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\newcommand{\1}{Nd$_{0.9}$Sm$_{0.1}$FeO$_{3}$}
\newcommand{\2}{Nd$_{0.7}$Sm$_{0.3}$FeO$_{3}$}
\newcommand{\3}{Nd$_{0.5}$Sm$_{0.5}$FeO$_{3}$}
\newcommand{\4}{Nd$_{1-x}$Sm$_{x}$FeO$_{3}$}

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\begin{document}

\title{Supplemental information for\\
Tuning of Spin Reorientation and Spin Switching in Sm-doped NdFeO$_3$}

\author{
Sk Samim Ahammad\thanks{Email: sksamima@iisc.ac.in}\\
Department of Physics, Indian Institute of Science, Bangalore, India
\and
C.M.N. Kumar\\
Institute of Nuclear Physics, Polish Academy of Sciences, Krakow, Poland
\and
Michał Rams\\
Jagiellonian University, Kraków, Poland
\and
Suja Elizabeth\\
Department of Physics, Indian Institute of Science, Bangalore, India
}

\date{\today}
\maketitle



\beginsupplement

\newpage
\begin{figure}[!t]
\centering\includegraphics[width=0.9\linewidth]{FigureS1.png}
\caption{Rietveld refinement results of PXRD patterns of \4\ ($x=0.1, 0.3, 0.5$) samples.}
\end{figure}

\FloatBarrier

\begin{table}[t]
\centering
\caption{Structural parameters, selected atomic coordinates, Fe--O bond distances, O--Fe--O bond angles, Rietveld refinement reliability factors, and Goldschmidt tolerance factor $t$  for Nd$_{1-x}$Sm$_x$FeO$_3$ compositions.}
\label{tab:combined_orthoferrite}

\setlength{\tabcolsep}{6pt}
\renewcommand{\arraystretch}{1.15}

\begin{tabular}{lccc}
\toprule
Composition ($x$) & 0.1 & 0.3 & 0.5\\
\midrule

\multicolumn{4}{l}{\textbf{Lattice parameters}} \\
$a$ (\AA) & 5.4426(2) & 5.4313(3) & 5.4226(2) \\
$b$ (\AA) & 5.5852(2) & 5.5865(3) & 5.5906(2) \\
$c$ (\AA) & 7.7531(3) & 7.7402(4) & 7.7307(3) \\
$V$ (\AA$^{3}$) &  235.68(2) & 234.85(2) & 234.36(2) \\

\addlinespace
\multicolumn{4}{l}{\textbf{Atomic coordinates (fractional)}} \\
Nd/Sm $x$  & 0.9862(7) & 0.9874(8) & 0.9853(7) \\
Nd/Sm $y$  & 0.0489(3) & 0.0507(3) & 0.0524(3) \\
O(1) $x$   & 0.097(3)  & 0.098(4)  & 0.089(3) \\
O(1) $y$   & 0.478(3) & 0.479(4) & 0.476(3) \\
O(2) $x$   & 0.715(3) & 0.717(4) & 0.722(4) \\
O(2) $y$   & 0.291(3) & 0.303(3) & 0.289(3) \\
O(2) $z$   & 0.033(2) & 0.040(3) & 0.041(2) \\

\addlinespace
\multicolumn{4}{l}{\textbf{Fe--O bond distances (\AA)}} \\
Fe--O1  & 2.013(6) & 2.010(6) & 1.997(4) \\
Fe--O2 (short) & 1.96(2) & 1.92(2) & 1.94(2) \\
Fe--O3 (long)   & 2.02(19) & 2.09(2) & 2.04(2) \\

\addlinespace
\multicolumn{4}{l}{\textbf{O--Fe--O bond angles (deg)}} \\
O1--Fe--O2  & 92.6(1) & 91.3(14) & 89.4(12) \\
O1--Fe--O2  & 85.5(2) & 86.9(11) & 87.4(10) \\
O1--Fe--O2  & 87.4(12) & 88.7(12) & 90.6(10) \\
O1--Fe--O2  & 94.5(13) & 93.1(13) & 92.6(11) \\
O2--Fe--O2  & 90.2(13) & 90.6(13) & 89.9(13) \\
O2--Fe--O2  & 89.8(15) & 89.4(13) & 90.1(15) \\

