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[Takao Tsumuraya](https://orcid.org/0000-0001-9063-9278), [Ikumu Watanabe](https://orcid.org/0000-0002-7693-1675), [Takahiro Sawaguchi](https://orcid.org/0000-0002-9405-002X)

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[Origin of phase stability in Fe with long-period stacking order as an intermediate phase in cyclic γ−ε martensitic transformation](https://mdr.nims.go.jp/datasets/66e4ce03-a040-40f9-8136-1015450d7669)

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Origin of phase stability in Fe with long-period stacking order as an intermediate phase in cyclic $\gamma$-$\epsilon$ martensitic transformationPHYSICAL REVIEW RESEARCH 3, 033215 (2021)Origin of phase stability in Fe with long-period stacking order as an intermediatephase in cyclic γ-ε martensitic transformationTakao Tsumuraya ,1,2,* Ikumu Watanabe ,3 and Takahiro Sawaguchi3,†1International Center for Young Scientists, National Institute for Materials Science, Tsukuba 305-0044, Japan2Priority Organization for Innovation and Excellence, Kumamoto University, Kumamoto 860-8555, Japan3Research Center for Structural Materials, National Institute for Materials Science, Tsukuba 305-0047, Japan(Received 14 October 2020; revised 10 August 2021; accepted 13 August 2021; published 7 September 2021)A class of Fe-Mn-Si–based alloys exhibit a reversible martensitic transformation between the γ phase witha face-centered cubic (fcc) structure and an ε phase with a hexagonal close-packed (hcp) structure. Duringthe deformation-induced γ -ε transformation, we identified a phase that is different from the ε phase. In thisphase, the electron diffraction spots are located at the 1/3 positions that correspond to the {0002} plane ofthe ε (hcp) phase with 2H structure, which suggests long-period stacking order (LPSO). To understand thestacking pattern and explore the possible existence of an LPSO phase as an intermediate between the γ and εphases, the phase stability of various structural polytypes of iron was examined using first-principles calculationswith a spin-polarized form of the generalized gradient approximation in density functional theory. We foundthat an antiferromagnetic ordered 6H2 structure is the most stable among the candidate LPSO structures and isenergetically closest to the ε phase, which suggests that the observed LPSO-like phase adopts the 6H2 structure.Furthermore, we determined that the phase stability can be attributed to the valley depth in the density of states,close to the Fermi level.DOI: 10.1103/PhysRevResearch.3.033215I. INTRODUCTIONAustenitic steel is an industrial structural material witha long history. Ever since wear-resistant Fe-Mn-C steel wasdeveloped at the end of the 19th century, much attention hasbeen paid to its superior mechanical properties. Among thesealloys, those with 28–32 mass percent (mass %) Mn and 4–7mass % Si are known to demonstrate a shape-memory effect.This effect is governed by a nondiffusive solid-to-solid phasetransformation from γ -austenite with a face-centered-cubic(fcc) structure to ε martensite with a hexagonal close-packed(hcp) structure [1]. Plastic deformation and subsequent shaperecovery upon heating are associated with the forward γ →ε and reverse ε → γ transformations, respectively [2–5].Figure 1 illustrates the atomic displacement during theγ ↔ ε transformation. The (111) planes in the fcc structureare parallel to the basal (0001) planes of the hcp lattice. Theformation of the ε phase from the γ phase is induced bystacking faults bounded by Shockley partial dislocations withan a/√6 shift on the (111) plane. These partial dislocationsoccur every two layers in the pathway from the fcc to thehcp structure [6]. This transformation is one of the notable*tsumu@kumamoto-u.ac.jp†sawaguchi.takahiro@nims.go.jpPublished by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Furtherdistribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.plastic deformation modes in austenite steels. The reversibletransformation between the γ and ε phases occurs duringheating and cooling cycles, and cyclic plastic deformation.Since researchers showed that a dual-phase magnesium-based alloy with a long-period stacking order (LPSO)structure and α-Mg (hcp) exhibited superior mechanical prop-erties and a tensile yield strength of approximately 600 MPa[7], there has been a growth in the development of novelalloys with LPSO structures [8,9]. In the 1960s, Lysak andNikolin discovered a phase in a disordered Fe-Mn-C alloysubjected to heating and cooling cycles of 400 � −196 ◦Cthat was distinct from the ε (hcp) phase [10–13]. This phasewas referred to as the ε′ phase. Many LPSO-like phaseswere later discovered in various Fe-Mn-(Al)-C–based alloys[4,11,14–18]. Although the existence of LPSO phases in Fe-Mn-Si–based alloys has also been verified by thermodynamicmodeling [19], the actual stacking pattern