## 4 Discussion  
Previous DFT studies on SrRuO\({}_{3}\) show different magnetic phases in 1 u.c. SRO including ferromagnetic and paramagnetic ones. Many-body simulations by dynamical mean-field theory (DMFT) show AFM insulator of 1 u.c. SRO in SRO/STO superlattice. Thus far, no insulating phases of 1 u.c. thickness SRO are reported by DFT calculations. Our study demonstrates that the antiferromagnetic insulating phase in 1 u.c. SRO can be obtained in the single particle picture. In addition, the lattice constant and on-site energy of Ti play an important role in this magnetic metal-insulator transition. More importantly, our results agree well with the experimental findings in which tensile strain can affect the Curie temperature in SRO thin film [51; 52].  
Note that the MIT of the SRO monolayer could have several sources such as the defect effect, inter-diffusion, layer thickness, as well as the strain effect and magnetic phase transition as studied in this work. Particularly, the SRO/STO interface exhibits intermixing on a length scale of 1\(\sim\)1.5 unit cells [53]. Thus, the impurities and intermixing may also play an important role in MIT of SRO monolayer. However, among the aforementioned several effects that could cause MIT of SRO monolayer, we focus on the strain induced magnetic phase transition associated MIT of SRO monolayer. Our study offers a route to tuning the electric and magnetic properties of SRO monolayer, and could be generalized to SRO multilayers.  
Our aforementioned conclusions are slightly different from those from previous study using DFT + DMFT [29]. In a previous DMFT study, [PERSON] et al. [29] claimed SRO thin films are always antiferromagnetic insulator even under the lattice stress of \(\pm\)0.5%. Differences in computational methods between our work and previous studies are discussed below. Firstly, we use the experimental STO lattice constant of 5.5225 while the previous study [29] uses the optimized lattice constant of \(3.95\times\sqrt{2}=5.5861\), differing from the former by 1.2%. Secondly, the lattice structure is optimized using the PBE+mBJ method in the previous study [29], even though mBJ method actually does not contribute to atomic forces. We use LDA + \(U\) method to optimize our atomic positions. Thirdly, all calculations in the previous studies do not consider the on-site Hubbard energy of Ti \(d\) orbitals.  
To realize the room temperature ultra-thin spin-polarized two-dimensional electron gas is an important issue [25; 29]. From our calculations, the intersite exchange field \(J_{1}\) can be modified by lateral strain through tuning the lattice parameter \(a\) and \(b\). Therefore, the FM half-metallic phase in monolayer SRO can be induced by applying the external tensile lateral strain. Experimentally, this can be achieved by growing SRO on another perovskite with similar lattice constants as SRO, i.e., larger lattice parameters than the STO ones. This pave a new route toward realizing the desirable ultra-thin spin-polarized 2 DEG for future spintronics.  
Owing to the inability of handling two-dimensional systems by the self-consistent Gaussian approximation (SCGA), the \(T_{\rm C}^{\rm SCGA}\) of the SRO layer in \(({\rm SRO})^{1}/({\rm STO})^{5}\) superlattice cannot be calculated. However, combining our MC and MF results, we can estimate the SCGA counterparts by \(\theta_{\rm C}=T_{\rm C}^{\rm MC}/T_{\rm C}^{\rm MF}\). As shown in Figure 3b and discussed previously, \(\theta_{\rm C}\simeq 0.536\). This parameter also gives a convenient method to estimate the two-dimensional square lattice such as the other two-dimensional magnetic perovskite.