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\begin{document}

\begin{center}
\textbf{\Large Supplementary Materials for\\
Tunable valley splitting and magnetic anisotropy in two-dimensional buckled honeycomb lattice $\text{Mn}_2\text{X}_2$ (X = F, Cl, Br)}\\[1em]

Mingyu Lü$^{1}$, Wendan Song$^{1}$, Yuping Tian$^{1}$, Chao-Bo Wang$^{1}$, Xuefeng Dai,$^{1}$, Xudong Su$^{1}$, Xiangru Kong$^{1,*}$, Wei-Jiang Gong$^{1,*}$\\[0.5em]

$^{1}$College of Sciences, Northeastern University, Shenyang 110819, China\\
$^{*}$Corresponding authors: \texttt{kongxiangru@mail.neu.edu.cn} and \texttt{gwj@mail.neu.edu.cn}
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\maketitle

\begin{table*}[htbp]
\centering
\small
\renewcommand{\arraystretch}{1.3}
\caption{Calculated optimized lattice constants ($a$), nearest Mn--Mn distances ($d$), Mn--X--Mn angles ($\alpha$), cohesive energy ($\Delta E_{\mathrm{coh}}$), and the energy differences of the FM and AFM2 states with respect to the AFM1 state (meV/unit cell).}

\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}lcccccc@{}}
\hline\hline
\textit{Material} & $a$ (\AA) & $d$ (\AA) & $\alpha$ ($^\circ$) & $\Delta E_{\mathrm{coh}}$ (eV/atom) & $\Delta E_{\mathrm{FM-AFM1}}$ & $\Delta E_{\mathrm{AFM2-AFM1}}$ \\
\hline
Mn$_2$F$_2$  & 3.07 & 2.64 & 91.7 & -4.180 & 374.972 & 281.069 \\
Mn$_2$Cl$_2$ & 3.37 & 2.66 & 84.9 & -3.297 & 583.703 & 480.695 \\
Mn$_2$Br$_2$ & 3.54 & 2.69 & 84.0 & -2.957 & 621.503 & 536.694 \\
\hline\hline
\end{tabular*}
\end{table*}





\begin{figure*}[htbp]
    \centering
    \includegraphics[width=0.95\textwidth]{FIG. S1.png}
    \caption{Phonon spectra of Mn$_2$Br$_2$ and Mn$_2$F$_2$.}
\end{figure*}

\begin{figure*}[htbp]
    \centering
    \includegraphics[width=0.95\textwidth]{FIG. S2.png}
    \caption{AIMD simulations of Mn$_2$Br$_2$ and Mn$_2$F$_2$.}
\end{figure*}

\begin{figure*}[htbp]
    \centering
    \includegraphics[width=0.95\textwidth]{FIG. S3.png}
    \caption{ELF of Mn$_2$Br$_2$ and Mn$_2$F$_2$.}
\end{figure*}

\begin{figure*}[htbp]
    \centering
    \includegraphics[width=0.95\textwidth]{FIG. S4.png}
        \caption{T$_N$ of monolayer Mn$_2$X$_2$ .}
\end{figure*}

\begin{figure*}[htbp]
    \centering
    \includegraphics[width=0.9\textwidth]{FIG. S5.png}
    \caption{PBAND of monolayer Mn$_2$Br$_2$ without SOC.}
\end{figure*}

\begin{figure*}[htbp]
    \centering
    \includegraphics[width=0.95\textwidth]{FIG. S6.png}
    \caption{Band structures of Mn$_2$Cl$_2$ and Mn$_2$F$_2$ with and without SOC.}
\end{figure*}

\begin{figure*}[htbp]
    \centering
    \includegraphics[width=0.9\textwidth]{FIG. S7.png}
    \caption{Band structure of Mn$_2$Br$_2$ in the FM state with SOC.}
\end{figure*}

\begin{figure*}[htbp]
    \centering
    \includegraphics[width=0.95\textwidth]{FIG. S8.png}
    \caption{Total energies of monolayer Mn$_2$X$_2$ for different magnetic configurations as a function of the Hubbard $U$, with the energies measured relative to the AFM1 state.}
\end{figure*}

\begin{figure*}[htbp]
    \centering
    \includegraphics[width=0.95\textwidth]{FIG. S9.png}
    \caption{Band structures of Mn$_2$Br$_2$ without SOC under different Hubbard $U$ values.}
\end{figure*}

\begin{figure*}[htbp]
    \centering
    \includegraphics[width=0.95\textwidth]{FIG. S10.png}
    \caption{Band structures of Mn$_2$Br$_2$ with SOC under different Hubbard $U$ values.}
\end{figure*}

\begin{figure*}[htbp]
    \centering
    \includegraphics[width=0.95\textwidth]{FIG. S11.png}
    \caption{Band structures of Mn$_2$Cl$_2$ without SOC under different Hubbard $U$ values.}
\end{figure*}

\begin{figure*}[htbp]
    \centering
    \includegraphics[width=0.95\textwidth]{FIG. S12.png}
    \caption{Band structures of Mn$_2$Cl$_2$ with SOC under different Hubbard $U$ values.}
\end{figure*}

\begin{figure*}[htbp]
    \centering
    \includegraphics[width=0.95\textwidth]{FIG. S13.png}
    \caption{Band structures of Mn$_2$F$_2$ without SOC under different Hubbard $U$ values.}
\end{figure*}

\begin{figure*}[htbp]
    \centering
    \includegraphics[width=0.95\textwidth]{FIG. S14.png}
    \caption{Band structures of Mn$_2$F$_2$ with SOC under different Hubbard $U$ values.}
\end{figure*}

\begin{figure*}[htbp]
    \centering
    \includegraphics[width=0.95\textwidth]{FIG. S15.png}
    \caption{Band structures of Mn$_2$Br$_2$ without SOC under different strain values.}
\end{figure*}

\begin{figure*}[htbp]
    \centering
    \includegraphics[width=0.95\textwidth]{FIG. S16.png}
    \caption{Band structures of Mn$_2$Br$_2$ with SOC under different strain values.}
\end{figure*}

\begin{figure*}[htbp]
    \centering
    \includegraphics[width=0.95\textwidth]{FIG. S17.png}
    \caption{Band structures of Mn$_2$Cl$_2$ without SOC under different strain values.}
\end{figure*}

\begin{figure*}[htbp]
    \centering
    \includegraphics[width=0.95\textwidth]{FIG. S18.png}
    \caption{Band structures of Mn$_2$Cl$_2$ with SOC under different strain values.}
\end{figure*}

\begin{figure*}[htbp]
    \centering
    \includegraphics[width=0.95\textwidth]{FIG. S19.png}
    \caption{Band structures of Mn$_2$F$_2$ without SOC under different strain values.}
\end{figure*}

\begin{figure*}[htbp]
    \centering
    \includegraphics[width=0.95\textwidth]{FIG. S20.png}
    \caption{Band structures of Mn$_2$F$_2$ with SOC under different strain values.}
\end{figure*}

\begin{figure*}[htbp]
    \centering
    \includegraphics[width=0.95\textwidth]{FIG. S21.png}
    \caption{Berry curvatures of monolayer Mn$_2$Br$_2$ and Mn$_2$F$_2$.}
\end{figure*}

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