⇦Chinese Physics Letters, 2017, Vol. 34, No. 1, Article code 016101 Optimal Bandgap of Double Perovskite La-Substituted Bi$_{2}$FeCrO$_{6}$ for Solar Cells: an ab initio GGA+$U$ Study B. Merabet1,2**, H. Alamri3, M. Djermouni2,4, A. Zaoui2, S. Kacimi2, A. Boukortt5, M. Bejar6 Affiliations 1Electrotechnics Department, Faculty of Technology, Mustapha Stambouli University, Mascara 29000, Algeria 2Computational Materials Physics Laboratory, UDL-SBA 22000, Algeria 3Physics Department-University College, Umm Al-Qura University, Makkah, Saudi Arabia 4Centre Universitaire Ahmed Zabana, Relizane 48000, Algeria 5Elaboration Characterization Physico-Mechanics of Materials and Metallurgical Laboratory ECP3M, Faculty of Sciences and Technology, Abdelhamid Ibn Badis University of Mostaganem, Mostaganem 27000, Algeria 6Laboratoire de Physique Appliquée, Faculté des Sciences, Université de Sfax, Sfax 3000, Tunisie Received 26 August 2016 **Corresponding author. Email: boualem19985@yahoo.fr Citation Text: Merabet B, Alamri H, Djermouni M, Zaoui A and Kacimi S et al 2017 Chin. Phys. Lett. 34 016101 Abstract The ab initio generalized gradient approximation (GGA)+$U$ study of multiferroic (La$_{0.5}$Bi$_{0.5}$)$_{2}$FeCrO$_{6}$ in pnma structure and ferri-magnetic order, including Hubbard corrections ($U=4.1$ eV) for transition metal/rare earth $d$-electrons with 20 atoms cell, shows optimum local magnetic moments of (Cr$^{3+}$, Fe$^{3+})$ equal to ($-$2.56, 4.14) μB and an ideal spin-down band gap of 1.54 eV. Tuned-band gap La-substituted double oxide perovskites BFCO should exhibit enhanced visible-light absorption and carrier mobility, thus could be convenient light absorbers and then efficient alternatives to wide-gap chalcopyrite absorber-based solar cells failing to achieve highest power conversion efficiencies, and even compete with their metal-organic halide perovskites counterparts. DOI:10.1088/0256-307X/34/1/016101 PACS:61.50.Ah, 72.40.+w, 73.22.Pr, 78.56.-a, 78.67.Wj © 2017 Chinese Physics Society Article Text Unlike traditional semiconductor-photovoltaic (SC-PV) devices, ferroelectrics (FEs) display spontaneous polarization (centrosymmetry breaking of crystallographic unit cells), built-in depolarization field to separate photoexcited electron-hole pairs.[1] Since FE response/other properties coupling is possible, FEs are applied in memory storage media,[2] field-effect transistors and random-access memories.[3] In hetero p–n and Schottky-junctions, the charge separation is realized by a built-in potential near interface and output photovoltage is limited by band gap energy ($E_{\rm g}$ of 1.34 eV as an ideal value[4]) of SC absorbers. However, most FEs are perovskite oxides with $E_{\rm g}$ above 3.3 eV. Wide $E_{\rm g}$ and small optical absorption coefficient are critical shortcomings of FE oxide perovskites (FEOPs) for PV applications. In terms of PV efficiency, poor visible light absorption and weak bulk conductivity negatively affecting the performance limit FE-PVs for such applications, though power conversion efficiencies (PCEs) that such based devices have are potentially beyond Shockley–Queisser limits[5] estimated in single junction devices. To reduce $E_{\rm g}$ of FEOPs (ABO$_{3}$) for absorbing more visible light, researchers have developed effective approaches such as cation (B) doping,[6] thus modifying B–O bonding could tune $E_{\rm g}$ in the range of 1.4–2.7 eV by tailoring the Fe/Cr cationic ordering and domain size.