Anisotropies and magnetic phase transitions in insulating antiferromagnets determined by a spin-hall magnetoresistance probe

ABSTRACT Antiferromagnets possess a number of intriguing and promising properties for electronic devices, which include a vanishing net magnetic moment and thus insensitivity to large

magnetic fields and characteristic terahertz frequency dynamics. However, probing the antiferromagnetic ordering is challenging without synchrotron-based facilities. Here, we determine the

material parameters of the insulating iron oxide hematite, α-Fe2O3, using the surface sensitive spin-Hall magnetoresistance (SMR). Combined with a simple analytical model, we extract the

antiferromagnetic anisotropies and the bulk Dzyaloshinskii-Moriya field over a wide range of temperatures and magnetic fields. Across the Morin phase transition, we show that the electrical

response is dominated by the antiferromagnetic Néel vector rather than by the emergent weak magnetic moment. Our results highlight that the surface sensitivity of SMR enables access to the

magnetic anisotropies of antiferromagnetic crystals, and also of thin films, where other methods to determine anisotropies such as bulk-sensitive magnetic susceptibility measurements do not

provide sufficient sensitivity. SIMILAR CONTENT BEING VIEWED BY OTHERS SPIN-POLARIZED WEYL CONES AND GIANT ANOMALOUS NERNST EFFECT IN FERROMAGNETIC HEUSLER FILMS Article Open access 24

November 2020 EXTRAORDINARY PHASE TRANSITION REVEALED IN A VAN DER WAALS ANTIFERROMAGNET Article Open access 31 July 2024 LONG-DISTANCE SPIN-TRANSPORT ACROSS THE MORIN PHASE TRANSITION UP TO

ROOM TEMPERATURE IN ULTRA-LOW DAMPING SINGLE CRYSTALS OF THE ANTIFERROMAGNET Α-FE2O3 Article Open access 10 December 2020 INTRODUCTION In recent years, various new effects have been

discovered which enable easy and efficient ways to probe and manipulate the antiferromagnetic order (or Néel vector) by electrical current1,2,3. The Néel vector can be manipulated by

electrical fields in magnetoelectric materials like Cr2O34 or multiferroics like BiFeO35, by bulk spin-galvanic effects in conducting antiferromagnets6,7, or by interfacial spin-orbit

torques in multilayers with the insulating NiO8,9,10. For magneto-transport measurements, effects that are even functions of the magnetic order parameter like anisotropic11,12 and spin-Hall

magnetoresistance (SMR)13,14,15 could be used to probe the antiferromagnetic state and detect switching events8,9,10. However, it is not obvious how one can extract the equilibrium state of

an antiferromagnet and in particular determine from the field dependence of the SMR key magnetic properties such as the anisotropy values that are otherwise difficult to ascertain. The SMR

technique can probe the magnetic state of bilayer systems consisting of a ferromagnetic or antiferromagnetic insulator and a heavy metal. In the simplest models [15,22,28], the longitudinal

SMR signal ΔR is proportional to \(\left( {1 - \left( {{\mathbf{m}}\cdot {\mathbf{\mu }}} \right)^2} \right)\) for a ferromagnet and \(\left( {1 - \left( {{\mathbf{n}}\cdot {\mathbf{\mu }}}

\right)^2} \right)\) for an antiferromagnet (where the unit vector Μ of the spin-accumulation is perpendicular to the charge current J in the heavy metal and M and N are the magnetization

and the Néel vector, respectively). As the magnetization in ferromagnets aligns along the external magnetic field, while the Néel vector of an antiferromagnet tends to align perpendicularly,

ferro- and antiferromagnets should exhibit a different field dependence of the SMR signal on the external magnetic field. When the magnetic field is applied parallel to the current, the SMR

contribution to the longitudinal resistance should increase for a ferromagnet (positive SMR) and should decrease for antiferromagnets (negative SMR). However, recent experiments show a more

