Magnetic anisotropy in single-crystalline antiferromagnetic Mn$_2$Au
Mebatsion S. Gebre Rebecca K. Banner Kisung Kang Kejian Qu Huibo Cao André Schleife Daniel P. Shoemaker





Department of Materials Science and Engineering  and Materials Research Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, United States

Department of Materials Science and Engineering  and Materials Research Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, United States
School of Materials Science and Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, United States
Department of Chemistry, Skidmore College, Saratoga Springs, New York 12866, United States

Department of Materials Science and Engineering  and Materials Research Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, United States

Department of Physics and Materials Research Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, United States

Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, United States

Department of Materials Science and Engineering  and Materials Research Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, United States
National Center for Supercomputing Applications, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, United States

Department of Materials Science and Engineering  and Materials Research Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, United States

 

Multiple recent studies have identified the metallic antiferromagnet Mn_2Au to be a candidate for spintronic applications due to apparent in-plane anisotropy, preserved magnetic properties above room temperature, and current-induced Néel vector switching.
Crystal growth is complicated by the fact that Mn_2Au melts incongruently. 
We present a bismuth flux method to grow millimeter-scale bulk single crystals of Mn_2Au in order to examine the intrinsic anisotropic electrical and magnetic properties.
Flux quenching experiments reveal that the Mn_2Au crystals precipitate below 550^∘ C, about 100^∘ C below the decomposition temperature of Mn_2Au.
Bulk Mn_2Au crystals have a room-temperature resistivity of 16 – 19 μΩ-cm and a residual resistivity ratio of 41. 
Mn_2Au crystals have a dimensionless susceptibility on the order of 10^-4 (SI units), comparable to calculated and experimental reports on powder samples. 
Single-crystal neutron diffraction confirms the in-plane magnetic structure.
The tetragonal symmetry of Mn_2Au constrains the ab-plane magnetic susceptibility to be constant, meaning that χ_100=χ_110 in the low-field limit, below any spin-flop transition. We find that three measured magnetic susceptibilities χ_100, χ_110, and χ_001 are the same order of magnitude and agree with the calculated prediction, meaning the low-field susceptibility of Mn_2Au is quite isotropic, despite clear differences in ab-plane and ac-plane magnetocrystalline anisotropy.
Mn_2Au is calculated to have an extremely high in-plane spin-flop field above 30 T, which is much larger than that of another in-plane antiferromagnet Fe_2As (less than 1 T).
The subtle anisotropy of intrinsic susceptibilities may lead to dominating effects from shape, crystalline texture, strain, and defects in devices that attempt spin readout in Mn_2Au.


Magnetic anisotropy in single-crystalline antiferromagnetic Mn_2Au
    Daniel P. Shoemaker
    ^1Division of Space Research and Planetary Sciences, University of Bern, Switzerland

    ^2Max Planck Institute for Astronomy, Heidelberg, Germany

    

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§ INTRODUCTION




Antiferromagnets (AF) have compensated, net-zero magnetization but still host a wealth of field- and current-dependent phenomena, notably spin canting in response to an applied field <cit.>, spin flop transitions <cit.>, and fundamentally higher resonance frequencies than ferromagnets <cit.>. 
These THz-frequency resonances especially have attracted interest for fast data storage,<cit.> but questions remain about the exact processes of domain rearrangement and the associated energy scales and barriers <cit.>. 
In antiferromagnet-based spintronics, Néel vector switching and anisotropic magnetoresistance (AMR) can potentially be used to write and read data, respectively <cit.>.








Mn_2Au is one of few room-temperature metallic antiferromagnets, along with CuMnAs and Cr_2Al, that have the crystal symmetry and collinear antiferromagnetic order that allows a staggered (Néel) accumulation of spins in the AF sublattices,<cit.>
which in principle allows the electronic manipulation of the Néel vector using Néel spin-orbit torques. <cit.>
Mn_2Au has been claimed to demonstrate successful current-induced Néel vector switching, though the switched volume fraction is small and the applied currents are large.<cit.>
The unit cell of Mn_2Au is shown in Fig. <ref> with arrows on Mn atoms to show this magnetic ordering.
Neutron diffraction and magnetization studies on powder Mn_2Au by Barthem et al. confirm the k=0 antiferromagnetic structure of Mn_2Au with spins in the ab-plane.<cit.> 
However, powder neutron diffraction data of a tetragonal compound does not permit unique identification of the easy axis within the ab-plane. The powder study also showed an approximate susceptibility of 5 × 10^-4 averaged over all crystal directions. Neutron data showed a Mn moment of about 4 μ_B, which is retained to about 85 % of its base value at 900 K <cit.>. 
Mn_2Au thus maintains its magnetic ordering until it decomposes upon heating into MnAu and an fcc solid solution phase, with a Néel temperature theorized up to 1300-1600 K <cit.>. 
The presence of gold in Mn_2Au should also lead to a significant contribution from spin-orbit coupling.<cit.>
This high thermal stability is a hallmark of strong exchange interactions and anisotropy, and makes the material ideal for fundamental studies at room temperature or future data storage applications,<cit.>
but no bulk crystal growth has been reported to date.


