On the sign of the linear magnetoelectric coefficient in Cr2O3

We establish the sign of the linear magnetoelectric (ME) coefficient, α, in chromia, Cr2O3. Cr2O3 is the prototypical linear ME material, in which an electric (magnetic) field induces a linearly proportional magnetization (polarization), and a single magnetic domain can be selected by annealing in combined magnetic (H) and electric (E) fields. Opposite antiferromagnetic (AFM) domains have opposite ME responses, and which AFM domain corresponds to which sign of response has previously been unclear. We use density functional theory (DFT) to calculate the magnetic response of a single AFM domain of Cr2O3 to an applied in-plane electric field at zero kelvin. We find that the domain with nearest neighbor magnetic moments oriented away from (towards) each other has a negative (positive) in-plane ME coefficient, α⊥ , at zero kelvin. We show that this sign is consistent with all other DFT calculations in the literature that specified the domain orientation, independent of the choice of DFT code or functional, the method used to apply the field, and whether the direct (magnetic field) or inverse (electric field) ME response was calculated. Next, we reanalyze our previously published spherical neutron polarimetry data to determine the AFM domain produced by annealing in combined E and H fields oriented along the crystallographic symmetry axis at room temperature. We find that the AFM domain with nearest-neighbor magnetic moments oriented away from (towards) each other is produced by annealing in (anti-)parallel E and H fields, corresponding to a positive (negative) axial ME coefficient, α∥ , at room temperature. Since α⊥ at zero kelvin and α∥ at room temperature are known to be of opposite sign, our computational and experimental results are consistent.


