A theoretical study of new polar and magnetic double perovskites for photovoltaic applications

Searching for novel functional materials has attracted significant interest for the breakthrough in photovoltaics to tackle the prevalent energy crisis. Through density functional theory calculations, we evaluate the structural, electronic, magnetic, and optical properties of new double perovskites Sn2MnTaO6 and Sn2FeTaO6 for potential photovoltaic applications. Our structural optimizations reveal a non-centrosymmetric distorted triclinic structure for the compounds. Using total energy calculations, antiferromagnetic and ferromagnetic orderings are predicted as the magnetic ground states for Sn2MnTaO6 and Sn2FeTaO6, respectively. The empty d orbitals of Ta5+-3d0 and partially filled d orbitals of Mn/Fe are the origins of ferroelectricity and magnetism in these double perovskites resulting in the potential multiferroicity. The studied double perovskites have semiconducting nature and possess narrow band gaps of approximately 1 eV. The absorption coefficient (α) calculations showed that the value of α in the visible region is in the order of 105 cm−1. The structural stability, suitable band gap, and high absorption coefficient values of proposed compounds suggest they could be good candidates for photovoltaic applications.


Introduction
Our society constantly needs technological innovations for a more sustainable future. Due to the continuous increase of energy demand, creating clean energy resources and designing multifunctional materials have become a global issue. Optimal use of clean solar energy as a renewable and abundant source through photovoltaic (PV) conversion of solar photons into electrical energy is an environmentally friendly solution to the current energy crisis. The discovery of the bulk PV effect in ferroelectric (FE) perovskites, which is quite different from that of the traditional p-n junction PVs, has paved the way for exploring new approaches for energy conversion. 1,2 In conventional p-n junction-based PV devices, the photo-generated charge carriers are driven by the electric eld created at the depletion region. Whereas, in FE materials without the need for p-n junction, electric polarization induces a strong internal electric eld, which can be used to separate photo-generated charge carriers, thus producing a photovoltage. FEs are typically wide band gap materials (∼3-4 eV), making them unsuitable for PV applications. Therefore, reducing the band gap without affecting the FE properties is necessary to obtain and develop competitive PV devices. Most recently, multiferroic (MF) materials with the coexistence of ferroelectricity and magnetism in the same phase show promise as candidates for harvesting energy, especially from sunlight. Due to the inuence of magnetic ordering on electron-electron interaction, MFs have relatively smaller band gaps which is one of the key parameters to harness electrical power from sunlight, as well as improved photocurrents compared to conventional FEs, making them applicable as active elements in PV devices. 3 MF systems have recently been explored as alternative materials for PV applications among the next generation of solar cell technologies. [4][5][6] For instance, BiFeO 3 , as one of the most studied MF systems with a band gap of ∼2.6-2.7 eV, has been intensively investigated in view of its use in PV devices. [7][8][9] Nevertheless, a lower band gap facilitates the improved PV performance of such compounds as recently Bi 2 FeCrO 6 with a tuneable band gap in the range of ∼1.4-2.1 eV is highly promising for PV applications. 4,10,11 These results draw an intense theoretical research interest to look over a broad range of possible MFs for solar cells. On the other hand, to solve the metal toxicity and instability issues of conventional perovskite solar cells, lead-free double perovskites (DPs) are a noteworthy choice because of their favourable PV properties, intrinsic chemical stability, and environmental friendliness. The lead-based MFs such as PZT, PLZT, and PbTiO 3 are extensively studied. [12][13][14] Nevertheless, due to health and environmental concerns, the practical applications of these materials are also limited. Oxide materials, including perovskites and DPs, can be cost-friendly and widely available. Their properties like electrical energy, ferromagnetic (FM) and FE effects, and bandgap energy are in a favorable zone. Given that MF materials for PV applications are still not thoroughly investigated, it is strategically important to focus on designing these materials from an electronic point of view to engineering their band gap as a powerful technique for designing new materials and devices. In this work, we theoretically explored the structural, electronic, magnetic, and optical properties of new DPs Sn 2 MnTaO 6 (SMTO) and Sn 2 FeTaO 6 (SFTO), based on density functional theory (DFT). In search of multiferroics, we have designed polar and magnetic double perovskites in which the polarization can be attributed to the second-order Jahn-Teller (SOJT) effect of Ta 5+ : d 0 ions and magnetism is due to the 3d transition metals Mn and Fe. Our calculations show that SMTO and SFTO possess signicant structural distortion and semiconducting behaviour with band gaps within the optimal range of PV devices. To the best of our knowledge, there is no experimental work, and this is the rst theoretical prediction for SMTO and SFTO compounds, so we hope this research serves as a reference and guidance for applying these new materials for PV applications.

