Electronic Energy Band Structure of the Double Perovskite Ba2MnWO6

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J. Phys. Chem. B 2008, 112, 6742–6746

Electronic Energy Band Structure of the Double Perovskite Ba2MnWO6 Yukari Fujioka,* Johannes Frantti, and Risto M. Nieminen Laboratory of Physics, Helsinki UniVersity of Technology, Post Office Box 1100, FI-02015 HUT, Espoo, Finland ReceiVed: NoVember 22, 2007; ReVised Manuscript ReceiVed: March 20, 2008

The electronic and magnetic structures of the double perovskite oxide Ba2MnWO6 (BMW) were determined by employing the density functional theory within the generalized gradient approximation (GGA) +U approach. BMW is considered a prototype double perovskite due to its high degree of B-site ordering and is a good case study for making a comparison between computations and experiments. By adjusting the U-parameter, the electronic energy band structure and magnetic properties, which were consistent with the experimental results, were obtained. These computations revealed that the valence bands are mainly formed from Mn 3d and O 2p states, while the conduction bands are derived from W 5d and O 2p states. The localized bands composed from Mn 3d states are located in the bandgap. The results imply that the formation of polarons in the conduction band initiate the resonance Raman modes observed as a series of equidistant peaks. 1. Introduction Electron correlation is a central factor to determine the electronic states of many solids. Because this effect is not treated explicitly within the one-electron approximation, the local density approximation for the exchange-correlation functional is too crude when, in particular, a magnetic compound is studied. On-site Coulomb repulsion is frequently taken into account in the calculations of correlated electron systems by the “+U method”, based on the Hubbard’s model. In principle, the U parameter can be estimated through the constrained density functional theory,1 while it is often adjusted to be in accordance with the experimental values, such as the band gap, cell volume, crystal structure, and magnetic ordering. In this work, particular attention is paid to the relationship between the U value and the magnetic properties of Ba2MnWO6 (BMW). The correlated electrons on Mn ions also play a crucial role in the dynamics of the lattice relaxation during the excitations and, therefore, the determination of an accurate U parameter and the correct energy band structure are important for the understanding of the light induced distortions. BMW is an insulator and is considered a prototype double perovskite with the lattice parameter a ) 8.1995 Å where Mn and W alternate in the B-site of the ABO3 perovskite structure.2 First signs of magnetic ordering appear at around 45 K, though a well-defined antiferromagnetic ordering takes place below 10 K.3 This ordering is of type II (Figure 1(b)), where the spins in planes parallel to the cubic (111) plane are reversed with respect to the neighboring planes below and above. A previous resonance-Raman scattering study of BMW and Sr2MnWO64,5 revealed two series of resonance modes induced at a specific photon energy. These modes had particular symmetries. This type of vibration, where the nuclear positions are readjusted in coupling electron excitations, customarily referred to as vibronic modes, are frequently observed in molecular materials and crystals containing color centers. Similar phenomena are rather rare in most crystalline materials because an electron decays back to its ground state before the nuclei readjust to their * To whom correspondence [email protected].

should

be

addressed.

E-mail:

Figure 1. Schematic figures of antiferromagnetic structures corresponding to (a) type-I and (b) type-II orderings in Ba2MnWO6. Black and gray octahedra indicate MnO6 with the up and the down spin directions. Gray circles display W ions. Ba ions are not shown.

equilibrium positions. It becomes possible, however, in the case that the electronic energy band is sufficiently narrow and the corresponding electron states are spatially localized. This type of electron-phonon coupling is linked to the idea of a polaron, which is an important charge transfer mechanism in an ionic insulator. This paper is organized as follows. We first give a short description of the exchange interactions, molecular field theory, and the determination of the Curie and Ne´el temperatures in section 2 and describe the computational details in section 3. In section 4, the magnetic structures and their stabilities are discussed together with the resonance phenomena. The conclusions are given in section 5. 2. Magnetic Ordering within Molecular Field Theory In a complex transition metal oxide, magnetic ordering is determined by exchange interactions between transition metal ions, often mediated by the intervening ions. Both direct and indirect interactions can be modeled by assuming an exchange constant Jij describing the interactions between spins Si and Sj by the equation

Uex ) -

∑ JijSi · Sj

(1)

i, j

where Uex is the interaction energy. It is usually sufficient to consider the interactions between the nearest neighbor and the