I. INTRODUCTION
Materials in which both time-reversal Θ and spaceinversion I symmetries are broken, while the product IΘ symmetry is preserved, have a term in their free energy of the form where E / H are electric / magnetic fields, α is the ninecomponent magnetoelectric tensor (SI units s/m) and V is the unit cell volume.This term reveals two distinctive and related material properties.First, there is a preferred magnetic domain orientation, determined by the sign and form of α, in simultaneous magnetic and electric fields, so that annealing in such a combination of fields, called magnetoelectric annealing, can be used to select for a specific magnetic domain.Second, by differentiating Eq. 1 with respect to electric (magnetic) field to obtain the polarization (magnetization), we see that and where µ 0 is the vacuum permeability.Eqs. 2 and 3 reveal a linear proportionality between an applied electric (magnetic) field and an induced magnetization M i (polarization P i ), with α the response tensor.Materials with non-zero α therefore show a linear magnetoelectric (ME) effect and are promising for spintronic applications since they enable voltage-control of magnetism [1].
Corundum-structure chromia, Cr 2 O 3 , is the prototypical linear magnetoelectric, and the first material in which the linear ME effect was predicted [2] and measured [3,4].In addition to its historical relevance, Cr 2 O 3 has a high Néel temperature compared to other ME materials, and continues to be the primary material of focus in theoretical, experimental and technological studies of the ME effect.
We show the primitive rhombohedral unit cell of Cr 2 O 3 in Fig. 1(a).Below its Néel temperature T N = 307 K [5], Cr 2 O 3 adopts a superexchange-mediated easy-axis antiferromagnetic (AFM) "up-down-up-down" ordering of the magnetic dipole moments on the d 3 Cr 3+ ions along the rhombohedral ⟨111⟩ direction [6,7].The R 3′ c ′ magnetic space group breaks both I and Θ while preserving IΘ, thus allowing a linear ME response [8].In Fig. 1 (b) we show the primitive unit cell of the opposite AFM domain, with "down-up-down-up" magnetic dipole ordering.While (a) and (b) are energetically degenerate in the absence of external fields, they correspond to opposite ME domains.As a result, the signs of their linear ME responses are opposite, and they are obtained by ME annealing in opposite combinations of E and H fields.In Fig. 1(c), we show the unit cell of Cr 2 O 3 in the hexagonal setting conventionally used in neutron diffraction, in which the hexagonal ⟨001⟩ axis is parallel to the rhombohedral ⟨111⟩ axis.
The symmetry of the R 3′ c ′ magnetic space group allows for a diagonal response tensor α, described by two independent components which we denote as α ∥ and α ⊥ [2]: α ∥ describes the magnetization (polarization) induced when E (H) is applied along the rhombohedral ⟨111⟩ axis, and α ⊥ refers to the perpendicular ME response when the field and induced property lie in the basal plane.Fig. 2 shows the measured temperature dependence of α ∥ and α ⊥ , extracted from the original experimental report [4].While α ⊥ , which results from the E-field induced canting of the magnetic dipole moments away from the easy axis [9,10], follows the   usual order-parameter onset below T N , α ∥ has a peak in magnitude just below T N before decreasing and switching sign at low temperature.This is understood in terms of the response of spin fluctuations at high temperature [11], with the orbital magnetization response [10] dominating at low temperature.Importantly, at T =0 K, relevant to first-principles calculations, α ⊥ and α ∥ have the same sign, whereas at room temperature, relevant to many experimental setups, α ⊥ and α ∥ have opposite signs.
While the relative signs of α ⊥ and α ∥ were established unambiguously in Ref. [3], it was not possible at the time to determine which set of α values correspond to the out-pointing or in-pointing magnetic domains of Figs. 1 a) and b).Instead, Ref. [3] showed that reversal of the AFM domain reverses the signs of α as required by symmetry, and that the measured magnitudes in multi-domain or poly-crystalline samples are substantially reduced due to cancellation effects.The experimental determination of the specific bulk AFM domain corresponding to a particular ME response is highly non-trivial and requires a generalized form of polarized neutron scattering called spherical neutron polarimetry; to our knowledge only four such experiment has been performed for Cr 2 O 3 [12][13][14].While in principle first-principles calculations based on density functional theory (DFT) yield this information directly, the AFM domain modeled is often not reported in the literature, and the sensitivity of the magnetic anisotropy to the details of the DFT parameters render an independent experimental determination desirable.To compound confusion, in both the theoretical and experimental literature the terms "magnetic moments" and "spins" have sometimes been used interchangeably, in spite of their being opposite in sign.The purpose of this paper is to establish unambiguously the signs of the ME effect corresponding to each of the two opposite AFM domains in Cr 2 O 3 .We achieve this goal by reviewing and reanalyzing the relevant computational and experimental literature, as well as presenting the results of our own new DFT calculations.In Sec.II, we begin by reviewing the DFT-based results for the zero-kelvin values of α ⊥ and α ∥ , computed both by us and by others in earlier publications.We then perform a comprehensive cross-check of the domain-dependent sign of α using four different codes, three different methods for applying the external fields, and different choices of DFT parameters.
We find that the ab-initio results give consistent signs for α across authors, DFT parameters, and codes used.
In Sec.III, we reanalyze the seminal neutron polarimetry experiments which provided the first experimental indicator for the sign of α [12][13][14][15].While the stated conclusion of the original polarimetry papers contradicts the DFT findings, we show that this is actually due to the assumed sample orientation with respect to the instrument axes during analysis in Refs.[13][14][15].When we account for and correct these inconsistencies, the raw polarimetry data indicate a room-temperature sign of α for a given AFM domain consistent with all DFT calculations (taking into account the experimental temperature dependence of α found by Astrov in Fig. 2).We hope that this paper clears up long-standing ambiguities and confusions in the literature, and facilitates future interpretations of theoretical and experimental data related to the ME effect in Cr 2 O 3 and other ME materials.