complicated behavior of SMR in antiferromagnets, with positive SMR in SrMnO316, negative SMR in the easy-plane NiO17,18, and both positive19,20 and negative21 SMR measured in the easy-axis

antiferromagnet Cr2O3. To explain the different SMR signs, there were suggestions of mechanisms such as contributions from proximity induced magnetization21 or canted moments22. Additional

complications stem from the possible strong dependence of the SMR response on the magneto-crystalline anisotropy resulting from structural symmetries23. However, such effects, can be

concealed in multi-domain states17,24 by the presence of magneto-elastic coupling. Hematite, α-Fe2O3, is not only the main component of rust, but also a prototypical insulating

antiferromagnet that lends itself to disentangle the origins of the SMR signals in antiferromagnets and identify the role of easy-axis and easy-plane symmetries in the magnetoresistance

response. This iron oxide has a hexagonal crystal structure \(R\bar 3c\), and at room temperature shows an easy-plane canted antiferromagnetic ordering in the basal crystallographic plane.

It undergoes a magnetic phase transition at the Morin temperature25,26,27 (at about 260 K), below which it exhibits an easy-axis antiferromagnetic ordering along the c-axis. It also

possesses a bulk Dzyaloshinskii-Moriya internal field28,29 along the _c_-axis, which is thus hidden in the easy-axis phase and which leads to a small (less than a millirad) canting angle

between the magnetic sublattices, generating a weak moment in the easy-plane phase. Furthermore, hematite shows an accessible spin-flop field below 10 Tesla so that the Néel vector fully

reorients perpendicular to external magnetic fields30. The size of the magnetic domains can exceed ten micrometers in both single crystals (see Supplementary Fig. 1) and thin films31

enabling long distance spin transport32, and domain redistribution in the easy-plane can be dominated by the coherent rotation of the Néel vector instead of domain wall motion. To understand

the spin structures and the switching mechanisms, knowledge of the anisotropies and Dzyaloshinskii-Moriya (DMI) fields is crucial, which is however challenging in antiferromagnets using

conventional approaches requiring large scale facilities. In this article, we determine the antiferromagnetic anisotropies and the bulk Dzyaloshinskii-Moriya field of the insulating iron

oxide hematite, α-Fe2O3, using a surface sensitive spin-Hall magnetoresistance (SMR) technique. We develop an analytical model that in combination with SMR measurements, allows for the

identification of the material parameters of this prototypical antiferromagnet over a wide range of temperatures and magnetic field values. The SMR measurements are thus directly related to

the equilibrium orientation of the Néel vector N calculated by minimizing the magnetic energy of the sample. The SMR response is shown to depend strongly on the direction of the charge

current with respect to the magneto-crystalline anisotropies axes. Both the small net magnetization induced by the external magnetic field H or by the bulk Dzyaloshinskii-Moriya HDMI above

the Morin transition, do not significantly contribute to the electrical response in the canted easy-plane phase. These results demonstrate that the SMR technique is a powerful and

alternative tool to the complex neutron scattering33 or X-ray29 based measurements to ascertain not only the orientation of the Néel vector as demonstrated previously, but also to extract

key parameters such as the magnetic anisotropies and DMI fields in antiferromagnets. RESULTS ANALYTICAL MODEL We first develop the necessary model that allows us to analyze the results of

the SMR transport experiments. We assume that the SMR signal depends only on the components of the Néel vector22,28: $$\frac{{{\mathrm{\Delta }}R_{jk}}}{{R_{jk}}} = \rho _0\left\{