[7] However, Bi-substitution that reduces $E_{\rm g}$ of FEOPs also produces structural disorder, nonstoichiometric defects or oxygen vacancies into systems, usually lowering carrier mobility and weakening transport properties in cation-doped FE oxides, thus most cation-doped semiconducting FE oxide solid-solutions still have low output photocurrent densities and PCE.[8] In multiferroic Bi$_{2}$FeCrO$_{6}$ (BFCO) materials with A$_{2}$BB$'$O$_{6}$ double perovskite (DP) structure in which ferromagnetism and ferroelectricity coexist,[9] a PCE of $\sim$8.1% was achieved through $E_{\rm g}$ engineering.[7] In bismuth-based multiferroics where Bi$^{3+}$ ions occupy the perovskite A site, stereo-chemically active 6$s^{2}$ lone pair induces in magnetic oxides a symmetry-lowering structural distortion that can lead to ferroelectricity,[10] due to the fact that non-centrosymmetric crystal structure Bi-multiferroics exhibit strong inversion symmetry and then are promising for PVs via spontaneous electric polarizations that promote required separation of photo-excited carriers and permit photo-voltage that can exceed $E_{\rm g}$ of such materials.[11] Also, large dielectric constant and polarization magnitude are innate advantages of such FEOPs for separating photo-excited charge carriers.[4] To enhance dielectric response in Bi-multiferroics, Bi could be substituted by lanthanum,[12] as the spin cycloid characterizing bulk Bi-based materials disappears upon La doping.[13] Since the photovoltage achievable from simple SC devices is mainly limited by $E_{\rm g}$ of the light absorber, we propose here to focus on (La$_{x}$Bi$_{1-x}$)$_{2}$FeCrO$_{6}$ (LBFCO) DPs, with La-concentration of $x=0.5$ as a representative example, trying to achieve possible conspicuous success in $E_{\rm g}$ engineering. Vijayanandhini et al.[14] investigated the orthorhombic ($Pnma$ space group) perovskite LBFCO, reporting susceptibility versus temperature measurement, a ferrimagnetic (FiM) or weak ferromagnetic (WFM) order for these systems, and WFM paramagnetic transition above 400 K. On the other hand, it is worth noting that magnetic properties of LBFCO are influenced by the strain effect and crystallographic orientation that such effect induces, thus (Bi$_{0.9}$La$_{0.1}$)$_{2}$FeCrO$_{6}$ epitaxial films exhibit large coherent compressive strains for inducing a substantial magnetic moment, and strain engineering of oxide perovskites may also be somehow employed for $E_{\rm g}$ tuning.[9] Providing that via a control of the cation ratio and distribution in (Bi, La)$_{2}$FeCrO$_{6}$ DPs $E_{\rm g}$ of these BLFCOs can be tuned to enhance the PCE, our density functional theory (DFT) study within the generalized gradient approximation (GGA) (+$U$: Hubbard parameter), mainly focuses on electronic properties of (La$_{0.5}$Bi$_{0.5}$)$_{2}$FeCrO$_{6}$ in the $pnma$ structure/FiM phase and La substituting effects on PCE via $E_{\rm g}$ tuning, which is crucial for optimizing performance characteristics of LBFCOs applied for solar cells.[7] Figure 1 shows our structure with the lattice parameters $a=55393$ Å, $b=78171$ Å and $c=55246$ Å.[14]
cpl-34-1-016101-fig1.png
Fig. 1. The $pnma$ (#62) crystal structure of DP La-substituted BFCO (50% La-substitution into the Bi site): an ideal ordered DP structure La$_{2}$BB$'$O$_{6}$ (B=Fe, B$'$=Cr). For the FiM state, the spin state of (B, B$'$) is (+ or $\uparrow$, $-$ or $\downarrow $).