II. COMPUTATIONAL STUDIES
Several ab initio studies of the magnitude and sign of the ME effect in Cr 2 O 3 have been performed previously [9-11, 16, 17].Three main techniques have been employed: Explicit inclusion of i) a static magnetic field or ii) a static electric field within the DFT Hamiltonian, and iii) the so-called "lattice-mediated" method, in which a polar displacement of the ions simulates the application of an electric field.Both spin and orbital contributions to the response have been calculated, and α has been resolved into so-called clamped-ion (the electronic response to an electric field with fixed ions) and latticemediated (in which the ions are displaced by the electric field) components.Since most DFT codes (in particular abinit [18,19], elk [20], quantum espresso [21,22] and vasp [23,24]) output magnetic moments rather than spins, we adopt this convention here.
First, we summarize the results of the various literature studies that report both the AFM domain studied and the sign of the calculated α.The technical details for each calculation are summarized in Table I.First, Malashevich et al. [10] found α ∥ and α ⊥ to have the same positive sign at 0 K for a domain with in-pointing moments.[as in Fig. 1(b)].They used the finite electric-field method so that both spin and orbital contributions and the full lattice-mediated and electronic responses were included.For the same domain, Íñiguez [9] used the 'lattice-mediated' method and obtained a positive 0 K lattice-mediated spin ME response α ⊥ ; since Ref. [9] did not include orbital contributions, α ∥ was zero.Also using the lattice-mediated approach but including the orbital contributions, Ye and Vanderbilt [16] found positive α ∥ and α ⊥ for the domain with in-pointing moments at 0 K. Bousquet et al. [17], using an explicitly applied magnetic Zeeman field, including both the lattice-mediated and clamped-ion spin contributions, find a positive 0 K α ⊥ for the in-pointing domain as well [26].Finally, Mostovoy et al. [11] considered the opposite domain (note that Fig. 1 of Ref. [11] shows spins) and calculated the finitetemperature spin contribution to α ∥ , using Monte-Carlo simulations of a DFT-derived model Hamiltonian containing Heisenberg exchanges and a magnetic momentpolarization coupling.They found a positive α ∥ in the temperature range of T = 60-400 K, consistent with a negative α ∥ at T = 0 K [Fig.2].Since their calculations modeled the out-pointing domain, these results are consistent with the other computational studies discussed earlier.
To supplement the literature results, we perform a comprehensive cross-check of the domain-dependent sign of α ⊥ using four different codes and three different meth-0.070.00 0.07 ods.First, we calculate the lattice-mediated spin contribution to α ⊥ using the lattice-mediated method, as described in Ref. [9], using the elk [20], vasp [23,24], and quantum espresso [21,22] codes, with the parameters listed in Table I.In all cases we find α ⊥ < 0 for the out-pointing domain at 0 K, consistent with the literature findings summarized above.In addition, we use the abinit code [18,19] to calculate the spin contribution to the ME effect by both explicitly applying an electric field as in Ref. [10], and a magnetic Zeeman field method as in Ref. [17].Both methods give the same positive value of α ⊥ for the in-pointing domain at 0 K. Computational details for the calculations in elk, vasp, quantum espresso, and abinit can be found in Appendices A-D.We list our calculated α ⊥ values in Table I, and in Fig. 3, we plot the induced in-plane magnetizations as a function of in-plane electric fields calculated here and from the literature.Although there is complete agreement on the sign of α, it is clear that there is some spread in the magnitude of calculated values.This distribution cannot be explained only by the different contributions to α that were taken into account, and is most likely also the result of the different choices in electronic structure code, electronic exchange parameters and convergence criteria.Considering these differences, the agreement on the magnitude of α is remarkable.
In summary, the calculated signs of α are consistent across DFT codes and methodologies, with the 0 K α ⊥ and α ∥ positive for the in-pointing domain, the 0 K α ⊥ negative for the out-pointing domain, and the room temperature α ∥ positive for the out-pointing domain.We summarize this result in Tab.II with the addition values at 0 K and room T extrapolated.
Overview the sign of alpha for the two domains, at 0 K and room temperate (RT), as determined by different ab-initio calculations (black, bold), and extrapolated from the DFT results (grey) using the temperature dependence measured by Astrov, see Fig. 2.