{\begin{array}{*{20}{c}} {1 - \left( {{\mathbf{n}}\cdot {\mathbf{\mu }}} \right)^2,j = k,} \\ {n_jn_k,j \, {\ne} \, k,} \end{array}} \right.$$ (1) where _j, k_ = _x_, _y_, are the

coordinates along and perpendicular to the current direction within the sample plane, the constant ρ0 depends on the spin-mixing conductance at the Pt-hematite interface and is considered as

a fitting parameter. The unit vector of spin-accumulation Μ lays within the _xy_ plane and is perpendicular to the current. The equilibrium orientation of the Néel vector N is calculated by

minimizing the potential energy (per unit volume) $$w_{{\rm{pot}}} = 2M_s\left[ {\frac{1}{2}H_{{\rm{ex}}}{\mathbf{m}}^2 + H_{{\rm{DMI}}}\left( {n_Xm_Y - n_Ym_X} \right) - {\mathbf{H}}\cdot

{\mathbf{m}}} \right] + w_{{\rm{ani}}}$$ (2) where _H_ex parametrizes the exchange field which keeps the magnetic sublattices antiparallel, _H_DMI is the Dzyaloshinskii-Moriya field, _M__s_

is the sublattice magnetization, N and M are the Néel vector and magnetization of the antiferromagnet related by the conditions \({\mathbf{{n}}}\cdot{\mathbf{m}}\) = 0 and \({\mathbf{n}}^2 +

{\mathbf{m}}^2 = 1\). The coordinate frame _XYZ_ is related to the crystallographic axes and differs from the _xyz_ frame defined by the surface plane of the R-cut hematite sample (_X_

being the projection of the easy axis Z in the sample plane, _Y_ the axis perpendicular to it) and its normal _Z_. The anisotropy energy of the antiferromagnet depends on two parameters:

$${w_{{\rm{ani}}} = 2M_s\left[ { - \frac{1}{2}H_{2||}(T)n_Z^2 - \frac{1}{6}H_ \bot \left( {n_X^2 - n_Y^2} \right)\left( {4\left( {n_X^2 - n_Y^2} \right)^2 - 3\left( {n_X^2 + n_Y^2}

\right)^2} \right)} \right]}.$$ (3) The temperature-dependent uniaxial anisotropy field \(H_{2||}(T)\) is positive at low temperature, where it stabilizes the easy-axis phase with

\({\mathbf{n}}||{\boldsymbol{Z}},\) and changes sign at the Morin temperature. The in-plane anisotropy field \(H_ \bot > \,0\) selects one of three equivalent easy axes in the easy-plane

phase. We keep the in-plane anisotropy also in the easy-plane phase to remove degeneracy of states and avoid multiple solutions. Based on the previously estimated values25,29,30,34 of the

parameters, we assume that \(H_{{\rm{ex}}} \gg H_{{\rm{DMI}}} \gg H_{2||},H_ \bot\), and, hence, \(m \ll n \approx 1.\) The external magnetic field induces a rotation of the Néel vector

towards a direction perpendicular to the magnetic field. In the easy-axis phase \(\left( {H_{2||} \ > \ 0} \right)\), the final state with \({\mathbf{n}} \bot {\mathbf{H}}\) can be

achieved either by a spin-flop or by a smooth reorientation, depending on the orientation of the magnetic field with respect to the easy axis. A spin-flop takes place at a critical value

\(H_{{\rm{sf}}} = \sqrt {H_{{\rm{ex}}}H_{{\rm{an}}}^{{\rm{eff}}}}\) when H is parallel to the easy-axis and competes with the effective anisotropy field \(H_{{\rm{an}}}^{{\rm{eff}}} =

H_{2||} - H_{{\rm{DMI}}}^2/H_{{\rm{ex}}}\). If, in contrast, H is perpendicular to the easy-axis, the DMI field pulls the magnetization along the applied field and simultaneously induces a

smooth rotation of the Néel vector into the state with \({\mathbf{n}} \bot {\mathbf{H}}\) and \({\mathbf{n}} \bot {\mathbf{Z}}\). In this case, the final state is achieved at the critical

field \(H_{{\rm{DMI}},{\rm{sf}}} = H_{{\rm{sf}}}^2/H_{{\rm{DMI}}}\). For a generic configuration with H forming an angle \(\chi _H\) with the easy axis, the critical field required for a

reorientation is $$H_{{\rm{cr}}}\left( {\chi _H} \right) = \frac{{H_{{\rm{DMI}},sf}^2 - H_{{\rm{sf}}}^2}}{{2H_{{\rm{DMI}},{\rm{sf}}}}}\left| {\sin \chi _H} \right| \\ +

