## Abstract

Optimal materials to induce bulk photovoltaic effects should lack inversion symmetry and have an optical gap matching the energies of visible radiation. Ferroelectric perovskite oxides such as BaTiO_{3} and PbTiO_{3} exhibit substantial polarization and stability, but have the disadvantage of excessively large band gaps. We use both density functional theory and dynamical mean field theory calculations to design a new class of Mott multiferroics–double perovskite oxides *A*
_{2}VFeO_{6} (*A* = Ba, Pb, etc). While neither perovskite *A*VO_{3} nor *A*FeO_{3} is ferroelectric, in the double perovskite *A*
_{2}VFeO_{6} a ‘complete’ charge transfer from V to Fe leads to a non-bulk-like charge configuration–an empty V-*d* shell and a half-filled Fe-*d* shell, giving rise to a polarization comparable to that of ferroelectric *A*TiO_{3}. Different from nonmagnetic *A*TiO_{3}, the new double perovskite oxides have an antiferromagnetic ground state and around room temperatures, are paramagnetic Mott insulators. Most importantly, the V *d*
^{0} state significantly reduces the band gap of *A*
_{2}VFeO_{6}, making it smaller than that of *A*TiO_{3} and BiFeO_{3} and rendering the new multiferroics a promising candidate to induce bulk photovoltaic effects.

## Introduction

The lack of inversion symmetry caused by ferroelectric ordering in certain transition metal oxides can separate electrons and holes generated by photo-excitation, making these materials promising candidates for photovoltaic devices^{1,2,3,4}. However, many known ferroelectric perovskite oxides including BaTiO_{3} and PbTiO_{3} have very large band gaps (~3–5 eV)^{5}, significantly limiting their absorption efficiency in the visible frequency range. The large band gap is intrinsic: it is set by the energy difference between the Ti-*d* and O-*p* levels, which is large because Ti and O have substantially different electronegativity. Intensive research in perovskite oxides has focused on reducing band gaps while maintaining ferroelectric polarization. One approach is to replace a fraction of transition metal ions with a different cation, with one transition metal species driving ferroelectricity and the other providing lower energy states that reduce the band gap^{6,7,8,9,10,11}. Using this approach, band gap reductions of ~1 eV have been attained^{10} and a high power conversion efficiency has been experimentally achieved in Bi_{2}FeCrO_{6}
^{11}. In another method, a class of layered double perovskite oxides *AA*′*BB*′O_{6} has been theoretically proposed, in which a large in-plane polarization is found via nominal *d*
^{0} filling on the *B*-site, *A*-site cations bearing lone-pair electrons, and *A*′ ≠ *A* size mismatch; the band gap is controlled by *B*/*B*′ electronegativity difference^{12}.

In this work, we propose a simple design scheme. We introduce a new class of double perovskite oxides *A*
_{2}VFeO_{6} where *A* is a divalent cation (*A* = Ba, Pb, etc) and demonstrate that a ‘complete’ charge transfer (nominally one electron transfer) between the two transition metal ions^{13,14,15,16,17,18} can induce desirable properties for bulk photovoltaics. First-principles calculations show that while neither bulk perovskite *A*VO_{3} nor *A*FeO_{3} is ferroelectric, a ‘complete’ charge transfer occurs from V to Fe, rendering the new double perovskite oxides a Mott multiferroic: at zero temperature a ferroelectric antiferromagnet and around room temperatures a ferroelectric Mott insulator. The ferroelectric polarization is substantial, comparable to *A*TiO_{3}, but the band gap is significantly lower, smaller than that of *A*TiO_{3} and BiFeO_{3}.

We first focus on Ba_{2}VFeO_{6} (similar results are obtained for Pb_{2}VFeO_{6} and Sr_{2}VFeO_{6}, see section 4). Figure 1a and b show the atomic and electronic structures for perovskite BaVO_{3} and BaFeO_{3}, respectively. Bulk perovskite BaVO_{3} has been recently synthesized at high pressure and has been found to remain cubic and metallic to the lowest temperature^{19}. Bulk BaFeO_{3} normally crystallizes in a hexagonal structure but cubic perovskite BaFeO_{3} can be stabilized in powders^{20} and in epitaxial thin films^{21,22,23,24} and exhibits a robust ferromagnetism^{20,21,22,23,24}. Both metallic^{20, 23} and insulating^{21, 22, 24} behaviors have been reported.