DFT band structure calculation[15] for bulk crystalline (La$_{0.5}$Bi$_{0.5}$)$_{2}$FeCrO$_{6}$ DP is carried out within Wien2K, implementing the FP-LAPW[16] and GGA-PBE approximations,[17] full relativistic effects are calculated with the Dirac equations for core states, and the scalar relativistic approximation is used for the other states.[18] The spin-orbit coupling is ignored because it only has a slight effect. The on-site Coulomb interaction is included in the GGA+$U$ approach[19] with $U=4.1$ eV for both Fe and Cr 3$d$ electrons[20] (exchange and correlation potential to treat such electrons). For simplicity, the same $U$ was used on both the Fe and Cr sites in La-substituted Bi$_{2}$FeCrO$_{6}$.[21] Our DFT calculations are performed by using the highly accurate full-potential projector augmented wave method,[22] whose potentials are used to describe the electron–ion interaction with 15, 14, 12 and 6 valence electrons for Bi(5$d^{10}$6$s^{2}$6$p^{3}$), Fe(3$p^{6}$3$d^{6}$4$s^{2}$), Cr(3$p^{6}$3$d^{5}$4$s^{1}$) and O(2$s^{2}$$p^{4}$), respectively, representing ionic cores. Brillouin zone (BZ) integrations are performed with the tetrahedron method in a $7\times11\times7$ Monkhorst–Pack $k$ point mesh centered at high symmetry point ${\it \Gamma}$, inside the BZ.[23] We adopted an orthorhombic $Pnma$ (#62) structure with 20 atoms in a primitive unit cell and an FiM phase[14] with Fe spin orienting up and Cr down. The muffin tin (MT) radii of La/Bi, Fe, Cr, O equal to 2.6, 1.95, 2.0, 1.6 Bohr, respectively. The parameter $R_{\rm MT}\times k_{\rm MAX}$ and separation energy between valence and core states are respectively set to 7 and $-$7 Ry. Thus ordered (La$_{0.5}$Bi$_{0.5}$)$_{2}$FeCrO$_{6}$ with Fe$^{3+}$/Cr$^{+3}$ lattice planes has a spin-down $E_{\rm g}$ of 1.54 eV near the ideal Shockley–Queissier limit,[5] known equal to 1.34 eV. The La substitution in LBFCO-DP could be a powerful tool to engineer the band structure and then impact optical properties. Motivated by the ordered La$_{2}$FeCrO$_{6}$ (LFCO) as an artificial superlattice of LaFeO$_{3}$/LaCrO$_{3}$ synthesized by Ueda et al.,[24] predicted to be FiM from GGA Bandstructure calculations,[25] Baettig et al.[20] proposed the analogous Bibased compound Bi$_{2}$FeCrO$_{6}$ (BFCO) exhibiting an FiM order,[26] although the magnetic coupling between Fe$^{3+}$ and Cr$^{3+}$ was predicted in terms of super-exchange to be either FiM or FM.[27] To further improve FM moments and significantly reduce leakage currents in bulk BFCO materials, Bi has been substituted in highly distorted BFCODP[28] structure by the rare earth metal La,[13] since La$^{3+}$ prefers to substitute for Bi$^{3+}$ due to valence and ion radius similarities, though with La-substitution many phase transitions may occur in Bi-based FEOPs as La content ($x$) increases,[29] like $R3c$ to $pnma$ occurring in the range 0.1–0.3 of $x$ for Bi$_{1-x}$La$_{x}$FeO$_{3}$.[30] Since the magnetic order of (La$_{x}$Bi$_{1-x})_{2}$FeCrO$_{6}$ (for $x=0$), with Fe spin orienting up and Cr spin down, turns out to be FiM[31-33] (as proved by experiment for (La$_{1-x/2}$Bi$_{x/2}$)(Fe$_{0.5}$Cr$_{0.5}$)O$_{3}$ samples[14]), and even the ground state of La$_{2}$FeCrO$_{6}$ ($x=1$) from band structure calculations might be FiM,[25] we adopted it throughout our calculations. So far, the ground state magnetic ordering of (La$_{0.5}$Bi$_{0.5}$)$_{2}$FeCrO$_{6}$ is FiM exhibiting optimum local magnetic moments of (Cr$^{3+}$, Fe$^{3+}$)=($-$2.56, 4.14) μB and a spin-down gap of 1.54 eV, compared with those of the FM order-based structure of (2.63, 4.14) μB and 2.01 eV, and those of the FiM pnma symmetry with 25% La-substitution into the Bi site of ($-$2.59, 4.09) μB and 1.53 eV. Though the lowest energy structure of (La$_{0.5}$Bi$_{0.5}$)$_{2}$FeCrO$_{6}$ reveals the structure to be $R3$ (like BFCO[20]), within the FiM $pnma$ symmetry an SC character has been obtained contrary to the metallic behavior of $R3$ structure may be related to Goldschmidt's tolerance factor.[33] For solar cell applications excellent photoelectronic properties for LBFCO such as optimal gap ($\approx$1.5 eV) could be predicted. Like BFCO, (La$_{0.5}$Bi$_{0.5}$)$_{2}$FeCrO$_{6}$ should also be an example of such a $d^{5}$–$d^{3}$ orbital combination, and the local magnetic moments are found to be (Cr$^{3+}$, Fe$^{3+}$)=($-$2.56, 4.14) μB, instead of ($-$2.18, 3.70) μB for BFCO,[33] as the magnetic order in such ordered DP structure could be controlled by the spin state of Fe$^{3+}$ and Cr$^{3+}$. Here it is worth mentioning that (La$_{0.5}$Bi$_{0.5}$)$_{2}$FeCrO$_{6}$ FiM could exhibit magnetic moments depending on the degree of Fe/Cr cation ordering in the films, an FiM SC phase should be more suitable for many applications mainly due to considering core electrons in our full-potential calculations, and substituting La at Bi sites could be related to a complicated magnetization. If ion spin states could be changed, applying GGA+$U$ will be crucial for a meaningful result to be obtained.
