Observe that in order to preserve full coverage, any given specification $\{\lx, p \land \pc_p \} f(x) \{ q \land \pc_q \}^{\sexp}$ has to maintain that $\pc_p \Rightarrow \pc_q$, meaning that the post-condition cannot introduce new pure facts about existing symbolic variables, that is, cut execution paths. This is a standard requirement which can easily be met for bounded code, but has to be handled via user-defined predicates for code with unbounded loops or recursive calls. While Gillian does support user-defined predicates, the related machinery will be introduced in subsequent versions of this technical report.

Finally, note that backward completeness no longer holds in general because it is possible to define over-approximating user-defined predicates that can lead to loss of information due to predicate folding. If the specifications are not over-approximating, either due to functions not having unbounded executions or the user-defined predicates not losing information, then backward completeness does hold and the \gil execution still does not have false positive bug reports.

\begin{lemma}[Forward Soundness: \gil One-Step]\label{lem:gilfspecone}
Given a concrete and a symbolic compositional state model, $\gstate(\vals,
\vals, \corePreds, \cmemory(\vals, \corePreds), \allocator(\vals)) =
\langle\sset{\gstate}, \ceval{~}{}, \kwT{ea} \rangle$ and $\gstate(\sexps,
\lxs, \corePreds, \cmemory(\sexps, \corePreds),$ $\allocator(\sexps)) = \langle\sset{\hat\gstate}, \ceval{~}{}, \hat{\kwT{ea}} \rangle$, respectively,
if the consumers and producers in  satisfy Requirements~\ref{req:mem:fs}, and assuming that procedure specifications do not strengthen the context, then the corresponding \gil one-step symbolic semantics satisfies the following property:
\begin{mathpar}
\inferrule[GIL: Forward Soundness: Interpretations (GIL-FS-CFG-I)]{}{
\prog \vdash \scf \rightarrow \ssscf'  \gand
\sintp(\prog) \vdash  \sintp(\scf)  \semarrow {\cf'}  \implies \exists \varepsilon' \geq \varepsilon, \scf' \in \ssscf'.~\cf' = \sintpe{\varepsilon'}(\scf')
\end{array}}}
\end{mathpar}
\end{lemma}
\begin{proof}
Given Proposition~\ref{cor:spec:simpl}, we only need to prove the desired property for the new \prooflab{Call - Spec Re-Use} rule. Note that since this rule uses the collecting semantics in its premise, only one of its two variants can be applied at any given time. We prove the \prooflab{Call - Spec Re-Use - Success} case; the error case is analogous.
\case{\prooflab{Call - Spec Re-Use - Success}}
\begin{proofpf}
\pflongnumbers
\step{s1} {Hypotheses:
\begin{pfenum}
\item[(H1)] $\prog \vdash \scf \rightarrow \ssscf'$
\item[(H2)] $\sintp(\prog) \vdash  \sintp(\scf)  \semarrow {\cf'}$
\end{pfenum}}
\step{s2} {[by H1, \prooflab{Call} case]
\begin{pfenum}
\item $\cmd(\prog, \hat\cs, i) = y := \e(\e') \text{ with } \subst$
\item $\eex{\sst}{\e} \rightsquigarrow f \quad \eex{\sst}{\e'} \rightsquigarrow \sexp'$
\item $\{\lx, P \} f(x) \{Q\}^{\sexp} \in {\prog}.\kwT{specs}$
\item $\ssubst' = \ssubst[\lx \mapsto \sexp']$
\item $\fcolactionx{\sst}{\fgetter{\ssubst'}}{P}{}{\{ (\sst_j, \true)^\osucc} |_{j \in J} \}$
\item $\lstxs = \textsf{svars}(Q) \setminus \textsf{svars}(P)$
\item $(\action{\sst_j}{\kwT{iSym}}{|\lstxs|}{(\sst_j', \lstxs')^\osucc})|_{j \in J}$
\item $(\ssubst_{j}' = \ssubst'[ \lstxs \mapsto \lstxs'])|_{j \in J}$
\item $(\fcolactionx{\sst_j'}{\fsetter{\ssubst_j'}}{Q}{}{\{ (\sst_{lj}, \true)^\osucc \mid_{lj \in L_j} \} })|_{j \in J}$
\item $(\action{\sst_{lj}}{\kwT{setVar}_y}{\subst_j'(\sexp)}{\sst_{lj}'})|_{j \in J, lj \in L_j}$ [note that this transition is deterministic]
\item $\ssscf' = \bigcup_{j \in J} \bigcup_{lj \in L_j} \{ \tup{\sst_{lj}, \hat\cs, i{+}1} \}$
\end{pfenum}}
\step{s3} {[by H2, \prooflab{Call - Spec Re-Use - Success} case]
\begin{pfenum}
\item $\cmd(\sintp(\prog), \cs, i) = y := \e(\e') \text{ with } \sintp(\subst)$
\item $\eex{\sintp(\sst)}{\e} \rightsquigarrow f \quad \eex{\sintp(\sst)}{\e'} \rightsquigarrow \sintp(\sexp')$ [by forward soundness and determinism of state actions, noting that no allocation happens, and so $\varepsilon$ can stay the same]
\item $\{\lx, P \} f(x) \{Q\}^{\sexp} \in {\sintp(\prog)}.\kwT{specs}$
\item $\subst' = \sintp(\ssubst)[\lx \mapsto \sintp(\sexp')] = \sintp(\ssubst')$ [by 2.4]
\item $\fcolactionx{\sintp(\sst)}{\fgetter{\sintp(\ssubst')}}{P}{}{\{ (\st_k, \true)^\osucc} |_{k \in K} \}$ [given 2.5, by forward soundness of state assertion consumers all concrete transitions must be successful, noting that no allocation happens, and so $\varepsilon$ can stay the same]
\item $k \in K$
\begin{pfenum}
\item $\exists j \in J.~\st_k = \sintp(\sst_j)$ [by 2.5, 3.5, and forward soundness of state assertion consumers]
\end{pfenum}
\item $\lstxs = \textsf{svars}(Q) \setminus \textsf{svars}(P)$
\item $\action{\sintp(\st_j)}{\kwT{iSym}}{|\lstxs|}{(\st_j', \lstvs)^\osucc}$ [note that this transition is not deterministic]
\begin{pfenum}
\item $\exists \varepsilon' \geq \varepsilon.~\st_j' = \sintpe{\varepsilon'}(\sst_j') \land \lstvs = \sintpe{\varepsilon'}(\lstxs')$ [by forward soundness of $\kwT{iSym}$, 2.7, and 3.8]
\item $\varepsilon' = \varepsilon[\lstxs' \mapsto \lstvs]$ [by 3.8a]
\end{pfenum}
\item $\subst_{j}' = \subst'[ \lstxs \mapsto \lstvs ] = \sintpe{\varepsilon'}(\ssubst_j')$ [by 2.8]
\item $\action{\sintp(\sst_j')}{\fsetter{\sintp(\ssubst_j')}}{Q}{(\sst_j'', \true)^\osucc}$
\begin{pfenum}
\item $\exists lj \in L_j.~\st_j'' = \sintpe{\varepsilon'}(\sst_{lj})$ [by forward soundness of state assertion producers, 2.9, and 3.10, noting that no allocation happens, and so $\varepsilon'$ can stay the same]
\end{pfenum}
\item $\action{\sintpe{\varepsilon'}(\sst_{lj})}{\kwT{setVar}_y}{\sintp(\ssubst_j'(\sexp))}{\sst'}$
\begin{pfenum}
\item $\sst' = \sintpe{\varepsilon'}(\sst_{lj}')$ [by forward soundness of $\kwT{setVar}$, 2.10, and 3.11, noting that no allocation happens, and so $\varepsilon'$ can stay the same]
\end{pfenum}
\end{pfenum}}