Numerical methods
We performed the rst-principles simulations using the SIESTA code 15 in which the wave functions of valence electrons, that is, the solutions to the Kohn-Sham Hamiltonian eigenvalue problem, were expanded by double zeta polarization (DZP) basis set. 16 SIESTA uses a linear combination of atomic orbitals (LCAO) with localized basis set from spherical functions. The generalized gradient approximation (GGA) in Perdew-Burke-Ernzerhof (PBE) form 17 was employed to approximate the exchange-correlation functional parameters. Following the convergence test, we sampled the Brillouin zone (BZ) by 7 × 7 × 5 Monkhorst-Pack k points, 18 and chose the cut-off energy 300 Ry. For the geometry optimization, the maximum atomic force was set to 0.01 eVÅ −1 and 1 × 10 −4 eV energy difference for the convergence criteria. [19][20][21] To investigate the electronic and magnetic properties of SMTO and SFTO DPs, including transition metals with localized d orbitals that need strong correlation electron systems, GGA + U calculations were carried out, which is a better description than GGA. 22 Here U was set to 4.0 eV and 5.0 eV for the 3d ions Mn and Fe, while the 5d Ta ions were weakly correlated with U = 1.0 and 2.0 eV, which are reasonable for 3d and 5d ions according to the literature. [23][24][25] To obtain the optical response, one needs to calculate the optical matrix element showing the transition probability of electrons from a lower-energy state to a higher state upon the absorption of an electromagnetic wave. This is possible by sandwiching the momentum operator of an electron between two Kohn-Sham electronic states and taking into account the energy conservation considerations and lled and empty states probabilities for electrons.
where j KS {i,j} are the Kohn-Sham Hamiltonian's eigenstates (wave functions), ħu is the incident photon energy, P is the electron momentum operator, f is the Fermi distribution, e is the electron's electric charge, V is the unit cell volume, m is the electron mass, and i and j denote lled initial and empty nal states, respectively. Furthermore, d indicates Dirac's delta distribution, u is the (angular) irradiation frequency, and E i and E j denote the energy of initial and nal states. The function 3 2 thereby denes the material's optical absorption behavior, and one denes the overall, frequency-dependent, optical response function by where the real part 3 1 relates to the imaginary part 3 2 via the Kramers-Kronig relationship, [26][27][28] and it describes the scattering power of a material. The absorption coefficient can be obtained from the real and imaginary parts of the dielectric function as a(u) = 2uk(u)/c, and k is the refractive index' imaginary part, which relates to the relative permittivity 3 as The optical properties, including dielectric functions and absorption spectrum, were computed using a denser k mesh of 30 × 30 × 30 elements.