Formal valence considerations imply that in BaVO_{3} the V adopts a *d*
^{1} configuration while in BaFeO_{3} the Fe is *d*
^{4}. In the double perovskite Ba_{2}VFeO_{6}, however, we expect that the large electronegativity difference between V and Fe leads to complete charge transfer from V to Fe, resulting in V-*d*
^{0} and Fe-*d*
^{5} configurations as illustrated in Fig. 1c. Similar phenomena have been predicted and observed in many different transition metal oxide heterostructures^{15,16,17,18, 25}. The particular relevance here is that the empty V-*d* shell and half-filled Fe-*d* shell are both susceptible to noncentrosymmetric distortions (for the empty *d* shell case, see refs 26 and 27 and for the half-filled *d* shell cases see refs 28–30) while Ba^{2+}-O^{2−} coupling stabilizes the ferroelectric phase over anti-ferroelectric phases, as in BaTiO_{3}
^{31}. The half filled Fe-*d* shell leads to magnetic ordering and Mott insulating behavior, while the position of the V-*d* level leads to a reduced band gap (a similar strategy to reduce band gap has been discussed in refs 12, 26, 27. Therefore as Fig. 1c shows, double perovskite Ba_{2}VFeO_{6} is predicted to be Mott multiferroic (paramagnetic ferroelectric at high temperatures and long-range magnetically ordered at sufficiently low temperatures). Furthermore, as illustrated in Fig. 1c, the band gap of double perovskite Ba_{2}VFeO_{6} is set by the filled lower Hubbard band of Fe-*d* states (strongly hybridized with O-*p* states) and empty V-*d* states (conduction band edge).

We note that the double perovskite structure is much more stable than the layered configuration proposed in ref. 12, because charge transfer generically results in substantial metal-oxygen bond disproportionation^{25}. Due to geometry consideration, the bond disproportionation inevitably induces internal strain in the layered structure but is naturally accommodated by the double perovskite structure, explaining the relative phase stability^{25}. Although previous work has suggested that rock-salt ordering of *B*-site atoms suppresses polarization in *A*
_{2}
*BB*′O_{6}
^{12, 32}, our work shows that it is possible to induce robust ferroelectricity in double perovskite oxides Ba_{2}VFeO_{6}.

In the rest of this paper we present calculations substantiating this picture. In Section II we outline the computational details. In Section III we present results for double perovskite Ba_{2}VFeO_{6}. Section IV extends the calculations to the double perovskite Pb_{2}VFeO_{6} and Sr_{2}VFeO_{6}, in which we discuss the similarities and differences. Section V is a summary and conclusion.

## Computational Details

Our first-principles calculations are performed using density functional theory (DFT)^{33} and dynamical mean field theory (DMFT)^{34}. Structural relaxation is performed within DFT. Gaps are calculated using both DFT and DFT+DMFT. It has been established that structural and magnetic properties of multiferroic oxides strongly depend on the choice of exchange correlation functionals^{5, 30, 35}. We use three exchange correlation functionals to test the robustness of our predictions: i) charge-density-only generalized gradient approximation with Perdew-Burke-Ernzerhof parametrization^{36} plus Hubbard *U* and Hund’s *J* corrections (PBE+*U*+*J*)^{37}, (ii) charge-only local density approximation with Hubbard *U* and Hund’s *J* corrections (LDA+*U*+*J*)^{37, 38}; (iii) spin-polarized generalized gradient approximation with Perdew-Burke-Ernzerhof parametrization revised for solids (sPBEsol)^{39}. In order to investigate Mottness and effects of long-range magnetic ordering, we use DMFT to study both paramagnetic and long-range magnetic ordered states.

The DFT calculations are performed using a plane-wave basis^{33}, as implemented in the Vienna Ab-initio Simulation Package (VASP)^{40, 41}. The Projector Augmented Wave (PAW) approach is used^{42, 43}. We use an energy cutoff of 600 eV. All the supercells of double perovskite oxides *A*
_{2}VFeO_{6} consist of 40 atoms to accommodate different magnetic orderings. We consider ferromagnetic ordering, [001] antiferromagnetic ordering, [010] antiferromagnetic ordering and [100] antiferromagnetic ordering (see the Supplementary Materials for their definitions). We note that since in *A*
_{2}VFeO_{6} the Fe ions form a face-centered-cubic lattice which has intrinsic ‘geometry frustration’, novel magnetism such as non-collinear magnetic ordering is possible in the ground state^{44, 45}. However, at finite temperatures, [001] antiferromagnetic ordering has been observed in various double perovskite oxides^{46,47,48,49}. In this study, we only consider collinear magnetic orderings. A 6 × 6 × 6 Monkhorst-Pack grid is used to sample the Brillouin zone. Both cell and internal coordinates are fully relaxed until each force component is smaller than 10 meV/Å and the stress tensor is smaller than 0.1 kbar.