III. EXPERIMENTAL STUDIES
To our knowledge, there exist four sets of data in which the magnetic structure of Cr 2 O 3 was measured using spherical neutron polarimetry (SNP), the generalized form of polarized neutron scattering [12][13][14][15].This technique allows for both the detection of the domain imbalance between the two different magnetic structures shown in Fig. 1, and for the determination of the magnetic moment configuration of the predominant domain [12].This is possible because with SNP, the polarization vectors of both the incident and scattered neutron beams are determined; in comparison, in conventional (uniaxial) polarized neutron scattering, the scattered neutron polarization information is only analyzed along the direction of the incident beam polarization [27].Therefore, SNP is an ideal method for elucidating which spin configuration shown in Fig. 1 is stabilized by the parallel or anti-parallel combination of electric and magnetic fields.
The SNP measurements reported in Refs.[12][13][14][15] were performed at the IN20 and D3 beamlines at the Institut Laue Langevin (ILL, Grenoble), using the CRYOgenic Polarization Analysis Device (Cyopad).The Cryopad consists of a zero-magnetic field sample chamber surrounded by magnetic fields manipulating the incident (P i ) and scattered (P f ) beam polarizations [28,29].The field regions are decoupled with a pair of concentric superconducting Meissner shields combined with µ-metal yokes and screens.The incident neutron beam polarization was controlled using a combination of a nutator and precession coil, and was oriented along one of three orthogonal experimental co-ordinates which were defined as x, which is along the direction of the scattering vector Q, z, which is perpendicular to the horizontal scattering plane, and y, which completes the right-handed coordinate set.The polarization of the scattered neutron beam was also analyzed along these three principal axes using another set of precession and nutator coils.In each of the four studies, the Cr 2 O 3 sample was aligned so that the crystal b-axis was perpendicular to the horizontal scattering plane, which allowed access to the (h 0 l) reflections (importantly, this introduces an ambiguity between b∥+z and b∥−z, which we will discuss in more detail in the following section).Here, the Miller indices correspond to the hexagonal setting of the rhombohedral (R 3c) unit cell of Cr 2 O 3 adopted in Refs.[12][13][14][15].In the three most recent studies, prior to installing the sample in the Cryopad for the SNP measurements, the Cr 2 O 3 sample was cooled through the Néel temperature (T N ∼310 K) in a combination of electric and magnetic fields oriented along the crystallographic c axis to achieve an imbalance of 180 • domain population [12][13][14].Brown et al. reported that this annealing process stabilized a single AFM domain [12][13][14], and that the type of AFM domain (Fig. 1) could be chosen based on the relative orientation of the external magnetic (H) and electric fields (E).The experimental determination of which magnetic domain is favoured then boils down to the determination and interpretation of the sign of the polarization matrix element P zx .
Experimentally, P zx is determined by measuring two quantities, namely n zx and n z x, which are the number of scattered neutrons with the polarization parallel and antiparallel to +x for the incident neutron polarization along +z.The experimental P zx matrix element is in turn obtained by taking the ratio, for a given Bragg reflection Q = (h k l).As such, the quantity P zx is bounded between -1 and 1.
In order to determine which AFM domain is favoured, the authors in Refs.[12][13][14][15] expressed P zx in terms of three dimensionless quantities, The η term defines the population imbalance between the two magnetic domains, and is given by η = (v + − v − )/(v + + v − ), where v + and v − are the volumes of the two magnetic domains.Hence, the value of η is bounded between 1 and -1.If the two magnetic domains are equally populated, the factor η becomes 0. The term q y is determined by the orientation of the crystal with respect to the experimental set up, with the sign of q y depending on whether the crystallographic +b axis is along the +z or −z direction of the experimental geometry; for example q y is +1 (-1) if the magnetic interaction vector M ⊥ (Q) is parallel (anti-parallel) to the +y axis of the experimental geometry.Hence, it is crucial to determine whether +b is along the +z or −z direction.Finally, the term γ is associated with the magnetic structure, with the sign of γ being positive (negative) for the out-pointing (in-pointing) magnetic domain.
Based on this discussion, we identify three inconsistencies across the four Refs.[12][13][14][15], which we clarify here.(Note that the measurements in Refs.[12][13][14][15] were made with the same crystal by the same group of coauthors so we expect the underlying physics to be consistent).

A. Spin vs magnetic moment
The first discrepancy is between Ref. [14] and Ref. [12], regarding the definition of spin and magnetic moments.In Ref. [14], the authors propose that the antiparallel E and H fields favor the 'out-pointing' arrows [Fig.1(a)] and designate the arrows as spin directions.On the other hand in Ref. [12], the authors present 'in-pointing' arrows [Fig.1(b)], which they designate as magnetic moments, and state that this magnetic structure is stabilized by parallel E and H fields.Since the Cr spin direction and magnetic moment direction are anti-parallel, these two statements are incompatible.
In the neutron scattering community, however, the terms spin and magnetic moment are often used interchangeably to mean magnetic moment direction.We should therefore assume that the arrows in Ref. [14] actually indicate magnetic moments, rather than spins as stated.This resolves the apparent discrepancy between Ref. [12] and Ref. [14].