Geometry optimization and stability of perovskite structures
We started our calculations by nding the ground state structures of SMTO and SFTO DPs. In that light, we considered three different crystal structures of cubic (space group Pm 3m, a = 8.09), tetragonal (space group I4/m; a = b = 5.50, c = 8.00), and monoclinic (space group P21/n; a = 5.43, b = 5.74, c = 7.90; 85.59, 90.00, and 90.00), leaning on literature reports for similar double perovskites. [30][31][32] Aer performing structural optimizations, including atomic positions and lattice parameters, we found that the considered monoclinic structure yields lower total energy than the tetragonal and cubic structures. The calculated structural parameters listed in Table 1 indicate that aer optimization, the symmetry is further decreased to triclinic symmetry. Therefore, the triclinic (space group P1) was determined to be the ground-state crystal structure of both compounds. Fig. 1(a) displays the relaxed conventional unit-cell of SMTO and SFTO containing 20 atoms having two formula units. For application analysis, the studied materials must be structurally and thermodynamically stable. To assess the geometrical stability and perovskite formability of A 2 BB ′ O 6 system, we calculated the octahedral factor m and the Goldschmidt tolerance factor t presented in eqn (3) (ref. 33). The octahedral factor m is the ratio between the ionic radius of B-site cation (r B ) and that of oxygen ion (r O ), predicting the stability of the BO 6 octahedra. The tolerance factor t applies the ionic radii of A, B, B ′ and O ions (r A , r B , r B ′, and r O ) to estimate the tolerable size mismatch of ions and predict the formability of the stable perovskite crystal.
Calculating the octahedral factor m and tolerance factor t, derived from simple geometrical principles, is an easy way to predict the basic formability of any new material with targeting perovskite structure based on chemical composition. The conditions proposed for these two parameters suggesting a stable perovskite structure are 0.410 < m < 0.895 and 0.813 < t < 1.107. [34][35][36] The tolerance factor t value gives an initial idea of the crystal structure type. The ideal cubic symmetry (space group Pm 3m) with B-O-B ′ bonding angles of 180°is usually predictable when the value of t is close to unity (1.0 < t < 1.05). The deviation of t from 1.0 causes different crystal structures as for t > 1.05, a hexagonal symmetry (space group R 3m) is expected. If 0.97 < t < 1.0, the DP compounds have a tetragonal symmetry (space group I4/m), whereas with a decreasing tolerance factor t < 0.97, the materials tend to crystallize into distorted perovskite structures such as an orthorhombic symmetry (space group Pnma or Pbnm) or a monoclinic symmetry (P21/n). 25,37 The deviation from the cubic symmetry occurs due to the octahedral distortion, which is oen accompanied by tension and compression in the three bonds A-O, B-O, and B ′ -O due to the mismatch compensated by tilting and rotation of BO 6 and B ′ O 6 octahedra. These distortions occur to obtain a more favourable conguration from the energetic point of view. Using available ionic radii, 38 our calculations gave the value of 0.46 and 0.89 for m and t, respectively for SMTO and SFTO, which are within the recommended range for stable perovskite structures. Such a value for t is generally observed for signicantly distorted perovskites. 39 Since the Mn 3+ and Fe 3+ ions have identical ionic radii (0.645Å), for the same cations of Sn and Ta at A and B ′ sites, respectively, SMTO and SFTO DP have the same value of m and t.
The calculated t values suggest distorted structures for our investigated compounds. The regularity of the octahedra distortion can be quantied using the octahedral bond-length distortion parameter that can be obtained as where the individual B(B ′ )-O bond distances are given by d n and hd n i is the average of B(B ′ )-O bond lengths. 40 The obtained distortion parameter values for Mn/FeO 6 and TaO 6 octahedra in SMTO and SFTO are listed in Table 2. The results show that MnO 6 octahedra undergo a higher distortion than FeO 6 , which can be attributed to the Jahn-Teller (JT) effect due to the electron conguration of Mn 3+ : 3d 4 . The JT theorem predicts that removing the two-and three-fold electronic degeneracy of highenergy e g (d z 2,d x 2 −y 2) and low-energy t 2g (d xy , d xz , d yz ) orbitals, respectively, provide an energetic stabilization geometry resulting in an octahedral distortion. 41 The value of Dd for signicantly distorted octahedra is in the range of 3.3 × 10 −3 to 5.0 × 10 −3 (ref. 42). Thus, as shown in Table 2, the calculated TaO 6 octahedral distortion in SMTO and SFTO compounds are comparatively high due to the non-centrosymmetric displacement of Ta 5+ (3d 0 ) atoms in their respective octahedra (SOJT effect). This off-centre shi of the Ta atom within TaO 6 octahedral may form a permanent dipole moment in the crystal and provide the main deriving force for ferroelectricity. The  Table 2).