\step{4}{$\exists \varepsilon' \geq \varepsilon, \scf' \in \ssscf'.~\cf' = \sintpe{\varepsilon'}(\scf')$ [for $\varepsilon' = \varepsilon'$ and $\scf' = \tup{\sst_{lj}', \hat\cs, i{+}1}$, given 2.11 and the fact that $\sintpe{\varepsilon'}(\cs) = \sintp(\cs)$ and therefore $\cf' = \tup{\sintpe{\varepsilon'}(\sst_{lj}'), \sintpe{\varepsilon'}(\hat\cs), i{+}1}$]}
\end{proofpf}

\end{proof}

From this Lemma, we can derive the following verification theorem, which has the same formulation as Theorem~\ref{lem:gttcsfs:mod} for partial verification, but given the ability to use procedure specifications, can handle code with recursive procedure calls and unbounded looping (which can be transformed to recursive procedure calls) and therefore captures full verification.

\begin{theorem}[\gil Verification]
\label{lem:verif}
\gil execution satisfies the following property:
\begin{mathpar}
\inferrule[GIL Verification]{}{
\prog \vdash \{ \scf \} \csemarrowfin \ssscf' \gand \cf \in \modls({\scf}) \gand~\prog \vdash  \cf  \csemarrowfin_C {\cf'} \\ \qquad \implies \exists \scf'.~\scf' \in \ssscf' \gand \cf' \in \modls(\scf') \gand \prog \vdash \scf \csemarrowfin_C \scf'
\end{array}}}
\end{mathpar}
\end{theorem}

\myparagraph{Correctness: Frame Preservation}
To prove that Theorem~\ref{thm:exfp} extends to the \gil execution with specification use, we only need to prove frame preservation of the two rules introduced in Figure~\ref{fig:specrules}. The error case is a composition of expression evaluation and an assertion consumer, which are all frame-preserving, and is, therefore, frame-preserving itself. The successful case is a composition of expression evaluation, followed by a consumer for the pre-condition, generation of post-condition existentials, and the production of the post-condition. The critical point is the production of the post-condition, which may have additional resource with respect to the pre-condition. However, this resource has to be existentially quantified, which means that fresh symbols will be created to represent it. This, in turn, means that the production of the post-condition, much like the production of allocated memory, will be frame-preserving.

