Impact of 3d-transition metal [T = Sc, Ti, V, Cr, Mn, Fe, Co]
on praseodymium perovskites PrTO
3: standard spin-polarized GGA and GGA+U investigations
M MUSA SAAD H-E
Department of Physics, College of Science and Arts, Qassim University, Al-Muthnib 51931, Saudi Arabia For correspondence (141261@qu.edu.sa)
MS received 8 August 2021; accepted 30 November 2021
Abstract. Generalized gradient approximation (GGA) computations based on the first-principles density functional theory (DFT) are executed to gain insight into the structural stability and physical properties of the 3d-transition-metal- based praseodymium series of perovskite compounds PrTO3. Correspondingly, to investigate the effect of on-site Cou- lomb repulsion energy, the exchange-correlation version of GGA is implemented via utilizing the (GGA?U) functional.
The computed ground state energies (E0) and equilibrium structural parameters of the varied T-site [T = Sc, Ti, V, Cr, Mn, Fe, Co] in the unit cell of PrTO3reveal a cubic symmetry (Pm-3m) in all compounds, in a good match with the few existing DFT and experimental literature. Besides, the computed spin-polarized band structures and partial and total density of states (DOS) within GGA predict a half-metallic (HM) behaviour for [T = Sc] perovskite and a metallic nature for the rest [T = Ti, V, Cr, Mn, Fe, Co]. The analysis ofE0results, DOSs and spin magnetic moments indicates that all perovskites PrTO3are stable in a ferromagnetic (FM) phase via the double exchange interaction T3?–O2-–T4?. PrTO3 show FM order with fractional values of their total spin magnetic moment per unit cell (MPrTO3), except [T = Sc]
perovskite that gives integer value (MPrScO3&2.0lB) with HM-FM property. Conversely, it is found that PrTO3exhibit HM-FM properties when [T = Sc, V, Cr, Mn, Fe] plus GGA?U is applied. Due to the cation–anion hybridizations, Pr3?–O2–and T3?–O2–, both Pr3?and T3?ions contribute to the largest part ofMPrTO3 with minor effects coming from O2– ions and interstitials. Furthermore, the three-dimensional and two-dimensional electronic charge density plots of PrTO3along the (110) plane confirm strong ionic nature along the Pr3?–O2–bonds, whereas the other O2––T3?–O2–bonds have strong covalent character.
Keywords. Magnetic materials; praseodymium perovskite compounds; physical properties; DFT?U.
1. Introduction
The perovskite oxides (ATO3) system was discovered as an earth mantle mineral more than 180 years ago, after that, it continued to attract the attention of scientists and researchers and became a hot topic in many fields. As a result, various inorganic ATO3 compounds were synthe- sized by replacing their A, T and O sites with different atoms from the periodic table of elements, except for the noble gases. This process has led to identifying many dif- ferent ATO3 compounds with exclusive physical and chemical characteristics among this group of inorganic systems. 3d-, 4d- and 5d-transition-metal perovskites, like alkaline-earth titanates (ATiO3) [1], zirconates (AZrO3) [2]
and hafnates (AHfO3) [3], respectively, having the con- ventional crystalline formula ATO3 are a very important class of inorganic materials, which affiliates to the per- ovskite oxides group of crystal structure. In the common
formula of perovskite oxide ATO3, sites A and T are two different cations, and site O is an anion, which connects with both A and T through the 1808-path (–O–T–O–). The cationic pair A and T must have a different ionic sizes and charge states with electric possibilities A1?-T5?, A2?-T4?
and A3?-T3? and positively total charge sums (q = ?6).
Besides, due to the simplicity of their synthesizing and the ease of their preparation methods, various solid compounds within ATO3group have been designed and studied over the past five decades. This nature motivated researchers to devote great attention to explore diverse systems of ATO3 in many vital fields such as solid-state physics, solid-state chemistry, materials science, materials engineering, etc.
[4,5]. The relatively simple chemical composition of these materials gives them distinctive characteristics; they exhibit varied structural, elastic, magnetic, electronic, thermal, optical and other physical properties. Currently, ATO3
materials are gaining considerable significance in many Supplementary Information: The online version contains supplementary material athttps://doi.org/10.1007/s12034-021-02645-6.
Bull. Mater. Sci. (2022) 45:69 ÓIndian Academy of Sciences
https://doi.org/10.1007/s12034-021-02645-6Sadhana(0123456789().,-volV)FT3](0123456789().,-volV)
modern technological fields like engineering constructions, spintronics, optoelectronics, solar cells, fuel cells, etc.
[4–12].
Moreover, during the past 10 years, researchers interest have shifted towards synthesizing and studying many compounds of perovskite oxide containing rare-earth elements that crystallize in the special chemical formula (RTO3), where (R3?= lanthanide atom) and (T3?= metal atom). They form one of the most studied classes of inorganic materials because they can deliver a broad range of functional physical and chemical properties.
Therefore, several members of single RTO3 and their doped compounds have been studied by many authors using different experimental and theoretical methods [11–15]. While a few researchers devoted their efforts to study the chemical composition and physical properties of different series of rare-earth-based perovskites. They substituted the two cationic sites R and T in the unit cell of RTO3by some suitable rare-earth from (R3?= La–Lu) and 3d-transition-metal from (T3? = Sc–Cu) elements, respectively, such as in LaTO3(T = Fe, Co, Ni, Cu) [15], CeTO3(T = Co, Ni, Cu) [7], CeTO3(T = Cr, Fe) [8] and SmTO3 (T = Mn, Fe) [16]. Also, there are some sporadic studies on individual compounds within RTO3, such as NdFeO3 [17], GdCrO3 [18], EuCrO3 [19], DyMnO3 [20]
and LuMnO3 [21].
On the other hand, there are very few studies on pra- seodymium perovskites (PrTO3), either in individual or within a series, compared to their well-known counter- parts. Aquino et al [22] synthesized two metallic com- pounds PrTO3(T = Co, Ni) by using gelatin powder, and characterized their crystal structures and thermal beha- viour. By using first-principles density functional theory (DFT), Sabir et al [12] systematically investigated the physical properties of PrTO3 (T = V, Cr). In their DFT computations, structural, electronic, magnetic, optical and thermoelectric properties of PrTO3 were evaluated via applying a combined method of PBE and mBJ under GGA and confirmed that PrTO3 have cubic structure and half-metallic ferromagnetic (HM-FM) nature in both cases of (T = V, Cr). In another DFT study, Ullah et al [23]
directed their goal to investigate the physical properties of a second type of Pr-based perovskites PrTO3(T = Al, Ga) that crystalize in a cubic structure. In this work, GGA and GGA?U approximations quite successfully predicted an HM-FM (T = Al) and a paramagnetic phase and metallic nature when (T = Ga) combined with elastic nature of brittle and ductile, respectively.