B. Orientation of the crystal b axis
Second, the labelling of the Miller indices (h0l) across the four reports is inconsistent.In the first report [15], the two reported reflections, namely (1 0 2) and ( 1 0 4), are in fact forbidden by the R 3c space group in the hexagonal setting of Cr 2 O 3 .In the subsequent study, the two reported reflections, ( 1 0 2) and (1 0 2) are both allowed by R 3c.In the following two reports [13,14], where forty reflections were reported in total, thirty-two are in fact forbidden by the R 3c space group of Cr 2 O 3 .The h and l Miller indices of the remaining eight reflections are both multiples of 3, e.g.(3 0 6) and (3 0 6), and are hence allowed.
Parallel H and E Pi P obs.

f
) polarization matrices with 'out-pointing' and 'in-pointing' magnetic domain for the data collected with E and H parallel.The data is consistent with the 'out-pointing' magnetic domain, as shown in Fig. 1(a).
Anti-parallel H and E Pi P obs.

f
) polarization matrices with 'out-pointing' and 'in-pointing' magnetic domain for the data collected with E and H anti-parallel.The data is consistent with the 'in-pointing' magnetic domain, as shown in Fig. 1(b).
Given that the magnetic propagation vector of Cr 2 O 3 is Q m =(0 0 0), the magnetic scattering intensity occurs at the same reciprocal space location as the structural Bragg peaks of Cr 2 O 3 .As such, the Miller index of the magnetic/nuclear reflections should follow the general condition of the R 3c space group, where −h + k + l=3n.Since the four reports were concerned with reflections in the (h 0 l) plane, the observed reflections should obey the rule −h + l=3n, given that k=0.In figure 4(a) we plot the calculated reciprocal space maps for Cr 2 O 3 in the (h 0 l) scattering plane, assuming that the +b crystal axis is along the +z direction as stated in the original papers.Here, the allowed reflections, such as ( 1 0 2) and (1 0 4), are denoted by the black filled circles, and the reciprocal space location of the forbidden reflections that do not obey −h + l=3n are shown by the crosses (×).
The observed reflections in Refs.[13][14][15] are denoted by the arrows in the reciprocal space map.Indeed, many of the observed reflections, including (1 0 2) and (1 0 4), are in fact forbidden by the R 3c space group.
If instead, we assume that the +b crystal axis was oriented along the −z direction (rather than +z), then the reciprocal space location of all forty-two observed reflections reported in [13][14][15], is fully compatible with the R 3c space group.This scenario is very plausible, due to a possible mix up between the +b and −b crystal axes, which are in-equivalent in Cr 2 O 3 .As shown in Fig. 4(b), where we plot the calculated reciprocal space maps for Cr 2 O 3 in the (h 0 l) scattering plane, assuming that the +b crystal axis is along the −z direction, the reciprocal space location of the observed reflections denoted by all of the arrows can now be accounted for.
Changing the direction of the +b axis has two main consequences for the interpretation of the results in Refs [13][14][15].First it swaps the h Miller index of the reflections, such that the observed peaks which were designated as (h 0 l) should be assigned as ( h 0 l) instead.This would allow the thirty-four reflection which are originally forbidden now be compatible with R 3c, i.e. to obey the −h+l=3n condition.The remaining eight reflections which have the h and l Miller indices both being multiples of 3, still obey this condition.The second is that in Ref [12] the sign of q y changes, which means that the interpretation of which magnetic domain is favoured also changes.
Therefore, we conclude that the conjugate field with E and H parallel favours the 'out-pointing' domain, as shown in Fig. 1(a).By the same token, the antiparallel E and H field favours the 'in-pointing' domain.This is opposite to the interpretation in Ref. [14].