Magnetic properties
Aer ensuring the ground state crystal structure stability, we calculated the magnetic and electronic properties of the predicted compounds using the optimized structural information. As mentioned before, to capture the correct magnetic and electronic properties, it is crucial to include the electronelectron correction beyond GGA at Mn/Fe (3d) and Ta (5d) sites by applying Hubbard correction. We utilized spin-polarized DFT calculations with the GGA + U techniques to assess the magnetic and electronic properties of the chosen compounds. Here, we tested values of U = 4, 5 eV for Mn/Fe-3d and U = 1, 2 eV for Ta-5d. We note that the magnetic and electronic behavior do not strongly depend on the choice of U values, as there is a convergence for total magnetic moments and band gaps with respect to the U values. Therefore, for the sake of simplicity, we only show the results of U 41 (U Mn,Fe = 4 eV and U Ta = 1 eV).
To conrm the stable magnetic state of SMTO and SFTO DPs in their triclinic crystal phase, total energy calculation was performed in which we assume different magnetic ordering of FM, antiferromagnetic (AFM1, 2), and ferrimagnetic (FiM) by the spin alignments of the Mn/Fe and Ta ions Fig. 2. The total energy calculations show the stability of AFM2 state over FM state by a relatively small energy difference of 0.20 meV for SMTO, while the FM phase has the minimum energy for SFTO. Table 3 summarizes the calculated total energy differences between FM and AFM orderings (DE = E FM − E AFM ), total and local magnetic moments for the investigated compounds. The Mn and Fe atoms dominate the magnetic behaviour, with magnetic moments of 5.03 m B and 3.98 m B in SMTO and SFTO, respectively. The zero-values for the total magnetic moments and local magnetic moments with different signs conrm that the Mn ions order antiferromagnetically with respect to the Ta ions in SMTO. The Mn 1 spin moment is completely compensated by the Mn 2 spin moment. In contrast, the magnetic moments of Fe and Ta are parallel in SFTO. From the formability point of view, the charge balance between cations and anions must be maintained to form a perovskite structure. The calculated magnetic moments at Mn/Fe and Ta sites indicate the stabilization of nominal +3 and +5 states with d 4 /d 5 and d 0 occupancies, respectively, keeping the charge neutrality in the ionic picture of Sn 2+ 2 (Mn/Fe) 3+ Ta 5+ O 6 2− . It puts the transition metals in the electron conguration of Mn 3+ (3d 4 : t 3 2g e 1 g ), Fe 3+ (3d 5 : t 3 2g e 2 g ), and Ta 5+ : 5d 0 , resulting in an AFM superexchange between Mn-3d half-eld states and a FM superexchange between Mn/Fe-3d half-eld and Ta-5d empty orbitals following the Goodenough-Kanamori rule. 46,47 The dstates of B (B ′ )-site cations split into low-energy t 2g triplet and high-energy e g doublet subsets driven by the crystal eld in the octahedral environment. In the +5-oxidation state, Tad states remain empty, and the small value of magnetic moment comes from the strong hybridization of Ta-5d with O-2p states. Total energy calculations show that AFM2 and FM orderings are the magnetic ground states for SMTO and SFTO, respectively, resulting from a superexchange interaction between Mn/Fe-3d half-field and Ta-5d empty orbitals following the Goodenough-Kanamori rule.

Electronic properties
We calculated the spin-polarized band structures along the high symmetry direction of BZ, total and partial DOS (PDOS) (Fig. 3).
Both spin channels comprise band gaps, resulting in a semiconducting nature for both compounds. For SMTO, there are indirect band gaps for the spin-up and spin-down channels, of 1.02 eV and 1.17 eV, respectively, as the valence band maxima   Table 3 for details.
(VBM) and conduction band minima (CBM) are located at the V 2 and T 2 symmetry points of BZ, respectively. SFTO has the same behavior with indirect band gaps of 1.08 eV and 1.06 eV for the spin-up and spin-down channels, respectively (along the V 2 / T 2 path).
For SMTO, we nd that the most contribution in the valence band from −7.0 to −2.0 eV below the Fermi level comes from Mn-3d and O-2p states, while the Ta ion does not contribute signicantly to PDOS in this region. Above the Fermi level, in the energy region between 0.0 and 2.0 eV, the Ta-5d states have the most contribution in the conduction band. In this energy region, the hybridization between O-2p and Ta-5d orbitals causes the non-centrosymmetric displacement of Ta 5+ ions in their respective octahedra. Below the Fermi level, Ta-5d states are relatively empty, conrming the reported Ta

Electron charge density analysis
We calculated the electron charge density, to describe the chemical bonding associated with the SMTO and SFTO DPs and charge sharing between the atoms, using the GGA + U approximation. The charge density plots describe the charge distribution difference between the optimized and initial structures. They show the loss or gain of the electron on a specic atom, blue (−0.01) and red (+0.01) in the color contour, respectively (Fig. 4). The maximum density is located in the oxygen atoms' vicinity, due to their large electronegative nature. It can also be observed that Ta atoms lose more electrons compared to the Mn/Fe atoms, agreeing with the assigned +5 and +3 valence states, respectively. The shared electrons between oxygen and Mn/Fe and Ta atoms show the covalent nature of the Mn/Fe-O and Ta-O bonds. Furthermore, the overlapping Mn, Fe, Ta and O charge contours evidence the p-d hybridization. The nonspherical shapes of the transition metal atoms Mn/Fe and Ta indicate that the d states are partially lled. Moreover, Fig. 4 visualizes the distorted and tilted Mn/FeO 6 and TaO 6 octahedra.