In the PBE+*U*+*J*/LDA+*U*+*J* as well as DMFT calculations, we use *U*
_{Fe} = 5 eV, *J*
_{V} = *J*
_{Fe} = 0.7 eV, following previous studies^{50, 51}. The choice of *U*
_{V} needs comment. While *U*
_{V} of about 5 eV has been accepted in literature^{50}, we find that *U*
_{V} = 5 eV induces an off-center displacement *δ*
_{VO} in perovskite BaVO_{3}, while in experiment perovskite BaVO_{3} (which experimentally is stable only at pressures P > 15GPa) is a cubic structure^{19}. The calculated off-center displacement of V is closely related to orbital ordering (\({d}_{xy}^{1}{d}_{xz}^{0}{d}_{yz}^{0}\)) stabilized by a large *U*
_{V} in the DFT+*U* method. Therefore we use a smaller *U*
_{V} = 3 eV which stabilizes a cubic structure in perovskite BaVO_{3} in our calculations of double perovskite Ba_{2}VFeO_{6}. This ensures that a non-zero *δ*
_{VO} in Ba_{2}VFeO_{6} is not a consequence of a large *U*
_{V}, but rather is induced by charge transfer. We repeated all the DFT calculations on Ba_{2}VFeO_{6} using *U*
_{V} = 5 eV and found qualitatively similar results in structural properties. On the other hand, *U*
_{V} controls the energy level of V-*d* states, which may affect the band gap of Ba_{2}VFeO_{6}. Therefore, in our DMFT calculations, we also studied a range of *U*
_{V} (from 3 to 6 eV) to estimate the variation of energy gap in the spectral function.

We perform single-site DMFT calculations with Ising-like Slater-Kanamori interactions. The impurity problem is solved using the continuous-time quantum Monte Carlo algorithm with a hybridization expansion^{52, 53}. The correlated subspace and the orbitals with which it mixes are constructed using maximally localized Wannier functions^{54} defined over the full 10 eV range spanned by the *p*-*d* band complex, resulting in a well-localized set of *d*-like orbitals. All the DMFT calculations are performed at the temperature of 290 K. For each DMFT iteration, a total of 3.8 billion Monte Carlo steps is taken to converge the impurity Green function and self energy. In double perovskite oxides, since V-*d* states are empty, we treat V-*t*
_{2g
} orbitals with the DMFT method and V-*e*
_{
g
} orbitals with a static Hartree-Fock approximation. Because the Fe-*d* states are half-filled, we treat all the five Fe-*d* orbitals with the DMFT method. The two self energies (one for V sites and the other for Fe sites) are solved independently and then coupled at the level of self-consistent conditions.

To obtain the spectral functions, the imaginary axis self energy is continued to the real axis using the maximum entropy method^{55}. Then the real axis local Green function is calculated using the Dyson equation and the spectral function is obtained following:

where *i* is the label of a Wannier function. **1** is an identity matrix, *H*
_{0}(**k**) is the DFT-PBE band Hamiltonian in the matrix form using the Wannier basis. Σ(*ω*) is understood as a diagonal matrix only with nonzero entries on the correlated orbitals. *μ* is the chemical potential. *V*
_{
dc
} is the fully localized limit (FLL) double counting potential, which is defined as in ref. 56:

where *N*
_{
d
} is the *d* occupancy of a correlated site.

