# In-plane magnetic structure and exchange interactions in the high-
temperature antiferromagnet Cr2Al

Chengxi Zhao  Kisung Kang  Department of Materials Science and Engineering and
Materials Research Laboratory, University of Illinois at Urbana-Champaign,
Urbana, IL 61801, USA  Joerg C. Neuefeind  Neutron Scattering Division, Oak
Ridge National Laboratory, Oak Ridge, TN 37831, USA  André Schleife
Department of Materials Science and Engineering and Materials Research
Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
National Center for Supercomputing Applications, University of Illinois at
Urbana-Champaign, Urbana, IL 61801, USA  Daniel P. Shoemaker
<EMAIL_ADDRESS>Department of Materials Science and Engineering and
Materials Research Laboratory, University of Illinois at Urbana-Champaign,
Urbana, IL 61801, USA

###### Abstract

The ordered tetragonal intermetallic Cr2Al forms the same structure type as
Mn2Au, and the latter has been heavily investigated for its potential in
antiferromagnetic spintronics due to its degenerate in-plane Néel vector. We
present the single crystal flux growth of Cr2Al and orientation-dependent
magnetic properties. Powder neutron diffraction of Cr2Al and first-principles
simulations reveal that the magnetic ordering is likely in-plane and therefore
identical to Mn2Au, providing a second material candidate in the MoSi2
structure type to evaluate the fundamental interactions that govern spintronic
effects. The single ordering transition seen in thermal analysis and
resistivity indicates that no canting of the moments along the $c$ axis is
likely. Magnetometry, resistivity, and differential scanning calorimetry
measurements confirm the Néel temperature to be $634\pm 2$ K. First-principles
simulations indicate that the system has a small density of states at the
Fermi energy and confirm the lowest-energy magnetic ground state ordering,
while Monte Carlo simulations match the experimental Néel temperature.

##  I Introduction

Antiferromagnets have attracted considerable attention due to fundamentally
different proposed magnetization switching mechanisms and optically-probed
spin relaxation dynamics in the THz range,Kimel  _et al._ (2005); Little  _et
al._ (2017); Kampfrath  _et al._ (2011) compared to GHz for ferromagnets and
ferrimagnets.Viala  _et al._ (2004); Lee  _et al._ (2016); Chai  _et al._
(2013); Sharma  _et al._ (2016) Recent studies have demonstrated partial
switching of the Néel vector of antiferromagnetic CuMnAs and
Mn2AuBhattacharjee  _et al._ (2018); Barthem  _et al._ (2013); Jourdan  _et
al._ (2015), perhaps due to current-induced spin-orbit torques.Olejník  _et
al._ (2018); Saidl  _et al._ (2017); Wadley  _et al._ (2015, 2016, 2018) The
magnetic-field-induced spin rotation and spin-flop behavior of epitaxial
CuMnAs thin films has been probed using X-ray magnetic linear dichroism and
signatures of the Néel vector reorientation have been seen in the anisotropic
magnetoresistance with multiterminal devices.Wang  _et al._ (2020); Wadley
_et al._ (2016); Olejník  _et al._ (2018) Néel vector reorientations have also
been observed in femtosecond pump-probe magneto-optical Kerr effect
experiments.Saidl  _et al._ (2017)

Intriguingly, Chien _et al._ have shown that the purported spin-orbit torque
magnetoresistance effects may be artifacts of excessive current-induced
heating through multiterminal devices, thus unequivocal detection of the Néel
vector before and after SOT switching at low currents is required to
effectively prove magnetic switching.Chiang  _et al._ (2019) The in-plane
current-pules switching experiments on Mn2Au quantified the anisotropic
magnetoresistance and planar Hall effect of thin films.Bodnar  _et al._ (2018)
A spin-orbit-torque-driven antiferromagnetic resonance of Mn2Au has not been
observed to date by time-domain THz spectroscopy, despite early
reports.Bhattacharjee  _et al._ (2020)

