# Advanced First-Principle Modeling of Relativistic Ruddlesden--Popper Strontium Iridates
## 4 Results and Discussions
### Magnetic Properties
The magnetic coupling parameters can be extracted by fitting the first-principles calculated data using the model Equation (25), yielding \(JS^{2}=-0.32\) meV, \(K=-0.1\) meV, and \(D_{z}S^{2}=0.25\) meV, comparable with available phenomenological data in literature [159; 21]. The AFM isotropic exchange, favoring a collinear alignment of the  
Figure 7: Magnetic orderings of three iridates predicted by the LSDA+\(U\)+SOC method: (**a**) Sr\({}_{2}\)IrO\({}_{4}\) with \(U=1.82\) eV and \(\lambda=1.0\); (**b**) Sr\({}_{3}\)Ir\({}_{2}\)O\({}_{7}\) with \(U=1.67\) eV and \(\lambda=1.0\); and (**c**) SrIrO\({}_{3}\), for which the magnetic structure depends strongly on the values of \(U\) and \(\lambda\) in the magnetic regime (see Figure 2). The arrows denote the magnetic moment of each Ir atom. Sr and Ir are shown in green and blue, respectively. O atoms are not shown. spins, aids the stabilization of the AFM ordering. The DM interaction, which prefers an orthogonal coupling of the spins, assists the formation of a canted spin arrangement. The SSA is smaller than isotropic and DM interactions but favors in-plane magnetism at small canting angles. As expected, the evolution of the DM and isotropic exchange energies follows a different trend with respect to \(\theta_{s}\). The formation of the in-plane canted AFM state at \(\theta_{s}=14.4^{\circ}\) is the result of the subtle competition between these three terms: \(JS^{2}\), \(D_{z}S^{2}\), and \(K\). The \(ab\)-canted AFM state in Sr\({}_{2}\)IrO\({}_{4}\) represents a stable magnetic ground state, but it has been found that it is susceptible to external perturbation such as small changes in the octahedral rotation angle [160], tetragonal distortions [10; 14], and doping [142]. Two different types of magnetic phase transition have been predicted: \(ab\)-canted to \(ab\)-collinear by reducing the octahedral rotation angle and a transition to a \(c\)-collinear ordering by increasing the octahedral tetragonal distortion (\(c/a\approx 1.1\)). The former has been confirmed by high-pressure experimental findings [14; 160]. Next, we turn to the \(c\)-collinear AFM magnetism of Sr\({}_{3}\)Ir\({}_{2}\)O\({}_{7}\). Based on the microscopic model Hamiltonian approach, [PERSON] [33] showed that the spin-flop transition in Sr\({}_{3}\)Ir\({}_{2}\)O\({}_{7}\) is due to strong competition among intra- and interlayer bond-directional pseudodipolar interactions of the spin-orbit-entangled \(J_{\text{eff}}=1/2\) moments. Hence, a natural interesting question is how the magnetic ordering changes with respect to the interlayer distance, which directly controls the competition and balance between intra- and interlayer magnetic couplings. Therefore, we performed an theoretical experiment by artificially increasing the interlayer distance while keeping the other local structures fixed, followed by noncollinear self-consistent total energy calculations adopting two different spin configurations: \(ab\)-canted AFM and \(c\)-collinear AFM. Figure 9 shows the total energy difference between the  
Figure 8: The spin-dependent total energy as a function of the in-plane spin canting angle \(\theta_{s}\) decomposed over the isotropic AFM exchange (EX), SSA, and DM contributions.The zero is set such that the DM energy becomes zero for \(\theta_{s}=0^{\circ}\). Reproduced with permission from Reference [14]. Copyright 2015 by the American Physical Society. _ab_-canted AFM and _c_-collinear AFM phases as a function of the interlayer distance. In the ground state of Sr\({}_{3}\)Ir\({}_{2}\)O\({}_{7}\), the interlayer distance is about 4.088 A, where the _c_-collinear AFM phase is significantly favored compared to the _ab_-canted AFM phase. However, as the interlayer distance increases to 4.38 A, the _ab_-canted AFM phase becomes the most energetically stable phase due to the reduced the interlayer magnetic coupling (see Figure 9). Our theoretical experiment thus suggests the important role played by interlayer coupling in the formation of the _c_-collinear AFM state of Sr\({}_{3}\)Ir\({}_{2}\)O\({}_{7}\).