## 3 Results
### Monolayer SRO in SRO/STO Superlattice
For \(U^{\mathrm{Ru}}=3.5\) eV and \(U^{\mathrm{Ti}}=5.0\) eV (square), the \(T_{\mathrm{C}}^{\mathrm{MF}}\) decreases with increasing \(\lambda\), similar to previous conclusions of the lattice effect on the SRO magnetic phase (cf. [51; 52]). Precisely for \(\lambda=0\), the \(J_{1}\simeq 2.53\,\)meV and \(T_{\text{C}}^{\text{MF}}\simeq 39.2\,\)K. However, with \(\lambda=1\), as described in the previous paragraph, the energy difference between FM and AFM phases becomes fairly small and the \(T_{\text{C}}^{\text{MF}}\) is only \(0.5\,\)K with \(J_{1}\simeq 0.059\,\)meV. Figure 2: Side (**a**) and top (**b**) view of the \((\text{SRO})^{1}/(\text{STO})^{5}\) superlattice structure used in the calculations. (**c**,**e**,**g**) Band structures of three magnetic phases of \((\text{SRO})^{1}/(\text{STO})^{5}\) superlattice. The red (blue) dots present the component of Ru \(t_{2g}\) and \(e_{g}\) orbitals. The orange circle and arrow highlight the band gap. (**d**,**f**,**h**) The total density of states of each magnetic phase. The strong correlations in Ti \(d\) orbitals also play an important role in this system. As also shown in Figure 3a, by tuning the order parameter \(\lambda\) with \(U^{\text{Ti}}=0\) (circle), the energy difference between FM and AFM phases becomes larger and hence makes the \(T_{\text{C}}^{\text{MF}}\) much higher, strongly stabilizing the FM half-metallic phase. The \(T_{\text{C}}^{\text{MF}}\simeq 78\,\text{K}\) for \(\lambda=0\) decreases to 59.7 K with \(\lambda=1\). In other words, the strong correlations in Ti ions help stabilize the AFM insulating phase and thus achieve the FM-AFM metal-insulator phase transition upon lateral strain. The Monte Carlo calculations of Curie temperature are also accomplished for the \((\text{SRO})^{1}/(\text{STO})^{5}\) superlattice, as presented in Figure 3c,d. Different from the zero magnetism at high temperature for the bulk systems (Figure 1f), the magnetism does not vanish at high temperatures for the superlattice cases. This is due to the Heisenberg model and the Mermin-Wagner theorem for two dimensional systems which conclude that there is no phase transition in a two-dimensional system. To simulate the Curie temperature, the magnetism of Monte Carlo results with \(M(T)>0.2\) are fitted by \(M(T)=M(0\,\text{K})\left(1-T/T_{\text{C}}^{\text{MC}}\right)^{p}\) with the critical exponents ranging from 0.42 to 0.50 (Figure 3c,d). As shown Figure 3a, the MC and MF results both show the same trend with more or less a constant ratio \(T_{\text{C}}^{\text{MC}}/T_{\text{C}}^{\text{MF}}\) by tuning \(\lambda\) and \(U^{\text{Ti}}\). The only difference is that the MF method always overestimates the Curie temperature for all the considered systems. Figure 3: **(a)** Mean field and Monte Carlo \(T_{\text{C}}\) of different lattice constants and \(U^{\text{Ti}}\) in SRO/STO superlattice. \(\lambda=0\) indicates using the SRO lattice constants \(a\), \(b\), while \(\lambda=1\) indicates that the STO lattice constants \(a\), \(b\) are used. The solid (dashed) line indicates the Monte Carlo (mean field) result. The circle (square) symbol indicates the \(U^{\text{Ti}}=0.0\,\text{eV}\) (5.0 eV) cases. **(b)** The ratio \(T_{\text{C}}^{\text{MC}}/T_{\text{C}}^{\text{MF}}\) of all considered cases. The dashed line represents the average ratio of 0.536. **(c,d)** The Monte Carlo and the fitting magnetization of SRO/STO superlattice.