Mn2Au remains antiferromagnetic until it forms a disordered (Mn,Au) solid
solution at 953 K,Cahn (1991) with a Néel temperature predicted to be above
1600 K, due to its large anisotropy energy, calculated based on first-
principles calculations.Khmelevskyi and Mohn (2008) Such large anisotropy may
not be desired in all spintronic applications. In particular, the ability to
observe a Néel-order-driven effect vanish at $T_{N}$, without excessive
current flow and Joule heating, would be a strong confirmation of true
spintronic effects. To understand the underlying physics and capabilities,
there is a pressing need to expand the library of intermetallic
antiferromagnets that share features with CuMnAs (which itself has not been
grown as bulk single crystals, likely due to a competing orthorhombic
phase)Uhlířová  _et al._ (2019) and Mn2Au. Single crystals of such materials
are needed to study the orientation dependence of their magnetic dynamics.

Figure 1:  The $k$=0 magnetic structure of Cr2Al is shown in (a), with Al
light and Cr dark. The reported magnetic structure from Atoji in (b) has a 65∘
canting away from the $c$ axis.Atoji (1965) No data in our study point to a
reduction in symmetry from our in-plane model to that of Atoji.

Cr2Al is an incongruently-melting intermetallic antiferromagnet with the MoSi2
structure type, tetragonal space group $I4/mmm$, isostructural to Mn2Au and
Au2Mn. There is no reported single crystal growth or measurements of its
anisotropic properties. Atoji presented variable-temperature powder neutron
diffraction of Cr2Al in 1965. Atoji (1965) In that study, the (001) neutron
diffraction peak intensity was used to determine the $T_{N}=598\pm 5$ K. The
magnetic structure was claimed to be inclined by an angle $65\pm 2.4^{\circ}$
toward the $c$ axis, though no explanation of that determination was given. We
show here that the in-plane (zero canting) spin configuration is far more
plausible.

In 1988, Semukhin _et al._ investigated the magnetic phase transition of Cr2Al
by high-temperature X-ray diffraction and calorimetry.Semukhin  _et al._
(1988) Sharp changes in the $a$ and $c$ axis lattice parameters and a heat
capacity anomaly occurred $655\pm 5$ K. Variation in the reported $T_{N}$ may
be due to a compositional width of the Cr2Al phase, which has been reported to
be about 10 %.Mahdouk and Gachon (2000); Rank  _et al._ (2019) Variable
$T_{N}$ can also arise from further alloying, as when Susner _et al._ showed
that the resistivity anomaly at $T_{N}$ shifts to lower temperatures and is
more pronounced with increased Fe substitution for Cr.Susner  _et al._ (2015)

Here we show the first steps of understanding the potential of Cr2Al as a
spintronic material, with single crystal growth via tin flux and melt
decanting. We consider the allowed magnetic orderings and examine the
sensitivity to a $c$-axis component of the local moments. Magnetometry
measurements with varying temperature and external field show the predicted
magnetic susceptibility and anisotropy, while resistivity and calorimetry
confirm the presence of a single $T_{N}$, without a second metamagnetic
transition that would be expected in the case of spin canting as proposed by
Atoji.Atoji (1965) First-principles simulations test multiple magnetic
configurations, with the experimental ordering found to be the predicted
ground state. Exchange coefficients and band structures show how Cr2Al has a
small density of states near the Fermi energy, and confirm the experimental
susceptibility and $T_{N}$.

##  II Methods

Tetragonal single crystals with edge lengths around 0.5 mm and mass $\sim$1-2
mg were grown from Sn flux. Cr ($>$99.99% metals basis) and Al (99.9999%
metals basis) powders were mixed in 2:1.5 molar ratio along with 95 at% Sn
powder (99.98% metals basis) inside an Ar filled glove box and vacuum sealed
inside a 13 mm inner diameter quartz tube. A stainless-steel mesh was inserted
inside the tube just above the powder mixture. The tube was heated to 1173 K
at 10 K/min and held for 12 hours, slowly cooled to 933 K at 2 K/hr, and then
the tube was taken out, flipped, and centrifuged to filter the grown crystals
with the inserted mesh, followed by an air quench to room temperature. The
collected crystals were sonicated in 1 M nitric acid for 2 hours to remove
residual Sn. Separately, Cr2Al powder was prepared by annealing a mixture of
Cr and Al powders with molar ratio of 2:1 that was pressed into a pellet and
heated at 1073 K for 72 hours.