of LPSO phases hasyet to be experimentally identified. In addition, even in pureiron, the phase stability and magnetic properties of the struc-tural polytypes have not yet been studied using first-principlescalculations.In the present work, transmission electron microscopy(TEM) measurements were performed on Fe-Mn-Si–basedalloys subjected to cyclic deformation. Electron diffractionspots at the 1/3 positions that correspond to the {0002} planeof the ε (hcp) phase with 2H structure suggest the exis-tence of an LPSO structure. However, for Fe-Mn-Si–basedalloys under cyclic deformation, the observed LPSO phasewas unstable during room temperature aging (probably dueto sensitivity to temperature variations), and so the structuraland magnetic properties of the phase are still unavailable.2643-1564/2021/3(3)/033215(12) 033215-1 Published by the American Physical Societyhttps://orcid.org/0000-0001-9063-9278https://orcid.org/0000-0002-7693-1675http://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevResearch.3.033215&domain=pdf&date_stamp=2021-09-07https://doi.org/10.1103/PhysRevResearch.3.033215https://creativecommons.org/licenses/by/4.0/TSUMURAYA, WATANABE, AND SAWAGUCHI PHYSICAL REVIEW RESEARCH 3, 033215 (2021)FIG. 1. Atomic displacement and formation of partialdislocation-stacking fault units associated with (a) γ → (b) εmartensitic transformation.Therefore, we aimed to determine the most stable stackingconfiguration of the LPSO structure. The structural and mag-netic phase stability of structural polytypes in pure iron wereinvestigated using first-principles calculations based on den-sity functional theory (DFT) [20,21]. The relative stabilitiesof the fcc and hcp phases were examined by structural opti-mization to understand the possible realization of an LPSOphase as an intermediate phase in the γ -ε transformation.Antiferromagnetic (AFM) order is crucial in the stabiliza-tion of Fe-Mn–based alloy phases with fcc and hcp structures[22–26]. Therefore, we consider all the possible AFM spinstructures with collinear spin order, and compare the totalenergies obtained from non-spin-polarized calculations. Thetotal energies of the candidate LPSO structures are comparedto the more stable structure of hcp Fe at 0 K. We discuss theorigin of structural stabilities based on the difference in thetotal density of states (DOS) near the Fermi level.For the sake of simplicity, pure iron is used as a model forthose Fe-rich Fe-X alloys; the amount of alloying element (X )is relatively low (�25 atom %), and the Fe content is large.The Fe ground state at ambient pressure is widely known tobe ferromagnetic with a body-centered-cubic (bcc) structure;hcp Fe with a nonmagnetic (NM) ground state only appearsunder high pressure [27,28]. In contrast, various hexagonalFe-rich Fe-X alloys exhibit a wide variety of magnetic states atboth ambient and high pressures. Fe-rich Fe-Mn–based alloyswith the hcp structure (ε phase) exhibit a Néel temperatureof 230 K [29]. ε-Fe-X alloys (X = Ru and Os) have AFMground states at ambient pressure with Néel temperatures ofapproximately 100 K. The Fe2Ta alloy exhibits a paramag-netic state in which either an excess of Fe or Ta can induceferromagnetic ordering at low temperatures (approximately150 K) [30,31]. The study of structural polytypes of pureiron will also facilitate investigations into the possibility ofmetastable phases in such Fe-rich alloys and the new ε phasefound in pure iron under extremely high pressures [32,33].It is also notable that determination of the ground statefor both fcc and hcp Fe from first-principles calculations hasproved challenging because the energy difference betweendifferent magnetic ordering patterns is constrained to a narrowenergy window [34–39]. Magnetic frustration occurs in hcpFe, where spins are expected to be geometrically frustratedwith respect to AFM order [40,41]. Each spin is shared by theeight tetrahedra of the hcp lattice, and the nearest neighbor(a)BA1 234(b)1 234 �[0001]hcp[0001]hcp(c) [0001]hcpBA[1210]hcp[2110]hcpFIG. 2. (a) Atomic configuration of the hcp lattice. The tetrahe-dron marked in bold green lines is the spin frustration unit in the hcplattice. (b) An example of spin frustration in a tetrahedron into whichan hcp lattice can be decomposed. Arrows show the spin orientationon an isolated tetrahedron in the hcp lattice. (c) Spin configurationof the AFM-II phase in hcp-Fe, which is projected on the (0001)hcpplane. Up and down arrows indicate the spin orientation. The solidline connects in-plane sites, and the dashed line connects sites tothose in the adjacent layer. The open circles with red arrows showthe internal atomic coordinates at z = 1/4, and the filled circles withblue arrows show those at z = 3/4.bonds are shared by the two tetrahedra, as shown in Fig. 2(a).The ground state spin configuration of the system is formed bythe stacking of adjacent tetrahedra. The following two