## Results for Ba_{2}VFeO_{6}

### Structural properties

We first discuss the fully relaxed atomic structure of double perovskite Ba_{2}VFeO_{6}, obtained using DFT calculations with three different exchange correlation functionals (PBE+*U*+*J*, LDA+*U*+*J* and sPBEsol). For each exchange correlation functional, we test ferromagnetic (*F*), [001] antiferromagnetic, [010] antiferromagnetic and [100] antiferromagnetic orderings (see the Supplementary Materials for precise definitions). For each case, we start from a crystal structure with rotations and tilts of VO_{6} and FeO_{6} (space group *P*2_{1}/*n*) and then perturb the V and Fe atoms along [001] or [011] or [111] directions. Next we perform atomic relaxation with all the symmetry turned off. After atomic relaxation, we find that the rotations and tilts of VO_{6} and FeO_{6} are strongly suppressed while the polarization along [001] or [011] or [111] direction is stabilized. Comparing the total energy of the three polarizations, we find the ground state of Ba_{2}VFeO_{6} has the polarization along the [001] direction. The ground state structure has tetragonal symmetry (space group *I*4/*m*). We note that based on the symmetry analysis^{57} and all the available experimental data for double perovsite oxides compiled in the review^{49}, there are altogether seven tilting patterns which are allowed in a double perovskite structure *A*
_{2}
*BB*′O_{6} and have been observed in experiment. They are: *a*
^{0}
*a*
^{0}
*a*
^{0}(*Fm*-3*m*), *a*
^{+}
*b*
^{−}
*b*
^{−}(*P*2_{1}/*n*), *a*
^{0}
*a*
^{0}
*c*
^{−}(*I*4/*m*), *a*
^{−}
*a*
^{−}
*a*
^{−}(*R*-3), *a*
^{0}
*b*
^{−}
*b*
^{−}(*I*2/*m*), *a*
^{0}
*a*
^{0}
*c*
^{+}(*P*4/*mnc*) and *a*
^{−}
*b*
^{−}
*c*
^{−}(*I*-1) (the last two tilting patterns are much rarer in experiment). Among them, the most common tilting is *a*
^{+}
*b*
^{−}
*b*
^{−}(*P*2_{1}/*n*) with over 300 compounds^{49}. We tested different initial guesses with these and other allowed symmetries, perturbed the system with ferroelectric distortions, and after relaxation we always obtained similar results. On the magnetic properties, given the *U* and *J* values, we find that the ground state is always of the [001] antiferromagnetic ordering (among the collinear magnetic orderings). Using the same methods and parameters, perovskite BaVO_{3} and BaFeO_{3} have cubic symmetry. The resulting lattice constant *a*, tetragonality *c*/*a* ratio and cation-displacement *δ*
_{
BO
} along the [001] direction (see in Fig. 1c) are shown in Table 1 for each exchange correlation functional. The full crystal structure data are provided in the Supplementary Materials. We observe that the reason that rotations and tilts of VO_{6}/FeO_{6} octahedra are strongly suppressed in Ba_{2}VFeO_{6} is due to the large ionic size of Ba ions, which is known to prohibit rotations and tilts of oxygen octahedra in perovskite Ba-compounds and to induce robust ferroelectricity in BaTiO_{3} and BaMnO_{3}
^{29, 58}.

For comparison, we also calculated the atomic structure of fully relaxed tetragonal BaTiO_{3}, a known ferroelectric perovskite. Since BaTiO_{3} is a *d*
^{0} band insulator with no magnetic properties, we do not add Hubbard *U* and Hund’s *J* correction to PBE/LDA and we use PBEsol instead of spin-polarized PBEsol (sPBEsol). We find that the calculated *c*/*a* ratio and ion-displacement (*δ*
_{VO} and *δ*
_{FeO}) of Ba_{2}VFeO_{6} are comparable to those of BaTiO_{3}. The ground state of tetragonal double perovskite Ba_{2}VFeO_{6} is an insulator (we will discuss the gap properties in details in the following subsections). The ground state of high-symmetry cubic double perovskite Ba_{2}VFeO_{6} is also an insulator (see Table 1). Therefore a switching path for ferroelectric polarization is well-defined and we can use the Berry phase method^{54} to calculate the polarization of the tetragonal structure. We find that for each exchange-correlation function the calculated polarization of Ba_{2}VFeO_{6} is comparable to that of BaTiO_{3} (see Table 1).

We comment here that our recent studies^{30, 35} of perovskite manganites show that PBE+*U*+*J* and sPBEsol yield the most accurate predictions of structural and magnetic properties of magnetic ferroelectrics, while LDA+*U*+*J* sets a conservative estimation for the lower bound of polarization. Therefore we believe that the polarization of Ba_{2}VFeO_{6} is larger than 18 *μ*C/cm^{2}, which is substantial enough to induce bulk photovoltaic effects^{4}.

### Electronic properties

The results of the previous subsection indicate that the double perovskite Ba_{2}VFeO_{6} has a noncentrosymmetric tetragonal distortion not found in the component materials bulk BaVO_{3} and BaFeO_{3}. In this section we consider the electronic reconstruction arising in this double perovskite.