Scanning electron microscopy was performed in a JEOL 6060 LV SEM, and X-ray
diffraction was performed in a Bruker D8 ADVANCE diffractometer with Mo
K$\alpha$ radiation. Powder neutron diffraction was performed on the NOMAD
instrument at the Spallation Neutron Source at Oak Ridge National
Laboratory.Calder  _et al._ (2018) Nuclear and magnetic structures were
refined using GSAS-IIToby and Von Dreele (2013) and structures were visualized
using VESTA.Momma and Izumi (2008) After alignment by XRD, single crystal
magnetometry was conducted using a standard quartz rod (2 to 400 K) and oven
attachment (up to 750 K) on a Quantum Design MPMS3 vibrating sample
magnetometer. High-temperature resistivity up to 773 K was measuring using a
two-point pressed contact configuration shown in Figure S1.sup  A separate
thermocouple was placed on the sample stage to measure the temperature and the
measurement was performed under flowing nitrogen atmosphere. Differential
scanning calorimetry was performed with 2.5 mg of crystals on a TA DSC 2500.

First-principles density functional theory (DFT) simulations were performed
using the Vienna _Ab-Initio_ Simulation Package Kresse and Furthmüller (1996);
Kresse and Joubert (1999) (VASP). In solving the Kohn-Sham equation, we used
the generalized-gradient approximation (GGA) formulated by Perdew, Burke, and
Ernzerhof Perdew  _et al._ (1996), to describe exchange and correlation. The
electron-ion interaction was described by the projector-augmented wave Blöchl
(1994) (PAW) method. For the Brillouin zone sampling,
$21\,\times\,21\,\times\,7$ Monkhorst-Pack (MP) Monkhorst and Pack (1976)
$\mathbf{k}$-points were used. The kinetic energy cutoff of the plane-wave
basis was chosen as 600 eV by convergence testing. Phonon dispersion
calculations within the finite displacement method were performed using the
phonopy package Togo and Tanaka (2015), a supercell of
$3\,\times\,3\,\times\,1$, and a $4\,\times\,4\,\times\,4$ MP
$\mathbf{k}$-point grid. We performed all energy dispersion calculations
accounting for noncollinear magnetism and spin-orbit coupling Steiner  _et
al._ (2016).

Exchange coefficients are calculated using the spin polarized relativistic
Korringa-Kohn-Rostoker (SPR-KKR) code Ebert  _et al._ (2011) and the relaxed
atomic structure from the DFT calculations described above. To integrate over
the Brillouin zone, 1000 randomly chosen $k$-points are used in our energy
convergence test with a criterion of $0.01$ meV/atom. Next, the exchange
coefficients are extracted by Lichtenstein’s approach, as implemented in the
SPR-KKR code Liechtenstein  _et al._ (1984). The exchange coefficients follow
from the isotropic exchange term in a Heisenberg model,

$\mathcal{H}_{ex}=-\sum_{i\neq j}J_{ij}e_{i}e_{j}.$ (1)

The exchange Hamiltonian ($\mathcal{H}_{ex}$) consists of isotropic exchange
coefficients ($J_{ij}$) and the unit vector of all magnetic moments at site
$i$ and $j$ ($e_{i}$ and $e_{j}$). $J_{ij}$ is calculated up to an relative
interaction distance $d=4.0a$, where $a$ is the $a$ axis lattice parameter of
Cr2Al.

We calculate atomistic spin dynamics using a Monte Carlo approach implemented
in UppASDEriksson  _et al._ (2017) to estimate the Néel temperature. To
simulate microscopic magnetism at finite temperature, the stochastic Landau-
Lifshitz-Gilbert (LLG) equationEriksson  _et al._ (2017) is solved,