choicesare present. Suppose that two spins are antiparallel (between1 and 2) [Fig. 2(b)], in which case, the two remaining spins(between 3 and 4) should also be antiparallel. However, theseaxes can be chosen to form an arbitrary angle (an infinitenumber of ways) with respect to the axis of the first two spins;the ground state is thus infinitely degenerate. If the first twospins form an angle α, then the other two spins must form thesame angle and be antiparallel with the first two spins [41].As a result of the geometrical frustration of the spinmoments arranged in a tetrahedral configuration in hcp Fe,noncollinear spin order (α �= 0) and spin-intensity modulated(spin-smectic) phases arise as local minima or saddle pointsby consideration of an isolated tetrahedron [36–39]. How-ever, previous DFT studies predicted that a collinear AFMstate known as type II (AFM-II) is the lowest energy spinconfiguration of ε Fe at ambient pressure [39,42,43]. Thiscollinear spin structure is represented by α = 0 [37], and eachatom has eight antiferromagnetically coupled and four fer-romagnetically coupled neighbors, as illustrated in Fig. 2(c).Their calculated bulk moduli and lattice parameters show bet-ter agreement with the recent experimental equation of state(EoS) [44–47], compared to the values obtained from NMcalculations. In this study, we have systematically searched forcollinear AFM ordering of various LPSO phases and comparethe stability over the total energy of the hcp AFM-II phase.The paper is organized as follows. The details of the first-principles DFT method are given in Sec. II. Experimental033215-2ORIGIN OF PHASE STABILITY IN Fe WITH … PHYSICAL REVIEW RESEARCH 3, 033215 (2021)observations of LPSO-like phases in Fe-Mn-Si–based alloysare presented in Sec. III. Section IV A describes the calcu-lated structural models for the structural polytypes of Fe. Theoverall procedure for exploration of the stable AFM patternsis given in Sec. IV B. The calculated structural and magneticstabilities are discussed using the EoS in Sec. IV C. Finally,Sec. IV D presents an analysis of the DOS for AFM statesof LSPO structures to discuss the electronic origin of phasestabilities, followed by our conclusions.II. CALCULATION METHODSFirst-principles DFT calculations were performedusing the all-electron full-potential linearized augmentedplane wave (FLAPW) method implemented in theQMD-FLAPW12 code [48–50]. This method is knownas the most accurate among the first-principles methods. Theexchange-correlation functional used was a spin-polarizedform of the generalized gradient approximation (GGA)proposed by Perdew, Burke, and Ernzerhof (PBE) [51].For NM phases, integration of the Brillouin zone wasperformed using k-point grids of 16 × 16 × 8 for hcp(2H) Fe, 12 × 12 × 12 for fcc (3C), 16 × 16 × 4 for dhcpFe, and 16 × 16 × 4 for 6H1, 6H2, and 10H structures.The symbols nH and 3C represent structural polytypes inRamsdell notation, where n refers to the stacking period(total number of close-packed planes in the unit cell), andthe letters H and C denote the hexagonal and cubic latticetypes. The number 3 in 3C refers to the stacking periodin close-packed layers (ABC), and 3C is the only possiblecubic polytype. The subscript in the 6H structures specifies adifferent stacking configuration of the close-packed planes.The k-point mesh used for orthorhombic cells with an AFMorder of 2H, 6H1, and 6H2 is 10 × 6 × 4, while the meshesfor tetragonal unit cells with AFM-S and AFM-D statesin fcc (3C) structures are 12 × 12 × 8 and 12 × 12 × 4,respectively. To accurately obtain the electronic structure andthe EoS, the cutoff energies for the LAPW basis functionsin the interstitial were set to 36 and 310 Ry for the planewaves and potentials, respectively. The common muffin-tin(MT) sphere radius of Fe was set to 1.16 Å for all structuralpolytypes of Fe. The angular momentum expansion inside theMT sphere was truncated at l = 8 for Fe atoms. Finally, themethod of explicit orthogonalization was used in the presentstudy [52].III. EXPERIMENTA. Sample preparation and setup for fatigue testA 10-kg ingot of Fe-33Mn-4Si (mass %) alloy was pre-pared by induction furnace melting in an argon atmosphere.The ingot was hot forged and rolled at an initial heating tem-perature of 1273 K into a 20-mm-thick plate. The plate wasannealed at 1273 K for 1 h and then quenched in water. Thedog-bone-shaped fatigue specimen shown in Fig. 3(a) wasmachined from the plate, and Fig. 3(b) shows a photograph ofthe test setup. The axial-strain controlled tension-compressionfatigue test was conducted at a total strain amplitude of 0.01with a triangular wave at a strain rate of 0.004 s−1 until failureat room temperature. 