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Kota Hasegawa, [Takao Shimizu](https://orcid.org/0000-0001-9508-7601), [Naoki Ohashi](https://orcid.org/0000-0002-4011-0031)

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[Lattice deformation and phase transition of aluminum nitride studied by density functional theory calculations](https://mdr.nims.go.jp/datasets/85ff19fe-deb9-421e-a066-51d070ef706f)

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Lattice deformation and phase transition of aluminum nitride studied by density functional theory calculationsFULL PAPERLattice deformation and phase transition of aluminum nitride studiedby density functional theory calculationsKota Hasegawa1,2, Takao Shimizu2,3 and Naoki Ohashi1,2,4,³1Interdisciplinary Graduate school of Engineering Sciences, Kyushu University,6–1 Kasuga-koen, Kasuga, Fukuoka 816–8580, Japan2National Institute for Materials Science, 1–1 Namiki, Tsukuba, Ibaraki 305–0044, Japan3JST-PRESTO, Japan Science and Technology Agency, 4–1–8 Honcho, Kawaguchi, Saitama 332–0012, Japan4Materials Research Center for Element Strategy (MCES), Tokyo Institute of Technology,4259 Nagatsuta, Midori-ku, Yokohama 226–8503, JapanThe density functional theory (DFT) was employed to understand the ferroelectric behaviors of wurtzite (WZ)-type aluminum nitride (AlN). To explain the decrease in the coercive field (Ec) due to lattice deformation, thetotal energy and enthalpy of the strained WZ phase were compared to those of the non-polar (NP) phase, whichacted as a transition state during polarity switching. The shrinkage of the c-axis length and elongation of the a-axis length were favorable for reducing Ec. In addition, the calculated residual stress in the transient NP phasewas as high as 30GPa, suggesting that such a high residual stress may be related to the polarity switchingbehavior under a very high electric field.©2022 The Ceramic Society of Japan. All rights reserved.Key-words : Aluminum nitride, Ferroelectricity, Density functional theory, Lattice deformation[Received December 31, 2021; Accepted March 14, 2022]1. IntroductionWurtzite-type (WZ-type) nitrides, that is, aluminumnitride (AlN), gallium nitride (GaN), indium nitride (InN),and their alloys, have been extensively developed asmaterials for optoelectronic devices.1) Furthermore, nitridesemiconductors are used in switching devices for electricpower management because they possess a very highdielectric strength due to their relatively wide electronenergy bandgap.2) Consequently, most previous studies onWZ-type semiconductors have focused on charge carrierinjection and carrier transport behaviors that provide thebasis for efficient electronics with low energy consump-tion. On the other hand, these WZ-type nitrides are alsoattracting increased attention for their piezoelectricity,because they belong to the non-centrosymmetric P63mcspace group. For instance, WZ-type aluminum-scandiumnitride [(Al1¹xScx)N] has been used in electromechanicalcoupling devices.3),4) In particular, WZ-type (Al1¹xScx)Nhas been employed as a piezoelectric material in film bulkacoustic resonators for high-frequency filter applications intelecommunication technologies5) because of its relativelyhigh piezoelectric performance. However, the spontaneouspolarization of WZ-type nitrides has emerged as a criticalissue not only for piezoelectric applications but also forelectronic and optoelectronic applications, because thecharge transport and distribution are significantly altered inthese semiconductor devices due to polarization. Hence,many experimental and theoretical studies relating to thepolarization and polar surfaces/interfaces of WZ-typecompounds have been performed.6) In particular, the sur-face termination of epitaxial WZ-type semiconductorfilms, which can either be anion(Xq¹)-terminated or metal-cation(Mq+)-terminated, is regarded as a key issue in thefabrication of electronic devices.Here, it must also be noted that many metal nitrides,such as scandium nitride (ScN), possess cubic rather thanpolar symmetry, i.e., they exist in a rock-salt type struc-ture. Consequently, a miscibility gap exists in some nitridealloy systems, such as the pseudo-binary system formedby WZ-type AlN and rock-salt-type ScN.7) With the aimof discovering high-performance piezoelectric materials innitride systems, many studies have been conducted