$\begin{split}\frac{d\mathbf{m}_{i}}{dt}=&-\gamma_{L}\mathbf{m}_{i}\times(\mathbf{B}_{i}+\mathbf{B}_{i}^{\mathrm{fl}})\\\
&-\gamma_{L}\frac{\alpha}{m_{i}}\mathbf{m}_{i}\times\left[\mathbf{m}_{i}\times(\mathbf{B}_{i}+\mathbf{B}_{i}^{\mathrm{fl}})\right],\end{split}$
(2)

where $\alpha$ is an isotropic Gilbert damping constant and
$\gamma_{L}=\gamma/(1+\alpha^{2})$ is the renormalized gyromagnetic ratio.
$\mathbf{m}_{i}$ is the magnetic moment at site $i$ and here we use the value
from our DFT ground state simulations. $\mathbf{B}_{i}$ is the effective
magnetic field containing exchange, anisotropy, and magnetic dipolar
interactions at magnetic site $i$. The stochastic term is introduced by the
thermally fluctuating magnetic field term $\mathbf{B}_{i}^{\mathrm{fl}}$. This
term is shaped based on a central limit theorem as a form of Gaussian
distribution with zero mean and a variance that depends on temperature. All
atomistic spin dynamics calculations use a $12\times 12\times 4$ supercell.

##  III Results and Discussion

Figure 2:  SEM micrographs of Cr2Al crystals grown from Sn flux, after
centrifuging and (a) before and (b) after removal of Sn with 1 M nitric acid.
Figure 3:  X-ray diffraction data for Cr2Al crystals are shown for (a)
multiple crystals collected simultaneously, along with a Rietveld-refined
model for a powder. The asterisked peak is from the paraffin wax mount. In
(b), data are shown for a single crystal oriented in three orthogonal
directions and collected with the sample stationary. Insets show the
measurement configurations with crystals loaded in kapton capillaries.

The crystals obtained from melt centrifuging were metallic silver and had
rectangular prismatic shapes, with a typical appearance shown in Figure 2,
before and after etching excess Sn in nitric acid. The phase purity was
confirmed by powder XRD of several aligned crystals in a capillary and
measured while rotating, which gives peaks that more fully reproduce the
powder pattern of Cr2Al in Figure 3(a), with some expected difference in peak
intensity due to preferred orientation. Collection of XRD data on stationary
single crystals was used to determine the direction of the $c$ axis, with XRD
shown for a typical crystal measured on three orientations shown in Figure
3(b). Energy-dispersive X-ray spectroscopy gave an average Cr:Al ratio of
2.0(2), shown in Figure S2.sup

Figure 4:  Time-of-flight neutron diffraction patterns and Rietveld-refined
patterns from NOMAD detector banks with average collection angles
$2\theta=65^{\circ}$ (a) and $2\theta=15.1^{\circ}$ (b). The two banks are
optimized for collection on different ranges of $d$-spacings. Blue and red
tick marks indicate nuclear and magnetic peaks, respectively. Magnetic peaks
are indicated by arrows and indexing.

Figures 4(a) and 4(d) show the neutron diffraction pattern of a Cr2Al powder
at 298 K. The nuclear peaks give a clear fit to the ordered $I4/mmm$
structure, with the magnetic peak (100) clearly visible at $d=8.65$ Å. The
magnetic propagation vector $k=(000)$ reproduces all magnetic peaks, which
confirms the magnetic unit cell exhibits the same dimension as the
crystallographic unit cell. The magnetic space group of Cr2Al with the Néel
vector along [100] is $P_{I}nnm$ in Belov-Neronova-Smirnova (BNS) notation and
$I_{P}m^{\prime}m^{\prime}m$ in Opechowski-Guccione (OG) notation, and the
refinement in Figure 4 is thus constrained with Cr1 moments on the body center
sites opposite to the Cr2 moments on the edges of the cell.

