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\title{Supplementary Material: Is Cr$_2$AlN the Highest Temperature Superconductor Possible in the M$_2$AX family?}% Force line breaks with \\   


\author{Karaca, E.$^{1,2}$; Byrne, P. J. P.$^{1}$; Hasnip, P.J.$^{1}$ and  Probert, M.I.J.$^{1}$}

\affiliation{
    $^1$ Department of Physics, University of York, York YO10 5DD, United Kingdom\\
}
\affiliation{ 
    $^2$ Sakarya University, Biomedical, Magnetic and Semiconductor Materials Research Center (BIMAS-RC), 54187, Sakarya, Turkey\\
} 
\maketitle
%\tableofcontents

\subsection{Structural, electronic and Fermi surface  properties of M$_2$AC (M: Ti, Mo and Lu; A: Ge and Sn) compounds}

The structural, electronic, Fermi surface, phonon, and electron-phonon properties of M$_2$GeC (M: Mo and Lu) and M$_2$SnC (M: Ti, Mo and Lu) have been examined by \emph{ab initio} pseudopotential calculations within the generalized gradient approximation (GGA) of the density functional theory (DFT). Of these, only Lu$_2$SnC was previously known to be superconducting in experiment. We have recently performed theoretical studies of superconductivity in Ti$_2$GeC, Nb$_2$GeC, and Nb$_2$SnC\cite{karaca2021electron, karaca2022prediction}. In the previous paper\cite{karaca2021electron}, $\mu^{*}$ for Nb$_2$GeC and Nb$_2$SnC was set at 0.10, which was determined by matching the Fr\"ohlich model to a series of Nb-based MAX phases; when studying a wider range of MAX-phase materials\cite{ karaca2022prediction} and this work, it was found that $\mu^{*}$=0.13 was a better fit, and hence this latter value was used throughout this study. This change causes a small decrease in the predicted $T_c$ for Nb$_2$GeC and Nb$_2$SnC compared to ref. \cite{karaca2021electron} but all other results are the same. Hence in this SI we will focus on the 5 materials which have been studied here using the full Eliashberg approach and whose results have not previously been reported.

There are two formula units and eight atoms in a primitive cell for these crystals, and they crystallise in the hexagonal structure (P6$_3$/mmc), as shown in Figure~\ref{fig_Scrystal} (repeated from figure~2 in main text). The crystal structure properties of Cr$_2$AlN are discussed in the main text. Here we give the properties of M$_2$GeC(M: Mo and Lu) and M$_2$SnC(M: Ti, Mo and Lu). The calculated equilibrium lattice constants ($a$,$c$), internal parameter ($z$), bulk modulus, and its pressure derivative are listed in Table~\ref{TabS1}, together with experimental data and previous theoretical results. The calculated results are in good agreement with previous experimental \cite{du2021unlocking,barsoum1997fabrication,kuchida2013superconductivity} and theoretical\cite{cover2009comprehensive,ali2020physical} although there have been no publications on Lu$_2$GeC material to yet.

The electronic band structure, the total and partial density of states, and the Fermi surface of M$_2$GeC(M: Mo and Lu) and M$_2$SnC(M: Ti, Mo and Lu) are  presented  in figure~\ref{figS1} and figure~\ref{figS3}, respectively. It can be seen from the figures that all of them have metallic properties, with the density of states at the Fermi energy N(E$_F$) is dominated by the M(M: Ti, Mo and Lu) d states. The calculated values for N(E$_F$) are listed in table~\ref{TabS2}. Also, Ge and Sn p states in Lu$_2$GeC and Lu$_2$SnC materials significantly contribute to N(E$_F$).

