# Advanced First-Principle Modeling of Relativistic Ruddlesden--Popper Strontium Iridates
## 4 Results and Discussions
### Electronic Structures  
#### 4.2.1 Interplay between \(U\) and SOC  
For the RP series of iridates, the spin-orbit coupling, electron-electron correlations, and spin-exchange interactions operate with comparable strengths, and depending on their subtle interplay, different phases can develop. Therefore, it is essential to first study how the interplay between \(U\) and SOC influences the electronic and magnetic states as well as the transition from one phase to another. Figure 2 shows the metal-insulator transition (MIT) and magnetic phase diagrams of three iridates as a function of Hubbard \(U\) and SOC strength. To characterize the strength of SOC, we introduced a factor \(\lambda\) in the SOC term of the Hamiltonian. In this way, one can enhance the effect of the SOC by taking \(\lambda>\)1 or can reduce it by taking \(\lambda<\)1, where \(\lambda=\)1 refers to the self-consistent reference value. The phase diagram displayed in Figure 2 show that the cooperation of \(U\) and SOC and dimensionality (\(n\)) lead to the formation of different phases: (i) In the limit \(U=0\), all compounds are nonmagnetic metals (NM-M), It can be seen from Figure 2 that, without \(U\) (\(U=0\)), the three compounds are nonmagnetic metals (NM-M), regardless of the SOC strength. (ii) In the limit of full quenching of SOC strength \(\lambda=0\), a MIT induced by the electron correlation is only achieved for nonphysically large \(U>3\) eV. For smaller \(U\), the systems enter a transient magnetically ordered metal state (AFM-M for \(n=1,2\) and a more complex magnetic-metal state for \(n=\infty\), generally denoted M-M, which will be discussed later on). (iii) The opening of the gap in the Dirac-Mott insulator, \(n=1\) and \(n=2\), is clearly driven by the cooperative action of SOC and \(U\): The general tendency is that, the stronger the SOC strength, the smaller the critical interaction \(U_{c}\) required for opening the gap. For \(\lambda=\)1, a moderate \(U\approx 1\) eV is sufficient to initiate the MIT in the \(n=1\) and \(n=2\) compounds, whereas a substantially larger value of \(U\approx 2\) eV is required for SrIrO\({}_{3}\) (\(n=\infty\)), the only non-insulating member of the considered series. As we will see, the overall predicted ground state, marked by a cross and corresponding to \(\lambda=1\) and \(U\) set to the calculated cRPA value, is in excellent agreement with expectations for each member of the series. Our calculated phase diagram for SrIrO\({}_{3}\) is consistent with the one reported by [PERSON] et al. [148]. Figure 2: Metal–insulator transition (MIT) and magnetic phase diagrams of three iridates in the \(U-\lambda\) space (\(\lambda\) characterises the spin–orbit coupling (SOC) strength, with \(\lambda=1\) being the self-consistent SOC strength). The filled circles are the computed points. The crosses correspond to the constrained random phase approximation (cRPA)-calculated \(U\) and therefore mark the first-principle-derived ground state. The right colored bar indicates the value of the band gap (in eV) in the insulating regime. NM-M, AFM-M, AFM-I, M-M, and M-I represent the nonmagnetic metal, antiferromagentic metal, antiferromagnetic insulator, magnetic metal, and magnetic insulator, respectively. To further examine the combined effects of \(U\) and SOC, we analyzed the band structures shown in Figure 3. In the nonrelativistic (\(\lambda=0\)) DFT (\(U=0\)) case (Figure 3(a1,b1,c1)), all three iridates are metallic due to partial filling of the \(t_{\rm 2g}^{5}\) bands that crosses the Fermi level. When the SOC is taken into account (Figure 3(a1,b2,c2)), the \(t_{\rm 2g}\) bands are split into the upper \(t_{\rm eff}=1/2\) and low-lying \(t_{\rm eff}=3/2\) bands (blue curves). Due to the similar crystal structures, Sr\({}_{2}\)IrO\({}_{4}\) and Sr\({}_{3}\)IrO\({}_{7}\) show overall analogous electronic dispersions: the band energies are strongly SOC split at \(\Gamma\) and lifted at the \(M\) point.