# Advanced First-Principle Modeling of Relativistic Ruddlesden--Popper Strontium Iridates
## 4 Results and Discussions
### Electronic Structures
#### 4.2.2 Quantification of Relevant Energy Scales  
To quantify the degree of electronic correlations of three iridates and to determine the ground state of each compound in the phase diagram, we calculated the Hubbard \(U\) and exchange \(J\) by cRPA. The Ir-\(t_{\text{2g}}\) states are chosen as a target correlated subspace. This choice has been justified by a good match between the nonmagnetic DFT bands and the corresponding ones for the Ir-\(t_{\text{2g}}\) manifold obtained by Wannier interpolation (see Figure 4). Table 1 summarizes the obtained matrix elements of \(U_{ij}\) and \(J_{ij}\). According to Equation (13), we obtained averaged \(U\) values of 1.82, 1.67, and 1.37 eV for Sr\({}_{2}\)IrO\({}_{4}\), Sr\({}_{3}\)Ir\({}_{2}\)O\({}_{7}\), and SrIrO\({}_{3}\), respectively. This is expected, since the gradual increase in bandwidth associated with the increase in dimensionality leads to enhanced screening. With these cRPA values of \(U\), the corresponding DFT+\(U\)-predicted ground state for \(n=1,2\), and \(\infty\), marked with a cross in the phase diagram shown in Figure 2, very well reproduces the experimentally detected electronic ground state: insulating for \(n=1,2\) and being a metal for \(n=\infty\).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & \(\mathbf{U}_{ij}\) & & & \(\mathbf{J}_{ij}\) & \\ \hline Sr\({}_{2}\)IrO\({}_{4}\) & \(d_{\text{yz}}\) & \(d_{\text{zx}}\) & \(d_{\text{xy}}\) & \(d_{\text{yz}}\) & \(d_{\text{zx}}\) & \(d_{\text{xy}}\) \\ \(d_{\text{yz}}\) & 2.30 & 1.72 & 1.57 & – & 0.23 & 0.22 \\ \(d_{\text{zx}}\) & 1.72 & 2.30 & 1.57 & 0.23 & – & 0.22 \\ \(d_{\text{xy}}\) & 1.57 & 1.57 & 2.03 & 0.22 & 0.22 & – \\ \hline Sr\({}_{3}\)Ir\({}_{2}\)O\({}_{7}\) & \(d_{\text{yz}}\) & \(d_{\text{zx}}\) & \(d_{\text{xy}}\) & \(d_{\text{yz}}\) & \(d_{\text{zx}}\) & \(d_{\text{xy}}\) \\ \(d_{\text{yz}}\) & 2.16 & 1.58 & 1.43 & – & 0.23 & 0.22 \\ \(d_{\text{zx}}\) & 1.58 & 2.16 & 1.43 & 0.23 & – & 0.22 \\ \(d_{\text{xy}}\) & 1.43 & 1.43 & 1.85 & 0.22 & 0.22 & – \\ \hline SrIrO\({}_{3}\) & \(d_{\text{yz}}\) & \(d_{\text{zx}}\) & \(d_{\text{xy}}\) & \(d_{\text{yz}}\) & \(d_{\text{zx}}\) & \(d_{\text{xy}}\) \\ \(d_{\text{yz}}\) & 1.78 & 1.21 & 1.19 & – & 0.22 & 0.22 \\ \(d_{\text{zx}}\) & 1.21 & 1.73 & 1.22 & 0.22 & – & 0.22 \\ \(d_{\text{xy}}\) & 1.19 & 1.22 & 1.74 & 0.22 & 0.22 & – \\ \hline \end{tabular}
\end{table}
Table 1: On-site Coulomb and exchange interactions (in eV) calculated by cRPA for three RP iridates. \(U_{ij}=U_{ijij}(\omega=0)\) and \(J_{ij}=U_{ijij}(\omega=0)\) with \(i\) and \(j\) representing \(t_{\text{2g}}\)-like Wannier orbitals. See Equation (12) for the notations used. Reproduced with permission from Reference [100]. Copyright 2018 by the American Physical Society.  
Figure 4: The nonmagnetic DFT bands (black line) superposed with Wannier interpolated bands (red dashed line). Reproduced with permission from Reference [100]. Copyright 2018 by the American Physical Society.  
To complete the quantification of the other relevant interactions, we also estimated the crystal field energies \(10 Dq\), the SOC constant \(\lambda_{\text{soc}}\), and the band gap. The data are shown in Table 2. The \(10 Dq\) is evaluated from the central mass of density of states between the \(e_{g}\) and \(t_{\text{2g}}\) bands without \(U\) and SOC. The \(\lambda_{\text{soc}}\) is approximated as \(\lambda_{\text{soc}}\simeq 2/3\xi_{soc}\), with \(\xi_{soc}\) being the SOC energy, which is estimated from the total energy difference with and without the inclusion of SOC. One can observe that crystal field energies are the largest energy scale (over 3 eV), and it is thus reasonable to assume the \(t_{\text{2g}}\) states as the effective total angular momentum \(L_{\text{eff}}=-1\). The SOC constant \(\lambda_{\text{soc}}\) for three iridates are estimated to be about 0.5 eV, in line with electron-spin resonance measurements (0.40-0.5 eV) [149]. Estimation of the band gap is discussed in the following session, where we assess and compare \(GW\) and DFT+\(U\).