The Fermi surfaces of M$_2$GeC (M: Mo and Lu) and M$_2$SnC (M: Ti, Mo and Lu) are roughly similar to those of Cr$_2$AlN, which is discussed in the main text. The calculated Fermi surfaces agree with previous theoretical results\cite{hadi2019electronic,pinek2021fermi,kanoun2009theoretical}. The properties of M$_2$GeC(M: Mo and Lu) and M$_2$SnC(M: Ti, Mo and Lu) are mostly set by the d orbitals of the M (M:Ti, Mo and Lu)  atoms at the Fermi level N(E$_F$).  The Fermi surface of Ti$_2$SnC has six Fermi surface sheets, whilst Mo$_2$GeC and Mo$_2$SnC have 5 and Lu$_2$GeC and Lu$_2$SnC have 4. As the mass of the M atom increases, the number of bands that cross the Fermi levels tends to decrease, reducint N(E$_F$). As with Cr$_2$AlN, hole-like sheets occur in the corners of the Brillouin zone in the $H$-$K$ and $L$-$M$ directions, whilst the Fermi surface is completely prismatic and cylindrical in the $\Gamma$-$A$ direction and exhibits electron-like behaviour. We also see that Mo$_2$GeC and Mo$_2$SnC materials exhibit strong Fermi nesting, similar to that of Cr$_2$AlN, which can increase electron-phonon interaction.

\subsection{Phonon and electron-phonon interaction of  M$_2$AC (M: Ti, Mo and Lu; A: Ge and Sn) compounds}

The calculated phonon spectrum, total and partial vibrational density of states (VDOS), and electron–phonon spectral function are presented in figure~\ref{figS2} and  figure~\ref{figS4}. All of them are dynamically stable as only positive phonon modes are exist. Lu$_2$GeC and Lu$_2$SnC has 2 distinct frequency regions: a low-frequency area up to 7~THz with three acoustic and fifteen optical phonon modes, and a high-frequency region between 11 and 17~THz with six optical modes. These two regions are separated by a large gap. Both the high-frequency and low-frequency regions of Lu$_2$GeC are divided into two regions by small gaps. The phonon band structure of all other materials can be divided into three distinct regions. For M$_2$GeC (M: Mo and Lu) and M$_2$SnC (M: Ti, Mo and Lu), there is overlap and hybridization of M - A modes in the low-frequency region. The high frequency region has two peaks separated by a small gap for Mo$_2$GeC, M$_2$SnC(M:Ti or Mo) and Lu$_2$GeC, but just a single peak for Lu$_2$SnC. 

The frequency dependence of the Eliashberg spectral function ($\alpha^{2}F(\omega)$) and the electron-phonon interaction parameter ($\lambda$) are shown in figure~\ref{figS2} and figure~\ref{figS4}, respectively, and can be used to investigate the impact of electron-phonon interaction on the superconducting properties of M$_2$GeC(M: Mo and Lu) and M$_2$SnC(M: Ti, Mo and Lu). The spectral function ($\alpha^{2}F(\omega)$) significantly contributes to the vibrational modes at low frequencies. Then, using the evaluated values of $\lambda$,  the reduced Coulomb parameter ($\mu^{*}$=0.13) and $\omega_{\ln}$ in the Allen-Dynes modified McMillan formula, the value of $T_c$ of Mo$_2$GeC, Lu$_2$GeC, Ti$_2$SnC, Mo$_2$SnC and Lu$_2$SnC is estimated to be 9.607, 2.504, 5.897, 7.635 and 5.963, respectively. The calculated $T_c$ value of Lu$_2$SnC agrees well with the known experimental result\cite{kuchida2013superconductivity}.