\multicolumn{4}{l}{\textbf{Rietveld refinement reliability factors}} \\
$\chi^{2}$ & 1.67 & 1.26 & 1.91\\
$R_{wp}$ & 17.2 & 18.3 & 15.8 \\
$R_{exp}$ & 13.3 & 16.3 & 11.4 \\

\multicolumn{4}{l}{\textbf{Goldschmidt tolerance factor}} \\
$t$ & 0.9222 & 0.9201 & 0.9180 \\

\bottomrule
\end{tabular}

\vspace{2mm}
\footnotesize
\textit{Notes:} The crystal structure was refined in the orthorhombic \textit{Pbnm} setting (No.~62).
The Wyckoff positions are as follows:
A-site (Nd/Sm) at 4$c$ $(x, y, 1/4)$, Fe at 4$b$ $(0, 0.5, 0)$, O(1) at 4$c$ $(x, y, 1/4)$, and O(2) at 8$d$ $(x, y, z)$.
\end{table}

\clearpage
\begin{figure}[!t]
\centering\includegraphics[width=0.8\linewidth]{FigureS2}
\caption{(a--c) XRD of the oriented crystals of (\4) cut into discs with the crystallographic ($h$00) plane perpendicular. Insets show Laue images of the ($h$00) plane and photographs of the grown crystals.
(d--f) Rocking curve of (100) plane; black circles are experimental data and red curves are pseudo-Voigt fits.}
\label{laue1}
\end{figure}

\begin{table}[!t]
\centering
\caption{
{FWHM of rocking curves shown in figure \ref{laue1}(d-f).}
}
\begin{tabular}{lccc}
\hline
\hline
Composition $x$ & 0.1 & 0.3 & 0.5 \\
\hline
FWHM & 0.092(4) & 0.092(1) & 0.104(2) \\
\hline
\hline
\end{tabular}
\end{table}


\FloatBarrier  

\clearpage

\subsection*{Supplementary note 1: Explanation of different magnetization protocols.}

\textbf{ZFC (Zero-Field Cooled):} The sample is cooled from a high temperature to the lowest measurement temperature (2~K for $x = 0.5$) without applying any magnetic field. At 2~K, a magnetic field is applied, and the magnetization data are recorded during subsequent heating under that applied field.

\textbf{FCC (Field-Cooled Cooling):} The sample is cooled from a high temperature down to 2~K under an applied magnetic field, and the magnetization data are recorded during the cooling process.

\textbf{FCW (Field-Cooled Warming):} After cooling the sample from a high temperature to 2~K under an applied magnetic field, the sample is heated with the same field, and the magnetization data are recorded as a function of temperature.

\textbf{NFC (Negative Field Cooled):} The sample is cooled from a high temperature to 2~K under a negative magnetic field. At 2~K, the field is switched to a positive value, and the magnetization data are collected during the subsequent heating process.

\FloatBarrier  

\begin{figure}[t]
\centering\includegraphics[width=0.5\linewidth]{FigureS3.pdf}
\caption{Temperature derivative of the magnetization along the $a$-axis under field-cooled cooling (FCC) conditions for \4.}
\label{tsr_diff}
\end{figure}

\begin{figure}[t]
\centering\includegraphics[width=0.5\linewidth]{FigureS4.pdf}
\caption{Temperature derivative of the magnetization along the $a$-axis under zero-field-cooled (ZFC) conditions.}
\label{zfc_diff}
\end{figure}

\begin{figure}[t]
\centering\includegraphics[width=0.5\linewidth]{FigureS5.pdf}
\caption{Temperature derivative of the magnetization along the $a$-axis under negative-field-cooled (NFC) conditions.}
\label{nfc_diff}
\end{figure}