The refined magnetic moments on Cr positions are 1.06(19)$\mu_{B}$. The
obtained magnetic structure resolved is visualized in Figure 1. Powder
diffraction is not sensitive to the orientation of the Cr moments within the
$ab$ plane. The refined structure, with the nearest-neighbor Cr spins
antiferromagnetic, which are in turn ferromagnetic across the Al plane, gives
a significantly improved fit over a model where the Cr bilayers have
ferromagnetic alignment within the layers, or antiferromagnetic alignment from
layer to layer. These fits to the neutron diffraction data are shown Figures
S4-5.sup  The latter configurations relax to a nonmagnetic state in first-
principles simulations. Likewise, a uniaxial configuration with spins along
$c$ does not reproduce the neutron data (Figure S6-8).sup  A small canting of
the moments toward $c$, as suggested by Atoji,Atoji (1965) does not
significantly improve the fit, even with the addition of an extra free
parameter, shown in Figure S9.sup  Small canting is, thus, unlikely due to the
additional irreducible representation that would be required to observe it,
along with the presence of a single Néel transition, which we will discuss
subsequently. Canting the moment along $c$ also leads to a calculated increase
in the total energy, as shown in Figure S10.

First-principles density functional theory simulations for antiferromagnetic
Cr2Al are implemented for ground state calculations, electronic band
structure, phonon dispersion, magnetic susceptibility, and exchange
coefficients. The calculated ground state confirms the $I4/mmm$ structure with
$a=2.98$ Å and $c=8.63$ Å, in good agreement with the room temperature lattice
parameters $a=3.00$ Å and $c=8.65$ Å from neutron diffraction in Figure 4.
Magnetic moments on Cr sites shown in Figure 1 converged to 1.311 $\mu_{B}$.
The discrepancy of 23 % likely arises from decrease of the low-temperature
moment upon heating to 300 K. Ground state DFT simulations for other possible
magnetic orderings (see Figure S4-5)sup  conclude that there is no stable
state for those.

Figure 5:  The temperature dependence of (a) Cr2Al dimensionless magnetic
susceptibility $\chi$ under an applied field $H=1~{}$T, measured to 750 K.
Zero-field-cooled (ZFC) and field-cooled (FC) measurements, represented as
open and filled circles, overlap in both crystal orientations. In (b), DSC
measurements of multiple crystals confirm $T_{N}$, with a small anomaly at the
Sn melting temperature 505 K.

The temperature dependence of the magnetic susceptibility of an aligned single
crystal from 430 to 750 K with $H=1$ T is shown in Figure 5(a). A small hump
is observed at $T_{N}=634$ K for measurements with $H||a$ and $H||c$. The
splitting of the two orientations and the lower susceptibility with field
along $a$ indicates that the magnetic ordering is in the $ab$ plane. Further
confirmation of $T_{N}$ is seen in calorimetry, shown in Figure 5(b). The
small event at $T=505$ K is the melting of a small amount of residual Sn.  The
fraction of residual Sn is estimated to be 2 wt% based on peak integration of
the DSC measurement (see Figure S11).sup ; Grønvold (1993)

Figure 6:  Field-dependent magnetization of Cr2Al at $T$ = 4, 300, 600, and
750 K (a-d), shown for field along the $a$ (blue) and $c$ axes (red).
Anisotropy in the susceptibility vanishes above $T_{N}\sim 634K$ (a) T = 4 K,
(b) T = 300 K, (c) T = 600 K, and (d) T = 750 K, with $H$ along the $a$-axis
(blue line) and $H$ along the $c$ axis (red line).

The magnetization as a function of applied field and orientation is shown in
Figure 6. The anisotropy consistently shows that the susceptibility along $c$
is greater than along $a$, which is to be expected for moments lying in the
$ab$ plane. No evidence of a spin flop transition is observed up to $H=7$ T.
When the temperature of the experiment is raised to 750 K, the anisotropy
between $c$ and $a$ directions vanishes, as expected above $T_{N}$. This is
the first measurement of the magnetic anisotropy of a bulk MoSi2-type
antiferromagnet in the vicinity of its $T_{N}$, and understanding the
temperature dependence will be a crucial topic of future work.

The phenomenon that the measured moment of Cr2Al is greater with $H||c$ over
the entire field and temperature range can be compared to similar measurements
on another in-plane degenerate antiferromagnet Fe2As.Yang  _et al._ (2020a) In
Fe2As, the low-field susceptibility mirrors that of Cr2Al. Above approximately
0.7 T, the susceptibility of Fe2As with $H||a$ becomes greater than with
$H||c$, indicating rotation of the antiferromagnetic domains into a single-
domain state with all moments likely along $b$. This domain rotation is absent
in Cr2Al for the fields we are able to achieve here. This lack of domain
rotation is most likely due to the small intrinsic susceptibility of Cr2Al
versus Fe2As.

