Supplementary Information: Magnetic, Electronic and Optical Properties of Double Perovskite Bi2FeMnO6

Double perovskite Bi2FeMnO6 is a potential candidate for the single-phase multiferroic system. In this work, we study the magnetic, electronic, and optical properties in BFMO by performing the density functional theory calculations and experimental measurements of magnetic moment. We also demonstrate the strain dependence of magnetization. More importantly, our calculations of electronic and optical properties reveal that the onsite local correlation on Mn and Fe sites is critical to the gap opening in BFMO, which is a prerequisite condition for the ferroelectric ordering. Finally, we calculate the x-ray magnetic circular dichroism spectra of Fe and Mn ions (L2 and L3 edges) in BFMO.


I. COMPUTATIONAL METHOD
To study the electronic, optical and magnetic properties of double perovskite Bi 2 FeMnO 6 , we applied density functional theory (DFT) as implemented in the full-potential LAPW code Wien2k. 1 The unstrained crystal structure of BFMO is shown in Fig. 1(a) in the main text. Lattice parameters considered for this calculation were taken from the experimental values. 2 The lattice parameters for the double perovskite BFMO unit cell are a = b = 5.5579Å, c = 8.06Å, and α = β = γ = 90 • . Electronic relaxation was achieved self-consistently using a 10×10×10 kpoint mesh in the Brillouin zone. Spin polarized calculations were performed using an exchangecorrelation functional within the generalized gradient approximation. 3 The spin-orbit coupling for the valence electrons, was taken into account within a second variational approach. 4,5 In our calculation, we used the muffin-tin radius of 2.5a 0 for Bi, 1.99a 0 for Fe and Mn, and 1.71a 0 for O, and the cutoff RK max = 7.0.
For the x-ray XMCD calculations, 6 the core-states were obtained fully relativistically by solving the Dirac equations. 7 For L 2 and L 3 edge XMCD calculations of Fe and Mn sublattices, the P 1/2 and P 3/2 core level energies and core-hole lifetime broadening were obtained from experimental values. 8 For Fe, P 1/2 and P 3/2 core level energies are -49.95470 eV and -50.873489 eV, while the core-hole broadening is 0.37 eV and 0.2 eV, respectively. For Mn, P 1/2 and P 3/2 core level energies are -45.04057 eV and -45.80827 eV, while the core-hole broadening is 0.34 and 0.34 eV, respectively.
To account for the onsite Coulomb interaction in Fe and Mn ions, we used the mean-filed approximation of GGA+U method. To study the electronic structure of BFMO, we calculated the spin resolved band structure and DOS. The band structure is calculated along the high symmetry convention. The Brillouin zone and high symmetry lines are shown in Fig. 1(b) in the main text.
For the optical conductivity calculations, the momentum matrix elements were constructed and integrated over the brillouin zone following the prescription of Draxl et al.. 10 The interband and intraband contributions to the imaginary part of the dielectric tensors (ε 2 ) for both spin up and down were then computed. The real part of ε (ε 1 ) is then computed using Kramers-Kronig transformation. These quantities were then used to calculate the optical conductivity. 10 In this work, we calculated the σ xx = σ yy and σ zz contribution of the optical conductivity ignoring all the off-diagonal terms.

II. EFFECTS OF THE ONSITE COULOMB U ON MAGNETISM AND BAND GAP
Corresponding to Fig. 2(b-c) in the main text, we present in Tables S-I and S-II the net moment and gap in BFMO, as a two-dimensional parameter space formed by (U Fe ,U Mn ). These results provide a path forward on the optimization of magnetic moment through the strain in insulating BFMO crystals as well as how it can be affected by the electronic correlation of Fe and Mn ions.

III. XMCD SUM RULES
The magnitude of spin and orbital moments can be obtained by the well known XMCD sum rules: [11][12][13] These sum rules are expressed as, and where and n 3d is the number of 3d electrons in Fe and Mn ions. From the measured XMCD spectra, these quantities can be easily extracted. In Figs. S1 and S2, we present theoretical L 2 and L 3 XMCD spectra as well as the sum rule components of Fe and Mn ions.
These XMCD sum rules give rise to spin and orbital moment of Fe and Mn. In Table S-III, we list all of our calculated sum rule components p, q, r as well as the orbital and spin magnetic moments along with other quantities. * To whom correspondence should be addressed.