10.1021/jp711115v CCC: $40.75  2008 American Chemical Society Published on Web 05/09/2008

Double Perovskite Ba2MnWO6

J. Phys. Chem. B, Vol. 112, No. 22, 2008 6743

next nearest neighbor transition metal ions. It is worth to note for the forthcoming treatment that Uex does not explicitly depend on the configuration of the other spins. If Jij is negative (positive), it favors antiparallel (parallel) spins. Assuming that the total energy of the magnetically ordered crystal can be divided into nonmagnetic (E0) and magnetic energy parts, with E0 independent of the magnetic ordering, and the magnetic part described by eq 1, one can give quantitative estimations for the exchange constants by evaluating the total energy expressions per formula unit for different magnetic orderings. Within the mean field approximation it is also possible to estimate the transition temperatures between the paramagnetic and ordered states, such as the Curie (TC) and Ne´el temperature (TN). To describe the exchange constants, we recall that manganese and tungsten ions form the rocksalt structure. In terms of the cubic axes, manganese ions are at the cube corners and at the face centers. We now take account of the two types of interactions by introducing exchange constants J1, which describes the interaction between the Mn ion and its 12 nearest neighbor Mn ions, and J2, which is the exchange constant for Mn ion and its six next nearest Mn neighbors. The J1 exchange coefficient is “direct” in the sense that there is no intervening ion between the Mn-Mn ion pair along the cubic diagonal, whereas the J2 is a type Mn-O-W-O-Mn “superexchange” coefficient along the cubic axes. However, the given model treats J1 and J2 as model parameters and does not specify the exact nature of the exchange interaction. Therefore, it only gives the net exchange energy for a given spin pair. The total energies per formula unit for the ferromagnetic (EFM), antiferromagnetic type I (EAFI), and antiferromagnetic type II (EAFII) orders are 2

EF ) E0 - 12J1S - 6J2S

(2)

EAFI ) E0 + 4J1S2 - 6J2S2

(3)

EAFII ) E0 + 6J2S2

(4)

2

kBTη ng2µB2λS

) BS(η)

(9)

The condition for ferromagnetism (i.e., the condition for the existence of roots η * 0) to occur is

[ ] dBS dη

η)0

>

kBT

(10)

ng2µB2λS

In the case of η , 1, BS(η) ≈ 1/3(S + 1)η, and then the temperature range at which ferromagnetic phase is stable is given by the condition

∑ Jij

1 2 kBT < S(S + 1)ng2µB2λ ) S(S + 1) 3 3

(11)

j

The same ideas can be extended for the present antiferromagnets by dividing the magnetic structure to spin-up and spindown sublattices. The two sublattices are equivalent except for the direction of the spins. One can estimate the Ne´el temperature by setting a similar stability condition, as given above for the sublattices. The only difference is that λ should be replaced by its absolute value. Thus, one can derive the following expressions for the Curie (TC) and Ne´el temperatures (TN)

TC )

2S(S + 1) (-12J1 - 6J2) 3kB

TN(AFI) )

(12)

2S(S + 1) |4J1 - 6J2| 3kB

(13)

2S(S + 1) |6J2| 3kB

(14)

TN(AFII) ) 3. Computational Methods

(8)

The calculations, based on the density functional theory within the generalized gradient approximation (GGA), were carried out using the VASP (Vienna Ab-initio Simulation Package) code7 employing the projected-augmented-wave (PAW) method.8,9 The valence states included 5s5p6s, 3p4s3d, 5p6s5d, and 2s2p for Ba, Mn, W, and O, respectively. The on-site Coulomb repulsion between localized 3d electrons of Mn was taken into account through the simplified +U approach scheme of Dudarev10 in which U and the exchange parameter J are treated together as a single effective parameter, Ueff ) U - J. Hereafter, we label Ueff by U. In the case of tungsten, we did not consider the U parameter because the electronic state of W is hexavalent, the electronic configuration being 4f145d06s0. This means that the W d electrons are essentially delocalized as they hybridize with the surrounding oxygen 2p states, which was confirmed by the present computations. The energy cutoff value was 600 eV. The number of bands was set so that a reasonable account of the conduction bands was also given. To describe magnetic structure corresponding to the antiferromagnetic I (AFI) and II (AFII) orderings, simple cubic and rhombohedral unit cells were used (Figure 1), respectively, with appropriate constraints so that the nuclear symmetry (chemical unit cell) was equivalent to the space group Fm3jm. The k-point meshes were 9 × 9 × 9 for the paramagnetic and ferromagnetic phases, 6 × 6 × 6 for the AFI phase, and 8 × 8 × 8 for the AFII phase. The Brillouin zone integrations were carried out using the tetrahedron method.11,12

where BS(η) is the Brillouin function, η ) gµBHm/(kBT) and n is the number of magnetic atoms per unit volume. By reducing eq 7 to the form Hm ) λM, where λ ) 2/(ng2µ2B)ΣjJij, one obtains