0.2-mm-thick and 3-mm-diameter disks91 mm24 mm 24 mm16 mm� = 8 mm�(a)(b)FIG. 3. (a) Dimensions of a low-cycle fatigue specimen. (b) Pho-tograph of the fatigue test setup.were obtained from the fatigue-failed specimen using a low-speed cutter and chemical polishing in a solution consistingof hydrofluoric acid, hydrogen peroxide, and water (1:10:2by volume). Thin foils for TEM (JEM-4000EX, JEOL; ac-celeration voltage of 400 kV) observations were prepared bytwo-step electrolytic polishing using an electrolyte composedof acetic acid and perchloric acid (10:1 by volume) duringwater cooling.B. TEM observation of LPSO phaseHere, we present experimental evidence for the presenceof an LPSO-like structure in the Fe-Mn-Si alloys after beingsubjected to low-cycle fatigue failure in a series of studies in asearch for a seismic damping alloy [53–55]. Figures 4(a) and4(b) show a bright-field image and a selected area electrondiffraction (SAED) pattern, respectively. The TEM image ofthe Fe-33Mn-4Si (in mass %) alloy was captured after thesample was subjected to cyclic tensile-compressive loadingat room temperature and a constant strain amplitude of 0.01until fatigue failure; the image depicts a fatigue failed alloy.FIG. 4. (a) Bright-field image and (b) SAED pattern of theLPSO–like (ε ′) phase found in the fatigue-failed Fe-33Mn-4Si (mass%) alloy [encircled area in (a)]. The red arrows indicate the extraspots at the 1/6 positions that correspond to the (1̄1̄1)γ plane [the1/3 positions corresponding to the {0002} plane of the ε phase],the latter of which corresponds to the (0006)ε ′. (c) Dark-field imageof (a).033215-3TSUMURAYA, WATANABE, AND SAWAGUCHI PHYSICAL REVIEW RESEARCH 3, 033215 (2021)FIG. 5. (a) Bright-field image for crossing variant plates in anLPSO-like (ε ′) phase of the Fe-33Mn-4Si alloy, and both observedin the SN orientation. (b) The (0006) planes of the variants, where ε′1and ε ′2 are parallel to the (11̄1)γ and (1̄1̄1)γ planes, respectively.The red arrows indicate the extra spots at the 1/6 positions thatcorrespond to the (11̄1)γ plane (the 1/3 positions correspond to the{0002} plane of the ε phase), the latter of which corresponds to the(0006)ε ′1. Dark-field images of the (c) (011̄3)ε ′1 and (d) (011̄3̄)ε ′2spots. The parallel plates of the ε ′1 and ε ′2 phases are shown in brightcontrast in those figures. The streaks along the (0006)ε ′ spots suggesta high concentration of stacking faults in the ε ′ planes.The SAED pattern in Fig. 4(b) was taken from the encircledarea shown in Fig. 4(a).The γ austenite and LPSO-like phase (ε′ phase) are ori-ented [011]γ and [21̄1̄0]ε to the electron beam direction. Thered arrows in Fig. 4(b) show the periodic spots observed atthe 1/6 positions between the {111}γ spots, which is equiva-lent to the 1/3 positions that correspond to the {0002} planeof the ε (hcp) phase. These spots show a hexagonal-typestructure with a six-layer periodicity of a close-packed plane(6H). The (0006)ε′ plane is parallel to the (1̄1̄1)γ plane,and the extra spots are aligned in the c∗ axis in the recip-rocal space. This orientational relationship between the γaustenite and the ε′(6H) phases is identical to the so-calledShoji-Nishiyama (SN) orientational relationship between theγ and ε (2H) phases [6]. Figure 4(c) shows a dark-field imageof the fatigue-failed alloy. The ε′ phase is shown in brightcontrast. This image was taken using the (01̄12̄)ε′ spot, whichis encircled and indicated by the double-headed blue arrow inFig. 4(b). The ε′ phase is highly defective, and the fringes inthe bright-field image indicate stacking faults.Figure 5(a) shows another TEM image for the LPSO struc-ture taken in a different location of the specimen. In this area,crossing variant plates of the ε′ phase are observed in theSN orientation. As shown in Fig. 5(b), the (0006) planes ofthe variants ε′1 and ε′2 are parallel to the (11̄1)γ and (1̄1̄1)γplanes, respectively. The dark-field images in Figs. 5(c) and5(d) are taken at the (011̄3)ε′1 and (011̄3̄)ε′2 spots, respectively.The parallel plates of the ε′1 and ε′2 phases are shown in brightcontrast in the figures. The streaks and the (0001)ε′ spotsTABLE I. Various structural polytypes of Fe. Stacking sequencesare characterized by three different notations: Ramsdell, ABC, andhc notations. The symbol − between layers indicates the position ofa stacking fault. β represents the hexagonality parameter.Type ABC hc [59] β (%)2H (hcp) AB− h 1.04H (dhcp) ABC−B− hc 0.56H1 ABCA−C−B− hcc 1/36H2 ABC−BC−B− hchhhc 2/310H ABC−BC−B−A−CA−C cchhh 0.33C (fcc) ABC c 0suggest a high concentration of stacking faults in the ε′ planes.The LPSO structure is induced by cyclic deformation; there-fore, the crystallographic orientation of the LPSO structurewith respect to the parent γ phase should be determined bythe deformation axis. The LPSO structures shown in Figs. 4and 5 have the same six-layer periodicity and differ only intheir relative orientation in the γ matrix.Notably, extra spots were not observed in the initial mi-crostructure, but only in the fatigue-failed specimen aftercyclic loading at room temperature. Unlike the Mg alloyswith