on(Al1¹xYx)N,8) (Al1¹xBx)N,8) (Al1¹xYbx)N,9) (Al1¹xCrx)N,10)(Al1¹xTax)N,11) (Al1¹xErx)N,12) [Al1¹x(Mg1/2Zr1/2)x]N,13)[Al1¹x(Mg1/2Hf1/2)x]N,14) [Al1¹x(Mg2/3Nb1/3)x]N,15) and(Ga1¹xScx)N16) systems, focusing on the stability of theWZ form and piezoelectric performance.Investigations on WZ-type compounds entered a newera when Fichtner et al.17) discovered that WZ-type(Al,Sc)N exhibited ferroelectricity. It was found that the³ Corresponding author: N. Ohashi; E-mail: ohashi.Naoki@nims.go.jp‡ Preface for this article: DOI http://doi.org/10.2109/jcersj2.130.P7-1Journal of the Ceramic Society of Japan 130 [7] 452-457 2022DOI http://doi.org/10.2109/jcersj2.21190 JCS-Japan©2022 The Ceramic Society of Japan452This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.http://doi.org/10.2109/jcersj2.130.P7-1http://doi.org/10.2109/jcersj2.130.P7-1http://doi.org/10.2109/jcersj2.21190https://creativecommons.org/licenses/by/4.0/N-terminated ð000�1Þ surface transforms into the (Al,Sc)-terminated (0001) surface and vice versa through theapplication of a high electric field. Subsequently, a similarferroelectric behavior was reported in other WZ-typecompounds, including (Al1¹xBx)N,18),19) (Zn1¹xMgx)O,20)(Ga1¹xScx)N,21),22) and AlN.19) The ferroelectric propertiesof WZ-type (Al1¹xScx)N are characterized by an extremelyhigh remnant polarization (Pr = 80­120¯C/cm2) and anunusually high coercive field (Ec = 3­5MV/cm).17),23)The Ec of (Al1¹xScx)N is one order of magnitude higherthan those of PbZr1¹xTixO3 (PZT) and BaTiO3, which isunfavorable for practical applications. The properties ofWZ-type (Al1¹xScx)N strongly depend on its composition.It has been reported that increasing the Sc concentration(x) in (Al1¹xScx)N can decrease Ec but leads to a degra-dation in the dielectric properties, such as the breakdownvoltage.24) Therefore, reducing Ec without degrading thedielectric properties is essential for the development ofWZ-type ferroelectric materials with optimal properties.The mechanism of ferroelectricity in WZ-type com-pounds at the atomic scale has been investigated mostlythrough theoretical calculations, particularly by the densityfunctional theory (DFT).25)­31) The most likely polarizationreversal motion in WZ-type compounds under an electricfield is the movement of Xq¹ and Mq+ in opposite direc-tions along the c-axis. As shown in Fig. 1, the polar WZ-type structure [Figs. 1(a) and 1(c)] transforms to a non-polar (NP) structure with the P63/mmc space group[Fig. 1(b)] as a transition state during polarization rever-sal.25)­27) Under this assumption, the transition state inFig. 1(b) is the saddle point in the energy diagram forpolarization reversal. Therefore, the enthalpy differencebetween the WZ and NP structures corresponds to the acti-vation energy for the polarity change of the WZ-typelattice.As noted by Moriwake et al., the M­X distance in theNP structure is a highly important parameter for the occur-rence of polarization reversal.27) Yazawa et al. concludedfrom their experimental study that the Ec of ferroelectric(Al1¹xScx)N films is strongly correlated with their latticeparameters (c/a ratio).32) Hence, it is essential to correlatethe lattice constants to the ferroelectricity in WZ-typestructures. In experimental investigations, the most typicalapproach for changing lattice parameters is through ele-mental substitution, e.g., the substitution of Sc for Al inAlN. While such a substitution effectively modifies the lat-tice parameters, it also simultaneously changes the natureof the chemical bonding (e.g. covalency or ionicity) of theM­X bonds because of the different electronegativities andionicities of Al and Sc. Therefore, the contributions of thelattice parameters and characteristics of the constituentelements to the ferroelectric behavior cannot be distin-guished by experimental measurements.In this context, visualization of the deformation behav-ior of WZ-type compounds is necessary to understand theferroelectricity in WZ-type compounds. Therefore, in thisstudy, DFT calculations were employed to investigate thestructural features of AlN crystallized in the WZ phasewith the P63/mmc space group and the NP phase with theP63/mmc space group. Briefly, we investigated a stronglystrained pure AlN lattice in order to separate the effectsof elemental substitution and lattice deformation. Riahet al.33) investigated the piezoelectricity