4. Results and Discussion 4.1. Geometry and Magnetic Properties. Figure 2 shows the calculated total energy E as a function of cell volume for U

where S is the spin moment of the Mn ion. Then one can solve for the J1 and J2 as

1 (EAFI - EF) 16S2

(5)

1 (4EAFII - 3EAFI - EF) 48S2

(6)

J1 ) J2 )

The temperature at which magnetic ordering (ferromagnetic or antiferromagnetic) takes place can be estimated in terms of the mean field approximation. In what follows, we give a summary of the standard treatment of the Weiss molecular field approximation in the absence of external field. The molecular field Hm, in terms of the exchange interactions of the Mn ion at the site i, is given by the equation

Hm )

2 gµB

∑ JijSj

(7)

j

where g is the Lande´ g-value and µB is the Bohr magneton. Following a standard thermodynamical derivation (see, for example, ref 6), one can calculate the mean magnetic moment per unit volume and the magnetization M

M ) gnµBSBS(η)

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Fujioka et al.

Figure 2. Total energies for different magnetic orderings as a function of volume per formula unit V with (a) U ) 0 eV, (b) U ) 2 eV, (c) U ) 3 eV, and (d) U ) 6 eV, fitted by the third-order Birch’s equation. The energies of the paramagnetic structures were significantly higher and are not shown.

TABLE 1: Total Energies per Formula Unit and Calculated Lattice Parameters aa E/eV

a

a/Å

Ueff/eV

PM

FM

AFI

AFII

PM

FM

AFI

AFII

0 2 3 6

-78.840 -76.982 -76.111 -73.788

-80.022 -79.471 -79.234 -78.648

-80.046 -79.487 -79.246 -78.656

-80.055 -79.490 -79.249 -78.658

8.028 8.061 8.084 8.134

8.290 8.304 8.310 8.321

8.284 8.300 8.307 8.320

8.285 8.301 8.308 8.321

To allow a comparison, a refers to the conventional cubic lattice.

TABLE 2: Exchange Constants J1 and J2 (Eqs 5, 6), Calculated Curie TC and Ne´el Temperatures TN (Eqs 12–14)a J/meV

TC/K

TN/K

m/µB

Ueff/eV

J1

J2

FM

AFI

AFII

FM

AFI

AFII

0 2 3 6

-0.53 -0.34 -0.26 -0.18

-0.42 -0.20 -0.17 -0.10

318 207 162 118

14 6.3 0.8 5.1

90 47 40 26

4.4 4.5 4.5 4.7

4.3 4.5 4.5 4.7

4.3 4.5 4.5 4.7

a Estimated using the calculated spin moments S and magnetic moments m on the Mn ion.

) 0, 2, 3, and 6 eV. The data were fitted by the third order (n ) 3) Birch’s function E ) Σknν-2n ⁄ 3,13 where V is the cell volume. The equilibrium total energies and lattice parameters and exchange integrals, as well as the magnetic transition temperatures, and magnetic moments m of the Mn ions are summarized in Tables 1 and 2, respectively. The ordering temperatures (TC and TN) were calculated using eqs 12–14 with 2S(S + 1) ) m2/2. The lattice parameters and the magnetic moments increase with increasing U, but are not strongly dependent on magnetic orderings. The calculated lattice parameters for the magnetic orderings agreed within 1.5% with the experimental results, as it is known that GGA slightly overestimates volume.