\FloatBarrier  

\begin{table}[t]
\centering
\caption{$T_{\mathrm{SR}}$ extracted from the extrapolation of the $dM_a/dT$ curve measured with the FCC protocol.}
\begin{tabular}{lcc}
\toprule
Composition & $T_{\mathrm{SR1}}$ (K) & $T_{\mathrm{SR2}}$ (K) \\
\midrule
0.1 & 233.2 & 152.9 \\
0.3 & 313.3 & 247.6 \\
0.5 & 369.2 & 336.3\\
\bottomrule
\end{tabular}
\end{table}

\begin{table}[t]
\centering
\caption{$T_{\mathrm{ZFC}}$ extracted from the peak position of the $dM_a/dT$ curve measured with the ZFC protocol.}
\begin{tabular}{cc}
\toprule
Applied Field (Oe) & Temperature (K) \\
\midrule
20 & 174.7 \\
50 & 164.6 \\
\bottomrule
\end{tabular}
\end{table}

\begin{table}[t]
\centering
\caption{$T_{\mathrm{ZFC}}$ extracted from the peak position of the $dM_a/dT$ curve measured with the NFC protocol.}
\begin{tabular}{cc}
\toprule
Applied Field (Oe) & Temperature (K) \\
\midrule
20 & 333.3 \\
50 & 271.5 \\
75 & 182.9 \\
100 & 125.3 \\
150 & 45.4 \\
200 & 19.4 \\
\bottomrule
\end{tabular}
\end{table}

\FloatBarrier  

\begin{figure}[t]
\centering\includegraphics[width=0.5\linewidth]{FigureS6.pdf}
\caption{Magnetization of \3\ at low temperatures. The field where $M$ is temperature independent gives the molecular field of Fe moments at the rare-earth site (see main text).}
\label{mh-high-field}
\end{figure}

\begin{figure}[t]
\centering\includegraphics[width=0.5\linewidth]{FigureS7.pdf}
\caption{Temperature dependence of the switching energy barrier in \3.
This was calculated as $\Delta E = H \Delta M/2$, at the switching of magnetization of the hysteresis loop, and at the switching of magnetization during warming the sample at different fields following NFC (data in Figures 4 and 6).
}
\label{fig:dez}
\end{figure}


\clearpage
\subsection*{Supplementary note 2: Magnetization conservation across the spin-switching transition}

To verify the classification of the observed transition as type-I spin switching, the magnetization values immediately before and after the switching event were evaluated for both zero-field-cooled (ZFC) and negative-field-cooled (NFC) measurement protocols.

Tables \ref{table:deltaM-ZFC} and \ref{table:deltaM-NFC} summarize the magnetization measured just below and just above the spin-switching temperature T$_{SSW}$, together with the corresponding change $\Delta${M}. In all cases, the absolute value of the magnetization remains nearly conserved across the switching transition, with only a reversal of its sign.

This behavior is a defining characteristic of type-I spin switching, arising from the abrupt reconfiguration of two antiparallel magnetic sublattices when the Zeeman energy overcomes the anisotropy energy barrier.

\begin{table}[H]
\centering
\caption{$\Delta M$ in the ZFC protocol.}
\begin{tabular}{c c c c}
\hline\hline
Applied field & Magnetic moment before switching & Magnetic moment after switching &
$\Delta M$ \\
(Oe) & (emu\,g$^{-1}$) & (emu\,g$^{-1}$) & (emu\,g$^{-1}$) \\
\hline
20  & $-1.031$ & $1.028$ & $2.059$ \\
50  & $-1.038$ & $1.041$ & $2.079$ \\
\hline\hline
\end{tabular}
\label{table:deltaM-ZFC}
\end{table}


\begin{table}[H]
\centering
\caption{$\Delta M$ in the NFC protocol.}
\begin{tabular}{c c c c}
\hline\hline
Applied field & Magnetic moment before switching & Magnetic moment after switching &
$\Delta M$ \\
(Oe) & (emu\,g$^{-1}$) & (emu\,g$^{-1}$) & (emu\,g$^{-1}$) \\
\hline
20  & $-0.905$ & $0.875$ & $1.78$ \\
50  & $-0.956$ & $0.949$ & $1.905$ \\
75  & $-1.002$ & $0.998$ & $2.00$ \\
100 & $-1.003$ & $1.003$ & $2.06$ \\
150 & $-0.916$ & $0.928$ & $1.844$ \\
200 & $-0.822$ & $0.856$ & $1.678$ \\
\hline\hline
\end{tabular}
\label{table:deltaM-NFC}
\end{table}


\clearpage
\subsection*{Supplementary Note 3: Benchmarking of $T_{\mathrm{SSW}}$ tunability}

Table~\ref{tab:benchmark_condensed} summarizes representative reports of spin-switching tunability in rare-earth orthoferrites. Because the experimental protocols vary across studies (e.g., ZFC, FCC, FCW, and protocol variants involving field changes during cooling), the quantity $\Delta T_{\mathrm{SSW}}/\Delta H$ should be regarded as an effective tuning metric extracted from the reported field/protocol variation rather than a universal material constant.