LSPO, the LPSO structure observed in the cyclicallydeformed Fe-33Mn-4Si (mass %) alloy cannot include the or-dering of solute atoms due to their relatively low diffusion rateat room temperature. Therefore, the periodicity can only becaused by the stacking sequence of the close-packed planes,ABC. The extra spots are located at the 1/3 positions thatcorrespond to the {0002} plane of the ε phase with the 2Hstructure, which indicates that the unit cell has a hexagonal(H) type structure with six-layer periodicity (6H). It has beenwidely accepted that there are only two such structures with-out chemical ordering, 6H1 and 6H2, of which the stackingsequences are ABCACB and ABCBCB, respectively [56,57].However, from the TEM measurements, it is still unclearwhich stacking pattern of 6H is realized in the experimentalLPSO phase.Therefore, first-principles calculations were performed todetermine the structural and magnetic stabilities of the 6Hstructures. In addition, the stabilities of other structural vari-ants, 4H and 10H, were also investigated. As a result, wefound that the 6H2 structure, which has an AFM order, wasclosest in energy to the hcp AFM-II structure. Therefore,it was concluded that the ε′ phase most likely has the 6H2structure. The structural and magnetic properties obtainedfrom the DFT calculations and the microscopic origin of thephase stability are reported in Sec. IV. It should be noted thatplastic deformation modes in the deformation-induced γ -εtransformation can be found in a recent review [55].IV. CALCULATION RESULTSA. Structural models of LPSOWe considered six different stacking sequences. Thecandidate structural polytypes are listed in Table I. The se-quences of the 2H (hcp), 4H (dhcp), 6H1, and 6H2 structuresare ABAB, ABCB, ABCACB, and ABCBCB stacking ofclose-packed layers, respectively. The stability of the 10H033215-4ORIGIN OF PHASE STABILITY IN Fe WITH … PHYSICAL REVIEW RESEARCH 3, 033215 (2021)TABLE II. Equilibrium lattice parameters for various structural polytypes of Fe with NM and AFM ordered states alongside optimizedlattice constant ratios (c/a) in a hexagonal cell. Wyckoff positions and their internal atomic coordinates (x, y, z) for the NM and AFM statesare shown in Tables III and IV, respectively. The equilibrium lattice volume (zero pressure volume) V0 is given per Fe atom. B0 and B′ are thebulk modulus and its pressure derivative, respectively. �E is the relative total energy with respect to that of the AFM-II phase in hcp Fe. Thesquare bracketed items of 3C indicate fixed values in the fit. c/a is the axis ratio normalized to a 2H structure (hcp Fe).Crystal a b c c/a V0 B0 �EType Ordering system Space group (Å) (Å) (Å) (bohr3/atom) (GPa) B′ (meV/atom)2H (hcp) NM hex P63/mmc 2.46 3.89 1.580 68.7 255 8.58 27AFM-II ortho Pmcm 2.47 4.28 3.98 1.616 71.0 199 4.95 0Expt. [47] 74.8 191 4.52Expt. [46] 74.8 180 4.91Expt. [45] 74.8 202 4.5Expt. [44] 75.8 165 4.976H1 NM hex P63/mmc 2.45 11.85 1.613 69.2 279 4.12 80AFM1 ortho Pmnm 2.47 4.29 12.02 1.619 71.7 175 5.41 53AFM2 ortho Pm2m 2.47 4.29 12.04 1.620 72.0 183 7.21 286H2 NM hex P6̄m2 2.45 11.78 1.600 69.0 285 5.54 59AFM ortho Pm2m 2.47 4.29 11.97 1.611 71.5 197 4.50 154H NM hex P63/mmc 2.45 7.89 1.612 69.1 286 4.35 8210H NM ortho Cmcm 2.45 4.25 19.61 1.600 69.0 283 4.43 603C (fcc) AFM-D tetra Pmm2 2.50 2.50 7.08 74.8 126 2.71 56AFM-S tetra P4/mmm 2.47 2.47 3.49 71.8 200 7.19 62NM cubic Fm3̄m 3.45 69.2 279 4.51 106Expt. [67] (293 K) [79.3] 133 [5]Expt. [68] (1273 K) 82.7 111 5.3Expt. [66] 146 4.67bcc FM cubic Im3̄m 2.83 76.7 191 4.53 −57Expt. [65] 166 5.29Expt. [66] 164 5.50structure—an LPSO structure with a longer period than the6H structures—was also investigated. The number of possiblepatterns of the 10H polytype was too large to calculate all thepossible structures; therefore, an ABCBCBACAC stackingconfiguration was used, which has been observed in Mg-Zn-Yalloys [58]. In addition to the ABC notation, we introducea configurational notation called hc notation to characterizethe different stacking variants. In the hc notation, the sym-bol h represents a local set of three layers in a hexagonalpattern (layers with identical neighbors, i.e., ABA or ACA),and the symbol c represents a set of three layers in an fcc-likepattern (layers with different neighbors, i.e., ABC or BCA)[59]. To characterize the various polytypes, the hexagonalityparameter β was employed to represent each LPSO phaseas an fcc and hcp composite multilayer using hc notation[60]. This parameter is defined as the ratio of the number ofhexagonal layers (nh) to the total number of layers per unitcell: β = nh/(nh + nc), where nh and nc are the respectivenumbers of h and c blocks in each structure. Therefore, theβ values for the end members of the fcc and hcp structuresare 0 and 1, respectively, while those of the intermediatepolytypes 4H, 6H1, 6H2, and 10H are 1/2, 1/3, 2/3, and 0.3,respectively. Later, we discuss