Detecting quadrupole: a hidden source of magnetic anisotropy for Manganese alloys

Mn-based alloys exhibit unique properties in the spintronics materials possessing perpendicular magnetic anisotropy (PMA) beyond the Fe and Co-based alloys. It is desired to figure out the quantum physics of PMA inherent to Mn-based alloys, which have never been reported. Here, the origin of PMA in ferrimagnetic Mn$_{3-{\delta}}$Ga ordered alloys is investigated to resolve antiparallel-coupled Mn sites using x-ray magnetic circular and linear dichroism (XMCD/XMLD) and a first-principles calculation. We found that the contribution of orbital magnetic moments in PMA is small from XMCD and that the finite quadrupole-like orbital distortion through spin-flipped electron hopping is dominant from XMLD and theoretical calculations. These findings suggest that the spin-flipped orbital quadrupole formations originate from the PMA in Mn$_{3-{\delta}}$Ga and bring the paradigm shift in the researches of PMA materials using x-ray magnetic spectroscopies.


INTRODUCTION
Perpendicular magnetic anisotropy (PMA) is desired for the development of high-density magnetic storage technologies. Thermal stability of ultrahigh density magnetic devices is required to overcome the superparamagnetic limit 1−3 . Recently, research interests using PMA films have focused on not only magnetic tunnel junctions 4−7 toward the realization of spin-transfer switching magneto-resistive random-access memories but also antiferromagnetic or ferrimagnetic devices 8,9 . To design PMA materials, heavy-metal elements that possess large spin-orbit coupling are often utilized through the interplay between the spins in 3d transition-metals (TMs) and 4d or 5d TMs. The design of PMA materials without using the heavy-metal elements is an important subject in future spintronics researches. Recent progress has focused on the interfacial PMA in CoFeB/MgO 10 or Fe/MgO 11,12 . However, a high PMA of over the order of MJ/m 3 with a large coercive field is needed to maintain the magnetic directions during device operation 13 . Therefore, the materials using high PMA constants and without using heavy-metal atoms are strongly desired.
Mn-Ga binary alloys are a candidate that could overcome these issues. Mn 3−δ Ga alloys satisfy the conditions of high spin polarization, low saturation magnetization, and low magnetic damping constants 14−18 . Tetragonal Mn 3−δ Ga alloys are widely recognized as hard magnets, which exhibit high PMA, ferromagnetic, or ferrimagnetic properties depending on the Mn composition 15 . Two kinds of Mn sites, which couple antiferromagnetically, consist of Mn 3−δ Ga with the D0 22 -type ordering. Meanwhile, the L1 0 -type Mn 1 Ga ordered alloy possesses a single Mn site. These specific crystalline structures provide the elongated c-axis direction, which induces the anisotropic chemical bonding, resulting in the anisotropy of electron occupancies in 3d states and charge distribution. There are many reports investigating the electronic and magnetic structures of Mn 3−δ Ga alloys to clarify the origin of large PMA and coercive field 19−21 . To investigate the mechanism of PMA and large coercive fields in Mn 3−δ Ga, site-specific magnetic properties must be investigated explicitly.
X-ray magnetic circular/ linear dichroism (XMCD/ XMLD) may be a powerful tool to study the orbital magnetic moments and magnetic dipole moments of higher order term of spin magnetic moments 22,23 . However, the difficulty in the deconvolution of two kinds of Mn sites prevents site-selected detailed investigations. Within the magneto-optical spin sum rule, using the formulation proposed by C.T. Chen et al. 24 , the orbital magnetic moments are expressed as proportional to r/q, where q and r represent the integral of the x-ray absorption spectra (XAS) and XMCD spectra, respectively, for both L 2 and L 3 edges. In the cases of two existing components, the orbital moments are not obtained from the whole integrals of spectra; by using each component r 1 , r 2 , q 1 , and q 2 , the value of (r 1 /q 1 ) + (r 2 /q 2 ) should be the average value. The value of (r 1 + r 2 )/(q 1 + q 2 ) does not make sense as an average in the case of core-level atomic excitation, leading to the wrong value in the XMCD analysis. As a typical example, for the mixed valence compound CoFe 2 O 4 , the Fe 3+ and Fe 2+ sites can be deconvoluted by the ligand-field theory approximation 25 . However, the deconvolution of featureless line shapes in a metallic Mn 3−δ Ga case is difficult by comparison with the theoretical calculations. To detect the site-specific anti-parallel-coupled two Mn sites, systematic investigations using Mn 3−δ Ga of δ = 0, 1, and 2 provide the information of site-specific detections. In previous reports, although Rode 26 , which enables discussion of the electronic structures of Mn 3−δ Ga in δ = 0, 1, and 2. We adopted this growth technique and performed XMCD. By contrast, the XMLD in Mn L-edges enables detection of the element of quadrupole tensor Q zz by adopting the XMLD sum rule 27 . Although many reports of XMLD for in-plane magnetic easy axis cases exist, perpendicularly magnetized cases are attempted firstly in Mn 3−δ Ga using perpendicular remnant magnetization states.
The first-principles calculations based on the density functional theory (DFT) suggest the unique band structures in the Mn sites of mixed spin-up and -down bands at the Fermi level (E F ), which allow the spin transition between up and down spin states, resulting in the stabilization of the PMA 28−30 . Usually, the PMA originates from the anisotropy of orbital magnetic moments in the large exchange-split cases, such as Fe and Co, using second-order perturbation for spin-orbit interaction 31,32 . Meanwhile, for Mn compounds, the contribution from orbital moment anisotropy for PMA is smaller than the spin-flipped contribution to PMA 33−35 . However, this picture has not been guaranteed completely from an experimental viewpoint until now.