\frac{1}{{2H_{{\rm{DMI}},{\rm{sf}}}}}\sqrt {(H_{{\rm{DMI}},{\rm{sf}}}^2 - H_{{\rm{sf}}}^2)^2\sin ^2\chi _H + 4H_{{\rm{DMI}},{\rm{sf}}}^2H_{{\rm{sf}}}^2}.$$ (4) By fitting the angular

dependence of the critical field with Eq.(4), we can then determine the parameters \(H_{{\rm{sf}}}\) and \(H_{{\rm{DMI}},{\rm{sf}}}\), and calculate the anisotropy and DMI fields, as it will

be discussed below. SPIN-HALL MAGNETORESISTANCE BELOW THE MORIN TRANSITION Experimentally, we access the antiferromagnetic properties of hematite using Hall bar devices (Fig. 1a), aligned

along the projection of the _c_-axis in the R-plane (_X_-axis) or perpendicular to it (_Y_-axis). To determine the role of the crystal anisotropies in the SMR response, we first perform

measurements at _T_ = 200 K below the Morin temperature (_T_M = 260 K) in the easy-axis phase with zero magnetization (M = 0) and measure the SMR signal by rotating the magnetic field in the

sample plane. For devices directed along _Y_, we measure the field-dependence of the SMR signal when applying a magnetic field in three directions: parallel (_X_ and _Y_ directions) and

perpendicular (_Z_ direction) to the sample plane, as shown in Fig. 1b. We define the SMR zero signal to be the value at zero field. When the magnetic field is applied in the sample plane

perpendicular to the current Jy, H||_X_, the SMR signal increases monotonically until it reaches the critical field \(H_{cr}\left( {{{\chi }}_{{H}} = 33^{\mathrm{o}}} \right) = 6\) T, which

corresponds to the transition into the “spin-flop” state with \({\mathbf{n}} \bot {\mathbf{H}}\). In this state the Néel vector is parallel to the current and does not contribute to the SMR

signal, which reaches its maximal value. When the applied magnetic field is perpendicular to the easy axis, H||_Y_, the SMR signal shows a non-monotonic field dependence, which can be

interpreted as a rotation of the Néel vector in the _xz_ plane according to theoretical predictions (solid line in Fig. 1b). In this configuration, the magnetic field induces a small

magnetization along _Y_ and the equilibrium orientation of the Néel vector is defined by the competition of the DMI and the easy-axis anisotropy. Above the critical field

\(H_{{\rm{cr}}}\left( {90^{\mathrm{o}}} \right) = H_{{\rm{DMI}},{\rm{sf}}} \approx 10\) T, the Néel vector reaches its final state perpendicular to the easy axis (and almost perpendicular to

the sample plane) with a minor projection on the direction of the spin accumulation Μ. This is seen as the saturation of the SMR signal. Note that we can interpret the experimental data

using only the model of pure SMR14,15, without the need to resort to any proximity effect-induced anisotropic magnetoresistance12, which would not lead to a change of magnetoresistance for

this magnetic transition in the plane perpendicular to the applied current. Finally, for H || _Z_, we observe a monotonic field dependence with an intermediate value of the critical field

\(H_{cr}\left( {57^{\mathrm{o}}} \right) = 6\) T, as expected for an angle of 57° between the applied field and the easy-axis. The agreement between the theoretical model (1)–(3) and

experiment indicates that the main contribution to the SMR signal originates from the Néel vector N and the contribution of the small induced magnetic moment M to the SMR is not significant.