Several studies of thin films and polycrystals have examined the magnetic and electrical properties of Mn_2Au. <cit.>
Thin film XMLD-PEEM studies by Sapozhnik et al. found a magnetic easy axis of ⟨110⟩ and calculate an in-plane anisotropy term in the range 1 – 17 μeV/f.u., corresponding to lower and upper limits for the predicted spin flop fields H_SF of 7 to 30 T.<cit.>
While the upper limit of 17 μeV/f.u. is evidenced by domain growth from  ≈1 μm along the equivalent [ 110]  and [ 11̅0]  easy axis at zero field to several microns perpendicular to the field at an applied field of 30 T, the lower limit was determined using domain wall widths near the resolution of the images.
Two studies by Bodnar et al. <cit.> showed evidence of current-induced and magnetic-field induced Néel vector switching in Mn_2Au thin films separately. They also demonstrated AMR with a field along the hard ⟨100⟩ direction to be 6 %, which is much larger than that along the easy ⟨110⟩ crystal direction of -0.15 %.
The apparent switching behavior in magnetoresistance along ⟨110⟩ and ⟨100⟩ directions was discussed to be consistent with an extremely small/negligible in-plane anisotropy. 
This is consistent with an in-plane magnetocrystalline anisotropy energy (MAE) constant of 10 μeV/f.u. calculated by Shick et al.<cit.>


While most recent work has focused on Mn_2Au thin films, the fundamental properties of the material remain to be seen in bulk crystal form.
Notably, anisotropy may be strongly influenced by large stresses in epitaxial films and texturing from high applied currents.
In order to fully understand the material properties of Mn_2Au, we sought to synthesize and characterize bulk single crystals. 
According to the Mn-Au phase diagram, Mn_2Au decomposes into solid MnAu and Mn phases at 680^∘ C before melting <cit.>. This makes crystal growth by directly cooling from melt impossible. To grow a bulk single crystal, a flux method must be used to precipitate Mn and Au at lower temperatures. Considering its binary phase diagrams with Au and Mn each, Bi was the most logical flux candidate <cit.>. 
Here we describe the flux synthesis and physical characterization of the obtained crystals.  





§ METHODS






Synthesis was performed using solid-state methods and metal flux techniques.
As-received Mn pieces with oxide-tarnished surfaces were cleaned by heating in vacuum at 1000^∘ C for 12 h, after which the Mn pieces are bright silver and the inner walls of the tube are dark.
Flux growth of Mn_2Au crystals was conducted using elemental precursors of Mn granules (99.98 %, 0.8 – 12 mm),  Au powder (99.99 %, 100 mesh), and   Bi chunks (99.99 %, ≤15 mm) with an optimized nominal molar ratio of Mn:Au:Bi = 7:1:12.
The precursors were loaded into alumina crucibles, topped by alumina fritted disks<cit.> and a glass rod spacer in a fused silica ampoule. This setup, prepared in an argon-filled glovebox, was then evacuated and sealed at about 50 mTorr of pressure for reaction in an inert environment.
The precursors were heated to a maximum temperature of 750^∘ C for 12 h, quickly cooled and held at 650^∘ C for up to 24 hours below the dissociation temperature of Mn_2Au, slowly cooled to 470 – 480^∘ C for crystal growth, and from this temperature inverted and centrifuged to remove excess flux.
The synthesis is shown schematically in Fig. <ref>.
Variations of this recipe used to determine the precipitation temperature of the Mn_2Au phase and to optimize relative nominal amounts of precursors are summarized in Fig. <ref> and tabulated in the Supporting Information Table S1,<cit.> along with a list of the largest crystal sizes obtained from each reaction.
Mn_2Au single crystals were manually isolated from remnant Bi flux by mechanically scrapping or cleaving samples to isolate clean parts. Where remnant flux could not be easily removed mechanically, samples were acid etched with a 2:1 ratio of HNO_3 and HCl for less than 2 minutes. 