In this study, we performed the deconvolution of each Mn site using the systematic XMCD and XMLD measurements for different Mn contents in Mn 3−δ Ga. We discuss the site-specific spin and orbital magnetic moments with magnetic dipoleterm, which corresponds to electric quadrupoles. These are deduced from the angular-dependent XMCD and XMLD and compared with the DFT calculations to understand the PMA microscopically. To deconvolute the MnI and MnII sites in the XMCD spectra, we performed the subtraction of XMCD between Mn 1 Ga and Mn 3 Ga. Figure   Mn 1 Ga, if m Tz is negative, resulting in Q zz > 0 in the notation of m Tz = −Q zz ·m s , which exhibits the prolate shape of the spin density distribution; the second term favors PMA because of the different sign for the contribution of orbital moment anisotropy in the first term. Since 7m Tz is estimated to be in the order of 0.1 µ B from angular-dependent XMCD between surface normal and magic angle cases, Q zz is less than 0.01, resulting that the orbital polarization of less than 1% contributes to stabilize PMA. In this case, the contribution of the second term in eq.(1) is one order larger than the orbital term, which is essential for explaining the PMA of Mn 3−δ Ga. Third, in a previous study 19 , quite small ∆m orb and negligible m Tz were reported for Mn 2 Ga and Mn 3 Ga. Their detailed investigation claims that ∆m orb of 0.02 µ B in MnI site contributes to PMA and MnII site has the opposite sign.
These are qualitatively consistent with our results. The difference might be derived from the sample growth conditions and experimental setup. Fourth, the reason why H c in Mn 1 Ga is small can be explained by the L1 0 -type structure, due to the stacking of the Mn and Ga layers alternately, which weakened the exchange coupling between the Mn layers. Finally, we comment on the XMCD of the Ga L-edges. This also exhibits the same sign as the MnI component, suggesting that the induced moments in the Ga sites were derived from the MnI component ( Fig. S1), which was substituted by the MnII for Mn 2 Ga and Mn 3 Ga.
To determine the effect of m Tz , we performed XMLD measurements. Figure 3 shows the difference between the L 2 x and L 2 z terms through the spin-flipped transitions between the occupied (o) to unoccupied (u) states is significant for the gain of the PMA energy.
The matrix elements of u↑|L x 2 |o↓ were enhanced in the spin-flipped transition between yz and z 2 , and those of u|L z 2 |o were enhanced in the spin-conserved case between xy and x 2 − y 2 28 . These transitions favor the magnetic dipole moments of prolate shapes ( Q zz = 3L 2 z −L 2 >0) described by the Mn 3d each orbital angular momenta. We emphasize that the signs of ∆m orb and Q zz are opposite, which is essential to stabilize the PMA by the contribution of the second term in Eq. (1). The PMA energy of FePt exhibits around MJ/m 3 and the contribution of the second term in Pt is four times larger than the Fe orbital anisotropy energy 28 . Therefore, MnGa has a specific band structure by crystalline anisotropy elongated to the c-axis and intra-Coulomb interaction in Mn sites to enhance the PMA without using heavy-metal atoms.
In conclusion, we investigated the origin of PMA in Mn 3−δ Ga by decomposing into two kinds of Mn sites for XMCD, XMLD, and the DFT calculation. The contribution of the orbital moment anisotropy in Mn 3 Ga is small and that of the mixing between the Mn 3d up and down states is significant for PMA, resulting in the spin-flipped process through the electron hopping between finite unforbidden orbital symmetries in the 3d states through the quadratic contribution. Composition dependence reveals that the orbital magnetic moments of the two antiparallel-coupled components in Mn sites were too small to explain the PMA.
These results suggest that the quadrupole-like spin-flipped states through the anisotropic

XMCD and XMLD measurements
The XMCD and XMLD were performed at BL-7A and 16A in the Photon Factory at the High-Energy Accelerator Research Organization (KEK). For the XMCD measurements, the photon helicity was fixed, and a magnetic field of ±1.2 T was applied parallel to the incident polarized soft X-ray beam, defined as µ + and µ − spectra. The total electron yield mode was adopted, and all measurements were performed at room temperature. The XAS and XMCD measurement geometries were set to normal incidence, so that both the photon helicity and the magnetic field were parallel and normal to the surface, enabling measurement of the absorption processes involving the normal components of the spin and orbital angular momenta 36 . In the XMLD measurements, the remnant states magnetized to PMA were adopted. For grazing incident measurements in XMLD and XLD, the angle between incident beam and sample surface normal was kept at 60 • tilting as shown in the inset of Fig. 3.  constructed the scenario of quadrupole physics. All authors discussed the results and wrote the manuscript.

COMPETING INTERESTS
The authors declare no competing financial interests.

DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.    The inset shows an illustration of the XMLD measurement geometry. The angle between sample surface normal and incident beam is set to 60 • . All measurements were performed at RT.