Figure 1c–d, show the angular dependences of the longitudinal and transverse SMR signal measured for the fields around the minimal critical field (6 T), above the maximal critical field (11

 T), and also for an intermediate field value (8 T). The magnetic field is rotated within the sample plane, the in-plane angle α is measured relative to the current direction JY, as shown in

Fig. 1a. In all cases, the longitudinal SMR signal shows a negative sign (i.e a maximum resistance is achieved for the magnetic field applied perpendicular to the injecting current). The

amplitude of the SMR angular dependence peaks at the spin-flop field (H = 6 T), when the Néel vector spans all possible states within the sample plane _xy_. In contrast, for the rotation at

H = 11 T, the Néel vector varies between two “spin-flop” states (along _X_ and _Y_ crystallographic axes) with minor projections on the spin-accumulation axis _X_. This explains the lower

value of the SMR signal. The transverse SMR signal (Fig. 1d) shows similar behavior with a maximal amplitude at 6 T. Reasonable agreement between experimental data and theoretical modelling

for all angular and field dependences of the SMR signal shows that our single-domain approximation is sufficient to describe the system and allows us to estimate the values for the DMI

constant and \(H_{{\rm{an}}}^{{\rm{eff}}}\) based on the minimal and maximal values of critical fields (in combination with the reported value of the exchange field (_H__ex_ = 1040 T34).

However, these parameters can be determined directly with higher accuracy from fitting the angular dependence of the critical field HCR measured by rotating the magnetic field in the _xz_

plane, see Fig 2. ANGULAR DEPENDENCE OF THE CRITICAL FIELDS IN PRESENCE OF DMI Figure 2 shows the results of such measurements for a Pt Hall bar directed along _Y_. In Fig. 2a. we

demonstrate typical field dependences of the longitudinal SMR for different orientations of the magnetic field. When the magnetic field is almost parallel to the easy axis (β = −30°), the

SMR curve changes step-wise, thus reflecting the spin-flop reorientation of the Néel vector. The decrease of the signal in the near-spin-flop region originates from small (3°) misalignment

of the field which induces tilting of the Neel vector from _z_ axis. Such an abrupt transition is typical for easy-axis antiferromagnets, which can possess only indistinguishable 180o

domains. This behavior is in contrast with the smooth variation of the SMR signal in easy-plane antiferromagnets, where the magnetic field can induce domain wall motion between non-180o

domains17,24. For other field directions, the SMR signal varies smoothly, in accordance with theoretical predictions (solid lines). The critical field in these cases is associated with the

points of saturation of the SMR signal (vertical arrows). We can further exclude a contribution from a magnetic proximity effect in the platinum due to both the sign of the

magnetoresistance, the absence of a spin-reorientation at low magnetic fields, and the absence of XMCD signals (Supplementary Fig. 1 & Note 1). In Fig. 2b. we present the full

field-angular dependence of the longitudinal SMR signal calculated according to Eqs. (1)–(3). The black solid line delineates the region of the spin-flop state (with \({\mathbf{n}} \bot

{\mathbf{H}}\)) and the corresponding saturation of the SMR signal. This line, determined from Eq. (4), coincides with the experimental angular dependence of the critical field

\(H_{{\rm{cr}}}\left( {\chi_H} \right)\). By fitting the experimental data \(H_{{\rm{cr}}}\left( \beta \right)\) (dots) with the predicted dependence \(H_{{\rm{cr}}}\left( {\chi _H}

\right)\) (taking into account that \(\chi _H = \beta + 33^\circ\)), we extract the values of the internal DMI and effective anisotropy fields as _H_DMI = 2.72 T and