For the present work, $\Delta T_{\mathrm{SSW}}$ is defined as the difference between the switching temperatures obtained under the ZFC and NFC protocols at the same warming field ($H_{\mathrm{warm}}=20$~Oe), i.e. $\Delta T_{\mathrm{SSW}}\approx 158$~K. $\Delta H$ corresponds to the magnitude of the negative cooling field used to imprint the bias ($|H_{\mathrm{cool}}|=20$~Oe), yielding $\Delta T_{\mathrm{SSW}}/\Delta H \approx 7.9$~K\,Oe$^{-1}$.

\begin{table}[H]
\centering
\caption{Condensed comparison of spin-switching tunability reported for representative orthoferrites.
$\Delta T_{\mathrm{SSW}}$ and $\Delta H$ are extracted from the protocol/field variation described in each reference.}
\label{tab:benchmark_condensed}
\begin{tabularx}{\textwidth}{%
>{\raggedright\arraybackslash}X
c c c
>{\raggedright\arraybackslash}X}
\hline\hline
Material (Ref.) & $\Delta H$ (Oe) & $\Delta T_{\mathrm{SSW}}$ (K) & $\Delta T_{\mathrm{SSW}}/\Delta H$ (K\,Oe$^{-1}$) & Brief protocol note \\
\hline
Nd$_{0.5}$Sm$_{0.5}$FeO$_3$ (this work) & 20 & 158 & 7.9 & ZFC vs NFC at $H_{\mathrm{warm}}=20$~Oe \vspace{10pt} \\ 
DyFeO$_3$ (1) & 200 & 100 & 0.5 & Switching enabled by applying $-200$~Oe at 300~K prior to cooling \vspace{10pt} \\ 
SmFeO$_3$ (2) & 50 & 77.5 & 1.55 & ZFC: 356~K (250~Oe) to 278.5~K (300~Oe) \vspace{10pt}\\ 
ErFeO$_3$ (3) & 20000 & 25 & 0.00125 & FCW at 50~Oe; $\pm 10$~kOe applied at 85~K \vspace{10pt}\\ 
NdFeO$_3$ (4) & 400 & 20 & 0.05 & ZFC: $\sim$20~K shift from 100 to 500~Oe \\



\hline\hline
\end{tabularx}
\end{table}





\clearpage
\begin{figure}[t]
\centering\includegraphics[width=0.7\linewidth]{FigureS8.png}
\caption{EDS spectrum of \3}
\label{EDS_pattern}
\end{figure}


\begin{thebibliography}{99}

\bibitem{SI-DyFeO3}

S.~Cao, L.~Chen, W.~Zhao, K.~Xu, G.~Wang, Y.~Yang, B.~Kang, H.~Zhao, P.~Chen,
A.~Stroppa, \textit{et al.},
``Tuning the weak ferromagnetic states in dysprosium orthoferrite,''
\textit{Scientific Reports}, vol.~6, p.~37529, 2016.


 \bibitem{SI-SmFeO3}

S.~Cao, H.~Zhao, B.~Kang, J.~Zhang, and W.~Ren,
``Temperature-induced spin switching in SmFeO$_3$ single crystal,''
\textit{Scientific Reports}, vol.~4, p.~5960, 2014.

\bibitem{SI-ErFeO3}

I.~Fita, R.~Puzniak, E.~Zubov, P.~Iwanowski, and A.~Wi{\'s}niewski,
``Temperature-driven spin switching and exchange bias in the ErFeO$_3$ ferrimagnet,''
\textit{Physical Review B}, vol.~105, p.~094424, 2022.


\bibitem{SI-NdFeO3}

S.~Yuan, W.~Ren, F.~Hong, Y.~Wang, J.~Zhang, L.~Bellaiche, S.~Cao, and G.~Cao,
``Spin switching and magnetization reversal in single-crystal NdFeO$_3$,''
\textit{Physical Review B}, vol.~87, p.~184405, 2013.


\end{thebibliography}
\end{document}