Figure 2a shows the band structure of double perovskite Ba_{2}VFeO_{6} with the [001] antiferromagnetic ordering (only one spin channel is shown here), calculated using the PBE+*U*+*J* method. We see that a gap is clearly opened in Ba_{2}VFeO_{6} while using the same method with the same parameters, perovskite BaVO_{3} and BaFeO_{3} are found to be metallic with V-*d* and Fe-*d* states at the Fermi surface (see Section II in the Supplementary Materials for details). The gap opening in Ba_{2}VFeO_{6} is a strong evidence of a nominally “complete” charge transfer from V to Fe. A similar charge-transfer-driven metal-insulator transition is predicted^{14} and observed^{17} in LaTiO_{3}/LaNiO_{3} superlattices.

For comparison, we also calculated the band structure of tetragonal BaTiO_{3} using PBE (Fig. 2b). We note that while the polarization of double perovskite Ba_{2}VFeO_{6} is comparable to that of BaTiO_{3}, the band gap of Ba_{2}VFeO_{6} (0.78 eV) is significantly smaller than that of BaTiO_{3} (1.75 eV). Using other exchange correlation functionals, we find similar properties that the band gap of Ba_{2}VFeO_{6} is smaller than that of BaTiO_{3} by about 1 eV (see ‘fundamental gap’ Δ_{0} in Table 1).

For photovoltaic effects the relevant quantity is the optical gap Δ_{optical}. We calculate the optical conductivity of both Ba_{2}VFeO_{6} and BaTiO_{3} using standard methods^{59} and show the results in Fig. 2c. Due to the tetragonal symmetry, the off-diagonal matrix elements of the optical conductivity vanish and only two diagonal elements are independent (*σ*
_{
xx
} = *σ*
_{
yy
} and *σ*
_{
zz
}). For BaTiO_{3} the minimum optical gap is in the *xx* channel and is given by the direct (vertical in momentum space) gap (shown for BaTiO_{3} as the blue arrow in Fig. 2b). In BaTiO_{3} the optical gap is larger than the fundamental gap, which is indirect (momentum of lowest conduction band state differs from momentum of highest valence band state; the green arrow in Fig. 2c shows the size of the fundamental gap). The optical conductivity of Ba_{2}VFeO_{6} is also larger than its fundamental gap, which can be understood in a similar manner. If we consider (VFe) as a pseudo-atom *X*, the hypothetical single perovskite Ba*X*O_{3} would have an indirect gap (between Γ and *R*). However, the reduction in translational symmetry due to the V-Fe alternation leads to band folding which maps the original *R* point to the Γ point, leading to a direct gap of 0.8 eV at the Γ point. However the calculated optical gap is 1.1 eV (blue arrow in Fig. 2a). The difference between the direct and optical gaps is a matrix element effect: the lowest back-folded conduction band state does not have a dipole allowed transition matrix element with the highest-lying valence band state (see the Supplementary Materials for more details).

It is well-known that DFT with semi-local exchange correlation functionals substantially underestimate band gaps. Here we argue that since Ba_{2}VFeO_{6} and BaTiO_{3} have similar electronic structures (gap separated by metal *d* and oxygen *p* states), the DFT band gap underestimation with respect to experimental values is approximately a constant for BaTiO_{3} and Ba_{2}VFeO_{6}. The experimental optical gap of BaTiO_{3} is 3.2 eV and the DFT calculated value is 2.3 eV, about 0.9 eV smaller. The DFT calculated optical gap of Ba_{2}VFeO_{6} is 1.1 eV, hence we estimate the experimental optical gap of Ba_{2}VFeO_{6} is 2.0 eV, which is smaller than the optical gap of intensively investigated BiFeO_{3} (2.7 eV)^{60}.