\(H_{{\rm{an}}}^{{\rm{eff}}} =\)23.8 mT. These values are consistent with the previous measurements35 and show that the simple surface-sensitive electrical measurements allow for a direct

and precise access to the magnetic state and properties of antiferromagnetic system. ROLE OF THE ANISOTROPY AXIS IN THE ELECTRICAL READ-OUT In antiferromagnets, the SMR response strongly

depends on the orientation of the anisotropy axis relative to the patterned devices. Contrary to ferromagnets with low anisotropies like YIG or NiFe, the projection of the Néel vector along

the spin-accumulation can even be zero as long as the applied magnetic fields are lower than the large critical fields. To identify and corroborate the role of anisotropy fields in the

transport measurements, we thus repeat the measurements for Pt Hall bars rotated by 90°, with injected currents parallel to the _X_-axis (Fig. 3a). For this geometry of the devices, the Néel

vector is perpendicular to the spin-accumulation at zero field. The field dependences of the longitudinal SMR measured for H||_X_ and H||_Z_ (Fig. 3b) are in good agreement with the

theoretical model (Eqs. (1)–(3)) and show the same critical fields (6 and 8 T respectively) as the devices with a _y_-oriented electrode. For H||_Y_ the SMR signal is nearly constant, as the

Néel vector rotates in the plane perpendicular to the spin-accumulation. In this case, the SMR response does not reflect the Néel reorientation observed for Pt devices directed along Y. The

small quantitative discrepancy between the theoretical model and the experiments at high magnetic fields, larger than the critical fields, could be associated with a residual contribution

from the emerging canted moment and from the limitations of our analytical model. The angular dependences of the longitudinal (Fig. 3c) and transverse (Fig. 3d) SMR are negative, in

agreement with the model predictions (Eq. 1). This further confirms that we indeed measure a pure SMR effect with no significant proximity induced magnetic moment contributions. Note that in

spite of a visual difference in the SMR dependences measured for _X_- and _Y_-oriented devices, all the data can be consistently interpreted within the same theoretical model that we

develop. This also means that SMR signal must be analyzed taking into account the device orientation relative to the crystallographic axis. In our case, information from the _X_-oriented

devices is limited, due to the absence of an electrical signature of the DMI induced reorientation for H||_Y_. SPIN-HALL MAGNETORESISTANCE THROUGH THE MORIN PHASE TRANSITION Finally, the SMR

technique allows one to probe the equilibrium states of hematite over a wide temperature range to study the (_T_, _H_) magnetic phase diagram. To illustrate this, we measure field

dependences of the longitudinal SMR for _X_-oriented devices at different temperatures (see Fig. 4a), below and above the Morin transition temperature which separates the easy-axis and

easy-plane phases. Extracted values of the critical fields (closed black dots) are shown in Fig. 4b as a function of temperature. Within the interval 160–255 K, the temperature dependence of

the critical field is well described by \(H_{{\mathrm{cr}}}(T) \propto {\sqrt{T_{\mathrm{M}} - T}}\), with \(T_{\mathrm{M}} = 255\, {\mathrm{K}}\). Such a temperature dependence is typical

for the second-order phase transitions within the Landau theory and was also reported earlier34,36. Below 160 K, \(H_{{\rm{cr}}}\left( T \right)\) stays around a constant value of 8 T, above

255 K its value is close to zero. Using Eq. (4) and the experimental dependence \(H_{{\rm{cr}}}\left( T \right)\), we then determine the temperature dependence of the effective anisotropy

\(H_{{\mathrm{an}}}^{{\mathrm{eff}}}\left( T \right)\) (blue line in Fig. 4b) assuming that _H_DMI = 2.72 T is temperature-independent (as previously found35,37). The last assumption is

consistent with the observed (open dots in Fig. 4b) and predicted (solid line) temperature dependence of the maximal critical field. Note, that as the SMR is a surface sensitive technique,

similar measurements could determine the magnetic anisotropies of high quality epitaxial hematite thin films31, for which magnetometry measurements such as SQUID do not provide sufficient

sensitivity. Indeed, one can notice here the excellent agreement between previous bulk measurements and the measurements and theory presented here with a surface sensitive technique. This

also indicates that the surface and the bulk of the crystal have similar anisotropies. According to the theoretical model, the effective anisotropy vanishes at the Morin temperature,