cpl-34-1-016101-fig2.png
Fig. 2. Spin-resolved energy bands of DP La-substituted BFCO (with 50% of Ba substituted by La), calculated with GGA+$U$ (4.1 eV). The left-side panel is for up-spin, and the right-side for down-spin.
Since $E_{\rm g}$ of most solid oxide FEs is of at least 3 eV, absorbing mainly in the ultraviolet region (8% of the solar spectrum only), new materials, such as Bi based FEOPs with a decreased $E_{\rm g}$ and large polarization would be highly desirable. Our DFT study with GGA+$U$ shows that the ground-state FiM SC phase of DP La-substituted BFCO, with La substituted for Ba till 50%, has an optimal spin-down gap of 1.54 eV near the ideal value of 1.34 eV. Through a corresponding spin-dependent band structure shown in Fig. 2, one can clearly notice the FiM phase of DP La substituting BFCO, whose electronic structure could be studied. In the up-spin channel, the bands originate mainly from the O 2$p$ and Cr 3$dt_{\rm 2g}$ states, the empty bands are almost Fe 3$dt_{\rm 2g}$ states, and the upper states are from Fe 3$de_{\rm g}$, Cr 3$de _{\rm g}$, and Bi 6$p$. In the down-spin channel, filled and empty bands are from the O 2$p$ states and Fe 3$d$ ones, and Cr 3$d$ and Bi 6$p$ states, respectively. Contrary to the up-spin channel, the Fe 3$d$ states hybridize with O 2$p$ and those of Cr 3$d$ interact strongly with the Bi 6$p$, and the formed gap should be between the filled Cr 3$d$ and the empty Fe 3$d$ bands, which mainly originate from a crystal field splitting due to a deformation of the O octahedrons plus a spin exchange splitting of the 3$d$ electrons. The principal spin interaction that an O atom between its nearest Fe and Cr atoms intermediates is as spins contribute opposite magnetic moments forming the ferrimagnetism. Such interaction becomes sensibly weaker from a nearest to next-nearest neighboring spin pairs.
cpl-34-1-016101-fig3.png
Fig. 3. Total spin-resolved DOS of DP La-substituted BFCO (with 50% of Ba substituted by La), calculated with GGA+$U$ (equal to 4.1 eV). The upper part in each panel is the majority-spin DOS result, and lower the minority one. The Fermi level ($E_{\rm F}$) has been taken as the energy zero.
cpl-34-1-016101-fig4.png
Fig. 4. Partial DOS projected in the atomic spheres of Fe $d$ and Cr $d$ of DP La-substituted BFCO (with 50% of Ba substituted by La).
