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Electronic structure and magnetic interactions of a Tb adatom on graphene are investigated from first principles using combination of density functional theory and multiconfigurational quantum chemistry techniques including spin-orbit coupling. We determine that the six-fold symmetry hollow site is the preferred adsorption site and we investigate electronic spectrum for different adatom oxidation states including Tb$^{3+}$, Tb$^{2+}$, Tb$^{1+}$, and Tb$^{0}$. For all charge states, the Tb $4f^8$ configuration is retained with other adatom valence electrons being distributed over $5d_{xy}$, $5d_{x2+y2}$, and $6s/5d_0$ single-electron orbitals. We find strong intra-site adatom exchange coupling that ensures that the $5d6s$ spins are parallel to the $4f$ spin. For Tb$^{3+}$, the energy levels can be described by the $J=6$ multiplet split by the graphene crystal field. For other oxidation states, the interaction of $4f$ electrons with spin and orbital degrees of freedom of $6s5d$ electrons in the presence of spin-orbit coupling results in the low-energy spectrum composed closely lying effective multiplets that are split by the graphene crystal field. Stable magnetic moment is predicted for Tb$^{3+}$ and Tb$^{2+}$ adatoms due to uniaxial magnetic anisotropy and effective anisotropy barrier around 440 cm$^{-1}$ controlled by the temperature assisted quantum tunneling of magnetization through the third excited doublet. On the other hand, in-plane magnetic anisotropy is found for Tb$^{1+}$ and Tb$^{0}$ adatoms. Our results indicate that the occupation of the $6s5d$ orbitals can dramatically affect the magnetic anisotropy and magnetic moment stability of rare earth adatoms.

Commencing from the centrosymmetric MnRMnSbO$_6$ compound, we explore the realm of magnetic polar double-double perovskite oxides characterized by significant ferroelectric polarization. Employing symmetry operations, first-principles methodologies, and Monte Carlo simulations, our investigation delves into the structural, magnetic, ferroelectric, and electronic attributes of the polar LaFeMnNiO$_6$ and LaTiMnNiO$_6$ compounds. The structural analysis uncovers that the paraelectric-ferroelectric phase transition is intricately linked to the Fe/Ti-displacement of square planar Fe/TiO$_4$. Notably, the magnetic LaFeMnNiO$_6$ and LaTiMnNiO$_6$ compounds demonstrate robust ferroelectric polarizations, measuring 20.0 $\mu$C/cm$^2$ and 21.8 $\mu$C/cm$^2$, respectively, accompanied by minimalist forbidden energy gaps of 1.40 eV and 1.18 eV using the GGA+U method. Furthermore, we pinpoint elevated magnetic transition temperatures for these compounds. Additionally, our study scrutinizes the energies associated with diverse spin configurations and identifies potential minimum decomposition pathways into stable oxides. This comprehensive analysis ensures the meticulous formation of the LaFeMnNiO$_6$ and LaTiMnNiO$_6$ compounds.

We employed first-principles density functional theory calculations guided by group-theoretical analysis and demonstrated the control of insulator-metal-insulator transition, polarization and two sublattice magnetization in (LaFeO$_3$)$_1$/(CaFeO$_3$)$_1$ superlattice via. multi structural mode coupling i.e., 'multimode coupling'. We have discovered a polar A-type charge disproportionation mode, Q$_{ACD}$ (analogous to the A-type antiferromagnetic ordering), and found that it couples with the trilinear coupling, $Q_{Tri}$ mode (common in $Pnma$ perovskite oxides and involves three structural modes), and lowers the symmetry further. By tuning the strength of the coupling between the participating modes, the polar metallic phase, polar zero bandgap semiconducting, and polar insulating phases can be obtained. Here, $Q_{Tri}$ switches the polarization direction, whereas, Q$_{ACD}$ can trigger insulator-metal-insulator transition along with the polarization switching. The mechanism is true for any transition metal superlattices constituted with $Pnma$ building blocks and with partially filled $e_g$ or $t_{2g}$ electron(s) at the transition metal sites.

This work investigates the origins of cation ordering of double perovskites using first-principles theory computations combined with machine learning (ML) and causal relations. We have considered various oxidation states of A, A', B, and B' from the family of transition metal ions to construct a diverse compositional space. A conventional framework employing traditional ML classification algorithms such as Random Forest (RF) coupled with appropriate features including geometry-driven and key structural modes leads to highly accurate prediction (~98%) of A-site cation ordering. We have evaluated the accuracy of ML models by entailing analyses of decision paths, assignments of probabilistic confidence bound, and finally introducing a direct non-Gaussian acyclic structural equation model to investigate causality. Our study suggests that the structural modes are the most important features for classifying layered, columnar and rock-salt ordering. For clear layered ordering, the charge difference between the A and A' is the most important feature which in turn depends on the B, B' charge separation. Based on the outputs from ML models, we have designed functional forms with these features to derive energy differences forming clear layered ordering. The trilinear coupling between tilt, rotation, and A-site antiferroelectric displacement in Landau free-energy expansion becomes the necessary condition behind the formation of A-site cation ordering.