\(H_{{\rm{an}}}^{{\rm{eff}}}\left( {T_{\rm{M}}} \right) = 0\), and changes sign and becomes negative in an easy-plane phase (above \(T_{\rm{M}}\)). However, the SMR signal at 260 and 300 K

still shows the features of the spin-flop transition at small field around 0.4 T (Fig. 4a), a feature also observed in neutron diffraction and resonance measurements [33, 37]. Moreover, the

SMR has the same sign below and above _T_M which indicates that the SMR is still dominated by the Néel vector and not by the small canted moment. Assuming that reorientation of the Néel

vector in the easy-plane phase (represented in Fig. 4c) is driven mainly by in-plane anisotropy (Eq. 3), we estimate the value of the in-plane anisotropy field \(H_ \bot\) from the angular

dependences of the SMR measured for small (0.5 T) and large (5.5 T) fields applied in the _xy_ plane and for the two orientations of the Pt stripes (Fig. 4d). As seen from Fig. 4d, the

amplitude of the SMR signal remains the same for small and large fields. According to theoretical predictions this means that the critical field in the easy-plane phase is below 0.4 T, which

gives \(H_ \bot \le 1.6\) μT. The weak field-dependence of the SMR amplitude also means that the field-induced “weak” magnetic moment only contributes negligibly. Note, that for geometrical

reasons, the SMR amplitude is larger for _X_-oriented device demonstrating again the role of the crystal and device symmetry in the electrical response. DISCUSSION We have combined theory

with experiments to analyze the properties of the insulating antiferromagnetic hematite, α-Fe2O3, using accessible transport measurements. We successfully model Spin-Hall magnetoresistance

measurements in α-Fe2O3 by a simple analytical model. From comparison and fitting of our electrical measurement results, we determine the values of anisotropy and Dzyaloshinskii-Moriya

fields, which drive the Néel vector reorientation. Measuring the SMR across the Morin transition, we determined the specific properties of the easy-axis and canted easy-plane phases, and

observed that the contribution of the weak moments is negligible for the electrical response, which is governed by the Néel vector. The shape of the SMR curves strongly depends on the

orientation of the devices with respect to the crystallographic axis in line with our model predictions. Our observations demonstrate that the Spin-Hall magnetoresistance technique is a

unique surface sensitive technique that allows us to easily access to the fundamental properties of insulating antiferromagnets without the need to resort to large scale facility

measurements as previously required. METHODS EXPERIMENTAL SETUP The devices were carried out based on a sample geometry that was defined using electron beam lithography and the subsequent

deposition and lift-off of a 7 nm platinum layer by DC sputtering in an argon atmosphere at a pressure of 0.01 mbar. The devices were contacted using a bilayer of chromium (6 nm) and gold

(32 nm). The sample was mounted to a piezo-rotating element in a variable temperature insert that was installed in a superconducting magnet capable of fields up to 12 T and cooled with

liquid helium. For the rotation measurements, the sample was rotated in a constant field. SINGLE CRYSTAL OF Α-FE2O3 The single crystal of hematite, α-Fe2O3, was obtained commercially and has

an R-cut orientation, which means that the c-axis of the hexagonal structure is tilted 34 degrees below the surface plane. This orientation was chosen because of the large in-plane

projection of the c-axis in addition to the stability of the sample terminating with the r-plane. DATA AVAILABILITY The data that support the findings of this study are available from the

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_Phys. Solid State_ 52, 112–116 (2010). Article  ADS  Google Scholar  Download references ACKNOWLEDGEMENTS R.L. acknowledges the European Union’s Horizon 2020 research and innovation

programme under the Marie Skłodowska-Curie grant agreement FAST number 752195. R. L., A. R. and M.K. acknowledge support from the DFG project number 423441604. A.R., J.S. and M.K.