Powder X-ray diffraction was conducted in transmission geometry with a Bruker D8 diffractometer with Mo-Kα radiation.
Reflection X-ray diffraction was performed with a Bruker D8 diffractometer with Cu-Kα radiation on single-crystalline samples to confirm crystal orientation. Rietveld analysis was performed using GSAS-II <cit.>.

Microstructure and composition were assessed using a ThermoFisher Axia ChemiSEM scanning electron microscope with electron dispersive X-ray spectroscopy (SEM-EDS) using an acceleration voltage of 20 kV.
A Shimadzu Differential Thermal Analyzer (DTA-50) was used to examine the thermal stability of the flux-synthesized Mn_2Au single crystals between room temperature and 750^∘ C, with samples in sealed quartz ampoules.
Electron backscatter diffraction (EBSD) mapping was conducted in a Thermo Scios2 Dual-Beam SEM/FIB instrument at an acceleration voltage of 30 kV. 
Magnetic susceptibility measurements were performed using a Quantum Design Magnetic Property Measurement System (MPMS).
Four-point resistivity measurements were made using a Quantum Design Physical Property Measurement System (PPMS).

Single-crystal neutron diffraction was collected on a 1×0.1×0.1 mm^3 Mn_2Au crystal on the HB-3A four-circle diffractometer at the High Flux Isotope Reactor (HFIR) at Oak Ridge National Laboratory. A neutron wavelength of 1.542 Å was used and measurements were conducted at 295 K. Magnetic symmetry analysis was done using the MAXMAGN tool in the Bilbao Crystallographic Server <cit.>. Single crystal neutron diffraction refinements of the nuclear and magnetic structure were done using the FULLPROF suite <cit.>.

First-principles density functional theory (DFT) simulations were carried out using the Vienna Ab-Initio Simulation Package (), employing the projector-augmented-wave (PAW) formalism <cit.> to describe the electron-ion interaction.
Kohn-Sham states were expanded into plane waves with a cutoff energy of 600 eV.
The exchange-correlation term in the Kohn-Sham equation was described using the Perdew, Burke, and Ernzerhof (PBE) formulation of the generalized-gradient approximation<cit.>.
We also used the DFT+U approach<cit.> and selected an effective on-site Coulomb interaction parameter U_eff=1.4 eV, resulting in good agreement between lattice parameters from our experiment and those from another X-ray diffraction experiment <cit.>, see details in Fig. S10 in the supplemental information.
The measured magnetic moments from powder samples <cit.> further substantiate the suitability of our choice of U_eff value.
However, the magnetic moments derived from this work are about 0.5 μ_B larger, see Fig. S10 in the supplemental information.
For a reliable representation of the in-plane and out-of-plane magnetocrystalline anisotropy energy, a Monkhorst-Pack 𝐤-point mesh <cit.> of 24×24×9 was meticulously chosen, as depicted in Fig. S8 of the supplemental information.
We ensure 𝐤-point convergence of the anisotropic energy difference within 540 kJ/m^3 for K_1 and 5 kJ/m^3 for K_22.

The influence of spin-orbit coupling is included within the non-collinear magnetism framework by applying a non self-consistent field (SCF) calculation for the spin-orbit coupling effect after a spin-polarized SCF calculation <cit.>.
Atomic geometries are relaxed using the SCF spin-polarized approach only, i.e. without spin-orbit coupling.
In all cases, the magnetic structure was included in the description.
In the calculation of magnetic susceptibility, we enforced the angular constraint on the magnetic moments using a penalty term in the total energy, as implemented in .





§ RESULTS AND DISCUSSION


Bulk single crystals of Mn_2Au, with dimensions on the mm-scale, were grown using Bi flux.
The decomposition temperature of Mn_2Au in the solid state is reported to be 680^∘ C <cit.>.
In bismuth flux, our quenching experiments, summarized in Fig. <ref>, reveal that Mn_2Au precipitated from flux below 550^∘ C, while MnAu formed between 550 and 700^∘ C. 