In Figs. 3 and 4, we present the spin-resolved total density of states (DOS) and partial DOS in GGA calculations projected in the $d$ states of Fe and Cr between $-$6 and 6 eV. La-substituted BFCO should undergo an octahedra tilting to optimize the local environment around the La/Bi site cation,[34] due to the crucial valance and size differences between the Fe$^{3+}$ and Cr$^{3+}$ cations for controlling the physical properties in such DPs,[35] octahedral tilts should change an overlap between Fe/Cr $d$ states and O 2$p$ states (Fig. 5) and significantly affect the FEOPs properties.[36] The $d$ states of Fe/Cr ion tend to split into three-fold lower in energy degenerate 3$dt _{\rm 2g}$ states and two-fold degenerate 3$de _{\rm g}$ states, lying higher in energy pointing respectively away from/directly towards O atoms, causing the Fe/Cr–O bonds to distort. These so-called Jahn–Teller distortions have a significant impact on the electronic and magnetic properties of such DPs.[37] Our DP has its $d$-states close to the Fermi level ($E_{\rm F}$), tending to form due to their small spatial narrow bands overlap with the O 2$p$ states. However, the lone pair orbital of Bi$^{3+}$(6$s^{2}$), namely outer orbital/energy level of Bi$^{3+}$ has a 5$d^{10}$6$s^{2}$ electron configuration, in which a lone pair exists. The stereochemical activity is responsible for a ferroelectric distortion that still occurs on substitution of Bi$^{3+}$ with La$^{3+}$, and is more likely caused by the diminishing lone pair activity of Bi.[38] If we overlap Figs. 4 and 5, one can see that the O 2$p$ orbital extending below $E_{\rm F}$ from $-$6 to $-$0.25 eV hybrids with the Fe 3$dt _{\rm 2g}$ orbital mainly in the up-spin channel, and Cr 3$dt _{\rm 2g}$ in the opposite one stretching from $-$6 to $-$0.75 eV also hybrids with O 2$p$. A bangap appears at the down-spin channel centered by $E_{\rm F}$, above which Fe 3$dt _{\rm 2g}$ and Cr 3$de _{\rm g}$ orbitals act almost alternatively: 1–1.8 eV for Fe (down-channel) and 3–6 eV for Cr (up-channel).
cpl-34-1-016101-fig5.png
Fig. 5. Partial DOS projected in the atomic spheres of O $p$ of DP La-substituted BFCO (with 50% of Ba substituted by La).
Figure 6 focuses on a typical polarized lone pair electron character due to La-substitution into the Bi site of 6$s^{2}$ electrons that strongly affect the crystal structure and magnetic order of our (La$_{0.5}$Bi$_{0.5}$)$_{2}$FeCrO$_{6}$ DP. Though Bi$^{3+}$ and La$^{3+}$ have close ionic radii, a prominent contribution is introduced by the highly polarizable Bi$^{3+}$ 6$s^{2}$ lone pair due (whose presence may play here an important role in lowing the symmetry) to its anisotropic local lattice distortion. In fact, as in Pb-based perovskites,[6] substituting the Bi site of BFCO-DP with elements (like La) whose bonds with oxygen are more covalent and less should reduce the band gap, and covalent/ionic characters are strongly related to the electronegativity difference between anion- and (Fe,Cr)-site with O. Figure 7 displays the charge density distribution of our DP La-substituted BFCO in the (110) plan showing that almost all atoms are covalently bonded.
cpl-34-1-016101-fig6.png
Fig. 6. Partial DOS projected in the atomic spheres of La $d$ and Bi $s$ of DP La-substituted BFCO (with 50% of Ba substituted by La).
cpl-34-1-016101-fig7.png
Fig. 7. Charge density of DP La-substituted BFCO, (with 50% of Ba substituted by La), at the (110) plane containing the five species.