Finally, given the frame preservation properties of the \gil execution, we can prove compositionality of function specifications, stated as follows:
\begin{itemize}[leftmargin=*]
\item if $\models\! \{\lx, P\} f(x) \{Q\}^{o(\sexp)}$ and $P * R$ has a successful production, then $\models_{} \{\lx, P \ast R \} f(x) \{Q \ast R \}^{o(\sexp)}$.
\end{itemize}

\bibliographystyle{ACM-Reference-Format}
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%%% ACM-Reference-Format-Journals [18-Jan-2012].

\begin{thebibliography}{4}

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\providecommand\showeprint[2][]{arXiv:#2}

\bibitem[\protect\citeauthoryear{Maksimovi\'c, Ayoun, Santos, and
Gardner}{Maksimovi\'c et~al\mbox{.}}{2021}]%
\bibfield{author}{\bibinfo{person}{Petar Maksimovi\'c},
\bibinfo{person}{Sacha{-}\'Elie Ayoun}, \bibinfo{person}{Jos\'e~Fragoso
Santos}, {and} \bibinfo{person}{Philippa Gardner}.}
\bibinfo{year}{2021}\natexlab{}.
\newblock \showarticletitle{{Gillian, Part II: Real-World Verification for
JavaScript and C}}. In \bibinfo{booktitle}{\emph{Computer Aided Verification
- 33rd International Conference, {CAV} 2020, Los Angeles, CA, USA, July
18-23, 2021}}. \bibinfo{publisher}{Springer}.
\newblock
\newblock
\shownote{Accepted for publication.}

\bibitem[\protect\citeauthoryear{O'Hearn}{O'Hearn}{2020}]%
\bibfield{author}{\bibinfo{person}{Peter~W. O'Hearn}.}
\bibinfo{year}{2020}\natexlab{}.
\newblock \showarticletitle{{Incorrectness Logic}}.
\newblock \bibinfo{journal}{\emph{Proc. {ACM} Program. Lang.}}
\bibinfo{volume}{4}, \bibinfo{number}{{POPL}} (\bibinfo{year}{2020}),
\bibinfo{pages}{10:1--10:32}.
\newblock
\urldef\tempurl%
\url{https://doi.org/10.1145/3371078}
\showDOI{\tempurl}

\bibitem[\protect\citeauthoryear{Raad, Berdine, Dang, Dreyer, O'Hearn, and
Villard}{Raad et~al\mbox{.}}{2020}]%
\bibfield{author}{\bibinfo{person}{Azalea Raad}, \bibinfo{person}{Josh
Berdine}, \bibinfo{person}{Hoang{-}Hai Dang}, \bibinfo{person}{Derek Dreyer},
\bibinfo{person}{Peter~W. O'Hearn}, {and} \bibinfo{person}{Jules Villard}.}
\bibinfo{year}{2020}\natexlab{}.
\newblock \showarticletitle{{Local Reasoning About the Presence of Bugs:
Incorrectness Separation Logic}}. In \bibinfo{booktitle}{\emph{Computer Aided
Verification - 32nd International Conference, {CAV} 2020, Los Angeles, CA,
USA, July 21-24, 2020, Proceedings, Part {II}}}
\emph{(\bibinfo{series}{Lecture Notes in Computer Science},
\bibfield{editor}{\bibinfo{person}{Shuvendu~K. Lahiri} {and}
\bibinfo{person}{Chao Wang}} (Eds.). \bibinfo{publisher}{Springer},
\bibinfo{pages}{225--252}.
\newblock
\urldef\tempurl%
\url{https://doi.org/10.1007/978-3-030-53291-8\_14}
\showDOI{\tempurl}

\bibitem[\protect\citeauthoryear{Santos, Maksimovic, Ayoun, and Gardner}{Santos
\bibfield{author}{\bibinfo{person}{Jos{\'{e}}~Fragoso Santos},
\bibinfo{person}{Petar Maksimovic}, \bibinfo{person}{Sacha{-}{\'{E}}lie
Ayoun}, {and} \bibinfo{person}{Philippa Gardner}.}
\bibinfo{year}{2020}\natexlab{}.
\newblock \showarticletitle{{Gillian, Part I: a Multi-language Platform for
Symbolic Execution}}. In \bibinfo{booktitle}{\emph{Proceedings of the 41st
{ACM} {SIGPLAN} International Conference on Programming Language Design and
Implementation, {PLDI} 2020, London, UK, June 15-20, 2020}},
\bibfield{editor}{\bibinfo{person}{Alastair~F. Donaldson} {and}
\bibinfo{person}{Emina Torlak}} (Eds.). \bibinfo{publisher}{{ACM}},
\bibinfo{pages}{927--942}.
\newblock
\urldef\tempurl%
\url{https://doi.org/10.1145/3385412.3386014}
\showDOI{\tempurl}

