, minority and majority spin, through the high symmetry directions for MnFe2O4 with calculated Ucal.

has half metallic behavior. Accordingly, semiconducting or insulating trend should has inversion in between zero (normal MnFe2O4) and 50% inversion for ðMn # Fe #Þ½Mn " Fe3 "O8 . The band structure diagram for ðMn2 #Þ½Fe4 "O8 by GGA + Ucal is shown in Fig. 3. The valance bands in these TM oxides are always formed from 2p for ligand and 3d for TM. Insulating behavior was observed for minority with 4 eV direct gap as shown in Fig. 3a. Direct semiconductor band gap of about 0.4 eV was observed for majority in Fig. 3b. Such value (0.4 eV) of gap is greater than the experimental value 0.04–0.06 eV but still has a small value as usual in spinel ferrite. The Mn 3d states were touching Fermi level at C-point as shown in majority channel. Fig. 3c and d shows the flat band in [1 0 0] direction. Such flat band describe constant energy for electrons and independed on momentum. Accordingly, these electrons should have the same spin. Hence, this flat band is significantly affect the magnetic properties of the compound. The flat band was observed at 2.26 and at 5.15 eV for up and down channel respectively. To check the effect of Ucal on lattice parameter, volume optimization was carried out with GGA + Ucal. A lattice parameter 8.58 Å was obtained which is about 0.07 Å greater than the experimental value. But, when the Mn inversion effect on lattice parameter, from A-site to B-site, is taken into account a 0.035 Å decrease in lattice parameter was obtained with ðMn # Fe #Þ½Mn " Fe3 "O8 as shown in Table 2. Accordingly the lattice parameter for this compound should approach 8.545 Å for this calculated values for U. Since U has significant effect on the Mn ferrite to be closer to the experimental results. More study about the effect of U on the

4. Conclusion The GGA + U method accounts for an orbital dependence of the Coulomb-exchange interaction was used for electronic structure calculation of MnFe2O4. The U values for 3d elements were calculated with new technique in Wien2k code. The decrease of the octahedral point group symmetry from Oh to D4h to C4v to C3v does not give tangible effect on the energy of formula unit which give rise to the ideal geometry for the octahedral site inside the spinel structure of MnFe2O4. Calculations show no change occur in the Brillouin zone due to the magnetic ordering. Where a distortion of the lattice occur sometimes as a result of the lower symmetry of the ferrimagnetic phase. The calculated energy per formula unit for the tetragonal phase for MnFe2O4 is higher than that for cubic phase by 6.3 m Ry, which means that the tetragonal phase is not the ground state for MnFe2O4. The calculated Hubbard parameter Ucal values are able to explain the magnetic properties in MnFe2O4. A small band gap of about 0.4 eV was obtained with Ucal values. Lattice parameters of order of experimental values were obtained after considering the effect of inversion (moving of Mn from A-site to B-site). MnFe2O4 is well recognized as mixed ferrimagnetic correlated system and strong Fe-3d band localization far away from Fermi level This is the reason for the insulating trend for this compound. High spin state is favorable for this compound. This can be attributed to the less crystal field which produces less d-splitting and unpaired electronic behavior becomes more stable. References [1] C.M.B. Henderson, J.M. Charnock, D.A. Plant, J. Phys.: Condens. Matter. 19 (2007) 076214.

406 [2] [3] [4] [5] [6] [7] [8]

A. Elfalaky, S. Soliman / Journal of Alloys and Compounds 580 (2013) 401–406

K.E. Sickafus, J.M. Wills, J. Am. Ceram. Soc. 82 (1999) 327992. A. Szytula, L. Gondek, Acta Phys. Pol. A 106 (2002) 583. E. Claude, M. Komelj, Phys. Rev. B 76 (2007) 064409. I. Masahiko, U. Yutaka, J. Alloy Compd. 383 (2004) 85. X. Zuo, A. Yang, C. Vittoria, V.G. Harris, J. Appl. Phys. 99 (2006) 08M909. S. Soliman, A. Elfalaky, G.H. Fecher, C. Felser, Phys. Rev. B 83 (2011) 085205. Z. Szotek, W.M. Temmerman, D. Ködderitzsch, A. Svane, L. Petit, H. Winter, Phys. Rev. B PRB 74 (2006) 174431. [9] A.G. Flores, V. Raposo, L. Torres, J. Iñiguez, Phys. Rev. B 59 (1999) 9447. [10] J.M. Hastings, L.M. Corliss, Phys. Rev. 104 (1956) 328. [11] J. Sakurai, T. Shinjo, J. Phys. Soc. Jpn. 23 (1967) 1426.

[12] John P. Perdew, B. Kieron, E. Matthias, Phys. Rev. Lett. 77 (October) (1996). [13] P. Blaha, K. Schwarz, M.G.K. H, D. Kvasnicka, J. Luitz, WIEN2k, An Augmented Plane Wave+Local Orbitals Program for Calculating Crystal Properties Karlheinz Schwarz, Techn. Universitat Wien, wien Austria, 2001. [14] B. Antic, A. Kremenovic, A.S. Nikolic, M. Stoiljkovic, J. Phys. Chem. B 108 (2004) 12646–12651. [15] C. Wende, K.H. Olimov, H. Modrow, F.E. Wagner, H. Langbein, Mater. Res. Bull. 41 (2006) 1530–1542. [16] D.J. Singh, M. Gupta, R. Gupta, J. Appl. Phys. 91 (2002) 7370. [17] C.A.P., Rep. Prog. Phys. 32 (1969) 633. [18] cf Table 8.3 of J.C. Slater, Quantum Theory of Atomic Structure, vol. 1.a.