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%
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% TABLE S1
\begin{table*}
\caption{Structural properties of M$_2$GeC (M: Mo and Lu) and M$_2$SnC (M: Ti, Mo and Lu) for this work in bold, and their comparison with previous experimental and theoretical results.}
\begin{ruledtabular}
\begin{tabular}{cccccccccccc}
Phase                                  &$a$(\AA)       &$c$(\AA)         &z             &d$_{M-M}$(\AA)  &d$_{M-A}$(\AA) &d$_{M-C}$(\AA)  &B(GPa)        &B$^{'}$  \\ \hline
{\bf Mo$_2$GeC}                        &{\bf 3.131}    &{\bf 12.632}     &{\bf 0.089}   &{\bf 2.888}     &{\bf 2.719}    &{\bf 2.129}     &{\bf 216.3}   &{\bf 4.67} \\ 
Exp\cite{du2021unlocking}              &3.022          &                 &              &                &               &                &              &       \\
GGA\cite{cover2009comprehensive}       &3.149          &12.640           &              &                &               &                &212           &       \\
GGA\cite{khazaei2014effect}            &3.135          &12.635           &0.089         &                &2.718          &2.134           &214.4         &       \\
GGA\cite{huang2017structural}          &3.130          &12.653           &0.089         &                &2.720          &2.130           &210           &4.75       \\
GGA\cite{shao2020first}                &3.120          &12.609           &              &                &               &                &              &       \\
LDA\cite{bouhemadou2009calculated}     &3.121          &12.332           &0.092         &                &               &                &225           &4.55       \\
{\bf Lu$_2$GeC}                        &{\bf 3.482}    &{\bf 14.577}     &{\bf 0.089}   &{\bf 3.518}     &{\bf 3.291}    &{\bf 2.475}     &{\bf 92.6}    &{\bf 4.11} \\ 
{\bf Ti$_2$SnC}                        &{\bf 3.176}    &{\bf 12.610}     &{\bf 0.081}   &{\bf 2.737}     &{\bf 2.816}    &{\bf 2.096}     &{\bf 136.9}   &{\bf 4.14} \\ 
Exp\cite{barsoum1997fabrication}       &3.163          &13.675           &              &                &               &                &              &       \\
Exp\cite{vincent1998chemical}          &3.163          &13.679           &              &                &               &                &              &       \\
Exp\cite{lu2021isotope}                &3.180          &13.860           &              &                &               &                &              &       \\
GGA\cite{ali2020physical}              &3.174          &13.781           &              &                &               &                &              &       \\
GGA\cite{kanoun2009theoretical}        &3.136          &13.641           &0.082         &                &2.923          &2.126           &170.0         &       \\
GGA\cite{hug2006electronic}            & 3.171         &13.857           &0.081         &                &               &                &167.0         &       \\
GGA\cite{bouhemadou2008prediction}     &3.096          &13.404           &0.083         &                &               &                &159.3         &4.29       \\
{\bf Mo$_2$SnC}                        &{\bf 3.191}    &{\bf 12.409}     &{\bf 0.080}   &{\bf 2.717}     &{\bf  2.796}   &{\bf 2.095}     &{\bf 187.7}   &{\bf 5.35} \\ 
GGA\cite{cover2009comprehensive}       &3.203          &13.520           &              &                &               &                &184           &       \\
{\bf Lu$_2$SnC}                        &{\bf 3.514}    &{\bf 15.377}     &{\bf 0.083}   &{\bf 3.268}     &{\bf 3.269}    &{\bf 2.399}     &{\bf 89.1}    &{\bf 4.51} \\ 
Exp\cite{kuchida2013superconductivity} &3.514          &15.159           &              &                &               &                &              &       \\
GGA\cite{hadi2019mechanical}           &3.546          &15.323           &0.085         &                &               &                &              &       \\
GGA\cite{shao2020first2}               &3.542          &15.315           &              &                &               &                &              &       \\
GGA\cite{hadi2022dft}                  &3.546          &15.323           &0.085         &3.314           &3.255          &2.428           &86            &       \\