Motivated and guided by the above, this article pre- sents a comprehensive and systematical study on a series based on rare-earth transition-metal perovskites PrTO3. Where T is taken as [T3? = Sc, Ti, V, Cr, Mn, Fe, Co], and further investigations on their ground-state energies and equilibrium structural parameters, as well as the
obtained magnetic and electronic properties, are performed by using DFT first-principles. All these investigations are carried out via using the first-principles DFT computations under the generalized gradient approximation (GGA), as well as its corresponding exchange-correlation functional (GGA?U). Also, this study aims to expose the correlative impact as a result of substituting the T-site and the application of GGA?U on the physical and chemical properties of the investigated perovskites PrTO3. We believe that this study will be an important addition to the existent data of inorganic per- ovskite materials, as its results provide detailed and useful information for PrTO3 compounds.
2. Materials of PrTO3
The unit cell of praseodymium perovskites Pr3?T3?O3
was constructed by systematically selecting the 3d-tran- sition-metal atom in the following sequence [T3? = Sc, Ti, V, Cr, Mn, Fe, Co]. It has been verified that the most published compounds of rare-earth transition-metal-based perovskites PrTO3 take the rock salt-type structure (Na?Cl–) with cubic symmetry (Pm-3m; no. 221) [12,24].
Accordingly, the unit cell of all perovskites PrTO3 con- tains five atoms with the following Wyckoff sites and positions; the Pr-cation at 1a (0, 0, 0), T-cation at 1b (,
, ) and O-anion at 3c with three different sites, (,
, 0), (, 0, ) and (0, ,), see figure 1. The crystal structure and nature of T3?-cation govern the main obtained properties for the studied series of perovskites PrTO3, this will be discussed in detail in Section 4. In addition to the effect of computational GGA functionals, the only difference between the compounds of this series is the chemical and physical nature of the T-atom, which have the electronic configurations Sc: [Ar] 3d1 4s2, Ti:
[Ar] 3d24s2, V: [Ar] 3d34s2, Cr: [Ar] 3d5 4s1, Mn: [Ar]
3d5 4s2, Fe: [Ar] 3d6 4s2and Co: [Ar] 3d7 4s2. It can be seen that the Cr atom has an odd electronic configuration among this series, which gives different physical prop- erties of its perovskite compounds ACrO3 compared to their counterparts [8,15,18,19]. In all perovskites, the rare-earth cation Pr and oxygen anion O show an ordi- nary electronic configuration in their bonds, [Xe] 4f3 6s2 and [He] 2s2 2p6, respectively. Therefore, the ionic con- figurations in unit cell of Pr3?T3?O3 are Pr3?: [Xe] 4f2 6s0, T3?, Sc3?: [Ar] 3d0 4s0, Ti3?: [Ar] 3d1 4s0, V3?: [Ar] 3d2 4s0, Cr3?: [Ar] 3d3 4s0, Mn3?: [Ar] 3d4 4s0, Fe3?: [Ar] 3d5 4s0 and Co3?: [Ar] 3d6 4s0. Whereas the outer 2p orbitals in O ion gain two shared electrons O2-: [He] 2s2 2p8 due to the double-exchange interaction between two neighbouring cations through the long-range path [–T3?–O2-–T4?–]. Furthermore, the second effect of T-cation is the ionic radius R (T3?) that look different in
their Pr3?T3?O3unit cell. Based on values of these radii, the structural stability and formability of the lattice in Pr3?T3?O3 crystals can be estimated by computing the tolerance factor per unit cell (TF) via the relation [25]:
TF¼ ffiffiffi2 p
2 ðRPr3þþRO2Þ RT3þþRO2
ð Þ
ð1Þ Ordinarily, crystal structure is stable in cubic symmetry (Pm-3m; No. 221) when (TF = 0.890–1.100) [26–29], beyond this range (TF\0.890) or (TF[1.100) the ideal crystal structure will reduce and transform to other sym- metry like orthorhombic (Pbnm; No. 62) or hexagonal (P6(3)/mmc; No. 194) [27,28], respectively. Another structural factor, i.e., octahedral factor (OF), probes the stability of octahedra [T3?O6] in crystal network, can be evaluated through the ratio:
OF¼RT3þ
RO2
ð2Þ where crystal structure is stable when the value of OFis in the range (OF = 0.4400–0.9000) [29]. Table1 summarizes the main physical properties of the constructed atoms for the perovskites Pr3?T3?O3. All these elements have a standard metallic solid state at room temperature (RT = 298
K), except O has standard non-metallic gas state at RT [26,30]. The symbol R represents the effective ionic radii of Pr3?-site cation, and T3?-site cation and O2–-site anion at XII and IV coordination systems, respectively [31]. Based on the obtained values ofTFandOFby means of these ionic radii, it can be seen that all crystal structures of Pr3?T3?O3 stabilize in a cubic symmetry.
3. Computational methods and details
The ground-state energies and equilibrium structural parameters, and magnetic and electronic properties of the praseodymium perovskite compounds PrTO3 were com- puted by WIEN2k-19.2 code [32]. The WIEN2k computa- tions are based on all-electron self-consistent full-potential linearized augmented plane-wave method within the framework of the most successful Kohn-Sham (KS) [33]
DFT [34]. The effect of exchange and correlation (XC) between the electrons in all PrTO3compounds was treated by using standard first-principles DFT computations based on the GGA [35–37]. Here, GGA is executed under the gradient-corrected exchange functional of Perdew, Burke Figure 1. (a) Two-dimensional and (b) 3D view of the crystal structures for the unit cells of perovskite
PrTO3; Pr (blue), T (green) and O (red).
Table 1. Main physical properties of Pr, T and O atoms in Pr3?T3?O3at RT; atomic number (Z), ionic radius (R), tolerance factor (TF), octahedral factor (OF) and Pauling electronegativity (P. electro.) [30].