Our results that Ba_{2}VFeO_{6} should have a smaller gap than that of BaTiO_{3} and BiFeO_{3} are also supported by physical arguments (see Fig. 3). The band gap for transition metal oxides is set by the energy difference between transition metal *d* states and oxygen *p* states. This *p*-*d* separation is a measure of the relative electronegativity of transition metal and oxygen ions. Ti and V are both first-row transition metals and in BaTiO_{3} and Ba_{2}VFeO_{6}, Ti and V both have a *d*
^{0} configuration. Because V has a larger nuclear charge than Ti, the V-*d* states have lower energies than the Ti-*d* states, which leads to a smaller band gap for Ba_{2}VFeO_{6} than for BaTiO_{3} (compare panels a and c of Fig. 3). On the other hand, the Fe *d* states are half-filled in both Ba_{2}VFeO_{6} and BiFeO_{3}, while V-*d* states are empty in Ba_{2}VFeO_{6}. Due to Coulomb repulsion and Hund’s coupling effects, adding one more electron in a half-filled *d* shell generically costs more energy than adding an electron in an empty *d* shell. Therefore the upper Hubbard band of Fe *d* states have higher energy than V *d* states, which results in a larger band gap for BiFeO_{3} than for Ba_{2}VFeO_{6} (compare panels b and c of Fig. 3).

### Estimation of critical temperatures

Double perovskite Ba_{2}VFeO_{6} is a type-I multiferroic^{61}, in which ferroelectric polarization and magnetism arise from different origins and they are largely independent of one another. This means that ferroelectric polarization and magnetism have their own critical temperatures and usually the critical temperature of polarization (*T*
_{
C
}) is higher than the critical temperature of magnetism (*T*
_{
N
})^{62}. In this subsection, we estimate *T*
_{
C
} and *T*
_{
N
} for Ba_{2}VFeO_{6}.

#### Estimation of *T*
_{
C
}

In order to estimate the ferroelectric Curie temperature *T*
_{
C
}, we use the predictor \({T}_{C}\propto {P}_{0}^{2}\) where *P*
_{0} is the zero-temperature polarization^{63}. This predictor has been successfully applied to a wide range of Pb-based perovskite ferroelectric oxides and it yields an accurate and quantitative estimation for ferroelectric *T*
_{
C
}
^{64}. We apply this predictor to our Ba-based ferroelectrics, i.e. BaTiO_{3} and Ba_{2}VFeO_{6}. Here we use tetragonal BaTiO_{3} as the reference system. The experimental Curie temperature *T*
_{
C
} for BaTiO_{3} is about 400 K^{65}. Using the DFT+Berry phase method^{54}, we can obtain the values of the zero-temperature polarization for both BaTiO_{3} and Ba_{2} VFe _{6} shown in Table 1. Therefore we estimate that *T*
_{
C
} for Ba_{2}VFeO_{6} is 473 K (PBE+*U*+*J*), 245 K (LDA+*U*+*J*) and 425 K (sPBEsol). While different exchange correlation functionals predict a range for *T*
_{
C
}, we find that *T*
_{
C
} is near or above room temperature.

#### Estimation of *T*
_{
N
}

We use a classical Heisenberg model \(E=\frac{1}{2}{\sum }_{\langle kl\rangle }{J}_{kl}{{\bf{S}}}_{k}\cdot {{\bf{S}}}_{l}\) to estimate the magnetic ordering transition temperature *T*
_{
N
}, where **S**
_{
k
} is a unit-length classical spin and 〈*kl*〉 denotes summation over nearest Fe neighbors. Here we only consider Fe-Fe exchange couplings. Because double perovskite Ba_{2}VFeO_{6} has a tetragonal structure, there are two exchange couplings of *J*
_{
kl
}: *J*
_{in} for the short Fe-Fe bonds and *J*
_{out} for the long Fe-Fe bonds. By calculating the total energy for the ferromagnetic ordering, [001] antiferromagnetic ordering and [100] antiferromagnetic ordering, we obtain that the in-plane exchange coupling *J*
_{in} is 2.5 meV (PBE+*U*+*J*), 3.7 meV (LDA+*U*+*J*) and 3.1 meV (sPBEsol); and the out-of-plane exchange coupling *J*
_{out} is 3.1 meV (PBE+*U*+*J*), 4.0 meV (LDA+*U*+*J*) and 3.7 meV (sPBEsol). The positive sign means that exchange couplings are all antiferromagnetic. Based on a mean-field theory, the estimated Néel temperature is *T*
_{
N
} = |4*J*
_{in} − 8*J*
_{out}|. The minus sign is because on a quasi face-centered-cubic lattice, every Fe atom has 8 nearest neighbors that are antiferromagnetically coupled and 4 nearest neighbors that are ferromagnetically coupled. Therefore *T*
_{
N
} is estimated to be 172 K (PBE+*U*+*J*), 200 K (LDA+*U*+*J*) and 200 K (sPBEsol). Since mean-field theories usually overestimate magnetic transition temperatures, the actual *T*
_{
N
} could be lower. An experimental determination of the magnetic ordering temperature would be of great interest.