\end{thebibliography}

\end{document}
\endinput


# Antiferromagnetic order in Co-doped Fe5GeTe2 probed by resonant magnetic
x-ray scattering

Xiang Chen<EMAIL_ADDRESS>Materials Science Division, Lawrence
Berkeley National Lab, Berkeley, California 94720, USA  Physics Department,
University of California, Berkeley, California 94720, USA  Enrico Schierle
Helmholtz-Zentrum Berlin für Materialen und Energie, BESSY II, D-12489 Berlin,
Germany.  Yu He  Department of Applied Physics, Yale University, New Haven,
Connecticut, 06511, USA  Physics Department, University of California,
Berkeley, California 94720, USA  Materials Science Division, Lawrence Berkeley
National Lab, Berkeley, California 94720, USA  Mayia Vranas  Department of
Physics, University of California, San Diego, California 92093, USA  John
William Freeland  Advanced Photon Source, Argonne National Laboratory,
Argonne, Illinois 60439, USA  Jessica. L. McChesney  Advanced Photon Source,
Argonne National Laboratory, Argonne, Illinois 60439, USA  Ramamoorthy Ramesh
Department of Materials Science and Engineering, University of California,
Berkeley, California 94720, USA  Materials Science Division, Lawrence Berkeley
National Lab, Berkeley, California 94720, USA  Physics Department, University
of California, Berkeley, California 94720, USA  Robert J. Birgeneau  Physics
Department, University of California, Berkeley, California 94720, USA
Materials Science Division, Lawrence Berkeley National Lab, Berkeley,
California 94720, USA  Department of Materials Science and Engineering,
University of California, Berkeley, California 94720, USA  Alex Frano
<EMAIL_ADDRESS>Department of Physics, University of California, San Diego,
California 92093, USA

###### Abstract

The quasi-two-dimensional van der Waals magnet Fe5-δGeTe2 has emerged as a
promising platform for electronic and spintronic functionalities at room
temperature, owing to its large ferromagnetic ordering temperature
$T_{\text{C}}$ $\sim$ 315 K. Interestingly, by cobalt (Co) substitution of
iron in F5GT, $i.e.$
$({\text{Fe}}_{1-x}{\text{Co}}_{x})_{5-\delta}{\text{GeTe}}_{2}$ (Co-F5GT),
not only can its magnetic transition temperature be further enhanced, but the
magnetic and structural ground states can also be tuned. Specifically, an
antiferromagnetic (AFM) order is induced beyond the Co doping level $x\geq
0.4$. Here, we investigate the magnetic properties of a Co-F5GT single crystal
at $x=0.45(1)$, by utilizing the element specific, resonant magnetic x-ray
scattering technique. Our study reveals an A-type, Ising-like AFM ground
state, with a transition temperature $T_{\text{N}}$ $\sim$ 340 K. In addition,
our work unveils an important contribution from Co magnetic moments to the
magnetic order. The application of the in-plane magnetic fields gradually
polarize the spin moments along the field direction, but without inducing
incommensurate spin texture(s).

The unique nature of the cleavable, quasi-two-dimensional (quasi-2D) van der
Waals (vdW) materials offers rich platforms for exploring both exotic physical
phenomena and technological applications [1, 2, 3, 4, 5, 6, 7, 8, 9]. A broad
variety of intriguing physical phenomena have been reported in recent years by
investigating the (atomically thin) vdW bonded compounds. Some exceptional
phenomena include unconventional superconductivity in twisted graphene [10],
nonlinear Hall effect in few-layer WTe2 [11, 12], 2D magnetism in monolayer
Cr2Ge2Te6 [13, 14], CrI3 [15, 16] and the quantum anomalous Hall effect in
insulating tellurides [17, 18]. Among the different materials, the layered
vdW, itinerant magnets represent ideal quasi-2D material systems which enable
the coupling between the electronic and magnetic degrees of freedom [1, 4, 5,
6, 7]. Therefore, vast opportunities arise because of the rich physical
properties and the abundant possibilities for functional devices [19, 20].

Very recently, some promising quasi-2D magnetic tellurides [21, 22, 23, 24,
25, 26, 27, 28, 29, 30, 31, 32, 33], such as Fe5GeTe2 (F5GT) and CrTe2, with
atomically thin nanoflakes for promising room temperature (RT) spintronics
have been reported. The F5GT compound has a ferromagnetic (FM) transition
above RT at $T_{\text{C}}$ $\sim$ 315 K [30, 31]. Intriguingly, with cobalt
(Co) substitution of iron (Fe) in F5GT
($({\text{Fe}}_{1-x}{\text{Co}}_{x})_{5-\delta}{\text{GeTe}}_{2}$, Co-F5GT),
the magnetic transition temperature is further increased up to $\sim$ 360 K,
and the magnetic ground state switches from FM to antiferromagnetic (AFM) when
the Co doping level reaches $x\geq 0.4$ [34, 35]. Interestingly, a novel
wurtzite-type polar magnetic metal, which hosts a metastable, zero-field
N$\acute{\text{e}}$el-type skyrmion lattice at RT, was discovered at $x$ = 0.5
of Co-F5GT [36, 37]. The aforementioned observations of the Co-F5GT system
highlight its immense tunability and capacity for unusual magnetic properties
and could render applications in next-generation spintronics.