Differential thermal analysis (DTA) of isolated Mn_2Au crystals was used to assess the dissociation temperature compared to the existing Mn–Au phase diagram<cit.>.
The DTA signal for pure Mn_2Au crystals, Fig. S3 in the Supporting Information, <cit.> shows phase changes with onset at ∼680^∘ C on heating and ∼630^∘ C on cooling.
This sets the upper temperature bound for applications using Mn_2Au, as heating a spintronic device past 630^∘ C could destroy the Mn/Au ordering.










In order to find optimal heating and composition protocols, many samples were prepared under varying conditions such as the nominal amounts of Mn:Bi and the heating profile. A variety of crystal shapes were obtained, as shown in Fig. <ref>, including cubes, thick needles, and flat plates on the mm-scale. There may be some significance of specific heating steps that were attempted:
As an example, a hold at 650^∘ C seems to favor a smaller quantity of larger single crystals, and larger Mn:Bi ratios favor plate-like morphology over thin needles.
However, a mechanistic understanding of these steps remains to be fully developed due to the complexity of the process.
Variations of flux syntheses are tabulated in the Supporting Information<cit.>.
The largest crystals at optimized conditions were used for further characterization.







Rietveld refinement of powder XRD for Bi-flux products caught after centrifuging is presented in Fig. <ref>.
The refinement shows 58 mol % Mn_2Au, 20 mol % Bi, and 22 mol % MnBi.
This distribution indicates that the Mn_2Au single crystals have residual MnBi and Bi on their surfaces.
The Mn_2Au single crystals used for further characterizations were picked out and cleaned mechanically.
However, a small amount of Bi and MnBi typically remains on the surface, which is subsequently removed (to the best of our ability) by etching as described in Sec. <ref>.
This phase distribution in the reaction products is visually apparent in SEM-EDS images shown in Fig. S1 <cit.>.







Reflection XRD (see Fig. <ref>) is used as a preliminary tool to check the orientation of Mn_2Au single crystals.
The diffraction data in green is obtained for plate-like crystals, showing that the (001) plane is the largest exposed plane.
The data in pink were obtained for crystals with horizontally lying needle-like crystal formation; the reflections correspond to ⟨ h0l ⟩.
The large plates have the c-axis pointing perpendicular to the widest exposed plane while the narrow needles have the c-axis lying in the plane perpendicular to the longest sample dimension.







Electron backscatter diffraction (EBSD) maps as in Fig. <ref> were used to confirm the orientation of mm-scale plate-like Mn_2Au crystals.
The map shows that the c-axis of the crystal is perpendicular to the largest exposed sample plane. The inverse pole figure shows that the entire face is red, corresponding to 001 orientation. 
Since EBSD is quite surface-sensitive, crystal facets and steps with different orientations appear dark as their Kikuchi patterns are not well-recorded. The EBSD data also offers 3-dimensional information of orientation within the surface plane. The vector difference between two points along the longest dimension (horizontal in Fig. <ref>) corresponds to the 100 crystal direction. Accordingly, the diagonal dark lines with 110 orientation are growth terraces. 
Anisotropic magnetic and transport measurements on this and other Mn_2Au crystals were performed after confirming the orientation of each with XRD and EBSD.

Four-point resistivity measurements with current applied along the length of a needle-like crystal along ⟨ 100 ⟩ confirm the metallic behavior of Mn_2Au with a room temperature resistivity of ∼ 16 μΩcm (see Fig. <ref>) and a low temperature (2 K) resistivity of ∼ 0.4 μΩcm. This compares to resistivity of ∼ 21 and ∼ 7 μΩcm reported for an epitaxial film at 300 K and 2 K respectively <cit.>. The residual resistivity ratio (RRR) of 41 in our bulk sample is an order of magnitude larger than that reported for epitaxial films <cit.> because of the small low-temperature resistivity in the bulk sample. 

Similar four-point resistivity measurements along other crystal directions are limited by the sample's aspect ratio. We attempted to conduct measurements of micron-scale lamellae by focused ion beam liftout, but a high concentration of implanted Ga led to a drastic decrease in the RRR. 












