## 3 Results  
### Bulk Properties of SRO  
Firstly, we discuss the electronic and magnetic properties of bulk SRO in this section. The counterpart for the single layer SRO in SRO/STO superlattice is discussed in the next section. SRO crystallizes in the perovskite structure similar to many transition metal oxides such as SrTiO\({}_{3}\) and LaRuO\({}_{3}\). Each Ru ion is surrounded by six O ions forming an RuO\({}_{6}\) octahedron. At low temperature, SRO transits from space group \(Pm\bar{3}m\) of the original \(1\times 1\) perovskite unit cell into \(Pbnm\) (\(\#62\), \(D_{2h}^{16}\)) with a lower symmetry and a larger \(\sqrt{2}\times\sqrt{2}\times 2\) supercell [13] by rotating the RuO\({}_{6}\) octahedrons, as shown in Figure 1a. Figure 1: **(a)** The crystal structure of SrRuO\({}_{3}\). **(b)** Calculated band structures of SRO from LDA + U. The colors indicate different orbital components. **(c)** The G-type antiferromagnetism spin configuration of SRO. **(d)** The mean field and SCGA. **(e)** Curie temperatures of bulk SRO with different \(U\) and \(J\). The experimental Tc is shown in red dashed line. **(f)** The Monte Carlo simulations of Curie temperatures. The colors indicate different \(U\) and \(J\). The lines are the fitted curves for the data points given from Monte Carlo simulations. The red dashed line indicate the experiment result. The LDA + \(U\) band structures of SRO in Figure 1b show the half-metallic behavior, that is, the majority spin channel shows a Mott insulating energy band gap at the Fermi level (\(E_{F}\)), while the spin minority channel is metallic. Therefore, only the spin down electrons can conduct, resulting in 100% spin polarization at \(E_{F}\) which is ideal for spintronics. The orbital projections of the band structure demonstrate that the band gap in the majority spin channel is between the Ru-\(t_{2g}(d_{xy},\,d_{xz},\,d_{yz})\) and \(e_{g}(d_{x^{2}-y^{2}},\,d_{z^{2}})\) bands. These half-metal characters are the same as those in previous experimental observations [17; 18; 19] as well as ab initio calculations including LDA + \(U\)[20], DMFT method [21], and \(GW\) method [45]. Based on these LDA + \(U\) results, we then simulate the Curie temperature \(T_{\rm C}\) by considering the Heisenberg model:  
\[H=-\frac{1}{2}\sum_{i,j}J_{i}\mathbf{S}_{j}\cdot\mathbf{S}_{i}. \tag{1}\]  
Here, \(J_{i}\) is the intersite exchange parameter between different magnetic (Ru) ions and \(\mathbf{S}_{i}\) indicates the unit vector of the spin direction. These \(J_{i}\) values can be calculated from the DFT total energies of different magnetic phases. Beside the above ferromagnetic (FM) phase, we have also calculated the DFT total energy of SRO in the G-type antiferromagnetic (AFM) phase, as shown in Figure 1c. The exchange parameter \(J_{1}\) between nearest-neighbor Ru ions is then calculated by  
\[J_{1}=(E_{\rm AFM}-E_{\rm FM})/(N_{\rm nn}\cdot N_{\rm mag}). \tag{2}\]  
Here, the \(E_{\rm AFM}(E_{\rm FM})\) is the total energy of the AFM (FM) phase, \(N_{\rm mag}\) is the number of magnetic Ru ions in the unit cell and \(N_{\rm nn}\) is the number of nearest-neighbor Ru ions of each Ru ion. Since the Ru sublattice is of the simple cubic structure, the number \(N_{\rm nn}=6\), while \(N_{\rm mag}=4\) because the SRO \(\sqrt{2}\times\sqrt{2}\times 2\) unit cell is composed of four simple perovskite \(1\times 1\times 1\) unit cells. Thus, we obtain \(J_{1}\simeq 7.455\,\rm meV\). Several approaches can be used to estimate \(T_{\rm C}\) based on the Heisenberg model. The simplest one is the mean field (MF) method:  
\[T_{\rm C}^{\rm MF}\simeq\frac{1}{3}N_{\rm nn}J_{1}. \tag{3}\]  
This gives \(T_{\rm C}^{\rm MF}\simeq 173.0\,\rm K\), which is somewhat larger than the experimental \(T_{\rm C,SRO}\simeq 150\,\rm K\)[13]. As in the usual cases, the mean field method overestimates the \(T_{\rm C}\) value of SRO. Thus, we also adopt the self-consistent Gaussian approximation (SCGA) method [30].