# Advanced First-Principle Modeling of Relativistic Ruddlesden--Popper Strontium Iridates
## 4 Results and Discussions
### Electronic Structures
#### 4.2.3 \(Gw\) vs. LSDA+\(U\)  
In this section, we focus on analyzing the ground-state electronic structures in more detail by comparing the LSDA+\(U\)+SOC band structure (computed using the cRPA values of \(U\)) with the quasiparticle energies obtained by \(GW\)+SOC. As shown Figure 5, both approaches reproduce the \(J_{\text{eff}}=1/2\) spin-orbital Mott insulating state of Sr\({}_{2}\)IrO\({}_{4}\) and Sr\({}_{3}\)Ir\({}_{2}\)O\({}_{7}\) well and correctly predict SrIrO\({}_{3}\) as a semimetal. The obtained fundamental gaps agree well with experimental measurements (see Table 2). However, residual differences are observed between the two methods: \(GW\) pushes down the O-\(2p\) states by about 0.5 eV, which in turn decreases the hybridizations between Ir-\(d\) states and O-\(2p\) states. As already mentioned, Sr\({}_{2}\)IrO\({}_{4}\) and Sr\({}_{3}\)Ir\({}_{2}\)O\({}_{7}\) exhibit very similar bands. The most noticeable difference is the splitting at the top of the valence band at \(\Gamma\) induced by the bilayer structure in Sr\({}_{3}\)Ir\({}_{2}\)O\({}_{7}\). The computed splitting energy for Sr\({}_{3}\)Ir\({}_{2}\)O\({}_{7}\) is about 0.23 eV, consistent with the one obtained by ARPES [28]. It is important to note that, for both Sr\({}_{2}\)IrO\({}_{4}\) and Sr\({}_{3}\)Ir\({}_{2}\)O\({}_{7}\), the \(J_{\text{eff}}=1/2\) LHB and \(J_{\text{eff}}=3/2\) states are not separated by a well-defined gap in contrast to the ideal \(J_{\text{eff}}=1/2\) picture. The band structure of SrIrO\({}_{3}\) is semi-metallic and clearly different from the other two compounds. The most important characteristic is the Dirac cone at the U point that is protected by the lattice symmetry [148; 150]. The Dirac cone is the only crossing between the conduction and valence band in the entire Brillouin zone and is associated with a pseudogap at the Fermi energy (see Figure 5f,l), in agreement with the experimentally measured small charge carrier density [151], proving the semi-metallic character of SrIrO\({}_{3}\). We note that, although both \(GW\)+SOC and LSDA+\(U\)+SOC capture the main features of three iridates, their descriptions on relative binding energies between the top of the valence band at \(\Gamma\) and \(X\) for Sr\({}_{2}\)IrO\({}_{4}\) and Sr\({}_{3}\)Ir\({}_{2}\)O\({}_{7}\) are not satisfactory. For instance, ARPES indicates that the maximum of the LHB at \(\Gamma\) lies 150-250 meV compared to \(X\) for both Sr\({}_{2}\)IrO\({}_{4}\) and Sr\({}_{3}\)Ir\({}_{2}\)O\({}_{7}\)[8; 25; 28; 153; 153]. However, LSDA+\(U\)+SOC predicts that the \begin{table}
\begin{tabular}{c c c c} \hline  & **Sr\({}_{2}\)IrO\({}_{4}\)** & **Sr\({}_{3}\)Ir\({}_{2}\)O\({}_{7}\)** & **SrIrO\({}_{3}\)** \\ \hline Space group & \(14_{1}/acd\) (142) & _Ccee_ (68) & _Pbnm_ (62) \\ \(U\) & 1.82 & 1.67 & 1.39 \\ \(J\) & 0.22 & 0.22 & 0.22 \\ \(10 Dq\) & 3.70 & 3.80 & 4.24 \\ \(\lambda_{\text{soc}}\) & 0.47 & 0.49 & 0.45 \\ Magnetic ordering & _ab_-canted AFM & _c_-collinear AFM & NM \\ DFT+\(U\)+SOC gap & 0.23 & 0.14 & metal \\ \(GW\)+SOC gap & 0.25 & 0.16 & metal \\ Expt. gap & 0.30 [18] & 0.13 [29] & metal [146; 147] \\ \hline \end{tabular}
\end{table} 
Table 2: A summary of the space group, Hubbard \(U\), exchange \(J\), crystal field energy \(10 Dq\), and SOC constant \(\lambda_{\text{soc}}\) (eV) as well as DFT+\(U\)+SOC- and \(GW\)+SOC-predicted band gaps. The energy unit is in eV. Experimental gaps are also given for comparison. \(\Gamma\) point is 20 and 70 meV higher in energy than the \(X\) points for Sr\({}_{2}\)IrO\({}_{4}\) and Sr\({}_{3}\)Ir\({}_{2}\)O\({}_{7}\), respectively. \(GW\) improves the description only marginally for Sr\({}_{2}\)IrO\({}_{4}\), where the top of the valence band at \(X\) is found 10 meV higher than at \(\Gamma\) but fails in reproducing the correct order for Sr\({}_{3}\)Ir\({}_{2}\)O\({}_{7}\), even though the QP difference \(\Gamma-X\) is reduced to 40 meV. A second drawback of the employed level of theory is the reduced degree of electronic correlation for SrIrO\({}_{3}\).