In summary, we have studied the electronic structure and magnetic stability of La-substituted BFCO, La substituted for Ba till 50%, in the $pnma$ phase and the FiM order. Our GGA+$U$ results suggest La-substitution as a powerful tool to band gap engineer, thus impacting optical properties. La$^{3+}$ tending to prefer O environments (contrasting with large off-centering and anisotropic O environments usual for Bi$^{3+}$) is explained in terms of a lone-pair mechanism. LBFCO, an SC with an optimal spin-down $E_{\rm g}$ of 1.54 eV and a magnetic moment $\sim$4 μB, should be a possible candidate for FE-PVs, while it remains to be verified if structure distortions in it are so large to originate a strong nearest neighbor super-exchange coupling in LBFCO containing 3$d$ ions. References Ferroelectric materials for solar energy conversion: photoferroics revisitedArising applications of ferroelectric materials in photovoltaic devicesRegular arrays of highly ordered ferroelectric polymer nanostructures for non-volatile low-voltage memoriesApplications of Modern FerroelectricsElectronics: Inside story of ferroelectric memoriesMultilevel Data Storage Memory Using Deterministic Polarization ControlPerovskites for photovoltaics: a combined review of organic–inorganic halide perovskites and ferroelectric oxide perovskitesDetailed Balance Limit of Efficiency of p-n Junction Solar CellsNew Highly Polar Semiconductor Ferroelectrics through d 8 Cation-O Vacancy Substitution into PbTiO 3 : A Theoretical StudyComputational design of low-band-gap double perovskitesBandgap tuning of multiferroic oxide solar cellsRuddlesden–Popper perovskite sulfides A3B2S7: A new family of ferroelectric photovoltaic materials for the visible spectrumControlling magnetism of multiferroic (Bi0.9La0.1)2FeCrO6 thin films by epitaxial and crystallographic orientation strainVisualizing the Role of Bi 6s “Lone Pairs” in the Off-Center Distortion in Ferromagnetic BiMnO 3Tip-enhanced photovoltaic effects in bismuth ferriteFirst-principles investigation of the structural phases and enhanced response properties of the BiFeO 3 -LaFeO 3 multiferroic solid solutionComposition-induced transition of spin-modulated structure into a uniform antiferromagnetic state in a Bi1−x LaxFeO3 system studied using 57Fe NMRZero magnetization in a disordered (La 1− x /2 Bi x /2 )(Fe 0.5 Cr 0.5 )O 3 uncompensated weak ferromagnetSpin glass to cluster glass transition in geometrically frustrated CaBaFe 4 − x Li x O 7 ferrimagnetsInhomogeneous Electron GasSelf-Consistent Equations Including Exchange and Correlation EffectsGeneralized Gradient Approximation Made SimpleDensity-functional theory and strong interactions: Orbital ordering in Mott-Hubbard insulatorsFirst principles study of the multiferroics Bi Fe O 3 , Bi 2 Fe Cr O 6 , and Bi Cr O 3 : Structure, polarization, and magnetic ordering temperatureElectron-energy-loss spectra and the structural stability of nickel oxide: An LSDA+U studyImproved tetrahedron method for Brillouin-zone integrationsProjector augmented-wave methodFerromagnetism in LaFeO3-LaCrO3 SuperlatticesElectronic and magnetic properties of La 2 FeCrO 6 : Superexchange interaction for a d 5 − d 3 systemEpitaxial thin films of the multiferroic double perovskite Bi[sub 2]FeCrO[sub 6] grown on (100)-oriented SrTiO[sub 3] substrates: Growth, characterization, and optimizationSuperexchange interaction and symmetry properties of electron orbitalsHigh pressure bulk synthesis and characterization of the predicted multiferroic Bi(Fe[sub 1∕2]Cr[sub 1∕2])O[sub 3]Lanthanum doped BiFeO3 powders: Syntheses and characterizationStructure, ferroelectric properties, and magnetic properties of the La-doped bismuth ferriteOn the Thermodynamic Stability of BiFeO 3On the Thermodynamic Stability of BiFeO 3Colossal nonlinear optical magnetoelectric effects in multiferroic Bi[sub 2]FeCrO[sub 6]Electronic structure and magnetic and optical properties of double perovskite Bi 2 FeCrO 6 from first-principles investigationOctahedral Tilting in Perovskites. I. Geometrical ConsiderationsDetermination of ionic valency pairs via lattice constants in ordered perovskites (ALa)(Mn2+Mo5+)O6 (A = Ba, Sr, Ca) with applications to (ALa)(Fe3+Mo4+)O6, Ba2(Bi3+Bi5+)O6 and Ba2(Bi3+Sb5+)O6Control of octahedral connectivity in perovskite oxide heterostructures: An emerging route to multifunctional materials discoveryJAHN-TELLER PHENOMENA IN SOLIDSPhysics and Applications of Bismuth Ferrite