and ferroelectric-ity of WZ-type nitrides in the form of heteroepitaxial thinfilms, and reported that the lattice mismatch between thesubstrate and film gives rise to a residual strain in thecrystalline lattice. Their results suggested that the varia-tions in the lattice parameters were induced by the latticemismatch between the films and substrates. These findingssupport our idea that investigating the lattice deformationis crucial for understanding the piezoelectricity and ferro-electricity in WZ-type compounds. For this purpose, thetotal energies of the strained WZ and NP phases werecalculated to explain the possible reduction in Ec in AlN.2. Calculation methodThe structural stability of AlN and the possible trans-formation from the WZ type to NP type were examined byDFT calculations carried out using the CASTEP simula-tion package.34) Conventional unit cells with two Al andtwo N atoms were employed as the model systems to studythe effect of deformation. Norm-conserving pseudopoten-tials were used, and the Perdew-Burke-Ernzerhof gener-alized gradient approximation optimized for solids (abbre-viated as PBEsol)35) was adopted as the exchange­correlation functional for self-consistent field (SCF) calcu-lations. The cutoff energy of the plane-wave basis was setto 1250 eV. The Monkhorst­Pack grid mesh36) was usedfor k-point sampling, and the number of mesh points wasvaried from model to model because the lattice parameterswere restricted in the strained lattice models. Typically, the5 © 5 © 3 mesh was used to sample the Brillouin zone ofthe conventional WZ-type AlN lattice. In SCF calculations,the ensemble density functional theory37) was employedinstead of the conventional density mixing scheme,38) anda considerable number of empty bands were included toFig. 1. Schematic of polarization reversal in a WZ-type crys-tal through a transient NP structure belonging to P63/mmcsymmetry.Journal of the Ceramic Society of Japan 130 [7] 452-457 2022 JCS-Japan453ensure convergence. The convergence tolerance for elec-tronic energy minimization was set to 5.0 © 10¹7 eV/atom.The lattice parameters and atomic positions were optimizedwith some constraints, namely, a0 = b0, ¡ = ¢ = 90°, and£ = 120°, in order to maintain a hexagonal unit cell. Thetwo-point steepest descent algorithm39) was employed forefficient convergence under these structural constraints.The following steps were used to study the polarizationreversal in WZ-type AlN by DFT calculations. First, therelaxation of the WZ phase under axial stress and externalpressure along the c-axis of the WZ-type unit cell (P33)was carried out. Here, the a0 and c0 lattice parameters andatomic positions were fully relaxed under the assumedpressure to minimize the total energy under the given pres-sure conditions, and the enthalpy of the relaxed lattice wascalculated. This step was performed to evaluate possiblelattice deformations, such as the deformation due to thelattice mismatch between AlN thin films and substratematerials in some heteroepitaxial systems. Second, theenthalpy in the virtual NP phase was calculated. The NPphase is likely to be a transition state for the polarizationreversal of the WZ phase; hence, the enthalpy differencebetween the transition state NP phase and original WZphase was considered to be the activation energy of polar-ization reversal. Here, it was assumed that the a-axislength remained unchanged during polarization reversaland the NP phase belonged to the P63/mmc space group.Hence, only the c-axis length of the NP phase was opti-mized to calculate the enthalpy, because the Al and Nfractional atomic positions were fixed under the symmetryconstraints of the P63/mmc space group. The a0 of thevirtual NP phase was varied by considering the values ofthe strained WZ phase shown in Fig. 2. The NP phasewith the same lattice parameters as the strained WZ phaseshown in Fig. 2 was also used to elucidate the effect of thedeformation on the enthalpy. Both a0 and c0 were relaxedonly when the structure of the unstressed NP phase wasoptimized. Based on the results of the above two steps, theactivation energy of polarization reversal was determinedand correlated with Ec. For all structural relaxation calcu-lations, the structural optimization convergence tolerancevalues were set as follows: electron energy of 5.0 © 10¹6eV/atom, force of 0.1 eV/nm, and atomic displacement