Among the considered structures, AFII has the lowest energy. The total energy differences between the AFI and AFII phases, for the same U, were less than 10 meV. The magnetic moments for the Mn ions can be estimated assuming orbital quenching (i.e., orbital angular momentum L ) 0), so that the effective magnetic moment µeff ) 2µB√S(S + 1). This gives the value 5.92 µB for Mn2+ ion. This is somewhat larger than the value given in Table 2, which is due to the fact that the d electrons are not completely localized but hybridize with O 2p states, as is discussed below. For comparison, the effective magnetic moment for the Mn3+ ion is 4.90 µB. To see how reasonable the estimated exchange constants are, we compare them with the corresponding values reported for manganese monoxide MnO. MnO has the rocksalt structure and is a well-known antiferromagnetic compound with TN ) 117 K.14 The Mn ion is in a divalent (3d5) high spin state in both BMW and MnO and is coordinated by six oxygens, the Mn-O distance being 2.2 Å in both cases. The LAPW calculation within GGA15 of MnO gave the values J1 ) 1.62 meV and J2 ) 2.84 meV. We note there seems to be rather large variation among the values reported in literature for J1 and J2, depending on the computational method (see Table 2 in ref 15). Considering that the shortest (nearest neighbor) Mn-Mn distance in

Double Perovskite Ba2MnWO6

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Figure 3. Densities of states (DOS) of BMW for AFII structure for (a) U ) 0 eV, (b) U ) 2 eV, (c) U ) 3 eV, and (d) U ) 6 eV. Fermi level, indicated by the dotted line, was put to zero. Positive (negative) DOS values correspond to the spin up (spin down) cases. Total DOS, indicated by the black line, is the sum of the contributions from all ions.

BMW is 5.9 Å and 3.1 Å in MnO and that there are no intervening ions in either case, the J1 value of MnO should be larger, as is the case. The difference between J2 values is greater, which is consistent with the fact that in BMW there is a -O-W-O- chain intervening the next-nearest neighbor (NNN) Mn ions, whereas there is only an oxygen between the NNN Mn ions in MnO. 4.2. Electronic Energy Band Structure. Figure 3 shows the total density-of-states (DOS), including all ions, and the partial DOS due to the Mn 3d, W 5d, and O 2p states for the U ) 0, 2, 3, and 6 eV parameter values in the case of AFII ordering. For all U values, BMW was found to be insulating. For U ) 0, the two bands derived from the Mn 3d states are above the band originated from the O 2p states. With increasing U, Mn 3d states are pushed down toward O 2p states, which together form the valence band (VB). Mn 3d states overlap fully with O 2p states for U ) 6 eV and there are no states within the band gap. This contradicts the previous Raman spectroscopic study4,5 discussed below. The electronic structure for U ) 2 eV, having the localized narrow band in between the bandgap, agreed with the experimental results as well as the data shown in Tables 1 and 2. The conduction band (CB) minimum is mainly formed from the W 5d and O 2p states and is, therefore, insensitive to the U parameter value. The electronic band structure for U ) 2 eV is shown in Figure 4. The band due to the Mn 3d states of dx2-y2 and dz2 orbitals is very narrow, consistent with the notion that there is almost no orbital overlap due to the large Mn-Mn distance. Below this narrow band is a wider band, formed from the O 2p and Mn 3d states. This splitting is due to the different interactions between the d-orbitals with the neighboring oxygen 2p states. With the Mn in the center of an octahedron, the dx2-y2 and dz2 orbitals point toward neighboring oxygens, while the lobes of the dxy, dyz, and dzx orbitals point between the O 2p orbitals. This results

Figure 4. Band structure of BMW for AFII for U ) 2 eV. The labeling of the Brillouin zone symmetry points refers to the rhombohedral symmetry. The vertical axis indicates the relative energy with respect to the Fermi level, set at zero. The conduction band edge is mainly formed from the W 5d and O 2p states. The valence bands are derived from the O 2p and Mn 3d states, and the narrow band, due to the localized Mn 3d electrons, is located in between the valence and conduction bands.

in the higher energy of the eg states. We note that the site symmetry in the AFII magnetic lattice is lowered to D3d (3jm) from Oh (m3jm) and, therefore, the lower energy Mn 3d states (t2g in Oh symmetry) split into eg and a1g states by crystal field (energy difference was about 0.1 eV), while higher energy states (eg) remain 2-fold degenerate at the Γ point. However, the calculated octahedral distortion angle was almost zero and the resonance modes are observed above the magnetic phase transition temperature. Therefore, in the forthcoming discussion, we assume the Oh symmetry. Our previous resonance-Raman scattering experiments revealed that two different sets of resonant modes can be excited by tuning the laser beam wavelength to be either in the red