the ground state energies of theLPSO phases for both NM and AFM phases as a function ofthe β parameter in Sec. IV C.The crystal structures for non-spin-polarized calculationswere first generated using periodic boundary conditionsto determine their structural stabilities. The theoreticallyoptimized lattice parameters and the internal atomic coor-dinates are listed in Tables II and III, respectively. In theNM case, the hcp, dhcp, and 6H1 structures of Fe belongto the same space group (P63/mmc); however, the stackingsequence along the z axis of the hexagonal lattice is differ-ent. This stacking sequence means that the lattice constantc in the hcp structure is approximately half that of the dhcpstructure. Therefore, the unit cell of the dhcp structure con-tains four Fe atoms and the 6H structures contain six, whileTABLE III. Internal atomic coordinates (x, y, z) of NM statesfor various structural polytypes of Fe. Site represents the Wyckoffpositions. The lattice constants that correspond to atomic coordinatesare listed in Table II.Atomic coordinatesType Space group Site x y z2H P63/mmc 2c 1/3 2/3 1/44H P63/mmc 2a 0 0 06H1 P63/mmc 2b 0 0 1/44 f 1/3 2/3 –0.08636H2 P6̄m2 1c 1/3 2/3 02i 2/3 1/3 0.1662g 0 0 1/31 f 2/3 1/3 1/210H Cmcm 4c 0 0.6666 1/48 f 0 0 0.84898 f 0 0.3330 –0.05153C Fm3̄m 4a 0 0 0033215-5TSUMURAYA, WATANABE, AND SAWAGUCHI PHYSICAL REVIEW RESEARCH 3, 033215 (2021)the hcp Fe structure contains two. Of these structures, onlythe 6H2 structure belongs to the noncentrosymmetric P6̄m2space group. Due to the limitation of symmetry operations, the10H structure has an orthorhombic cell with the space groupCmcm, and it contains 20 Fe atoms (the primitive cell containsonly ten atoms.).B. Search for the antiferromagnetic orderThe stability of the magnetic structure in LPSO phaseswas also computed from first-principles. Previous DFT studiesof hcp Fe have proposed two collinear AFM configurations[42,43]. First, in the AFM-I configuration, ferromagneticallyordered basal planes of hcp lattice alternate the spin direction.Second, in the AFM-II configuration, the spins alternate alongthe hcp lattice a axis, as shown in Fig. 6(a). The latter spinconfiguration is more energetically favorable than the former[38,42,43].In order to identify stable AFM states for the 6H1 and 6H2structures, we first use the same spin alternation pattern as theAFM-II [Fig. 6(a)] structure in hcp Fe for all (0001)hcp planes(either of type A, B, or C), which provides an in-plane AFMorder and offers two possibilities for spin arrangements onthis plane (deduced from each other by exchanging up spinsand down spins). In the framework of the spin-polarized DFTcalculations for collinear AFM order, the atomic coordinateswith different spins were designated as crystallographicallyindependent sites. Therefore, the in-plane AFM patterns ofthe hcp lattice must be described by an orthorhombic rep-resentation of the hexagonal unit cell [61]. Therefore, foreach six-plane LPSO structure, there are 64(= 26) possiblecombinations of AFM orders to arrange two types of spinin six basal planes with the in-plane AFM arrangements.Of the 64 possible configurations, the patterns with oppositespin signs were equivalent, so that we obtain 32 patterns perthe 6H structure as the possible initial spin configurations.32 orthorhombic structures containing 12 atoms were thusprepared and their total energies were minimized by spin-polarized DFT calculations. As a result, five and three typesof spin configurations were determined for the 6H1 and 6H2structures, respectively. The magnetic moments of all otherinitial spin configurations were relaxed to zero by the energyminimization process. Furthermore, to clarify which structuralpolytypes of iron could be taken as an intermediate phasebetween the fcc and hcp phases, the total energy of fcc Fe(AFM-D, which is defined in the final paragraph of this sub-section) was selected as a criterion to select stable AFM orderfrom the eight patterns obtained. As a result, two types of spinconfigurations were determined for the 6H1 structure; theseAFM phases are referred to as type 1 (AFM1) and type 2(AFM2), respectively. The space groups of orthorhombic cellsfor AFM1 and AFM2 are Pmnm and Pm2m, respectively,where the latter spin order configuration is more stable thanthe former. Therefore, the magnetic structure of the AFM2phase is mainly discussed below. A stable AFM order wasalso identified in the 6H2 structure. This AFM ordered 6H2phase has lower energy than the 6H1 structures, and it is moreenergetically close to the most stable structure of hcp Fe.The space group of the magnetic unit cell is identified fromthe combination of the symmetry operations in this system.(b)(0001)hcp(a)CBAO[112]fcc[101]fcc[011]fcc6H1  AFM2A1B1C1Pm2mba(111)fcc// (0001)hcp[2110]hcp[1210]hcp(c)6H2 AFMCBA   abO[112]fcc[101]fcc[011]fccA1B1C1Pm2mBA2H AFM-IIA1B1Pmcm   abOahcpahcpFIG. 