C. Sign of γ
Finally, the third inconsistency is between Refs.
[13] and [14].In these studies, the γ term was obtained by measuring the polarization P zx component of various reflections.Ref. [13] reports, in Table 3, the γ values for twelve reflections obtained on the IN20 instrument with thermal neutrons (λ=1.532Å).On the other hand, Ref. [14] reports the measurements of γ for a further fifteen reflections acquired on the D3 instrument with hot neutrons.Table 2 of Ref. [14] lists the γ data acquired from the D3 instrument along with those measured on the IN20 instrument, which were reported in [13].
The discrepancy lies in the sign of γ of the data collected on the IN20 instrument, which are reported both in Table 2 of Ref. [14] and also in Table 3 of Ref. [13].Although the Miller indices of the twelve reflections and their corresponding magnitude of γ are the same, the signs are different.Since the sign of γ is used to interpret whether the magnetic domain is 'out-pointing' or 'in-pointing', this discrepancy calls into question which sign of γ was measured.
To resolve the ambiguity, here we use the Mag2Pol software [30] to re-analyze the measured spherical neutron polarimetry data presented in Table 2 of Ref. [12].We choose this data set because the Miller indices are allowed by the R 3c space group, and the raw data are presented explicitly.Moreover, these measurements were performed on cooling the Cr 2 O 3 sample with a conjugate field of parallel or anti-parallel E and H fields through T N to T =290 K, where the measurements were performed.Tables III and IV tabulate the measured polarization matrices for the case where E and H are parallel and antiparallel, respectively, along with the results of our new analysis for the two cases where the magnetic domain is 'out-pointing' or 'in-pointing'.
Our analysis assuming an 'out-pointing' domain is consistent with the measured scattered neutron polarization for the case where E and H are parallel, contrary to the conclusions in Refs.[12][13][14][15].Similarly, for the case where E and H are anti-parallel, we find that the measured polarization matrices are consistent with an 'in-pointing' domain.