Single crystal neutron diffraction data refinements (see Fig. <ref>) show excellent agreement between calculated and observed structure factors for antiferromagnetic tetragonal Mn_2Au with a propagation vector k = [000]. Refinements were carried out using two different magnetic space groups corresponding to the two possible Néel vectors within the ab-plane for the commensurate AF phase in the nuclear I4/mmm space group. The Fm'mm magnetic space group (#69.523) was used to refine the AF phase with a Néel vector along ⟨ 110 ⟩.
The Im'mm magnetic space group (#71.535) was used to refine that with the Néel vector along ⟨ 100 ⟩.
Both refinements converge to a Mn moment of 4.56(2) μ_B, which is slightly larger than the DFT-obtained value of 3.98 μ_B and the one measured on powder samples of 4 μ_B <cit.>. 
Visually, it is apparent from Fig. <ref> that both models describe the experimental data equally well.
Both fits give R_F = 0.019 and χ^2=0.26, due to the symmetry equivalence of the two in-plane models for single-crystal neutron scattering.
Since the easy axis cannot be distinguished between ⟨ 110 ⟩ and ⟨ 100 ⟩,
a complementary confirmation of in-plane anisotropy is presented using the directional magnetic property measurements that follow.


Magnetization isotherms and temperature sweeps were used to characterize the magnetic response of samples over a large span of temperatures between 10 K and 700 K, but careful experimentation is required to separate the intrinsic response of Mn_2Au from dilute MnBi impurities. 
Magnetization isotherms at 300 and 700 K, as well as intermediate magnetization versus temperature scans, discussed in detail in the Supplemental Information <cit.>, revealed that amounts as low as 2 mol % of the MnBi-low temperature phase (LTP) dominate the magnetic response.
LTP-MnBi orders ferromagnetically at 630 K <cit.>, concurrent with its phase change to the high temperature phase (HTP). Room temperature scans are easily overwhelmed by the ferromagnetic contribution from the MnBi impurity exhibiting a large magnetization M, a non-linear saturation of M, and hysteresis. Above the ordering temperature of LTP-MnBi, however, M is linear with applied field H with a susceptibility on the order of 10^-4 as expected for Mn_2Au.<cit.>
The disappearance of ferromagnetic behavior above the Curie temperature of the MnBi impurity confirms that the FM observed at room temperature is not an intrinsic property of the Mn_2Au sample itself, but rather of the MnBi impurity. 

To obtain anisotropic susceptibility of the single crystals, this large FM contribution had to be reduced further. Reducing the ferromagnetic signal at room temperature, without having to break clean sections of samples, was explored via two ways. In the first approach inspired by high temperature magnetization measurements, a sample was heated to 700 K in vacuum and quenched quickly to avoid reformation of the LTP-MnBi phase. In the second approach, the Mn_2Au crystals were etched with a 1:2 HCl and HNO_3 solution for 2 minutes to remove the Bi flux. Whereas both helped reduce the FM contribution, they brought the risk of oxidizing the sample or leaching the Mn from the sample. Further details on these methods are discussed in the Supporting Information <cit.>.







The cleanest Mn_2Au samples were obtained by manually cutting the flux-free sections of as-synthesized single crystals. Magnetic isotherms measured on this sample at 10 and 300 K with the field along ⟨ 110 ⟩ and ⟨ 100 ⟩ are shown in Fig. <ref>. A mostly linear behavior in M with field H is observed for all measurements with the exception of a steeper region between ± 0.1 T. While the behavior near H=0 T may be due to trace amounts of MnBi, the predominantly linear behavior at higher fields represents the intrinsic Mn_2Au susceptibility. The dimensionless susceptibility χ is calculated using the slope of this linear section of the data for applied fields H > 0.1 T. The susceptibility in each case is on the order of 10^-4, matching the calculated value of 3.98×10^-4 for  Mn_2Au in the literature <cit.>. The in-plane diagonal susceptibility χ_110 is larger than χ_100 by a factor of 1.5 and 1.25 at 10 K and 300 K, respectively.
Different samples, acid-cleaned by different methods, consistently showed that χ_110 > χ_100, as discussed in the Supporting Information <cit.>.
No subtle features that would indicate a polarization of in-plane domains or a spin-flop field are apparent in the linear regions of susceptibility for Mn_2Au, unlike those observed in the more susceptible in-plane antiferromagnet Fe_2As.<cit.>
As we discuss below, the in-plane spin flop field is predicted to be greater than 30 T.