6. (a) Projection of the hcp lattice with AFM-II order onthe (0001) plane, (b) 6H1-AFM2, and (c) 6H2-AFM structures pro-jected on the (111) plane of the fcc lattice, which is parallel to the(0001) plane of the hcp lattice. The arrows denote the direction andmagnitude of magnetic moments, and the magnetic moments on theA, B, and C layers are shown with red, blue, and green arrows,respectively. The orthorhombic cell is shown as bold dashed lines.The spin alternation along the [01̄1] and [101̄] directions of the fcclattice is realized with this unit cell. The A layer is on top of the Blayer, and the C layer is on top of the B layer. The parallelogramshown in the background is the B layer. The planes shown in (b) and(c) correspond to the bottom three layers of the orthorhombic cellsshown in Figs. 7(a) and 7(b), respectively.The symmetry operations that describe the symmetry groupof the orthorhombic structures were searched for using theFLAPW method. In this DFT method, symmetry operationsare applied for the internal coordinates to reduce the numberof computations required. The symmetry operations are asso-ciated with the general positions (x, y, z) of the space group.033215-6ORIGIN OF PHASE STABILITY IN Fe WITH … PHYSICAL REVIEW RESEARCH 3, 033215 (2021)TABLE IV. Atomic coordinates (x, y, z) of AFM states forvarious structural polytypes of Fe. mspin represents spin magneticmoments per atom within the muffin-tin sphere. σ represents thespin index, ↑ or ↓. Site represents the Wyckoff positions. The latticeconstants that correspond to the atomic coordinates are listed inTable II.Ordering Atomic coordinates mspinType (space group) σ Site x y z (μB)2H AFM-II ↑ 2c 0 0.3333 1/4 1.17(Pmcm) ↓ 2c 1/2 0.8333 1/4 1.176H1 AFM1 ↑ 2a 1/4 –0.2407 1/4 0.95(Pmnm) ↑ 4 f 1/4 0.092 0.4138 1.47↓ 2b –1/4 0.2593 1/4 0.95↓ 4 f 1/4 0.4081 –0.4138 1.47AFM2 ↑ 1a 0 0.2757 0 1.33(Pm2m) ↑ 1d 1/2 0.7244 1/2 1.19↑ 2h 1/2 0.4088 0.6650 1.40↑ 2h 1/2 0.0912 0.8350 1.38↓ 1c 1/2 0.7757 0 1.33↓ 1b 0 0.2244 1/2 1.19↓ 2g 0 –0.0912 0.6650 1.40↓ 2g 0 0.5912 0.8350 1.386H2 AFM ↑ 1d 1/2 0.1492 0 1.39(Pm2m) ↑ 1c 1/2 –0.1657 1/2 1.09↑ 2h 1/2 –0.4904 –0.3359 1.28↑ 2h 1/2 –0.1681 –0.1671 1.38↓ 1b 0 –0.3508 0 1.39↓ 1a 0 0.3343 1/2 1.09↓ 2g 0 0.0096 –0.3359 1.28↓ 2g 0 0.3320 –0.1671 1.383C AFM-S ↑ 1a 0 0 0 1.36(P4/mmm) ↓ 1d 1/2 1/2 1/2 1.36AFM-D ↑ 1a 0 0 0 1.48(Pmm2) ↑ 1d 1/2 1/2 −0.25 1.48↓ 1a 0 0 0.5 1.48↓ 1d 1/2 1/2 0.25 1.48The Wyckoff (general) positions for the atomic coordinates ofthe magnetic unit cells were then identified using the interna-tional tables of crystallography [62].The orthorhombic cell represented a hexagonal unit cellwith an in-plane AFM pattern of the hcp lattice. Therefore,the b/a axis ratio was fixed at√3 when optimizing the lat-tice constants of the orthorhombic structures. The c/a ratiowas optimized by fixing the lattice volume at equilibrium.The orthorhombic cells are generated by transforming crys-tal axes from the parent hexagonal structure by specifyingthe 3 × 3 rotation matrix shown in the Appendix and thenmoving the origin according to the spin configuration. Thelattice constants are taken as a = ah, b = √3ah, and c = ch.The optimized lattice parameters and atomic coordinates aresummarized in Tables II and IV, respectively. It should benoted that the orthorhombic cell for the hcp structure in thisstudy is equivalent to the previously reported structure forAFM-II [42,43,61].Figures 6(b) and 6(c) show the spin arrangements on the abplane of the orthorhombic cells in the 6H1 and 6H2 structures,respectively. This plane is equivalent to the (111) plane of fcc,ABCABCBA(a) (b)Up-spinDown-spinABCBCAA1B1C1A1B1C1FIG. 7. Schematic illustrations of the spin arrangements for the(a) 6H1-AFM2 and (b) 6H2-AFM structures with the spin direction(red and blue circles represent Fe atoms with up and down spins,respectively). Orthorhombic cells are shown as bold dashed lines.The ab planes in (a) and (b) corresponds to the {111} plane of theγ phase. The relative position of hexagonal units due to Shockleypartial dislocations are displayed; In (a) and (b), the lattice vector bcorresponds to the direction of the Shockley partial dislocation along[112̄] of the fcc structure.which is parallel to the (0001) plane in the hexagonal lattice.Only the bottom three layers (ABC) of the magnetic unit cellshown in Figs. 7(a) and 7(b) are depicted in these figures toanalyze the difference in the spin alignment between adjacentplanes. As shown in Fig. 6, a specified Fe atom in each planeis assigned as A1, B1, and C1. When the A1 site has up spinin the 6H1 structure [Fig. 6(b)], the magnetic moments onthe B1 and C1 sites have down spins. On the one hand, themagnetic moments at the A1, B1, and C1 sites in the 6H2structure [Fig. 6(c)] have the same spin orientation.In both 6H AFM unit cells, the calculated magnetic mo-ments on the four crystallographically inequivalent atoms foreach type of spin are all different from each