The magnetic susceptibility of Cr2Al can be extracted from total energy
calculations within DFT for magnetic configurations with tilted magnetic
moments. Since an explicit magnetic field is not implemented in our DFT
simulations, we use the tilting of magnetic moments to mimic the magnetic
structure under an applied external field Kang  _et al._ (2020). Within this
approach, we obtain the lowest energy for a given tilting angle. Constraining
the tilting angle of magnetic moments changes the total energy changes due to
exchange interactions in the tilted state (see Figure S10 sup ). We compute
the magnetic susceptibility from a quadratic fit to the resulting total-energy
curve and the equation

$\chi_{v}=\frac{\mu_{0}}{2a-\mu_{0}},$ (3)

where $\mu_{0}$ is the vacuum permeability and $a$ the quadratic fit
coefficient Kang  _et al._ (2020). Since DFT implements the calculation at
$T=0$ K, $\chi_{a}$ should be zero and $\chi_{c}$ is non-zero for a single
crystal with a single domain. Thus, we assume an experiment in which the field
is oriented along the $c$ axis and moments are tilted within the $ac$-plane.
The calculated magnetic susceptibility $\chi_{c}^{\text{DFT}}$ is 6.88$\times
10^{-4}$, which shows good agreement with the scale of measured value of
1.07$\times 10^{-4}$ at $T=4$ K, two orders of magnitude smaller than 0.015
for Fe2As.Yang  _et al._ (2020b)

Figure 7:  Two-point resistivity measurement of an unaligned Cr2Al polycrystal
shows a clear transition at $T_{N}$, measured under flowing N2.

The two-point resistivity of Cr2Al from 298 to 733 K is shown in Figure 7.
Since contact resistance between the Al pads and the Cr2Al sample is present,
this data serves as an upper bound for the intrinsic resistivity of Cr2Al. A
clear second-order transition can be observed at 634 K where the slope
$d\rho/dT$ increases, indicating an increase in scattering due to the
disappearance of AF domains.

The calculated electronic band structure shown in Figure 8 demonstrates the
metallicity of Cr2Al, with a low electronic density of states near the Fermi
level, which leads to high resistivity, especially when some antisite defects
may be present due to the compositional phase width of Cr2Al. The number of
states at the Fermi level of Cr2Al is 0.0155 states/(eV$\cdot$Å), smaller than
that of Fe2As (0.0770 states/(eV$\cdot$Å))Yang  _et al._ (2019). This explains
why Cr2Al shows higher resistivity than Fe2As. The projected density of states
shows that a wide energy range of states originates from Cr $3d$ orbitals,
while $s$ or $p$ orbital contributions from Cr and Al are small.

Figure 8:  (a) Electronic band structure and total and projected density of
states of Cr2Al and (b) magnified band structure near the Fermi level.

The structural stability of Cr2Al is confirmed by the phonon dispersion,
computed using the finite difference method including the antiferromagnetic
configuration and the effect of spin-orbit coupling, shown in Figure 9. There
are no states with imaginary energy/frequency, which confirms the dynamic
stability. There are a total of 18 phonon bands, spanning the energy up to 50
meV. The phonon density of states shows two peaks around 30 and 45 meV.

Figure 9:  Phonon bands and density of states of Cr2Al calculated by the
finite difference method and DFT.

Exchange coefficients describe the interaction between magnetic moments and
can further explain the magnetic structure of Cr2Al. These exchange
coefficients can be utilized for atomistic spin dynamics simulations, as has
been done for Fe2As.Karigerasi  _et al._ (2020) The exchange coefficients
extracted in this work are shown as a function of relative distance ($d/a$) in
Figure 10. Positive and negative exchange parameters indicate ferromagnetic
and antiferromagnetic couplings, respectively.  The nearest-neighbor exchange
interaction ($J1=-18.5$ meV, shown as a yellow arrow in Figure 10) is the
primary driving force for the AF ordering in Cr2Al. Even though the third-
nearest neighbor exchange interaction also shows a negative value ($J3=-3.1$
meV), the corresponding moments show parallel alignment. This is because the
$J3$ interaction is overcompensated by $J5=+2.2$ meV, i.e., ferromagnetic
coupling, which has a multiplicity four, while $J3$ has only one interaction.
Interactions along $a$ and $b$ directions are all ferromagnetic ($J2=+0.9$ meV
and $J4=+0.9$ meV). Numerical uncertainties of the exchange integrals mainly
originate from k-point sampling and we determined these to be lower than 2.7%,
except for J4, where due to the small absolute value the relative error is
larger (14.2%).