### Effects of long-range order

The estimates for the ferroelectric and magnetic transition temperatures of Ba_{2}VFeO_{6} suggest that its actual ferroelectric Curie temperature *T*
_{
C
} is probably higher than its actual Néel temperature *T*
_{
N
}, as is the case for most type-I multiferroics^{61}. It is therefore important to ask if the magnetically disordered state remains insulating, so that the ferroelectric properties are preserved.

Here we use DFT+DMFT to study both the paramagnetic and magnetically ordered states. The spectral functions for the three magnetic states that we have considered are shown in Fig. 4 along with the spectral function for the paramagnetic state. We find that the paramagnetic state is insulating, with a gap only slightly smaller than that of the ground state with [001] antiferromagnetic ordering, indicating that double perovskite Ba_{2}VFeO_{6} is a promising candidate for Mott multiferroics^{62}. The calculated spectral functions are consistent with our schematics of Fig. 3.

We also use our DFT+DMFT methodology to investigate how the electronic structure of Ba_{2}VFeO_{6} evolves as the ferroelectric polarization is suppressed within the paramagnetic state. Figure 5 compares the spectral function of Ba_{2}VFeO_{6} in the cubic structure (i.e. no polarization) versus in the tetragonal structure (i.e. with polarization). We see that the suppression of polarization reduces the gap by about 0.2 eV. This behavior is very consistent with similar calculations on the nonmagnetic perovskite oxide SrTiO_{3} in which the presence of ferroelectric polarization can increase the band gap by up to 0.2 eV^{66}.

### Hubbard *U* dependence

Finally we discuss the Hubbard *U* dependence. As Fig. 4 shows, the conduction band edge is set by V-*d* states, which is consistent with Fig. 1c and our previous discussion of band gaps. If we change the Hubbard *U*
_{V}, it may affect the energy position of V *d* states and energy gap. To address this issue, we repeat the DMFT calculations on tetragonal Ba_{2}VFeO_{6} using several values of *U*
_{V}. The panels a of Fig. 6 show the spectral function of the double perovskite as a function of *U*
_{V}. All the calculations are performed in a paramagnetic state. We note that as *U*
_{V} increases from 4 eV to 6 eV, the band gap is almost unchanged. This is due to the fully localized limit double counting correction which nearly cancels the Hartree shift. Hence, the V-*d* and O-*p* energy separation is practically unaffected, which is very consistent with a previous DMFT study of SrVO_{3}
^{67}. If we apply the same method and same Hubbard *U* parameters to tetragonal BaTiO_{3}, the spectral functions of BaTiO_{3} (panels b of Fig. 6) show that the energy gap of BaTiO_{3} is slightly increased. Thus while we have some uncertainty relating to the appropriate values for the Hubbard *U*, our estimates for energy gap are robust: double perovskite Ba_{2}VFeO_{6} has an energy gap ~1 eV smaller than that of BaTiO_{3}. The underlying reason is the differing electronegativities of Ti^{4+} and V^{5+}.

## Related materials Pb_{2}VFeO_{6} and Sr_{2}VFeO_{6}

In this section we employ the same parameters and methods used for Ba_{2}VFeO_{6} to discuss double perovskite Pb_{2}VFeO_{6} and Sr_{2}VFeO_{6}.

We first discuss Pb_{2}VFeO_{6}. Pb has a lone pair of 6*s* electrons, which favors off-center displacements as was already shown for tetragonal PbTiO_{3}
^{68}. Due to the same mechanism, double perovskite Pb_{2}VFeO_{6} has substantial cation-displacements, tetragonality and ferroelectric polarization (see Table 2). All these values are comparable to, or even larger than those of tetragonal PbTiO_{3}. We note however that within sPBEsol the polarization of this tetragonal structure is not-well defined because the corresponding high-symmetry cubic structure is metallic and thus the obvious switching path is not available.

While tetragonal double perovskite Pb_{2}VFeO_{6} has similar structural properties to tetragonal PbTiO_{3}, the fundamental gap Δ_{0} and optical gap Δ_{optical} are both smaller than those of PbTiO_{3} by about 1 eV (all three exchange correlation functionals make qualitatively consistent predictions).