Site R-cation
T-cation
Anion
Symbol Pr Sc Ti V Cr Mn Fe Co O
Z 59 21 22 23 24 25 26 27 8
R(A˚ ) 1.3000 0.7450 0.6700 0.6400 0.6150 0.6450 0.6450 0.6100 1.3500
TF — 0.9166 0.9276 0.9416 0.9536 0.9393 0.9393 0.9560 —
OF — 0.5519 0.4963 0.4741 0.4556 0.4778 0.4778 0.4519 —
P. electro. 1.1300 1.3600 1.5400 1.6300 1.6600 1.5500 1.8300 1.8800 3.4400
Bull. Mater. Sci. (2022) 45:69 Page 3 of 18 69
and Ernzerh (PBE) [38] according to the spin-polarized integral:
EXCGGAq"ð Þ;r q#ð Þr
¼ Z
qð Þer XCq"ð Þr ;q#ð Þr ;rq"ð Þr ;rq#ð Þr
dr ð3Þ whereqð Þr is the electron density andeXCis the XC energy per electron. To achieve accurate results for ATO3 that agree with the experiments, some computational methods based on DFT are often utilized, such as hybrid XC func- tionals B3LYP and B3PW [39,40], and DFT?U [8,12,32].
In this study, the spin-polarized GGA?U functional is implemented to include the effect of XC in PrTO3 compounds that originates in 4f and 3d states, as [35]:
E¼E0þEGGAþUXC q"ð Þ;r q#ð Þr
ð4Þ This effect is treated by incorporating the on-site Coulomb interaction energy (U) and Hund’s rule exchange energy (J) in DFT computations, according to:
EXCGGAþUq"ð Þ;r q#ð Þr
¼EXCGGAq"ð Þ;r q#ð Þr þUq"ð Þ;r q#ð Þr
ð5Þ where, the XC term Uq"ð Þ;r q#ð Þr
represents the energy of the Hubbard functional and double counting term EHub nlrmm
Edc nlr for the localized orbitals with occupation number mand spinr. Also, the energy for the GGA?U functional can be defined as:
EXCGGAþUq"ð Þ;r q#ð Þr
¼UJ
2 NX
m;r
n2m;r
!
ð6Þ In WIEN2k, these parameters were set as optimized values in which the effective value (Ueff) is related to U and J by:
Ueff¼UJ ð7Þ
In GGA?U computations, theJvalue is kept at (J= 0) and the input values ofUeffparameter are varied from 2.0 to 6.0 eV according to the DFT?U literatures [8,12,35,37]. In view of that, and after many computational optimizations, it is found that the appropriate values of Ueff for 4f and 3d states, which give results like lattice constants close to the experiments values, are: Ueff = U–J for (Pr) = 6.0 eV, (Sc, Ti) = 3.0 eV, (V, Cr) = 4.0 eV, (Mn) = 5.0 eV and (Fe, Co)
= 6.0 eV for strongly correlated P (4f) and T (3d) states. The introduction of these energy parameters permits to clearly realize their effect on the GGA version of electronic and magnetic results for perovskites PrTO3compounds.
To ensure correct convergence for all computations, the cutoff energy, separation energy between valence and core states of (Ecutoff = -6.0 Ry; *-81.6342 eV), many k-points of (k-points = 1000) and a mesh of (159159 15) k-points in their first Brillouin zone were set. The
spin-polarized scheme and ferromagnetic (FM) state were selected to examine the DFT equilibrium structural parameters of PrTO3 compounds. Also, the joined plane- wave parameter, which includes the smallest radius of muffin-tin sphere (RMT), supplementary table S1, and the largest reciprocal lattice vector for the expansion of flat wavefunction (Kmax) [41,42] was set as (RMTKmax = 9.0).
SettingRMTat these values ensures that there is no charges- escape from the inner atomic core of Pr, T and O sites, besides to achieve an accurate for their energy eigenvalues convergency. Hence, the potentialV(r) and charge density q(r) within these muffin-tin spheres are expanded in terms of crystalline spherical harmonics up to the value of angular momenta (‘max= 10.0) and the plane-wave (PLW) expan- sion has been applied on the interstitial region sites.
Moreover, the Fourier expansion parameter was set as (Gmax= 14.0) to delimit the magnitude of the largest vector inq(r). The values of energy and charge convergence were chosen as (Econv. = 0.0001 Ry) and (qconv. = 0.001e), respectively, during the self-consistency computational cycles of GGA and GGA?U for all PrTO3compounds.
4. Results and discussion
The following subsections present the details of com- puted information of the concerned compounds that were developed in this work along with analyzation and dis- cussion of their obtained results for depicting the appli- cable conclusions of our study. In this regard, the structural properties of all perovskites PrTO3 are firstly presented and the accuracy of the performance of GGA and GGA?U functional is analysed according to their supplied results. Then, after securing the optimized crystal structures, the dominant results of structural, magnetic and electronic properties for perovskites PrTO3 are analysed and discussed.
4.1 Structural properties of PrTO3
4.1a Crystal structures optimization: Initially, the lattice parameters of the cubic (Pm-3m) unit cells for all perovskites PrTO3are fixed at (a=b=c= 4.00 A˚ ) and their crystal volumes are built by setting the general atomic positions as: Pr at (0, 0, 0), T at (,,), and O at (,, 0), (, 0,) and (0,,), figure1. Secondly, the volume optimization task, as available in Wien2k package, is systematically implemented in order to achieve the functional optimization volume of PrTO3 crystals in FM and AF phases. Where the values of unit cell volumes (V) are ranged along thex-axis with differentV, and then all the associated total energies (Etotal), along the y-axis, are fitted
to the equation of state proposed by the scientists Murnaghan and Birch (BMES) [43,44]:
Etotalð Þ ¼V E0þB0V B00
V0=V ð ÞB
0
0þB001 B001
" #
B0V0
B001
" #
ð8Þ
The computational processes of (Etotal vs. V) are executed via the exchange-correlation GGA. Figure 2 displays the computed plots of (Etotalvs. V) for the unit cells of PrTO3 perovskites in FM and AF phases. From the computed results in figure 2 and supplementary figure 1S, it is seen that the FM phase is the most favourable in Etotal with regard to the AF one, showing that FM is the stable ground state. Also, the negative values of Etotal and difference Figure 2. The optimized (Etotalvs. V) curves for the unit cells of PrTO3[T = (a) Sc, (b) Ti], computed via GGA, in FM phase (up panels) and AF phase (down panels). For other compounds (c) V, (d) Cr, (e) Mn, (f) Fe, (g) Co, see supplementary figure S1.
Table 2. Structural optimizations data, equilibrium volumes (V0), lattice constants (a0), bulk modulus (B0) and their derivative (B00), ground total energy (E0) and total energy difference (DE) per formula unit, for the cubic (Pm-3m symmetry) unit cell of FM perovskites PrTO3, obtained from GGA.