Figure 10:  Exchange coefficients $J_{ij}$ as function of the interaction
distance $d$ (in units of the lattice parameter $a$) between lattice sites $i$
and $j$. First to fifth-nearest neighbor interactions ($J1$, …, $J5$) are
illustrated in the subset figure with colored arrows.

Based on the calculated exchange coefficients, $T_{N}$ was estimated using
three thermodynamic observables that were computed from Monte Carlo (MC)
simulations, see details in Ref. Eriksson  _et al._ , 2017. The observables,
computed using the stochastic LLG equation, are shown as a function of $T$ in
Figure 11. Sub-lattice magnetization ($M_{\text{sub}}$, black solid line) is
not sharp enough to determine the critical temperature because of finite size
effects, but it does reproduce the 23% difference between the calculated $T=0$
K $M_{sub}$ and the experimental $M_{sub}$ at 300 K from neutron scattering.
Instead, we use the Binder cumulant $U_{\text{L}}$, which indicates $T_{N}$ as
a transition from the normalized value of 0.667 to 0.444. The isothermal
susceptibility $\chi_{\text{thermal}}$ and the heat capacity $C_{v}$ show a
peak at the critical temperature. This leads to predictions for the transition
temperature of 660, 650, and 630 K, respectively, which are all in good
agreement with the experimental value of $634\pm 2$ K.

Figure 11:  Normalized sublattice magnetization and three thermodynamic
observables for Cr2Al, computed from Monte Carlo simulations of the stochastic
Landau-Lifshitz-Gilbert equation. The temperature is changed using a step size
of 10 K. Heat capacity $C_{v}$ mirrors the heat flow while heating from DSC in
Figure 5, while the predicted $T_{N}$ range is 620 – 670 K.

##  IV Conclusions

Neutron diffraction and magnetometry are consistent with an in-plane magnetic
structure of Cr2Al, identical to that of Mn2Au, but with light elements that
minimize spin-orbit coupling. Magnetometry, calorimetry, and transport all
confirm $T_{N}$ to be $634\pm 2$ K, in agreement with previous studies. The
ability to access this $T_{N}$ without decomposing the compound enables a more
stringent test for evaluating spintronic phenomena that should vanish in the
paramagnetic regime. Aligned measurements provide some early evidence for
anisotropy in Cr2Al, which will be the focus of further studies. The ability
to engineer the transport and morphology, for example by thin film deposition,
should provide a promising platform for spintronic investigation.

##  V Acknowledgments

This work was undertaken as part of the Illinois Materials Research Science
and Engineering Center, supported by the National Science Foundation MRSEC
program under NSF Award No. DMR-1720633. The characterization was carried out
in part in the Materials Research Laboratory Central Research Facilities,
University of Illinois. This work made use of the Illinois Campus Cluster, a
computing resource that is operated by the Illinois Campus Cluster Program
(ICCP) in conjunction with the National Center for Supercomputing Applications
(NCSA) and which is supported by funds from the University of Illinois at
Urbana-Champaign. This research is part of the Blue Waters sustained-petascale
computing project, which is supported by the National Science Foundation
(Awards No. OCI-0725070 and No. ACI-1238993) and the state of Illinois. Blue
Waters is a joint effort of the University of Illinois at Urbana-Champaign and
its National Center for Supercomputing Applications. This research used
resources of the Spallation Neutron Source, a DOE Office of Science User
Facility operated by Oak Ridge National Laboratory. The authors thank Jue Liu
for additional assistance with the neutron scattering experiment.

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