model Hamiltonian of the system is $$\begin{aligned}
\hat{H}^{{\rm ext}} &=& \hat{H}^{{\rm TBH}} 
                       + \sum_{p, q, k}  V_k \hat{n}^k_{p,q} \biggl [ 
                         \Big ( \hat{n}^k_{p-1,q} + \hat{n}^k_{p+1,q} 
                       + \hat{n}^k_{p,q -1}  \nonumber\\
                    && + \hat{n}^k_{p,q+1}\Big ) +  V_{12} \hat{n}_{p,q}^k
                         \Big(\hat{n}^{3-k}_{p-1,q} + \hat{n}^{3-k}_{p+1,q} 
                       + \hat{n}^{3-k}_{p,q -1} 
                                     \nonumber \\
                    && + \hat{n}^{3-k}_{p,q+1}\Big ) \biggr],
\label{bhm_ext}\end{aligned}$$ here $V_k$ is the intra-species NN interaction strength for both the species, and $V_{12}$ is the inter-species NN interaction strength. In the experiments the relative strengths of the NN interaction is controllable through the Feshbach resonances. Thus, to relate with the experimental observations and predict possible phases we vary the inter- and intra-species interaction strengths. We use SGMF theory to obtain the ground state of the system, then, in this method the total Hamiltonian is the sum of the single-site mean field Hamiltonian $$\begin{aligned}
\hat{h}_{p,q}^{{\rm ext}} &=& \hat{h}_{p,q}^{{\rm TBH}} 
                       + \sum_{k}  V_k \hat{n}^k_{p,q} \biggl [ 
                         \Big ( \hat{n}^k_{p-1,q} + \hat{n}^k_{p+1,q} 
                       + \hat{n}^k_{p,q -1}  \nonumber\\
                    && + \hat{n}^k_{p,q+1}\Big ) +  V_{12} \hat{n}_{p,q}^k
                         \Big(\hat{n}^{3-k}_{p-1,q} + \hat{n}^{3-k}_{p+1,q} 
                       + \hat{n}^{3-k}_{p,q -1} 
                                     \nonumber \\
                    && + \hat{n}^{3-k}_{p,q+1}\Big ) \biggr].
\label{bhm_ext_ss}\end{aligned}$$ For eTBHM also, we diagonalize the Hamiltonian at each site separately, and obtain the ground state. The NN interaction term contribute to the diagonal matrix 