PrTO3 V0(A˚ )3 a0(A˚ ) B0ðGPaÞ B00ðGPaÞ E0ðRyÞ DEðmeVÞ
T = Sc 66.603 4.0535 150.12 4.8083 -20466.310 -0.1176
T = Ti 61.089 3.9384 181.26 3.8544 -20645.475 -0.1002
T = V 57.723 3.8647 176.40 8.7476 -20836.361 -0.0335
T = Cr 57.952 3.8698 199.35 3.4533 -21039.361 -0.1462
T = Mn 58.474 3.8814 174.82 4.3430 -21254.909 -0.0556
T = Fe 56.402 3.8350 165.98 5.1132 -21483.120 -0.0789
T = Co 55.411 3.8124 195.35 5.0602 -21724.399 -0.0769
Ref. [45];a= 3.8900A˚[T = V],a= 3.8520A˚[T = Cr],a= 3.8870A˚[T = Fe].
Ref. [46];a= 3.8200A˚,B0¼204.20 GPa [T = Mn],a= 3.7800A˚[T = Co].
Ref. [47];a= 3.7300–3.7700A˚;B0¼130.51 GPa;E0¼2:9553105eV [T = Cr].
Ref. [48];a= 3.8900A˚[T = Co].
Bull. Mater. Sci. (2022) 45:69 Page 5 of 18 69
between these phases (DE¼EFMEAF), per formula unit (table2), confirm that all unit cells of perovskites PrTO3are stable in FM phase. Accordingly, in the first stage, the FM phase is adopted in the optimizations to compute the structural parameters of all PrTO3compounds. Most tran- sition-metal-based RTO3perovskites have this tendency in their magnetic arrangement [8,12].
Favourably, the fitted curves ofEtotalvs. V enables us to compute many essential structural properties, comprise ground total energies (E0), equilibrium volumes (V0), lattice constants (a0), main interatomic bond-distances (dhPrOi, dhTOi), and bulk moduli (B0) plus their first pressure derivative values (B00). Besides, the GGA and GGA?U computations of magnetic, electronic and other main physical properties are based primarily on these optimized structural quantities, which will be discussed through their specific points in the next subsections. Tables 2and3tab- ulate the summary of the obtained data from structural optimizations for the studied compounds evaluated within GGA, along with the available experimental and other DFT results. Through comparing the optimized results of PrTO3 compounds within the FM and AF phases, it can be con- cluded that FM gives reasonable cubic values that are consistent with the previous theoretical and experimental data [12,45–48].
According to the main structural information summarized in tables2and3, and plotted in figure3, it can be seen that the values of a0 (figure 3a), likewise their corresponding V0 (- figure3b) anddhTOi(figure3c), decrease from T = Sc to Ti, V, Cr, Mn, Fe and Co. While, the E0 (figure 3d) decreases along this sequence, which confirms the effect of variation in charge and ionic radius of T-site on PrTO3 crystals. Here, there is a significant observation that the above values are in excellent agreement with the main trends that were concluded by a few earlier DFT studies when utilizing the GGA func- tional [12,45]. In addition, the GGA results are very close to the experimental and the previous DFT reports [45–48].
4.1b Mechanical stability of crystal structures: In addition, B0 parameter evaluates how crystal’s resistant to the compression force acting on it, that is,B0 indicates the
stiffness nature in crystal structures. It can be estimated by the semiempirical second-order derivation [44];
B0¼ Vo2E oV2
V¼V
0
ð9Þ whereEis the total energy of the crystal under strain effect, it can be written as:
Etotal¼E0þPappliedðVV0Þ þEelastic ð10Þ E0 andV0 denote the unit cell total energy and volume of unstrained, initial, crystal,V is the strained, final, volume, and Papplied and Eelastic are applied pressure and elastic energy, respectively, which can be obtained from:
Papplied¼ oE0
oV
V¼V0
ð11Þ
and
Eelastic¼Veiej
2V0 o2E oeioej
ð12Þ where, ei and ej are the relative elastic deformations with (i;j= 1, 2, 3).
It is well known that for cubic (Pm-3m) perovskites, the required condition for their mechanical stability is (B0[0).
The optimizedB0andB00results, which are fitted by BMES, are used to measure the effect of Papplied on the structural and elastic properties of crystalline solids. Generally, when comparing the values ofB0, it is found that they look dif- ferent among the compounds in PrTO3;B0for [T = Cr, Co and Ti] is larger than the others, and the ionic radii of T-site have no apparent influence on the values ofB0.
4.2 Magnetic properties of PrTO3
4.2a Magnetic moments: One of the most interesting aspects is the investigation of the magnetic properties of perovskites PrTO3 because apart from the major contribution of transition-metal ions (T3?), rare-earth ions (Pr3?) also contribute a large portion to the total magnetic moment of their unit cell. This feature unlike in other T- based perovskites ATO3with A? = alkali metal or A2?= alkaline earth metal, for which most of their total magnetic moment comes from the contribution of T5?or T4? ions, respectively. As summarized in table 4, the spin-polarized magnetic properties of PrTO3 are evaluated via GGA and GGA?U to compute the partial magnetic moment of Pr, T and O ions besides the magnetic moment contribution of interstitial sites. Also, the total magnetic moments per cubic unit cell of the investigated perovskites are displayed in a separate column and plotted in figure 4a and b. It can be seen from the results that the main contribution of total magnetic moment is due to Pr3? and T3? ions, while the anions O2– and interstitials have a minor effect. The main Table 3. Optimized interatomic bond-distances for the cubic
(Pm-3m symmetry) unit cell of perovskites PrTO3, obtained from GGA.
PrTO3 dhPrPri(A˚) dhPrOi(A˚) dhTPri(A˚) dhTOi(A˚)
T = Sc 4.0535 2.8863 3.5104 2.0268
T = Ti 3.9384 2.7849 3.4107 1.9692
T = V 3.8647 2.7328 3.3469 1.9323
T = Cr 3.8698 2.7363 3.3513 1.9349
T = Mn 3.8814 2.7446 3.3614 1.9407
T = Fe 3.8350 2.7118 3.3212 1.9175
T = Co 3.8124 2.6958 3.3016 1.9062
remark here is that the GGA?U functional enhances the value of the partial magnetic moment on T3? ions (MGGAþUT [MGGAT ), which increases their total magnetic moments per unit cell (MGGAþUPrTO
3 [MGGAPrTO
3), see figure 4b, except for [T = Co]-based metallic compound.
