## 3 Results
### Monolayer SRO in SRO/STO Superlattice  
To study the phase transition of the single layer (1 u.c.) SRO, we consider the SRO/STO superlattice containing 1 u.c. SRO and 5 u.c. STO in the (001) direction. The distorted TiO\({}_{6}\) oxide octahedron in bulk STO, which rotates in the antiferrodistortive structure [47], is used as the initial structure, as shown in Figure 2a,b. To take the strong correlation into account, we apply on-site \(U=3.5\) eV and \(J=0.6\) eV for Ru-4d orbital and \(U=5.0\) eV and \(J=0.64\) eV for Ti-3d orbital. To study the lateral strain induced metal-insulator transition, we tune the lattice constants \(a,b\) of the SRO/STO superlattice from the SRO \(\sqrt{2}\times\sqrt{2}\times 2\) lattice parameters \(a=5.5670\,,b=5.5304\) (denoted as \(\lambda=0\)) to the STO lattice parameters \(a=b=5.5225\) (or \(a/\sqrt{2}=3.905\) in \(1\times 1\) u.c.) (denoted as \(\lambda=1\)) with the parameter \(\lambda\) indicating the strain effect. We then optimize the \(c\)-axis as well as the ion positions with lattice \(a,b\) fixed at a given \(\lambda\). Three different magnetic phases, including paramagnetic, ferromagnetic (FM) and antiferromagnetic (AFM), of the SRO layer are considered as discussed below. The electronic structures of three different magnetic phase of \((\mathrm{SRO})^{1}/(\mathrm{STO})^{5}\) superlattice are shown in Figure 2c-h. For the paramagnetic phase (Figure 2c,d), the band structures of spin up and spin down channels degenerate owing to the time reversal symmetry and spin rotational symmetry. The orbitals near the Fermi level are dominated by Ru \(d\) orbitals. The Ru \(t_{2g}\) and Ru \(e_{g}\) bands cross each other showing metallic behaviors. The orbital decomposition also presents a spin-unpolarized two-dimensional electron gas (2 DEG) conducted by Ru \(t_{2g}\) orbitals. On the other hand, for the ferromagnetic phase shown in Figure 2e,f, the spin up Ru \(e_{g}\) and \(t_{2g}\) bands split with each other and hence open an energy gap of 0.52 eV at the Fermi level (\(E_{F}\)), whereas the spin down channel remains gapless. Therefore, the ferromagnetic phase shows the half-metallic character, in which one spin channel is conducting while the other is insulating, similar to the fully spin polarized metallic behavior in bulk SRO. More importantly, this half-metallic single u.c. SRO in \((\mathrm{SRO})^{1}/(\mathrm{STO})^{5}\) superlattice demonstrates the desired ultrathin fully spin-polarized (SP) 2 DEG within a single layer, same as previous studies. As for the antiferromagnetic phase shown in Figure 2g,h, there exists a small energy gap \(\sim\)0.12 eV between \(e_{g}\) and \(t_{2g}\) bands, leading to the antiferromagnetic insulator. Total energy analysis shows that the half-metallic FM phase is energetically most favorable and is thus the ground state among all the three phase, while the paramagnetic phase is of the highest total energy. However, the total energy of the AFM phase is only 0.47 meV higher than that of the FM phase, indicating that the insulating AFM phase could also be observed experimentally. Previous studies also show that the doping or interdiffusion of Ti ions may enhance the insulating phase [48; 49; 50] in real systems. To study the strain effect originated from the lattice mismatch between SRO and STO in the SRO/STO superlattice, we tune the lattice parameters \(a\) and \(b\) with the order parameter \(\lambda=0\) representing the bulk SRO lattice parameters \(a\) and \(b\), and \(\lambda=1\) corresponding to the bulk STO lattice parameters \(a,b\). The \(c\)-axis and ion positions are optimized for all cases. Similar to the previous section, we perform electronic structure calculations and Monte Carlo simulations for AFM and FM phases. To determine the intersite exchange parameters, we use \(N_{\mathrm{nn}}=4\) for the nearest-neighbor Ru ions and \(N_{\mathrm{mag}}=2\) for the two magnetic ions in the superlattice in Equation (2). Figure 3a shows the calculated \(T_{\mathrm{C}}^{\mathrm{MF}}\) with different lattice constants (strains).