of5.0 © 10¹5 nm. The convergence tolerance for stress wasset to 0.02GPa for the relaxation of the lattice parameters,but was not applied for the calculations on the NP phase,where a0 and c0 were the same as those of the WZ phase.3. Results and discussionFigure 2 shows the calculated lattice constants of AlNas a function of P33. Here, a0, c0, and all atomic positionswere relaxed to determine the most stable structure underuniaxial stress. Because no external pressure is appliedalong the a and b axes (P11 = 0 and P22 = 0), the latticeparameter a0 increases while c0 decreases with increasingexternal pressure. Moreover, a0 and c0 showed a jumpbetween P33 = 16 and 17GPa due to the transition in thelattice symmetry between the WZ phase at the lower P33and NP phase at the higher P33. The c-axis fractionalatomic positions of N are the same as those of Al for P33 =17GPa, but are different for P33 = 16GPa. These resultsindicate that the WZ/NP phase transition occurs betweenP33 = 16 and 17GPa under the present simulation condi-tions. The lattice parameters of the unstressed NP phase atP33 = 0 (a0 = 3.326¡ and c0 = 4.078¡) were also deter-mined. If the in-plane lattice mismatch between the NPand WZ phases is defined as the difference in the calcu-lated a0 parameters between the unstrained NP phase (a0 =3.326¡) and WZ phase (a0 = 3.1208¡), a lattice mis-match of 6.5% is obtained. It is quite large and is indica-tive of the presence of a larger mismatch in the materialduring polarization reversal: the presence of the inter-mediate state, such as nucleation of the relaxed NP phasein relaxed WZ-lattice, may result in the degradation of thematerial because of the very large lattice mismatch. There-fore, we discuss the possible lattice deformation involv-ing WZ and NP structures to reduce Ec for polarizationreversal.The results presented in Fig. 2 are useful for consid-eration of the compatibility between experimental andtheoretical investigations. Experimental application of auniaxial external pressure to the WZ phase along the c-axisin order to reproduce the calculated results is in progress inthe authors’ group. On the other hand, the calculated totalenergy and enthalpy of the NP phase must be carefullyconsidered, because charge neutrality is established inevery single [AlN] layer in the NP phase, which is similarto the case of hexagonal boron nitride (h-BN). As dis-cussed in the literature,40) the structural stability in h-BNcannot be investigated by conventional DFT calculationssince the van der Waals interaction needs to be consideredthrough the inclusion of the dispersion correction in theDFT functional in order to determine the physical proper-ties of h-BN by electronic structure calculations. There-fore, the current results in Fig. 2 indicate a general trend inthe stability of WZ and NP phases, but further detailedcalculations must be performed, including the dispersioncorrection. At this stage, we only discuss some qualitativetrends given by the conventional DFT calculations thatExternal uniaxial pressure (P33) / GPa0 2015105a-axis lattice constant (a0 ) / Å3.13.23.33.43.65.24.04.84.25.03.84.44.6c-axis lattice constant (c 0) / ÅFig. 2. Structural parameters of AlN crystallized in a WZ-typestructure, P63mc, or a NP structure, P63/mmc, under a uniaxialpressure along the c-axis. The open and closed symbols indicatethe resultant structure with polar and NP structures, respectively.Hasegawa et al.: Lattice deformation and phase transition of aluminum nitride studied by density functional theory calculationsJCS-Japan454allow us to understand the ferroelectric properties of WZ-type phases.Figure 3 shows the lattice deformation behavior whena0 was fixed at 3.1208¡ (unstressed WZ-type AlN) andc0 was varied. The unstressed c0 was calculated to be4.9969¡, and the shrinkage of c0 was considered. Thecalculated enthalpy of WZ-type AlN increased monoton-ically with decreasing c0 due to elastic compression. Bycontrast, the calculated enthalpy of the NP phase had aminimum value when c0 was 4.40¡. Furthermore, theenthalpies of the NP and WZ phases were approximatelythe same for a c0 of 4.20¡. For polarization reversal of theWZ-type phase, the difference between the enthalpies ofthe unstrained WZ phase and NP phase for c0 = 4.40¡corresponds to the activation energy. Our calculated valueof the activation energy (0.82 eV) is quite close