The partial magnetic moments of cations T3?and anions O2– are antiparallel and reveal the FM nature of PrTO3. Besides, due to the number of unpaired electrons in 3d orbitals of T3? ions, the value of total magnetic moment increases in the primary sequence, except for [T = Co]
Figure 3. The main optimized structural data: (a) lattice constant (a0), (b) equilibrium volume (V0), (c)dhTOibond-distance and (d) ground total energy (E0), as functions of T-site in unit cells of PrTO3within GGA.
Table 4. Magnetic properties (inlB units) of the cubic unit cell of perovskites PrTO3, obtained from GGA and GGA?U.
PrTO3 Functional MPr MT MO Minter: MPrTO3
T = Sc GGA 2.0108 0.0009 -0.0262 0.0532 1.9864
GGA?U 1.9990 0.0009 -0.0205 0.0621 2.0006
T = Ti GGA 2.3391 0.1178 -0.0136 0.2038 2.6198
GGA?U 2.3769 0.3414 -0.0333 0.3175 2.9360
T = V GGA 2.2240 0.7102 -0.0304 0.1773 3.0202
GGA?U 2.2895 1.3770 -0.0370 0.4383 3.9937
T = Cr GGA 2.2075 2.2302 -0.0405 0.1681 4.9366
GGA?U 2.0336 2.4550 0.0155 0.4611 4.9972
T = Mn GGA 2.1828 3.1727 0.0391 0.4639 5.9366
GGA?U 2.2987 3.4260 -0.0418 0.3990 5.9983
T = Fe GGA 2.0856 2.6249 0.0594 0.2684 5.1571
GGA?U 2.2667 4.0259 0.1128 0.3359 6.9669
T = Co GGA 1.9339 1.1244 -0.0545 0.0899 2.9847
GGA?U 2.1582 2.3124 0.0245 0.0841 4.6281
Ref. [12];MPr,MT,MOandMinter:are respectively 1.41, 2.27,-0.0034 and 0.347lB[T = V]; 2.34, 2.17, 0.019 and 0.472lB[T = Cr].
Bull. Mater. Sci. (2022) 45:69 Page 7 of 18 69
perovskite, it includes one antiparallel pair of electrons in its 3d6orbitals, as discussed in Section4.2b. This point can be well understood from the postulates of Hund’s theory, where due to the presence of CF, the occupation of partial orbitals of T3? ions in PrTO3 takes the following forms:
Sc3?[3d0: t2g0:
t2g0;
eg0:
eg0;
; S = 0]; Ti3?[3d1: t2g1:
t2g0;
eg0:eg0;; S = 1/2]; V3?[3d2: t2g2:t2g0;eg0:eg0;; S = 1];
Cr3?[3d3: t2g3:t2g0;eg0:eg0;; S = 3/2]; Mn3?[3d4: t2g3:
t2g0;eg1:eg0;; S = 2]; Fe3? [3d5: t2g3:t2g0;eg2:eg0;; S = 5/2] and Co3?[3d6: t2g3:t2g1;eg2:eg0;; S = 2]. It confirms the ordinary increase of partial magnetic moments from Sc3?to Fe3?sites as compared to the unusual Co3?ions.
Furthermore, as it is known that in transition-metal per- ovskites, there are strong hybridizations that emerge between their energetic orbitals in cations–anions sites, which govern the magnetic and electronic properties. This feature will be discussed in detail in Section 4.3b. Here, in the case of PrTO3compounds in this study, it can be found that the main hybridizations come from Pr3?, T3?and O2- states via the long-range exchange mechanism Pr3?–O2–
and T3?–O2–. Where the Pr3? and T3? cations transfer a fraction of their magnetic moments to two sites in crystal, unoccupied 2p orbital in anions O2– and interstitial sites, which causes a decrease in the partial magnetic moments of these ions. As an obvious point, it is noted that the GGA?U improves the values of partial and total magnetic moments compared to those obtained from GGA. Also, the unique- ness of [T = Sc3?] compound is evident with computed values of total magnetic moment per unit cell of 1.9864lB (GGA), figure 4a, close to the theoretical integer value (MPrTO3 & 2.0lB), which indicates the presence of half- metallic (HM-FM) property in this case. While the rest of the compounds have a metallic FM property due to the fractional value of their total magnetic moments. Con- versely, it is found that PrTO3compound exhibits a HM-FM property when [T = Sc, V, Cr, Mn, Fe] plus applying the GGA?U functional. These are consistent with the results of
previous GGA works in that the partial and total magnetic moments increase with increasing the number of valence electrons in T3?-sites in FM perovskites [12,47].
4.2b Magnetic interaction mechanism: As it is well known that the magnetic and electronic properties of perovskites PrTO3 are essentially governed by the electronic nature of transition metal T (3d), which locate in the centre of cation–anion corner-sharing octahedra [T3?O6]. In 3-D octahedral network, the ordered distribution of O2– (2p) around T3? (3d) leads to the appearance of cubic crystal field (CF), which causes splitting of the 3d-orbitals into three- and two-fold degenerate suborbitals 3d–t2g [3dxy, 3dxy, 3dxy] and 3d–eg [3dz2;3dx2y2] with lower and higher energy, respectively (supplementary figure S2). According to Hund’s coupling rules, the filling of 3d–t2gand 3d–egstarts from low energy states to high energy states that depends on the competition between CF and XC energy. For that reason, it can be expected that the n-electrons in T3?(3dn) orbitals take the electronic configuration: Sc3? [3d0: t2g0: t2g0; eg0: eg0;], Ti3? [3d1: t2g1: t2g0; eg0: eg0;], V3? [3d2: t2g2: t2g0; eg0:
eg0;
]; Cr3?[3d3: t2g3:
t2g0;
eg0:
eg0;
]; Mn3?[3d4: t2g3:
t2g0;
eg1:
eg0;
]; Fe3? [3d5: t2g3:
t2g0;
eg2:
eg0;
] and Co3? [3d6: t2g3:t2g1;eg2:eg0;].
Therefore, the physical properties of Pr3?T3?O2–3can be explained by the nearest neighbouring (nn) double exchange (DE) interaction that occurs as a result of the long atomic arrangement style existing in these crystals. Since PrTO3contains two nn cations having mixed-valence states T3?and T4?, one of the t2g or egelectrons, i.e. exchange electron, makes an actual hopping from the 3d suborbitals of T3?to 2p of O2–to 3d of T4?(supplementary figure S3), which takes the 180°-long-range path [–T3? (3dn)–O2–
(2p)–T4? (3dn–1)–], with n = 0, 1, 2, 3, 4, 5 and 6. This interaction depends on the alignment of the local spins of 3d-electrons in these two cations. In event that the spins of Figure 4. The total magnetic moments (MPrTO3) per unit cells of PrTO3 within (a) GGA and (b) GGA?U
functionals.