IV. CONCLUSION
We have combined a literature review, new ab-initio results, and a careful reanalysis of spherical neutron polarimetry data in an effort to resolve long-standing confusion regarding the domain-dependent sign of the ME coefficient in Cr 2 O 3 .We have shown that all ab-initio results to date are in agreement in the assignments of negative and positive low-temperature α to the out-pointing and in-pointing domains depicted in Fig. 1.These conclusions are remarkably consistent across multiple codes and methods.Gratifyingly, the room-temperature spherical neutron polarimetry data are consistent with the low-temperature ab-initio findings given that the roomtemperature sign of α ∥ is opposite to its low-temperature sign.The opposite interpretation in some of our literature experimental papers stems from a sign error due to subtle inconsistencies in the analysis which we discussed in Sec.III.The confusion and deceptive inconsistency have also been compounded in the past by ambiguous terminology from numerous authors related to the usage of "spin" versus "magnetic" moment.We summarize the relationship between the domains and the sign of α, as well as the necessary alignment of the E and H fields during magnetoelectric annealing, in Fig. 5.
We mention here an important consequence of our work for a related feature of Cr 2 O 3 ; the magnitude and sign of the uncompensated magnetization on the (001) surface for a given bulk domain [31,32].At the 0-K limit in the absence of thermal fluctuations, the direction of the (001) surface magnetization is unambiguously determined by the bulk domain which is selected in the ME annealing process.For example, with the in-pointing domain depicted for the hexagonal cell in Fig. 1(c), the surface magnetization from the dangling Cr at the (001) surface points outwards (positive).However, for any experimental characterizations performed at roomtemperature, the relation between the bulk domain and the sign of surface magnetization is much less clear.Indeed, recent DFT-Monte Carlo calculations performed by some of the authors [33] indicate that the (001) surface magnetic moments of Cr 2 O 3 are essentially paramagnetic at room temperature due to weak coupling to the bulk order parameter.Thus, it is likely that for a fixed domain, the surface magnetization is substantially reduced, or even switches sign, with respect to its 0 K value.Now that we have definitively determined which domain is selected by a given ME annealing at room temperature, it will be very interesting to re-examine, and perform new, experimental measurements of surface magnetization to determine its sign for an unambiguous selection of bulk domain.
We hope that our work convincingly demonstrates the previously questioned consistency of computational and experimental findings on the sign of the ME coefficient in Cr 2 O 3 , and that it may motivate new, updated polarimetry measurements to test and confirm existing experimental and theoretical results.We also hope that this paper will assist in the correct interpretation of future studies of Cr 2 O 3 , as well as providing a cautionary tale for similar investigations of other ME materials.
ing the modern theory of polarization, as implemented in vasp in the LCALCPOL routine.These calculations were performed for four displacements of different magnitudes, allowing us to assess the linear response regime.The final Z e were obtained from the average of the Z e for different atoms of the same species and different displacements within the linear regime.
t e x i t s h a 1 _ b a s e 6 4 = " v r n n J a w K c U k U W x i Z p X w D 7 K X b + 6 E = " > A A A B / 3 i c b V D L S g M x F M 3 U V 6 2 v U c G N m 2 A p u C o T 8 b U Q K b h x W c E + o B 2 G T J p p Q z O T I c k I Z Z y F v + L G h S J u / Q 1 3 / o 2 Z d h b a e u D C 4 Z x 7 y c n x Y 8 6 U d p x v q 7 S 0 v L K 6 V l 6 v b G x u b e / Y u 3 t t J R J J a I s I L m T X x 4 p y F t G W Z p r T b i w p D n 1 O O / 7 4 J v c 7 D 1 Q q 4 l C p i c k F a 3 l M N e / l 4 n 9 e L 9 H B p Z u y K E 4 0 j c j s o S D h U A u Y l w E H T F K i + c Q Q T C Q z W S E Z Y Y m J N p V V T A l o / s u L p H 1 S R + f 1 s 7 v T a g M W d Z T B I T g C x w C B C 9 A A t 6 A J W o C A R / A M X s G b 9 W S 9 W O / W x 2 y 1 Z B U 3 + + A P r M 8 f n x m V y A = = < / l a t e x i t > < 111 > rhomb < l a t e x i t s h a 1 _ b a s e 6 4 = " v r n n J a w K c U k U W x i Z p X w D 7 K X b + 6 E = " > A A A B / 3 i c b V D L S g M x F M 3 U V 6 2 v U c G N m 2 A p u C o T 8 b U Q K b h x W c E + o B 2 G T J p p Q z O T I c k I Z Z y F v + L G h S J u / Q 1 3 / o 2 Z d h b a e u D C 4 Z x 7 y c n x Y 8 6 U d p x v q 7 S 0 v L K 6 V l 6 v b G x u b e / Y u 3 t t J R J J a I s I L m T X x 4 p y F t G W Z p r T b i w p D n 1 O O / 7 4 J v c 7 D 1 Q q FIG. 1.The crystal structure of Cr2O3 showing the primitive rhombohedral unit cell, with the two AFM domains, the 'out-pointing' domain (a) and the 'in-pointing' domain (b).The goal of this work is to determine the absolute signs of the components of α for the two individual domains."The hexagonal setting, which we use in our experimental discussion, is shown in (c) for the in-pointing magnetic domain.Note that ⟨001⟩ hex ∥ ⟨111⟩ rhomb .

FIG. 2 .
FIG. 2. Temperature dependence of the parallel α ∥ and perpendicular α ⊥ ME responses in Cr2O3, extracted from Ref. [4].α is given in dimensionless units (by multiplying by the speed of light) multiplied by 10 −4 (left-hand y axis) and SI ps/m units (right-hand y axis). f f

1 FIG. 4 .
FIG. 4. Plan view of the horizontal scattering plane of the reciprocal space map of Cr2O3, with the crystal b axis (a) along the +z direction quoted in [13-15] and (b) -z directions, respectively.Here the filled circles indicate the allowed reflections, with the size of the circle proportional to the neutron scattering cross-section, whereas the cross (×) denotes forbidden reflections.(a), The arrows indicate the reciprocal space location of the observed reflections in Refs.[13-15], of which many are forbidden by the R 3c space group of Cr2O3.(b), Instead, if the crystal b axis were actually along the -z direction, then the observed reflections denoted by the arrows can be accounted for.
FIG. 3. Induced net magnetic moment per rhombohedral unit cell parallel to an applied electric field oriented perpendicular to the easy axis, as a function of electric field strength for the out-pointing domain.The three lines show the response calculated from literature α ⊥ values.Markers indicate our results using four different DFT codes.* Results obtained at T = 240 K.

TABLE III .
Comparison between the measured (P obs.

TABLE IV .
Comparison between the measured (P obs.