Despite the reports of exotic magnetic textures such as the chiral soliton
lattices and skyrmions in F5GT or Co-F5GT [36, 38, 39], there is a scarcity of
information from neutron scattering studies of these systems. The challenge of
growing single crystals large enough for neutron experiments hinders the
timely investigations of the magnetic properties of Co-F5GT. Here, we
demonstrate the applicability of utilizing the element specific, resonant
magnetic x-ray scattering (RMXS) technique [40, 41, 42, 43] to investigate the
bulk magnetic properties of a specific Co-F5GT single crystal at $x$ =
0.45(1). By tuning the x-ray energy $E=\hbar\omega$ to either the Fe $L$ edges
or Co $L$ edges, our study determines the contribution from both the Fe and Co
spin moments to the magnetic order. In addition, our scattering data affirms
the A-type AFM order, with a N$\acute{\text{e}}$el temperature $T_{\text{N}}$
$\sim$ 340 K and an out-of-plane moment direction. The influence of external
magnetic fields on the AFM ground state is also explored and found to be
consistent with magnetization measurements.

Figure 1: (Color online) (a) The schematic of the resonant magnetic x-ray
scattering (RMXS) experiment on a Co doped Fe5GeTe2 single crystal at
$x=0.45(1)$
($({\text{Fe}}_{0.55}{\text{Co}}_{0.45})_{5-\delta}{\text{GeTe}}_{2}$,
Co45-F5GT). (b) The crystal structure of Co45-F5GT, with the AA stacking order
[34, 36, 37]. (c) Temperature dependent magnetization of Co45-F5GT. (d) In-
plane isothermal magnetization of Co45-F5GT at varying temperatures. The
$H^{\ast}$ is the saturation field, beyond which the spin moments are fully
polarized along the field direction with a saturation magnetization
$M_{\text{sat}}$.  Figure 2: (Color online) Photon energy dependence of the
(0, 0, 0.5) peak at $T$ = 300 K or 350 K. (a) The (0, 0, 0.5) peak at $T$ =
300 K near the Fe $L$ edges. Maximum intensity is observed at Fe
$L_{\text{III}}$ edge ($E$ = 707.7 eV). (b)-(c) Energy dependent integrated
area of the (0, 0, 0.5) peak at both 300 K and 350 K, near the Fe $L$ edges
(b) and Co $L$ edges (c), respectively.

Single crystals of Co-F5GT at $x=0.45(1)$ (labeled here as Co45-F5GT) were
synthesized using the chemical vapor transfer technique [30, 31, 44]. The
chemical composition of the samples was verified by energy dispersive x-ray
spectroscopy, with the cation deficiency $|\delta|\leq 0.1$. The RMXS
experiments were performed at the UE46-PGM-1 beamline at Helmholtz-Zentrum
Berlin (HZB) and the 29-ID IEX beamline at Advanced Photon Source (APS) at
Argonne National Laboratory. A horizontal scattering geometry is utilized with
the sample lattice $c$ direction lying within the scattering plane (Fig.
1(a)). The Bragg peaks Q =
$(H\cdot\frac{2\pi}{a},K\cdot\frac{2\pi}{b},L\cdot\frac{2\pi}{c})$ are defined
in reciprocal lattice units ($r.l.u.$) with lattice parameters $a$ = $b$
$\approx$ 4.02 Å and $c$ $\approx$ 9.80 Å. The data are collected near the Fe
and Co $L$ edges to enhance the magnetic scattering signal [45, 46, 47]. The
incoming x-ray is either horizontally polarized ($H$-pol or $\pi$-pol) or
vertically polarized ($V$-pol or $\sigma$-pol), but the scattered x-rays are
not analyzed. Therefore, both outgoing $\sigma^{\prime}$-pol and
$\pi^{\prime}$-pol channels will contribute to the scattered intensity, which
can be written as [42, 43]:

$I_{\mu\nu}\propto|\sum_{j}e^{i\textbf{Q}\cdot\textbf{r}_{j}}(\textbf{e}_{\mu}\times\textbf{e}_{\nu}^{\ast})\cdot\textbf{m}_{j}F(\text{E})|^{2},$
(1)

to first order in the magnetic moment $\textbf{m}_{j}$ of the ion located at
site $\textbf{r}_{j}$ within the unit cell. $\textbf{e}_{\mu}$ ($\mu=\sigma$
or $\pi$) and $\textbf{e}_{\nu}$ ($\nu=\sigma^{\prime}$ or $\pi^{\prime}$) are
unit vectors along the polarization of the electric field component of the
incident and out-going x-ray beams, respectively. $F(\text{E})$ is the non-
local, photon energy dependent scattering tensor. Since the polarization of
the scattered light is not analyzed, the measured intensities are
$I_{\text{H}}$ = $I_{\pi}$ $\equiv$ $I_{\pi\sigma^{\prime}}$ \+
$I_{\pi\pi^{\prime}}$ and $I_{\text{V}}$ = $I_{\sigma}$ $\equiv$
$I_{\sigma\sigma^{\prime}}$ \+ $I_{\sigma\pi^{\prime}}$.