3d-electrons in T3? (3dn) and T4? (3dn–1) have a parallel alignment, the 2p-electron in O2–(2p) with spin parallel to 3d-electrons spins in T4?(3dn–1) hops to T4?(3dn–1). That is automatically followed by a hopping of t2gor egelectron from T3? (3dn) to fill this vacant site in 2p suborbitals of anion O2–(2p). In another case, if the spins of 3d-electrons in T3?(3dn) and T4?(3dn–1) have an antiparallel alignment, the t2gor egelectron of T3?(3dn) cannot hop to 2p because their spin is parallel to the remained 2p-electron in O2–(2p).
Therefore, the conduction state of all perovskites PrTO3 governs by the above itinerancy of 3d-electrons, which is strongly coupled with an FM alignment based on the long- range DE interaction [–T3?(3dn):–O2–(2p)–T4?(3dn–1):–].
The magnetic exchange interaction on the two cations Pr3?and T3?in PrTO3can be examined in terms of their total energy computations based on FM and AF spin con- figurations. From table 2, it is clear that the AF coupling turns out to be less stable in energy than the FM coupling by the amount of DE. Using these values of energy, it is pos- sible to compute the three main spin-exchange parameters Jij; JT–T and JPr–Pr for the nn spin pairs (T3?–T3?) and (Pr3?–Pr3?), and JT–Pr for nearest pair (T3?–Pr3?). Thus, this parameter can be estimated via the Heisenberg spin model [49,50],
H¼X
i;j
JijSiSj ð13Þ
whereSiSjdenote the theoretical spin magnetic moments of the cation pairi–j; here,SPr= 5/2lBandST= 1/2, 1/2, 2/2, 3/2, 4/2, 5/2 and 4/2lBfor [T3?= Sc, Ti, V, Cr, Mn, Fe, Co], respectively. Hence, the approximation relation between the exchange energy (eij) and exchange parameter (Jij) can be determined by:
eij¼JijSiSj ð14Þ
Accordingly, the computed values of three main FM spin-exchange parameters JT–T JPr–Pr and JT–Pr in per- ovskites PrTO3are computed and presented in (table 5).
It can be seen that the JT–Tis larger than JPr–PrandJT–Pr due to the stronger exchange coupling in the T3?–T3?
pair, through T3?–O–T3?, than that in the Pr3?–Pr3?
and T3?–Pr3? pairs. This also confirms the accuracy of the obtained spin magnetic moments on T3? ions that mainly govern the FM order in the perovskites PrTO3 via the DE interaction.
4.3 Electronic properties of PrTO3
As it is well known, the electronic properties of materials are among the most important features that should be computed and discussed especially for magnetic crystalline compounds. Therefore, in this section, the band structures, total density of states (TDOS) and partial density of states (PDOS) as functions of unit cell energy of the studied perovskites compound PrTO3 are computed using the two GGA and GGA?U functionals. In terms of their desired benefits, the obtained properties from the analysis and dis- cussion of the band structures and density of states results, besides their FM nature, will give an indication of the suitability of PrTO3 materials in designing various spin- tronics devices.
4.3a Spin-polarized electronic band structures: Figures5 and6show the plots of the computed spin-up and spin-down (spin-dn) band structures of PrTO3 compounds at their equilibrium lattice parameters within GGA and GGA?U, respectively. The energy scale (vertical axis) in all band structures is plotted in eV units, and the origin point of the unit-cell energy is randomly fixed at the valence band maximum that is denoted to the Fermi level (EF). Besides, the symmetry k-points are selected identically as R,K,C, D, X, Z, M, R and C along the horizontal axis. The collective effect of spin-up and spin-down bands in figures 5 and 6 (and supplementary figures S4 and S5) shows typical band structures of PrTO3via GGA and reveal the presence of a HM feature when [T = Sc] and metallic property in other compounds [T = Ti, V, Cr, Mn, Fe, Co]
with no bandgap (Eg). However, five of the PrTO3 compounds [T = Sc, V, Cr, Mn, Fe] exhibit a HM property when the on-site Coulomb repletion energy (U) is applied via GGA?U (figures 6a–g). The band structures indicate that the HM for the spin-up bands, while
Table 5. Electronic energy-gap (Eg), (in eV), of perovskites PrTO3, obtained from GGA and GGA?U functionals. The computed values of spin-exchange parameters (in meV).
Functional PrTO3 T = Sc T = Ti T = V T = Cr T = Mn T = Fe T = Co
GGA 2.5232 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
GGA?U 2.8028 0.0000 3.1565 3.3535 3.2654 2.0681 0.0000
JT–T 0.0196 0.0368 0.0447 0.0675 0.0741 0.1315 0.1025
JPr–Pr 0.0098 0.0084 0.0028 0.0092 0.0046 0.0066 0.0064
JT–Pr 0.0147 0.0125 0.0067 0.0121 0.0079 0.0099 0.0055
Bull. Mater. Sci. (2022) 45:69 Page 9 of 18 69
the spin-down bands show an electronicEgwith different values for PrTO3; where the extent ofEg;depends mainly on both the type of T-atom and the application of U energy, see table5. Generally, GGA?U functional yields a wider Eg;than that using GGA in the HM cases of PrTO3. This energy has a minor effect on the electronic properties of [T
= Ti, Co] compounds, figure6b and g, show semi-metallic and metallic behaviour within GGA?U, respectively, which can be attributed to the odd electronic configuration of Ti3?(3d1–t2g1) and Co3? (3d6–t2g4eg2) in their compounds, similar to that observed for their corresponding magnetic properties.
From the obtained HM property, it can be seen that theEF is situated at the Eg in spin-down orientation whereas it crosses some spin-up bands, which leads to full spin polarization (SP) of the conduction electrons in HM com- pounds. The electronic SP ratio of perovskite compounds can be evaluated by using the number of states per unit cell [D (E)] that occupy the energy range (-E??E) through EF (E = 0.0 eV) in spin-up (:) and spin-down (;) directions via the simple relative equation:
SP¼ D"ð Þ E D#ð ÞE
D"ð Þ þE D#ð ÞE
100% ð15Þ
Figure 5. The computed spin-up (left panels) and spin-down (right panels) band structures of PrTO3[T = (a) Sc, (b) Ti] via GGA. For other compounds (c) V, (d) Cr, (e) Mn, (f) Fe, (g) Co, see supplementary figure S4.