We note here that the polarization in Pb_{2}VFeO_{6} has different origin from the polarization in tetragonal PbVO_{3}
^{69}. In tetragonal PbVO_{3}, V atoms have a *d*
^{1} charge configuration and its off-center displacement *δ*
_{VO} and insulating properties are associated with orbital ordering (\({d}_{xy}^{1}{d}_{xz}^{0}{d}_{yz}^{0}\))^{70}. In double perovskite oxide Pb_{2}VFeO_{6}, charge transfer leads to a *d*
^{0} configuration on V sites and therefore the off-center displacement *δ*
_{VO} is due to hybridization between V-*d* and O-*p* states^{31}. More importantly, perovskite PbVO_{3} is not ferroelectric because along the switching path (from the tetragonal-to-cubic structure) an insulator-to-metal phase transition is observed^{71}.

Next we discuss Sr_{2}VFeO_{6}. Sr_{2}VFeO_{6} is more complicated because the ionic size of Sr^{2+} is smaller than Ba^{2+} and therefore rotations of oxygen octahedra (so-called antiferrodistortive mode, or AFD mode) can exist in Sr-compounds, such as in SrTiO_{3}. These rotations compete against ferroelectric polarization^{72}. For double perovskite Sr_{2}VFeO_{6}, even if we do not take the AFD mode into account, different exchange correlation functionals predict different structural and electronic properties. Table 3 shows that PBE+*U*+*J* predicts that the ground state is tetragonal and ferroelectric. The polarization is sizable (26 *μ*C/cm^{2}) and the DFT-calculated optical gap is 1.36 eV. On the other hand, the LDA+*U*+*J* method can not stabilize the tetragonal structure. This method predicts that ground state of Sr_{2}VFeO_{6} has a cubic structure with no off-center displacements of either V or Fe, and is metallic. The sPBEsol method can stabilize a tetragonal structure with non-zero off-center displacements *δ*
_{VO} and *δ*
_{FeO}, but the ground state is also metallic and therefore the polarization is not well-defined. We may impose epitaxial strain to induce ferroelectricity in Sr_{2}VFeO_{6}, but the critical strain strongly depends on the choice of exchange correlation functional^{30}: PBE+*U*+*J* does not require any strain to stabilize the ferroelectric state, while LDA+*U*+*J* requires a 3% compressive strain to open the gap and stabilize the tetragonal structure with a sizable polarization. A similar situation occurs for SrTiO_{3}. If we use the same methods and do not take into account the AFD mode, PBE predicts a ferroelectric ground state, while LDA and sPBE predict that the ground state is cubic (i.e. no polarization). Experimentally, SrTiO_{3} is on the verge of a paraelectric-to-ferroelectric transition^{73}. Thus we conclude that our DFT calculations indicate that double perovskite Sr_{2}VFeO_{6} is close to the paraelectric-to-ferroelectric phase boundary and probably is on the paraelectric side.

## Conclusions

In summary, we use first-principles calculations to design a new class of Mott multiferroics among which double perovskite oxide Ba_{2}VFeO_{6} stands out as a promising candidate to induce bulk photovoltaic effects because of its large polarization (comparable to BaTiO_{3}); its reduced optical gap (smaller than BaTiO_{3} by about 1 eV); and its environmentally friendly composition (Pb-free). Our work shows that charge transfer is a powerful approach to engineering atomic, electronic and magnetic structures in complex oxides. New charge configurations not found in bulk materials can occur in oxide heterostructures (including complex bulk forms such as double perovskites), and these charge configurations can produce emergent phenomena and properties not exhibited in constituent compounds. In particular, V^{5+} is very rare in single perovskite oxides (probably due to its small ionic size). We hope our theoretical predictions can stimulate further experimental endeavors to synthesize and measure these new multiferroic materials for photovoltaic applications.

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## Acknowledgements

H. Chen is supported by National Science Foundation under grant No. DMR-1120296. A.J. Millis is supported by National Science Foundation under grant No. DMR-1308236. Computational facilities are provided via Extreme Science and Engineering Discovery Environment (XSEDE), through Award No. TGPHY130003 and via the National Energy Research Scientific Computing Center (NERSC).

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H. Chen conceived the project and performed numerical calculations. A.J.M. supervised the project. H. Chen and A.J.M. analyzed the data, discussed the results and wrote the manuscript.

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Chen, H., Millis, A. Design of new Mott multiferroics via complete charge transfer: promising candidates for bulk photovoltaics.
*Sci Rep* **7, **6142 (2017). https://doi.org/10.1038/s41598-017-06396-5

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