Optical properties
Revealing the optical properties of materials is essential to predict their possible application and device performance in optoelectronics and PVs. The optical response of matter can be described by the frequency-dependent complex dielectric function, visualizing the optical transition between the occupied and unoccupied orbitals (see Section 2 for details).
To obtain a deeper insight into the electronic band prole, and to analyse the optical behaviour, including the dielectric functions and absorption coefficients, we calculated the spinpolarized optical parameters for the SMTO and SFTO DPs, using the GGA + U method (U Mn,Fe = 4.0 eV and U Ta = 1.0 eV). Due to the studied compounds' asymmetric geometrical structure, we calculated the optical properties along x-, y-and zdirections. Our results, however, revealed identical values in different directions, not revealing any signicant anisotropy. Hence, here we only represent the optical properties along the xaxis. The calculated spin-dependent real and imaginary dielectric constants, functions of energy in the energy range of 0.0 eV to 8.0 eV above the Fermi level, are plotted in Fig. 5(a). A material behaves as a dielectric for 3 1 > 0 and as a metal for 3 1 < 0. In Fig. 5, SMTO represents a similar dielectric behavior for the spin up and down channels due to its AFM nature. The value of the real dielectric function at zero energy shows the static dielectric constant 3 1 (0) that explains the reverse relation between the semiconductor band gap as 3 1 (0) = 1 + (ħu/E g ) 2 (ref. 49). Therefore, the higher value of 3 1 (0) reveals a smaller band gap. For SMTO, the spin up channel has a slightly higher value of 3 1 (0) than the spin down channel indicating a lower band gap value as reported in Table 3. The imaginary dielectric constant is small (nearly zero) for both compounds up to certain energy. The peaks in the imaginary part of the dielectric function reveal the interband transition from the valence bands to the respective conduction bands. In SMTO and SFTO systems, the imaginary dielectric function has three main peaks at around 1.5, 2.5, and 4.5 eV. The low energy peaks are associated with the transition between the O-2p valence and the Ta-5d conduction bands. While the transition between O-2p valence band and Mn/Fe-3d conduction band caused the peaks located in higher energies. The rst peak in 3 2 of both compounds at 1.5 eV is attributed to the transitions from O-2p in the energy range of −1.5 to −0.5 eV to the rst small peak of Ta-5d exactly above the Fermi level in the range 0.5 to 1 eV. The second peak at energy 2.5 eV in the 3 2 diagram, which is stronger than the rst one, is ascribed to the transitions from O-2p in the energy range of −1.5 to −0.5 eV to a strong Ta-5d peak in the energy range of 1.0 to 2.0 eV in the DOS. A signicant difference between the two materials is the presence of additional Fe-3d states in the spin down channel of SFTO in the energy range between −2.0 and −1.0 eV, causing the additional electron transitions to O-2p states above the Fermi level. It gives rise to a stronger contribution of the spin down channel in the 3 2 curve compared to the spin up channel at the energy ∼3.0 eV in SFTO. To investigate the visible-light harvesting capacity, we calculated the DPs' absorption coefficients, and display them alongside the AM1.5g solar irradiance reference spectrum 48 (Fig. 5(b)). The absorption coefficient is zero when the energy is in the range of 0.0-1.0 eV, and there is no electric transition because the photon energy is below the SMTO and SFTO band gaps (∼1.0 eV). For a photon energy larger than the band gap value (∼1.0 eV), the absorption coefficient increases, reaching the rst peak with the value of around 0.5 × 10 5 cm −1 at 1.8 eV. The secondary peak appears at the 2.5-3.0 eV energy range, with an absorption coefficient value of 2.0 × 10 5 cm −1 . The absorption spectrum increases with increasing photon energy, aer around 4.0 eV. The a value in the visible region is in the order of 10 5 cm −1 , conrming that SMTO and SFTO are promising candidates for PV applications. 50,51

Conclusions
We performed DFT calculations to study the crystal structure, electronic, magnetic, and optical properties of SMTO and SFTO DPs. Our structural optimizations, using total energy calculations under the GGA-PBE scheme, indicate triclinic crystal structures for both compounds, with signicant polar distortions of BO 6 and B ′ O 6 originating from the JT effect. Negative formation energies furthermore assure their thermodynamic stability. We studied the magnetic, electronic, and optical properties with the GGA + U method. The magnetic phase stability of SMTO and SFTO in different magnetic congurations has also been discussed, and an AFM and FM nature was predicted to be their magnetic ground states, respectively. We showed that these materials have a semiconductor behavior with an energy gap of around 1 eV according to the density of state calculations. The optical properties, including the dielectric functions and absorption coefficients, were studied. The structural stability, suitable band gap, and absorption coefficient values of studied compounds suggest they could be good candidates for PV applications.

Conflicts of interest
There are no conicts to declare.  5 (a) Spin-dependent real 3 1 and imaginary 3 2 parts of dielectric function, (b) absorption spectra, as a function of photon energy calculated for SMTO and SFTO systems using GGA + U, together with the AM1.5g solar irradiance spectrum. 48 All spin states exhibit a non-vanishing absorption behaviour within a wide, PV-relevant spectral range between 1.3 and 4.2 eV.