to thepreviously reported results obtained when a0 was fixed atthe value calculated for the unstrained WZ phase duringpolarization reversal.27)The results shown in Fig. 3 imply that polarization isreversed in the WZ phase with the assumed lattice param-eters (a0 = 3.1208¡ and c0 = 4.9969¡) through the NPphase with the same a0 (a0 = 3.1208¡ and c0 = 4.40¡)as that of the transition state. Based on these calculatedresults and assumed polarization reversal mechanism, achange in the film thickness is expected during the polar-ization reversal motion under an external electric field,and the estimated thickness change is approximately 12%.Considering the experimental resolution of the availablecharacterization tools, such as optical interferometers andX-ray diffractometers, this model should be verified byhigh-speed measurements.Figure 3 also shows the stress appearing in the WZphase under the assumed lattice parameters values. It isinteresting to note that the WZ-type phase (a0 = 3.1208¡and c0 = 4.30¡) emerges under isostatic pressure condi-tions at approximately 30GPa, as the values of the in-plane and out-of-plane stresses were nearly the same.Furthermore, when an external isostatic pressure of 30GPais applied to the WZ phase, the activation energy for polar-ization reversal is reduced to approximately 0.1 eV. This isalmost one order of magnitude lower than the activationenergy for the unstressed WZ phase. It is challenging toapply an electric field to a film maintained under an isosta-tic pressure of 30GPa. However, according to our model,a polarity change is highly likely to occur under theseconditions.Next, the variation in the c0 of the NP phase with differ-ent a0 was considered, as summarized in Fig. 4. Here, thea0 of the NP phase was varied within an assumed range,and c0 was relaxed by DFT calculations to minimize thetotal energy. These calculations were performed to obtaininsights into ferroelectricity in heteroepitaxial structures,where a0 is strongly constrained due to lattice mismatch.The lattice parameters of the WZ phase shown in Fig. 2are plotted again in Fig. 4 for reference. The relaxed c0 ofthe NP phase is much smaller than that of the WZ phasewith the same a0 because of planar atomic coordination. InFig. 4, the c0 of the NP phase decreases with increasing a0.As discussed for the results presented in Fig. 3, the pro-posed polarization reversal motion illustrated in Fig. 1may reduce the film thickness under the transition state.The magnitude of the thickness reduction during polar-ization reversal motion appears to be smaller in the filmwith a relatively large a0.Figure 5(a) shows the enthalpy of the WZ and NPphases, with the lattice parameters shown in Fig. 4. Withincreasing a0, the enthalpy of the WZ phase increasesbecause of the deformation caused by a reduced c0. Thetotal energy values of the WZ phase with these assumedlattice parameters are provided in the supporting informa-tion (Fig. S1), and show that the WZ phase with a0 =3.1208¡ and c0 = 4.9969¡ is the equilibrium structure.On the other hand, the enthalpy of the NP phase decreaseswith increasing a0 and decreasing c0. Interestingly, theenthalpy of the WZ phase exceeds that of the NP phasewhen a0 exceeds 3.14¡. This implies that AlN willcrystallize in the NP form when grown heteroepitaxially4.2 4.4 4.6 4.8 5.0-944.8-944.6-944.4-944.2-944.0-943.8-943.6-943.4-943.2Lattice parameter (c0) / ÅEnthalpy / eV01020304050607080Stress / GPaWZ-typeNP-typeIn-planeOut-of-plane0.82 eVFig. 3. Structural parameters of AlN crystallized in a WZ-typestructure, P63mc, or a NP structure, P63/mmc, when the a-axislength is fixed at 3.1208¡. The left axis shows the enthalpy inthe crystal, and the right axis shows the stress in the calculatedWZ-type lattice.3.124.24.44.64.85.0Lattice parameter (a0) / Å3.14 3.16 3.18Lattice parameter (c0) / ÅWZ-typeNP-typeFig. 4. Calculated c-axis length of AlN crystallized in a non-polar structure type (NP type) in the P63/mmc symmetry as afunction of the a-axis length. The lattice parameters of the WZ-type structure under axial stress in Fig. 2 are plotted again as areference.Journal of the Ceramic Society of Japan 130 [7] 452-457 2022 JCS-Japan455on a certain substrate, to restrict its in-plane lattice param-eter to greater than 3.14¡. However, crystallization of theNP phase may not be experimentally observed because ofthe extremely high internal stress, as discussed below. Wemust also discuss the difference between the calculatedresults shown in Figs. 2 and 5(a). The results shown inFig. 