# Advanced First-Principle Modeling of Relativistic Ruddlesden--Popper Strontium Iridates
## 4 Results and Discussions
### Magnetic Properties  
Now, we turn to a more detailed analysis of the magnetic properties. Figure 7 shows the representative magnetic orderings for the three iridates predicted by LSDA+_U_+SOC calculations. The computed local magnetic moments including the orbital and spin contributions have been given in Reference [100]. Sr\({}_{2}\)IrO\({}_{4}\) exhibits an in-plane canted antiferromagnetic ordering with a weak ferromagnetic moment within the \(ab\)-plane. The predicted weak ferromagnetic moment is about 0.08 \(\mu_{B}\)/Ir, in line with the magnetic susceptibility measurements 0.06-0.14 \(\mu_{B}\)/Ir [144, 157]. The canting angle of magnetic moment is calculated as 12.5\({}^{\circ}\), in accordance with the x-ray resonant scattering measurement (12.2(8)\({}^{\circ}\)) [158]. In contrast to Sr\({}_{2}\)IrO\({}_{4}\), Sr\({}_{3}\)Ir\({}_{2}\)O\({}_{7}\) displays an out-of-plane \(c\)-collinear antiferromagnetic ordering, indicating stronger interlayer couplings. which has been confirmed by the resonant x-ray diffraction [30, 32, 33]. The nearly zero in-plane magnetic moment is also in agreement with tiny hysteresis in the magnetic susceptibility experiments [30]. Although the magnetic ground state of SrIrO\({}_{3}\) is nonmagnetic, its magnetic ordering in the magnetic regime is more complicated due to its distorted three-dimensional crystal structure (see Figure 1c). As illustrated in Figure 7c, the magnetic ordering depends strongly on \(U\) and \(\lambda\). For instance, for \(U\) = 2.0 and \(\lambda\) = 0.4, the system shows a canted antiferromagnetic order with a large ferromagnetic component in the \(ab\) plane (Figure 7(c1)). As \(\lambda\) increases to 1.0, the system becomes a canted A-type antiferromagnet and the moments are largely aligned to the \(c\) axis (Figure 7(c2)). Keeping \(\lambda\) = 1.0 and increasing \(U\) (compare Figure 7(c2,c3)), the agenetic moments are aligned in a very different way, but the average magnetization per unit cell is zero. Figure 6: Collapse of the [PERSON]–[PERSON] gap in electron-doped Sr\({}_{2}\)IrO\({}_{4}\) (Sr\({}_{2-x}\)La\({}_{x}\)IrO\({}_{4}\)). Comparison between calculated bands (left, (**a**,**d**,**g**)) and the corresponding Fermi surface [142] (middle, (**b**,**e**,**h**)) with the corresponding experimental Fermi surface [153] (right, (**c**,**f**,**i**)). Reproduced with permission from Reference [142]. Copyright 2016 by the American Physical Society. Having obtained the magnetic ground states for three iridates, we are now in a position to explain why Sr\({}_{2}\)IrO\({}_{4}\) exhibits an unusual \(ab\)-canted AFM, whereas the structurally similar compound Sr\({}_{3}\)Ir\({}_{2}\)O\({}_{7}\) shows a \(c\)-collinear AFM. The former has been largely discussed in Reference [14]. Here, we just summarize the main points. To study the anisotropic magnetic couplings in Sr\({}_{2}\)IrO\({}_{4}\), we put forward a scheme by mapping fully relativistic constrained noncollinear LSDA+\(U\) calculations onto a general spin model Hamiltonian:  
\[\Delta E=-\sum_{i<j}\mathbf{J}_{ij}\mathbf{S}_{i}\cdot\mathbf{S}_{j}+\sum_{i} \epsilon_{am}^{i}(\mathbf{S}_{i})+\sum_{i<j}\mathbf{D}_{ij}\cdot[\mathbf{S}_{ i}\times\mathbf{S}_{j}], \tag{24}\]  
where the first, second, and last terms represent the isotropic AFM exchange, the single-site anisotropy (SSA), and Dzyaloshinskii-Moriya interactions, respectively. After taking the sum over all the Ir ions in the conventional unit cell, Equation (24) reduces to  
\[\Delta E=16J^{2}\cos(2\theta_{s})+8K\cos(4\theta_{s})-16D_{z}S^{2}\sin(2\theta _{s}) \tag{25}\]  
where \(J\), \(K\), and \(D_{z}\) are the nearest-neighbor isotropic exchange, SSA, and DM coupling parameters, respectively, and \(S\) and \(\theta_{s}\) are the magnitude and canted angle of the in-plane spin moment, respectively. Figure 8 shows the spin-dependent total energy as a function of the in-plane spin canting angle \(\theta_{s}\) decomposed over the isotropic AFM exchange, SSA, and DM contributions.