Here,D:(E) andD;(E) denote the number of states atEFin two spin directions. For HM compounds, such as [T = Sc]
case, the computed values within GGA are D: (E) = 1.0 andD;(E) = 0.0; give (SP = 100%), and this confirms the [T = Sc] perovskite is completely spin-polarized material.
This property has been detected in a few similar compounds of Ce- and Pr-based perovskites with [T = V, Cr, Fe] using GGA?U functional [8,12]. TheEgof HM compounds rises in spin-down channels, and their obtained values by uti- lizing GGA?U are greater than the GGA values, see
table 5. Conversely, the bands of metallic PrTO3 compounds, figure 5b–g, overlap through the EF in both spin-orientations with small difference in spin-down bands distribution at valence band maximum from [T = Ti] to [T = Co]. This small difference in band structures dominates by the value of binding energy between the hybridized states in cations T (3d) and anions O (2p) sites through the strong 3d-2p coupling [T3?–O2––T4?], which finally yields smaller values ofEg. Thus, besides the effect of U energy in GGA?U functional, there is the only factor affecting the Figure 6. The computed spin-up (left panels) and spin-down (right panels) band structures of PrTO3[T = (a) Sc,
(b) Ti] via GGA?U. For other compounds (c) V, (d) Cr, (e) Mn, (f) Fe, (g) Co, see (supplementary figure S5).
Bull. Mater. Sci. (2022) 45:69 Page 11 of 18 69
electronic band structures of PrTO3is the number of elec- trons occupying the 3d orbitals in T3?ion, see Section4.3b.
4.3b Spin-polarized electronic densities of states: Fur- thermore, the TDOSs per unit cell for perovskites PrTO3 and their PDOSs per atom of Pr (6s, 5p, 6d, 4f), T (4s, 3p, 3d) and O (2s, 2p) are computed via GGA and GGA?U functionals and presented in figure7,8and9. It can be seen from figure 7b–g that there is no energy-gap (Eg) in the TDOSs for all PrTO3 compounds when GGA is utilized, which confirms their metallic property. This observation does not apply to [T = Sc] compound (figure 7a), whose TDOSs within GGA contain Eg in spin-down direction, which indicates this perovskite exhibits a HM property. On the other hand, the GGA?U functional opens an energy gap in the TDOSs of PrTO3[T = Sc, V, Cr, Mn, Fe] and induces the 3d and 4f electrons in these compounds to produce a HM structure with an Eg in spin-down direction (table 5).
While, the two compounds [T = Ti] (figure7b) and [T = Co]
(figure 7g) show semi-metallic and metallic properties, respectively, within both GGA and GGA?U, which in agreement with previous detection for [T = Co] compound
[22]. In general, when comparing the computed electronic properties, it can be concluded that the obtained results from TDOSs per unit cell of PrTO3compounds (figure7a–
g) are completely identical to those obtained from the band structures (figures5and6). Moreover, it is also noticed that the overlapping of the conduction states through theEF, i.e., the bandwidth in spin-up and spin-down directions in metallic cases, and spin-up of the HM ones, increases linearly with the T-site type through the substitution [T = Ti, V, Cr, Mn, Fe, Co].
To discern the special distributions of different partial states, the spin-polarized PDOSs within these compounds are also computed by using the GGA and GGA?U.
Figures8,9and10show the spin-up and spin-down PDOSs of Pr, T and O atoms in their unit cells of PrTO3, which confirm their contribution to the TDOS. It is clear how the states of Pr atoms appear as high peaks that occupy the range around theEF, between-2.0 and?2.0 eV (figure8a–
g), in valence bands and conduction bands, respectively.
The emergence of the PDOS of T atoms depends mainly on the electronic nature of T-atom (figure 9a–g), as their presence and amount in this range increases from [T = Sc], Figure 7. The computed spin-up and spin-down total density of states per unit cell of perovskites PrTO3[T = (a) Sc, (b) Ti, (c) V, (d) Cr, (e) Mn, (f) Fe, and (g) Co] via GGA and GGA?U. The vertical dashed line atE= 0.0 eV represents the Fermi level (EF).
with negligible PDOS, to [T = Co], which perform con- siderable spin-up and spin-down peaks around the EF. While, the states of O atoms (figure 10a–g), contribute with slight PDOS to the conduction bands, where their effect exists beyond the EF in the valence bands, -8.0 to -2.0 eV. Thus, the large PDOS indicates that the Pr atoms have the major contribution, with slight contributions coming from T and O atoms, to perform the metallic and HM properties that possess by PrTO3 compounds. More- over, as expected, the applying of U energy within GGA?U computations opens the energy gap between the PDOSs of Pr and T atoms, which appears as separated spin-up and spin-down peaks near EF. It can be seen that both T-site substitution and U energy cause the shift of the PDOSs of T atoms from high-energy positions towards the conduction range near EF. Similarly, this effect can dis- tinguish also on the PDOSs of O atoms, but they shift towards EF in the valence range.
To gain a deeper insight and explore how the different states of Pr, T and O atoms contribute to the formation of HM and metallic properties of PrTO3 compounds, the
projected PDOSs of Pr (6s, 5p, 5d, 4f), T (4s, 3p, 3d) and O (2s, 2p) electrons are computed and plotted separately in supplementary figures 6S–8S, respectively. It can be seen that the spin-up metallic band in all PrTO3 compounds is formed mainly due to the contribution of Pr (4f) sub-states (supplementary figure 6S), which appear as high peaks between -2.0 and ?2.0 eV. While, for the effect of the other two atoms in this range, the localized electrons in T (3d) (supplementary figure 7S) and O (2p) (supplementary figure 8S) orbitals contribute a little to their metallic bands in PrTO3. The slight difference that is evident in the height of conduction peak is due to the effect of both T-site sub- stitutions in PrTO3and the application of GGA?U func- tional. In all PrTO3, the Pr (4f) states locate mainly in the range of-2.0 to?2.0 eV, which hybridise with T (3d) and O (2p) states. Similarly, in spin-down PDOSs, the Pr (4f) electrons (supplementary figure 6S) appear in the conduc- tion states between?1.0 and?2.5 eV in front ofEF, except for [T = Sc] within GGA. Besides, the T (3d) electrons (supplementary figure 7S) and O (2p) electrons (supple- mentary figure 8S) overlap throughEFfrom-2.0 to?2.0 eV.
Figure 8. The computed spin-up and spin-down partial density of states for Pr atom per unit cell of perovskites PrTO3[T = (a) Sc, (b) Ti, (c) V, (d) Cr, (e) Mn, (f) Fe and (g) Co] via GGA and GGA?U.