The magnetic susceptibility measurements in Fig. <ref> complement our neutron diffraction data by probing the in-plane Néel direction.
Mn_2Au is antiferromagnetic at both temperatures, 300 K and 10 K, and the susceptibility measurements show that the in-plane susceptibilities are approximately equal at room temperature, with a small anisotropy at 10 K.
These susceptibilities are very small, on the same order as the isostructural room-temperature AF Cr_2Al,<cit.> and about 100 times smaller than than Fe_2As.<cit.>
The susceptibility along the c-axis χ_001 is about 4.9 × 10^-4, on the same order of magnitude as χ_100 and χ_110 (shown in Supplemental Information Figure S7).<cit.>
For a tetragonal crystal, the low-field (below spin flop) in-plane susceptibility should be isotropic–there are only χ_11, χ_33, and χ_13 terms in the susceptibility (as discussed in the Supplemental Information<cit.> and in the texts by Nye<cit.> and Newhnam),<cit.> so the measured difference we see between χ_100 and χ_110 is likely due to shape anisotropy of the Mn_2Au or texturing of minuscule amounts of ferromagnetic MnBi impurity (discussed in detail in the Supplemental Information).<cit.>
The ⟨ 110 ⟩ direction was proposed to be the easy axis by thin film studies employing XMLD-PEEM for direct domain imaging <cit.>.
The thin film study shows easy-axis domain growth when fields above 30 T are applied perpendicular to ⟨ 110 ⟩.









To gain a deeper understanding of this anisotropy, we employed first-principles density functional theory to investigate the magnetocrystalline anisotropy energy (MAE) of Mn_2Au. 
In Fig. <ref>, we present the MAE results for Néel vector orientations that include c components (out of the (001) plane, which are unfavorable), and orientations within the easy (001) plane.
Energies are plotted as a function of angles ϕ and θ deviating from the c-axis and a-axis, respectively (see Fig. <ref>).
Visually, it is clear that in-plane spin configurations are preferred, while the in-plane angular dependence is comparatively weak. 
The origin of the MAE in Mn_2Au can be attributed to the spin-orbit interaction (SOI), which plays a more significant role than the magnetic dipole-dipole interaction (MDD).

The MAE coefficients were determined by fitting the calculated data from Fig. <ref> to 

    E_MAE/V=K_1sin^2 ϕ + K_2sin^4 ϕ + K_22sin^4 ϕcos(4θ).

From this we obtain the total in-plane coefficient K_1^tot = 6574 kJ/m^3 and the total out-of-plane coefficient K_22^tot = 26 kJ/m^3.
These MAE values are one or two orders of magnitude larger than for the metallic antiferromagnets Fe_2As<cit.> and Cr_2Al<cit.>.
Both spin-orbit interaction (SOI) and magnetic dipole-dipole (MDD) interaction contribute to the MAE, and their respective contributions to the in-plane and out-of-plane MAE coefficients are as follows:
K_1^SOI = 6144 kJ/m^3, K_1^MDD = 430 kJ/m^3, K_22^SOI = 26 kJ/m^3, and K_22^MDD = 0 kJ/m^3.
The magnitude of K_1^MDD is comparable to that of Fe_2As<cit.> and Cr_2Al<cit.>.
However, the absence of K_2^MDD is solely due to magnetic symmetry in the (001) plane.
On the other hand, the presence of the heavy element Au in Mn_2Au significantly enhances the spin-orbit coupling effect, resulting in remarkably large values for both K_1^SOI and K_22^SOI, about 20 and 100 times larger than K_1^SOI and K_22^SOI of Fe_2As<cit.>.
The positive sign of K_1 indicates easy-plane magnetism, while the positive sign of K_22 signifies an easy axis along ⟨ 110 ⟩.
We observe a small energy bump near ⟨ 110 ⟩ in the in-plane MAE. 
This bump appears to be a higher-order feature that persists even when the number of 𝐤-points in the calculation is increased, as illustrated in Fig. S9.