\label{TabS1}
\end{tabular}
\end{ruledtabular}
\end{table*}

%
% TABLE S2
\begin{table*}[ht]
\caption{Density of states at the Fermi level   (N(E$_{F}$)), logarithmic frequency  ($\omega_{\ln}$), the average  electron-phonon coupling  parameter ($\lambda$) and the superconducting transition temperature ($T_c$ in K) for the hexagonal M$_2$GeC (M: Mo and Lu) and M$_2$SnC (M: Ti, Mo and Lu) for this work in bold, and their comparison with available previous experimental and theoretical results.}
\begin{ruledtabular}
\begin{tabular}{ccccc}
  Phase                                 &N(E$_{F}$)(states/eV)       &$\omega_{\ln}$ (K)  & $\lambda$   &$T_{c}$ (K) \\ \hline
{\bf Mo$_2$GeC}                         &{\bf 4.125}                 &{\bf 205.320}       &{\bf 0.879}  &{\bf 9.607}  \\ 
{\bf Lu$_2$GeC}                         &{\bf 4.372}                 &{\bf 188.675}       &{\bf 0.402}  &{\bf 2.504}  \\ 
{\bf Ti$_2$SnC}                         &{\bf 3.427}                 &{\bf 362.703}       &{\bf 0.602}  &{\bf 5.897}   \\
GGA\cite{ali2020physical}               &3.360                       &                    &             &             \\
GGA\cite{kanoun2009theoretical}         &3.300                       &                    &             &             \\
GGA\cite{hug2006electronic}             &3.710                       &                    &             &             \\
{\bf Mo$_2$SnC}                         &{\bf 4.838}                 &{\bf 218.861}       &{\bf 0.774}  &{\bf 7.635}    \\

{\bf Lu$_2$SnC}                         &{\bf 5.177}                 &{\bf 105.345}       &{\bf 0.798}  &{\bf 5.963}     \\ 
Exp\cite{kuchida2013superconductivity}  &                            &                    &             &5.2
\label{TabS2}
\end{tabular}
\end{ruledtabular}
\end{table*}

\begin{figure*}
\includegraphics[width=0.9\columnwidth]{S0.eps}% Here is how to import EPS art
\vspace{0cm}
\caption{(a) The hexagonal crystal structure of Cr$_2$AlN, where blocks of Cr-N (formed by edge-shared Cr$_6$N octahedra) are sandwiched with Al atomic sheets. (b) The hexagonal Brillouin zone  for Cr$_2$AlN.}
\label{fig_Scrystal} 
\end{figure*}

\begin{figure*}[p]
\includegraphics[width=18cm]{S1.eps}
\vspace{0cm}
\caption{ The electronic band structure, the total and atomic projected electronic local density of states and Fermi surface for the hexagonal phase of M$_2$GeC (M: Mo and Lu).} 
\label{figS1} 
\end{figure*}


\begin{figure*}[p]
\includegraphics[width=18cm]{S2.eps}
\vspace{0cm}
\caption{Phonon dispersion curves, total, partial vibrational density of states and the calculated electron–phonon spectral function $\alpha^2  F(\omega)$ (red line) and the variation of the electron–phonon coupling parameter (blue line) with rising frequency $\lambda$($\omega$) of M$_2$GeC (M: Mo and Lu).} 
\label{figS2} 
\end{figure*}


\begin{figure*}[p]
\includegraphics[width=18cm]{S3.eps}
\vspace{0cm}
\caption{ The electronic band structure, the total and atomic projected electronic local density of states and Fermi surface for the Hexagonal phase of M$_2$SnC (M: Ti, Mo and Lu)} 
\label{figS3} 
\end{figure*}


\begin{figure*}[p]
\includegraphics[width=18cm]{S4.eps}
\vspace{0cm}
\caption{Phonon dispersion curves, total, partial vibrational density of states and the calculated electron–phonon spectral function $\alpha^2  F(\omega)$ (red line) and the variation of the electron–phonon coupling parameter (blue line) with rising frequency $\lambda$($\omega$) of M$_2$SnC (M: Ti, Mo and Lu)} 
\label{figS4} 
\end{figure*}



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