The F5GT compound is composed of three identical layers with the rhombohedral
layer stacking (space group R$\bar{3}$m), labelled as ABC-stacking [30]. It
was experimentally established that the stacking order in F5GT is susceptible
to external perturbations [34, 36, 48]. Upon replacing Fe with Co in F5GT,
when $x\geq 0.4$, the crystal structure undergoes a transition from ABC-
stacking to AA-stacking (space group P$\bar{3}$m1, as shown in Fig. 1(b))
[34]. Meanwhile, the magnetic ground state evolves from FM at $x<0.4$ to AFM
when $x\geq 0.4$. Our magnetization measurements on the Co45-F5GT sample
confirm an AFM order, with $T_{\text{N}}$ $=$ 341(2) K (Fig. 1(c)). It is
worth emphasizing that a unique type of stacking order, labelled as
AA′-stacking, was reported recently at $x=0.5$ [36, 37].
N$\acute{\text{e}}$el-type skyrmions were identified at this doping, because
of the AA′-stacking order which breaks the inversion symmetry and therefore
features a bulk Dzyaloshinskii–Moriya interaction [49, 50]. The spin
structures of Co-F5GT, however, have not yet been investigated by bulk
scattering techniques. Particularly, the magnetic ground state, including the
spin moment direction, shows a strong dependence on the Co doping level $x$
[30, 31, 36, 38]. Consequently, it is important to investigate the magnetic
properties of Co-F5GT by using a direct, bulk-sensitive, element-specific
scattering technique such as RMXS.

Figure 3: (Color online) Temperature dependence of the (0, 0, 0.5) peak, at
different energies: (a) Fe $L_{\text{III}}$ edge, $E$ = 708 eV, (b) Co
$L_{\text{III}}$ edge, $E$ = 778.2 eV. Solid lines in (a)-(b) are Lorentzian
fits to the peak intensity. (c) Power law fits to the integrated area of the
(0, 0, 0.5) peak at both Fe and Co $L_{\text{III}}$ edges. For better
comparison, the data is normalized to 1 at $T$ = 300 K. Figure 4: (Color
online) Magnetic field dependence of the (0, 0, 0.5) peak on resonance. (a)
The (0, 0, 0.5) peak at $T$ = 300 K and $E$ = 708 eV, collected under select
magnetic fields. Solid lines are Lorentzian fits to the peak intensity. (b)
Field dependence of the integrated area of the (0, 0, 0.5) peak at 300 K, at
both Fe and Co $L_{\text{III}}$ edges. The peak intensity is completely
suppressed beyond $H^{\ast}$ = 2.4(1) T. The green line is a fit to the data
by assuming the intensity is proportional to the $c$ component of the
magnetization squared $m_{\text{c}}^{2}$. For better comparison, the data is
normalized to 1 at zero field. (c) Phase diagram of the temperature dependent
$H^{\ast}$, from both the magnetization and RMXS measurements.

Our RMXS experiments were performed on Co45-F5GT single crystals with the AA-
stacking sequence (P$\bar{3}$m1, Fig. 1(b)). A strong peak is identified at
the structurally forbidden Bragg peak position $\textbf{Q}_{0}$ = (0, 0, 0.5)
below the magnetic onset temperature. To understand fully the nature of this
reflection, we present a thorough study of its dependence on x-ray energy,
polarization, temperature and magnetic field. Fig. 2 shows the x-ray energy
dependence of the $\textbf{Q}_{0}$ reflection. Resonant peak profiles were
recorded near both the Fe and Co $L_{\text{III}}$ or $L_{\text{II}}$ edges (Fe
$L_{\text{III}}$ $\sim 708$ eV, Fe $L_{\text{II}}$ $\sim 720$ eV, Co
$L_{\text{III}}$ $\sim 778$ eV and Co $L_{\text{II}}$ $\sim 793$ eV) [45, 46,
47]. A single energy dependent peak profile is evident near both the Fe and Co
$L_{\text{III}}$ ($L_{\text{II}}$) edges. This resonance in photon energy is a
strong signature of the magnetic nature of the $\textbf{Q}_{0}$ peak [42, 43,
45, 46, 47]. To support this argument further, the temperature dependence of
the $\textbf{Q}_{0}$ peak at the resonance energies (both Fe and Co
$L_{\text{III}}$ edges) was also examined, as shown in Figs. 2-3. With
increasing temperature, the peak intensity at either the Fe or the Co
$L_{\text{III}}$ edge is becoming vanishingly small and independent of
temperature above the transition temperature $T_{\text{N}}$ = 337(3) K (Fig.
3) [44].