[1] Butler K T, Frost J M and Walsh A 2015 Energy Environ. Sci. 8 838
Yuan Y, Xiao Z, Yang B and Huang J 2014 J. Mater. Chem. A 2 6027
[2] Hu Z, Tian M, Nysten B and Jonas A M 2009 Nat. Mater. 8 62
Scott J F 2007 Science 315 954
[3] Garcia V and Bibes M 2012 Nature 483 279
Lee D, Yang S M, Kim T H, Jeon B C, Kim Y S, Yoon J G, Lee H N, Baek S H, Eom C B and Noh T W 2012 Adv. Mater. 24 402
[4] Fan Z, Sun K and Wang J 2015 J. Mater. Chem. A 3 18809
[5] Shockley W 1961 J. Appl. Phys. 32 510
[6] Bennett J W, Grinberg I and Rappe A M 2008 J. Am. Chem. Soc. 130 17409
Berger R F and Neaton J B 2012 Phys. Rev. B 86 165211
[7] Nechache R, Harnagea C, Li S, Cardenas L, Huang W, Chakrabartty J and Rosei F 2014 Nat. Photon. 9 61
[8] Wang H, Gou G Y and Li J 2016 Nano Energy 22 507
[9] Khare A, Singh A, Prabhu S S and Rana D S 2013 Appl. Phys. Lett. 102 192911
[10] Seshadri R and Hill N A 2001 Chem. Mater. 13 2892
[11] Alexe M and Hesse D 2011 Nat. Commun. 2 256
[12] Gonzalez-V O E, WojdełJ C, Diéguez O and Íñiguez J 2012 Phys. Rev. B 85 064119
[13] Zalesskii A V, Frolov A A, Khimich T A and Bush A A 2003 Phys. Solid State 45 141
[14] Vijayanandhini K, Simon Ch, Pralong V, Bréard Y, Caignaert V, Raveau B, Mandal P, Sundaresan A and Rao C N R 2009 J. Phys.: Condens. Matter 21 486002
Vijayanandhini K, Simon Ch, Pralong V, Caignaert V and Raveau B 2009 Phys. Rev. B 79 224407
[15] Hohenberg P and Kohn W 1964 Phys. Rev. 136 B864
Kohn W and Sham L J 1965 Phys. Rev. 140 A1133
[16]Blaha P, Schwarz K, Madsen G K H, Kvasnicka D and Luitz J 2001 Wien2k An Augmented Plane Wave Plus Local Orbitals Program For Calculating Cryst. Properties Vienna University Technol. Austria (ISBN 3-9501031-1-2)
[17] Perdew J P, Burke K and Ernzerhof M 1996 Phys. Rev. Lett. 77 3865
[18]Singh D 1994 Plane Waves Pseudopotentials and the LAPW method (Boston: Kluwer Academic)
[19] Liechtenstein A I, Anisimov V I and Zaane J 1995 Phys. Rev. B 52 R5467
[20] Baettig P, Ederer C and Spaldin N A 2005 Phys. Rev. B 72 214105
[21] Dudarev S L 1998 Phys. Rev. B 57 1505
[22] Blochl E, Jepsen O and Andersen O K 1994 Phys. Rev. B 49 16223
[23] Blochl P E 1994 Phys. Rev. B 50 17953
[24] Ueda K, Tabata H and Kawai T 1998 Science 280 1064
[25] Miura K and Terakura K 2001 Phys. Rev. B 63 104402
[26] Nechache R, Harnagea C, Carignan L P, Gautreau O, Pintilie L, Singh M P, Ménard D, Fournier P, Alexe M and Pignolet A 2009 J. Appl. Phys. 105 061621
[27] Kanamori J 1959 J. Phys. Chem. Solids 10 87
[28] Suchomel M R, Thomas C I, Allix M, Rosseinsky M J, Fogg A M and Thomas M F 2007 Appl. Phys. Lett. 90 112909
[29] Garcia F G, Riccardi C S and Simoes A Z 2010 J. Alloys Compd. 501 25
[30] Cheng Z X, Li A H, Wang X L, Dou S X, Ozawa K, Kimura H, Zhang S J and Shrout T R 2008 J. Appl. Phys. 103 07E507
[31] Selbach S M, Einarsrud M A and Grande T 2009 Chem. Mater. 21 169
Goldschmidt V M 1926 Naturwissenschaften 14 477
[32] Ju S and Guo G Y 2008 Appl. Phys. Lett. 92 202504
[33] Song Z W and Liu B G 2013 Chin. Phys. B 22 047506
[34] Woodward P M 1997 Acta Crystallograllogr. B: Struct. Sci. 53 32
[35] Nakamura T and Choy J H J 1977 Solid State Chem. 20 233
[36] Rondinelli J M, May S J and Freeland J W 2012 MRS Bull. 37 261
[37] Goodenough J B 1998 Jahn-Teller Phenom. Solids Annu. Rev. Mater. Sci. 28 27
[38] Catalan G and Scott J F 2009 Adv. Mater. 21 2463