Bull. Mater. Sci. (2022) 45:69 Page 13 of 18 69
Where the strong hybridization between T (3d) states leads to a very wide energy distribution of 3d states, spreading between -2.0 and ?2.0 eV through EF. From the above results, it is possible to notice some electronic aspects that indicate the effect of the employing of GGA?U functional on the PDOSs around the EF (supplementary figures 6S–
8S), the peaks in the valence bands shift to the lower energies and the peaks in the conduction bands shift to the higher energies, causing an increase in the Eg. These observations are consistent with the previous predictions for similar perovskite compounds investigated by applying the above techniques [12,23,51]. The obtained FM-HM nature with 100% spin-polarized of conduction electrons in PrTO3 compounds suggests potential applications in the field of spintronics devices.
4.3c 3D and 2D electronic charge densities: In order to create insight into the electronic and magnetic properties and to determine the driving mechanisms that govern the bonding nature in perovskite compounds PrTO3, two- dimensional (2D) and three-dimensional (2D) forms of
charge density are computed along the (110) plane. Here, the (110) plane designs the diagonal plane of the cubic unit cell of perovskites PrTO3, where the centre (, , ) is occupied by transition-metal cation T3?, the face centres (,, 0), (, 0,) and (0,,) are filled with oxygen anions O2-and all corners of the unit cell (0, 0, 0) are filled with the rare-earth cation Pr3?. These two diagrams provide crucial evidence about the transfer and sharing of valence electrons between cations, Pr3? and T3?, and anions O2–; their contour shapes can help to distinguish the pattern of ionic and covalent bonding in PrTO3 crystals. Figure 11a and b displays the combined GGA and GGA?U plots of the 2D and 3D charge density per unit cell of PrTO3. The positions, number and height of peaks in 3D illustration confirm the distributions of partial and total charge density, whereas the 2D plots use the contour lines style. The computed 2D charge densities in figure11indicate metallic bonding between corner bonds Pr3?–Pr3? in all PrTO3 crystals with condensed contoured distributions around the cations Pr3?. Also, the contour distributions around cations Pr3?–4f and anions O2-–2p take the spherical shape, Figure 9. The computed spin-up and spin-down partial density of states for T atom per unit cell of perovskites PrTO3[T = (a) Sc, (b) Ti, (c) V, (d) Cr, (e) Mn, (f) Fe, and (g) Co] via GGA and GGA?U.
indicating the existence of Pr3?–4f–O2-–2p ionic bonding, where the charges transfer from Pr3?–4f to their neighbouring O2-–2p due to the large electronegativity difference. While the long-range bonds between cation T3?
and anion O2–take the covalent bonding character through the DE interaction [–T3? (3d)–O2-(2p)–T3?(3d)–]. The effect of T-site can be inferred from high peaks in 3D charge density, which gives evidence of how much the spin of T3? contributes to the total magnetic properties of PrTO3. In this type, it is observed that an increase in the number of valence electrons in 3d-orbital of cation T3?
leads to an increase in the concentration of the charge density between these cation T3?-3d and anion O2–-2p.
Thus, the change in charge density distributions reflects in the strength of T3?-3d–O2--2p covalent bonds in a unit cell of PrTO3. In the case of HM-FM compound, GGA?U also reveals the distribution of charge density across the unit cell which may lead to a drop in the metallic nature of PrTO3, which appears as low peaks in 3D charge density figure11b.
Moreover, the 2D lines around O2-–2p sites through GGA?U are more condensed than those via GGA.
Furthermore, to obtain a deeper understanding of the bonding properties in these compounds, Bader analysis of the atomic charge distribution based on his quantum theory of atoms in molecules [52] is performed by computing the total ionic charge density on all ions of Pr3?, T3?and O2–in their perovskites PrTO3. The Bader charge analysis is a topological technique that is often used to describe the amount of charge transfer from the cations Pr3?and T3?to their neighbouring anions O2-[52,53]. The results of partial charge transfer (in units of e per unit cell) are listed in table6. Where, the partial charges that transfer away from these cations to anions, or share between them, do not match well with the nominal partial charges on Pr3?, T3?
and O2– per unit cell of perovskites PrTO3. The amount charge difference between these sites represents the bond population in Pr3?–O2- and T3?–O2- bonds. According to these values, the total charges provide a general idea by evaluating the amount of transferred or shared charge through ionic or covenant bonding [54], where the ideal values are?3e,?3e and-2e for purely ionic and covenant bonding between cations and anions, Pr, T and O, Figure 10. The computed spin-up and spin-down partial density of states for O atom per unit cell of perovskites PrTO3[T = (a) Sc, (b) Ti, (c) V, (d) Cr, (e) Mn, (f) Fe and (g) Co] via GGA and GGA?U.
Bull. Mater. Sci. (2022) 45:69 Page 15 of 18 69
respectively. It can be seen that in all PrTO3compounds, the Pr3?cation gives the total charge of*?2.90e in their ionic bonding. T3?cations have similar behaviour with an average total charge of*?1.87e; however, their ionic bonding char- acters are strong compared to their adjacent Pr3? cations.
While the average total charge in O2–anions is *-1.80e in their bonds, this indicates that the bonding nature between cations T3?and anions O2–shows more covalent bonding than that for other cations Pr3? and anions O2–. These bonding features are mainly to the fact that the value of Pauling elec- tronegativity (table1) that increases in strength along with the series of T3?-cation, except for the Mn. The effect of U energy on the PDOSs and charge density can be noticed from the enhanced charge on O2–-2p sites through the FM exchange interaction path –T3?(3d)–O2-(2p)–T4?(3d)–, as discussed in the magnetic properties, Sections4.2aand4.2b.
Figure 11. The computed 2D (left) and 3D (right) charge densities per unit cell of perovskites PrTO3via (a) GGA and (b) GGA?U.
In 2D charge density, the atoms in PrTO3are represented as Pr (blue), T (green) and O (red).
Table 6. The partial Bader charge values (QB) on Pr3?, T3?and O2–ions (in the units of electron (e)) per unit cell of perovskites PrTO3computed using GGA and GGA?U.
QB
PrTO3 Ion GGA GGA?U
T = Sc Pr3? 2.8953 2.8919
Sc3? 2.5312 2.4148
O2- -1.7963 -1.8369
T = Ti Pr3? 2.9209 2.9230
Ti3? 2.3973 2.3424
O2- -1.8393 -1.8455
T = V Pr3? 2.9171 2.9283
V3? 2.0751 2.1305
O2- -1.8260 -1.8241