Both the energy and temperature dependent studies in Figs. 2-3 demonstrate the
magnetic nature of the (0, 0, 0.5) peak, which is in agreement with the AFM
behavior from the magnetization in Fig. 1(c). This is further supported by
studying the magnetic field $H$ dependence of the $\textbf{Q}_{0}$ peak on
resonance (at both Fe and Co $L_{\text{III}}$ edges), examined at select
temperatures (Fig. 4). The field was applied in the $ab$ plane of the sample
(Fig. 1(a)). With increasing magnetic field, the $\textbf{Q}_{0}$ peak
intensity is gradually suppressed and reaches almost zero when $H$ $\geq$
$H^{\ast}$. This can be easily understood since the applied magnetic fields
are polarizing the spins along the field direction. When the magnetic moments
are fully polarized under $H$ $\geq$ $H^{\ast}$, the magnetic structure enters
into the polarized FM state and hence the peak intensity at $\textbf{Q}_{0}$
disappears. By assuming that the measured peak intensity is proportional to
the $c$ component of the magnetic moment squared $m_{\text{c}}^{2}$, which
equals $M_{\text{sat}}^{2}-M_{\text{ab}}^{2}$ from the magnetization data in
Fig. 1(d), an excellent agreement is found between the field dependence of the
resonant peak intensity at Fe $L_{\text{III}}$ edge and the measured
magnetization squared (green line in Fig. 4(b)). Specifically, the $H^{\ast}$
values at different temperatures inferred from both the RMXS data and the
magnetization data also match each other excellently, which are summarized and
plotted as the phase diagram in Fig. 4(c).

Combining all evidence, our study confirms the AFM nature of Co45-F5GT with a
propagation vector $\textbf{Q}_{0}$ = (0, 0, 0.5) and a transition temperature
$T_{\text{N}}$ $\sim$ 340 K. The similar behavior of the magnetic peak at both
the Fe or Co $L$ edges indicates an important contribution from the Co spin
moments in Co-F5GT. This explains why the saturation moment of Co45-F5GT at 10
K (Fig. 1(d))–$M_{\text{sat}}\sim$ 8 $\mu_{\text{B}}$ per formula unit
($f.u.$)–is only slightly smaller than the value of $M_{\text{sat}}\sim$ 10
$\mu_{\text{B}}$ per $f.u.$ in F5GT ($x$ = 0). By assuming the average Fe
magnetic moment is unchanged ($\sim$ 2 $\mu_{\text{B}}$/Fe) [30, 31, 36, 48],
the average Co spin moment contribution to the magnetization is estimated to
be $\sim$ 1.1 $\mu_{\text{B}}$/Co in Co45-F5GT. Clearly, identifying the
presence of an ordered magnetic moment at the Co site is an important result
of our work.

Figure 5: (Color online) (a) The (0, 0, 0.5) peak, collected at 708 eV and
300K, with the incoming x-ray either horizontally ($\pi$-pol) or vertically
($\sigma$-pol) polarized. $I_{\sigma}$ = $I_{\pi}$ is observed. (b) The
intensity ratio $I_{\sigma}$/$I_{\pi}$ $\approx$ 1 in a wide range of
azimuthal angle $\Psi$, where $\Psi$ = 0 is defined when the [1 0 0] direction
is parallel to the scattering plane.

Another important piece of information is the magnetic moment direction,
specifically the magnetic structure below $T_{\text{N}}$. This can be inferred
from the representational analysis [51], the photon polarization and azimuthal
dependence of the RMXS data. The magnetic representation of the
crystallographic sites of Fe or Co can be decomposed in terms of the
irreducible representations (IRs) [51], with the propagation vector
$\textbf{Q}_{0}$ = (0, 0, 0.5) (Table I):
$\Gamma_{\text{Mag}}=1\Gamma_{2}^{1}+1\Gamma_{3}^{1}+1\Gamma_{5}^{2}+1\Gamma_{6}^{2}$,
where $\Gamma_{2}$ , $\Gamma_{3}$ are one-dimensional IRs with moments
pointing parallel to the $c$ axis and $\Gamma_{5}$ , $\Gamma_{6}$ are two-
dimensional IRs with basis vectors lying in the $ab$-plane.

Considering the AFM nature, only the solutions from $\Gamma_{3}$ and
$\Gamma_{5}$ are possible. Experimentally, the magnetic peak intensity at the
$\textbf{Q}_{0}$ position remains the same with either $\sigma$ or $\pi$
polarized incoming x-rays (Fig. 5(a)), which implies $I_{\pi}$ = $I_{\sigma}$:

$|\sum_{j}e^{i\textbf{Q}\cdot\textbf{r}_{j}}\textbf{k}_{f}\cdot\textbf{m}_{j}|^{2}=|\sum_{j}e^{i\textbf{Q}\cdot\textbf{r}_{j}}\textbf{k}_{i}\cdot\textbf{m}_{j}|^{2}+|\sum_{j}e^{i\textbf{Q}\cdot\textbf{r}_{j}}\textbf{e}_{\sigma}\cdot\textbf{m}_{j}|^{2}$
(2)

where $\textbf{k}_{i}$ and $\textbf{k}_{f}$ are the unit vectors along the
incoming and reflected photon wave-vector directions, respectively. The
azimuthal $\Psi$ dependence of the $\textbf{Q}_{0}$ peak was performed by
rotating the sample about the $\textbf{Q}_{0}$ direction, which is parallel to
the $c$-axis (Fig. 1(a)), and measuring the peak intensity for both
polarizations. Clearly, $I_{\sigma}$ $\approx$ $I_{\pi}$ is maintained over a
broad range of the $\Psi$ angle (Fig. 5(b)). This rules out the possible
contribution from $\Gamma_{5}$ since the presence of in-plane moments will
manifest as an angle-dependent azimuthal scan by virtue of rotating the
projections shown in Equation (2). It is evident that only the solution from
$\Gamma_{3}$ satisfies Equation (2) ($I_{\pi}$ = $I_{\sigma}$), regardless of
the $\Psi$ angle, since the magnet moments pointing along the $c$ direction
are parallel to the rotation axis along the $\textbf{Q}_{0}$ direction. From
these arguments, the experimental data determine an A-type AFM order with
Ising moments in Co45-F5GT, as depicted in Fig. 1(b).