6746 J. Phys. Chem. B, Vol. 112, No. 22, 2008 range (resonant modes between 500 and 750 cm-1) or green/ blue range (resonant bands between 1400 and 1650 cm-1).4,5 The intensity of these modes increases with decreasing temperature, which is related to thermal vibrations. The larger the atomic displacements are, the shorter is the lifetime of the excitations. The electronic band structure, shown in Figure 4, suggests that the lower energy (red light) excitations could correspond to electron transfer from the localized Mn 3d band to the CB, whereas the higher energy excitation could correspond to electron transfer from the VB to the CB. This interpretation agrees with the fact that the intensity of the A1g normal mode was drastically lowered and T2g modes were split by red light excitations.5 This is consistent with the notion that the dx2-y2 and dz2 orbitals, which are the basic functions for the localized Mn 3d band, hybridize with the ligand oxygen. The lattice distortion is accompanied with an electron (a polaron is formed), which implies that the electron mobility is significantly lower than for the case of a free electron. This was confirmed by the fact that the mode energies were integer multiples of the same fundamental frequency. Because the CB minimum is formed from the W 5d and O 2p states, the question arises if local lattice distortion occurs in WO6 octahedra. The spatial extent of W 5d orbitals is larger than that of 3d electrons, though, they are still rather localized. Further studies are necessary to understand the mechanism of the lattice distortions related to the polaron formation. 5. Conclusions The magnetic structures and electronic energy bands of the double perovskite Ba2MnWO6 were determined. Exchange constants and magnetic transition temperatures were estimated for different U values. The calculated band structure with this U value revealed that the localized band, derived from Mn 3d states, was located between the valence and the conduction bands edges, formed from the O 2p and Mn 3d and O 2p and

Fujioka et al. W 5d, respectively. The results suggest that polaron formation initiates the resonance Raman modes observed as a series of equidistant peaks. Acknowledgment. We are grateful to K. Johnston for helpful discussions. This study has been supported by the Academy of Finland (Project Nos. 207071 and 207501). Y.F. also wishes to thank the Japan Society for the Promotion of Sciences and the Finnish Cultural Foundation for their financial supports. The Finnish IT Center for Science provided the computation platform. References and Notes (1) Martin, R. M. Electronic Structure; Cambridge University Press: Cambridge, 2004. (2) Fujioka, Y.; Frantti, J.; Eriksson, S.; Azad, A. K.; Kakihana, M. NFL Annual Report; Uppsala University: Nyko¨ping, 2003. A copy is available upon request. (3) Azad, A. K.; Ivanov, S.; Eriksson, S.-G.; Eriksen, J.; Rundlo¨f, H.; Mathieu, R.; Svedlindh, P. Mater. Res. Bull. 2001, 36, 2215. (4) Fujioka, Y.; Frantti, J.; Kakihana, M. J. Phys. Chem. B 2004, 108, 17012. (5) Fujioka, Y.; Frantti, J.; Kakihana, M. J. Phys. Chem. B 2006, 110, 777. (6) Reif, F. Fundamentals of Statistical and Thermal Physics; McGrawHill, Inc.: Singapore, 1985. (7) Kresse, G.; Furthmu¨ller, J. Comput. Mater. Sci. 1996, 6, 15. (8) Blo¨chl, P. E. Phys. ReV. 1994, B50, 17953. (9) Kresse, G.; Joubert, J. Phys. ReV. 1999, B59, 1758. (10) Dudarev, S. L.; Botton, G. A.; Savrasov, S. Y.; Humphreys, C. J.; Sutton, A. P. Phys. ReV. 1998, B57, 1505. (11) Jepsen, O.; Andersen, O. K. Solid State Commun. 1971, 9, 1763. (12) Blo¨chl, P. E.; Jepsen, O.; Andersen, O. K. Phys. ReV. 1994, B49, 16223. (13) Birch, F. J. Geophys. Res. 1978, 83, 1257. (14) Massidda, S.; Posternak, M.; Balderesch, A.; Resta, R. Phys. ReV. Lett. 1999, 82, 430. (15) Pask, J. E.; Singh, D. J.; Mazin, I. I.; Hellberg, C. S.; Kortus, J. Phys. ReV. 2001, B64, 024403.

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