Using the energy surface method<cit.> and a first-principles approach, we further investigated the anisotropic magnetic susceptibility of Mn_2Au, revealing subtle variations among different orientations.
This method assumes a single-crystalline structure with a single domain, where the Néel vector orientation is aligned perpendicular to the external magnetic field beyond the spin-flop field<cit.>.
We believe that this is well justified because with only one symmetrical magnetic site Mn_2Au has a much simpler magnetic structure than Fe_2As which we studied before <cit.>.
Consequently, this approach allows us to determine the anisotropic magnetic susceptibility during the occurrence of a spin-flop transition.
Since Mn_2Au possesses only one symmetrical magnetic site, we only need to calculate the energy curve while varying net magnetization corresponding to the tilting angle of the magnetic moment on the Mn atom, as illustrated in Fig. S11.
The curvature a of these curves is associated with the perpendicular magnetic susceptibility χ_⊥, as described by <cit.>

    χ_⊥=μ_0/2a-μ_0,

where μ_0 is the vacuum permeability.


Our calculations yield values of χ^[100]_⊥=4.27×10^-4, χ^[110]_⊥=4.11×10^-4, and χ^[001]_⊥=4.10×10^-4 for the anisotropic magnetic susceptibility in the regime with a single in-plane magnetic domain (e.g. above a spin flop transition).
The value of χ^[001]_⊥ does not change in the single- or multi-domain case (above or below spin-flop, low or high field) for Mn_2Au since spins remain normal to the c-axis, and the obtained experimental value of  4.9 × 10^-4 (Supplemental Figure S7) is in very good agreement.<cit.>
We estimate an error bar due to 𝐤-point convergence of about 0.6×10^-4 for these values, which is low enough to reliably conclude a very small anisotropy of the susceptibility.
The subtle differences in our simulated values are consistent with a relatively weak influence of magnetocrystalline anisotropy (MCA), which is on the order of kJ to MJ per cubic meter, see Fig. <ref>.
The susceptibility is instead dominated by exchange interactions that exhibit a much larger magnitude on the order of GJ per cubic meter (see Fig. S10).
Because of this difference in order of magnitude, any anisotropic contribution from MCA is not visible in the susceptibility above the spin-flop field.
As we discuss next, the magnetic field used to measure susceptibility in this work is clearly below the spin-flop transition.
In this regime, the susceptibility can be anisotropic (consistent with Fig. <ref>) and, in addition, be influenced by domain effects or magnetic defects.

Finally, we note that the out-of-plane and in-plane spin-flop fields can be determined using<cit.>

    H_SF^out   = √(2K_1/χ_⊥-χ_∥), 
     H_SF^in   = √(16K_22/χ_⊥-χ_∥).

By using the DFT-calculated anisotropy coefficients and magnetic susceptibility values at 0 K with an assumption of χ_∥=0, we obtain the spin-flop fields of H_SF^out=201 T and H_SF^in= 36 T, respectively.
These results agree with our lack of observation of an in-plane spin flop up to 7 T, and the small amount of domain orientation at 30 T observed in thin films.<cit.>
Higher-field magnetization experiments on bulk Mn_2Au could confirm this prediction.





§ CONCLUSIONS

In summary, we report the bulk single crystal growth and anisotropic characterization of the metallic antiferromagnet Mn_2Au in Bi flux.
Four-point resistivity measurements on bulk single crystals confirm the metallic transport behavior with RRR = 41 attesting to the low defect concentration in the crystals.
Single crystal neutron diffraction analysis agrees with an easy ab-plane, while
the orientation within the plane cannot be determined by low-field susceptibility or neutron scattering measurements due to tetragonal symmetry.
The comparison between experiment and theory for the magnetic arrangement, Mn moment size, and overall susceptibility are consistent.
The computational result indicates an easy axis along ⟨ 110 ⟩, in agreement with prior experiments on thin films.
Rigorous convergence testing and multiple experimental trials confirm both results. 
Further investigation is necessary to determine if there are additional subtle effects that affect the Néel vector orientation in experimental and computational results, since the anisotropy is small and could be influenced by strain from substrates, surface features, or defects. 
High-field magnetometry and torque magnetometry are additional plausible next steps.
The absence of evidence of an easy-plane spin-flop transition within the range of magnetic measurements up to 7 T is corroborated by computational prediction of the spin-flop field to be 36 T. The high spin-flop fields are likely to be increased by defects and applied strains, and together with the rather isotropic magnetic susceptibility (even along the c-axis) pose a unique challenge to fundamental and applied studies that seek to control antiferromagnetic domain structures in predictable ways.


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. 
Single-crystal neutron diffraction was conducted at ORNL's High Flux Isotope Reactor, sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, U.S. Department of Energy.
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 OCI-0725070 and 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.