2 indicate that a WZ/NP transition occurs whena0 µ 3.19¡, while Fig. 5(a) indicates that the transitionoccurs at a0 µ 3.14¡. This inconsistency arises from thedifference in the constraints applied for the calculation ofthe NP phase and therefore, is acceptable.The results of the enthalpy calculation shown in Fig. 2indicate a decrease in c0 for a fixed a0. The results inFig. 5(a), where a0 is expanded, are compared to theresults presented in Fig. 5(b). Here, we assume that a0remains unchanged during polarization reversal, implyingthat Ec should be correlated with the enthalpy differencebetween the NP and WZ phases with the same a0. The en-thalpy difference shown in Fig. 5(b) was evaluated underthis assumption and plotted versus the c0/a0 ratio of theassumed WZ phase. The plot indicates that the decrease inthe c0/a0 ratio results in a reduction of the enthalpy differ-ence. In particular, when the c0/a0 ratio decreases, theexpansion of a0 leads to a steep decrease in the enthalpydifference.Finally, the residual stress was studied in the NP phaseto consider whether the NP/WZ transition is realistic.Figure 6 shows the calculated stress as a function of a0,where two major constraints (symmetry of the P63/mmcspace group and a fixed a0) were applied. Convergedresults were obtained with a very high residual stress. Asshown in Fig. 6, a compressive stress is generated to limita0, while a tensile stress emerges to maintain the expand-ed c0; the magnitude of both stresses exceeds 10GPa.According to the literature,41),42) the tensile and compres-sive strengths of AlN are approximately 0.1 and 1GPa,respectively, suggesting that the AlN film in Fig. 6 issubject to a very high residual stress. In experiments, AlNthin films often break due to dielectric breakdown underthe bias of a high electric field strength. The mechanismfor the breakdown behavior is currently unclear. The cal-culation results shown in Fig. 6 suggest that the very highstress during the NP/WZ transformation may be an originof the observed breakdown behavior. When polarizationreversal is triggered under a high electric field, the film isdamaged due to the high stress in the transient NP phase.4. ConclusionsThe ferroelectric behaviors of WZ-type compoundswere investigated by DFT calculations. It was confirmedthat the shrinkage of c0 and expansion of a0 are favorablefor reducing Ec, based on the assumption that the NP phasewith symmetries of the P63/mmc space group appeared asa transition state during polarization reversal. The activa-tion energy for polarization reversal was 0.8 eV, which issimilar to the value obtained in previous studies.27) Never-theless, the lattice strain for decreasing c0 and increasinga0 appears to be sufficient to lower Ec. In addition, theresidual stress in the NP phase during polarization reversalwas found to be as high as 30GPa. This study indicatesthat such a high residual stress may be essential for thedevelopment of electronic and ferroelectric WZ-typecompounds.Conflict of interests The authors declare that there is noconflict of interest.Part of this study was performed at the Tokodai Institute forElemental Strategy (TIES), supported by the grant numberJPMXP0112101001. This work was partly supported by theJST and PRESTO under grant number JPMJPR20B3.1.30.0Enthalpy difference / eVc0/a03.12Lattice parameter (a0) / Å3.14 3.16 3.18-944.5-944.0-943.5-943.0-942.5Enthalpy / eV1.4 1.5 1.6-0.5-1.01.00.5(a)(b)WZ-phaseNP-phaseExpansion of a-axisShrinkage of c-axis Fig. 5. (a) Enthalpy in AlN crystallized in WZ and NP types asa function of the assumed a-axis length, and (b) difference inenthalpy between WZ-type and NP-type AlN as a function of thelattice parameter ratio of c0/a0.Fig. 6. Calculated stress in the NP phase as a function of the a-axis length.Hasegawa et al.: Lattice deformation and phase transition of aluminum nitride studied by density functional theory calculationsJCS-Japan456References1) D. Jena, R. Page, J. Casamento, P. Dang, J. Singhal, Z.Zhang, J. Wright, G. Khalsa, Y. Cho and H. G. Xing,Jpn. J. Appl. Phys., 58, SC0801 (2019).2) I. Abid, R. Kabouche, C. Bougerol, J. Pernot, C.Masante, R. Comyn, Y. Cordier and F. 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