IR | BV | Atom | BV components
---|---|---|---
|  |  | $m_{\|a}$ | $m_{\|b}$ | $m_{\|c}$
$\Gamma_{2}$ | $\mbox{\boldmath$\psi$}_{1}$ | 1 | 0 | 0 | 1
|  | 2 | 0 | 0 | 1
$\Gamma_{3}$ | $\mbox{\boldmath$\psi$}_{2}$ | 1 | 0 | 0 | 1
|  | 2 | 0 | 0 | -1
$\Gamma_{5}$ | $\mbox{\boldmath$\psi$}_{3}$ | 1 | 0 | -1 | 0
|  | 2 | 0 | 1 | 0
| $\mbox{\boldmath$\psi$}_{4}$ | 1 | -2 | -1 | 0
|  | 2 | 2 | 1 | 0
$\Gamma_{6}$ | $\mbox{\boldmath$\psi$}_{5}$ | 1 | 2 | 1 | 0
|  | 2 | 2 | 1 | 0
| $\mbox{\boldmath$\psi$}_{6}$ | 1 | 0 | -1 | 0
|  | 2 | 0 | -1 | 0
Table 1: Basis vectors for the space group P$\bar{3}$m1 with the propagation
vector $\textbf{Q}_{0}$ = (0, 0, 0.5). The decomposition of the magnetic
representation for the Fe$1$/Co$1$ site is
$\Gamma_{\text{Mag}}=1\Gamma_{2}^{1}+1\Gamma_{3}^{1}+1\Gamma_{5}^{2}+1\Gamma_{6}^{2}$.
The two different atoms of the Fe1/Co1 site within the unit cell are defined:
atom1, $(0,~{}0,~{}0.2318)$; atom2, $(0,~{}0,~{}0.7682)$. The results from the
irreducible representations analysis for the other Fe/Co sites are similar.

Our RMXS study together with the magnetization data on the Co45-F5GT single
crystals confirm the long-range AFM ground state. Under moderate magnetic
fields ($H<H^{\ast}$), this AFM state is still maintained, since the (0, 0,
0.5) magnetic peak is only gradually weakened in magnitude, but without
becoming incommensurate or significantly broadened in peak width. This
suggests that it is unlikely to have the magnetic field induced exotic spin
textures in Co45-F5GT with the AA-stacking order, unlike the novel AA′-stacked
structure hosting N$\acute{\text{e}}$el-type skyrmion lattices in Co-F5GT at
$x$ = 0.50 [36, 37]. The contrasting magnetic textures in Co-F5GT, albeit with
the similar Co doping level $x$ = 0.45 or $x$ = 0.5, highlight the essential
role of the underlying lattice symmetry, as well as the contribution from the
Co spin moments. Additionally, the Co magnetic moments appear to play an
essential role for the enhancement of the magnetic transition temperature in
Co-F5GT, in contrast to the nonmagnetic dilution in Ni doped F5GT [48].

In summary, our RMXS study together with the magnetization measurements on the
Co45-F5GT sample confirms the Ising nature of the A-type AFM spin structure,
with a propagation vector $\textbf{Q}_{0}$ = (0, 0, 0.5) and a
N$\acute{\text{e}}$el temperature $T_{\text{N}}$ $\sim$ 340 K. The unique,
element specific characteristics of RMXS, observed through tuning the photon
energy, highlight the sizable contribution from the Co spin moments to the AFM
order. In addition, the magnetic ground state under the in-plane magnetic
fields has also been investigated, suggesting the critical role of the
underlying lattice symmetry for stabilizing unusual spin textures. Our work
highlights the applicability of the RMXS technique to study the magnetic
properties in Co-F5GT and other quasi-2D vdW magnets.

X.C. wishes to thank Fanny M. Rodolakis for assistance before and during the
RMXS experiment carried out at the Advanced Photon Source. Work at University
of California, Berkeley and the Lawrence Berkeley National Laboratory was
funded by the U.S. Department of Energy, Office of Science, Office of Basic
Energy Sciences, Materials Sciences and Engineering Division under Contract
No. DE-AC02-05-CH11231 within the Quantum Materials Program (KC2202). Work at
UC San Diego was supported by the National Science Foundation under Grant No.
DMR-2145080. This research used resources of the Advanced Photon Source, a
U.S. Department of Energy (DOE) Office of Science User Facility at Argonne
National Laboratory and is based on research supported by the U.S. DOE Office
of Science-Basic Energy Sciences, under Contract No. DE-AC02-06CH11357. We
thank HZB for the allocation of synchrotron radiation beamtime. A.F. was
supported by the Alfred P. Sloan Foundation (FG-2020-13773) and the Research
Corporation for Science Advancement via the Cottrell Scholar Award (27551).

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