acknowledge support from the Graduate School of Excellence Materials Science in Mainz (DFG/GSC 266). O.G. and J.S. acknowledge the support from the Humboldt Foundation, the ERC Synergy Grant

SC2 (No. 610115), the EU FET Open RIA Grant no. 766566, the DFG (project SHARP 397322108). L. B. acknowledges the European Union’s Horizon 2020 research and innovation programme under the

Marie Skłodowska-Curie grant agreement ARTES number 793159. All authors from Mainz also acknowledge support from both MaHoJeRo (DAAD Spintronics network, project number 57334897) and SPIN+X

(DFG SFB TRR 173). A.Q. and A.B. acknowledge support from the European Research Council via Advanced Grant number 669442 ‘Insulatronics’. S.A.B. and R.A.D. acknowledge support from Stichting

voor Fundamenteel Onderzoek der Materie (FOM) and the European Research Council via Consolidator Grant number 725509 ‘SPINBEYOND’. A.Q., R.A.D., M.K. and A.B. were supported by the Research

Council of Norway through its Centres of Excellence funding scheme, project number 262633 ‘QuSpin’. AUTHOR INFORMATION AUTHORS AND AFFILIATIONS * Institute for Physics, Johannes

Gutenberg-University Mainz, 55099, Mainz, Germany R. Lebrun, A. Ross, O. Gomonay, L. Baldrati, J. Sinova & M. Kläui * Graduate School of Excellence Materials Science in Mainz,

Staudingerweg 9, 55128, Mainz, Germany A. Ross, J. Sinova & M. Kläui * Utrecht University, Princetonplein 5, 3584 CC, Utrecht, The Netherlands S. A. Bender & R. A. Duine *

Helmholtz-Zentrum Berlin für Materialien und Energie, Albert-Einstein Str. 15, 12489, Berlin, Germany F. Kronast * Center for Quantum Spintronics, Department of Physics, Norwegian University

of Science and Technology, NO-7491, Trondheim, Norway A. Qaiumzadeh, A. Brataas, R. A. Duine & M. Kläui * Department of Applied Physics, Eindhoven University of Technology, P.O. Box

513, 5600 MB, Eindhoven, The Netherlands R. A. Duine Authors * R. Lebrun View author publications You can also search for this author inPubMed Google Scholar * A. Ross View author

publications You can also search for this author inPubMed Google Scholar * O. Gomonay View author publications You can also search for this author inPubMed Google Scholar * S. A. Bender View

author publications You can also search for this author inPubMed Google Scholar * L. Baldrati View author publications You can also search for this author inPubMed Google Scholar * F.

Kronast View author publications You can also search for this author inPubMed Google Scholar * A. Qaiumzadeh View author publications You can also search for this author inPubMed Google

Scholar * J. Sinova View author publications You can also search for this author inPubMed Google Scholar * A. Brataas View author publications You can also search for this author inPubMed 

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inPubMed Google Scholar CONTRIBUTIONS R.L. and M.K proposed and supervised the project. R.L and A.R. performed the transport and SQUID experiments. A.R patterned the samples. R.L, L.B and

F.K performed the XMLD measurements of the Supplemental. R.L, O.G, S. B, A.R. analysed the data. O.G. performed the analytical calculations with inputs from S.B, A.Q., J.S, A.B and R-A.D. 

R.L, O. G A.R. and M.K wrote the paper. All authors commented on the manuscript. CORRESPONDING AUTHORS Correspondence to R. Lebrun or M. Kläui. ETHICS DECLARATIONS COMPETING INTERESTS The

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transitions in insulating antiferromagnets determined by a Spin-Hall magnetoresistance probe. _Commun Phys_ 2, 50 (2019). https://doi.org/10.1038/s42005-019-0150-8 Download citation *

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