other. Moreover,two atoms in the same (0001) plane (the same z positionof atomic coordinates in Table IV) have the same magneticmoment with an opposite direction of spin.The spin intensity of the 6H AFM structures is modulatedalong the c axis, as listed in Table IV. The magnitude of themagnetic moment (0.95μB) in the 2a (2b) site of the 6H1structure in the AFM1 state is much smaller than that of the4 f site (1.45μB). The results show that the decrease in themagnetic moment is favored by the geometrical frustrationand the large spin degeneracy of Fe sites in the tetrahedralgeometry. In addition, the frustrated spin state of AFM1 canchange to the more stable spin order of AFM2, which suggeststhe effect of spin frustration.Figure 7 depicts the relative position of the close-packedlayers due to the Shockley partial dislocations in the 6H struc-tures. The partial dislocations occur along the b axis of theorthorhombic cells, parallel to the [112̄] direction of the fcclattice. An attractive feature is identified in the 6H1-AFM2structure, in that the direction of the magnetic moment isreversed only in the A layer located at the origin of the caxis, as shown in Fig. 7(a), which indicates that the spinstructure has a sixfold period with respect to the Shockley033215-7TSUMURAYA, WATANABE, AND SAWAGUCHI PHYSICAL REVIEW RESEARCH 3, 033215 (2021))b()a(cca aa0FIG. 8. Body-centered-tetragonal (bct) cells for the (001)-typeAFM order structures of fcc-Fe; (a) AFM single-layered (AFM-S)and (b) AFM double-layered (AFM-D) structures. Black sphereswith red up arrows represent Fe atoms with up spin, and thosewith blue down arrows represent those with down spin. The latticeconstants in the bct cell were used; (a) as a = a0√2 and c = a0[= 2a0 in (b)], where a0 is the cubic lattice constant. Solid and dashedlines indicate the original cell with an fcc lattice and the magneticunit cell with the bct lattice, respectively. The bct cell contains twoand four atoms to describe the (001) layered structure of AFM-S andAFM-D, respectively.partial dislocation. The change in the direction of the magneticmoments corresponds to the up spin of the A1 site, while theB1 and C1 sites have down spins, as shown in Fig. 6(b).The direction of spins does not change for the 6H2-AFMstructure with the partial dislocation, as depicted in Fig. 7(b);the magnetic moments at the A1, B1, and C1 sites have thesame spin orientation, as shown in Fig. 6(c). However, inthe stacking sequence of atomic layers, the A layer appearsevery six layers, and the other layers are stacked in a BCBCBpattern.For the fcc structure, two possible AFM states have beenreported previously using body-centered tetragonal (bct) lat-tices [63]. The relationship between fcc and bct cells with(001) type AFM spin patterns is shown in Fig. 8. In both spinarrangements, the magnetic moments are parallel to each otherin the (001) plane. However, the first arrangement, termed theAFM single layer (AFM-S), has alternating layers of spin upand spin down along the [001] axis. In the second arrange-ment, double layers with ferromagnetic interlayer couplingare AFM ordered along the [001] direction; therefore, thisarrangement is termed an AFM double layer (AFM-D). Themagnetic unit cells of AFM-S and AFM-D belong to the spacegroups P4/mmm and Pmm2, respectively. The c/a ratios forAFM-S and AFM-D are√2 and 2√2, respectively, whichcorresponds to the fcc structure, and these values are fixedduring the structural optimization. These calculations verifythe result that the latter AFM pattern is more energeticallyfavorable than the former.C. Structural and magnetic phase stabilityIn this section, we discuss the results for the structuraland magnetic phase stability of LPSO structures. Table IIhcp AFM-II bcc FM hcp NM 6H1 AFM-26H2 AFM6H1 AFM-16H2 NM6H1 NMfcc AFM–Dfcc AFM–S�E (meV/atom) 60 65 70 75 80 85Volume (a.u./atom)-100 -80 -60 -40 -20   0  20  40  60  80 100 120 140 160 180FIG. 9. Volume dependence of the total energy difference instructural polytypes of Fe with respect to the hcp AFM-II phase.The total energies of each phase are calculated as a function ofthe lattice volume per atom within the GGA-PBE functional. Openand solid squares on black lines represent the AFM-S and AFM-Dstates in the fcc structure. Solid orange and blue triangles on dashedlines represent the AFM-1 and AFM-2 states for the 6H1 structures,respectively. Open and solid triangles on red dashed lines show NMand AFM states for the 6H1 structures, respectively. Open squaresshow the fcc structure with the AFM-D ordered state. Open andsolid circles show NM and AFM-II states of the 2H (hcp) structures,respectively. Open green squares on the green dotted line representthe ferromagnetic (FM) state of bcc Fe.shows the optimized structural parameters and the total energydifferences with respect to the AFM-II state of hcp Fe. Amongthe NM states of the LPSO candidates, the 6H2 and 10Hstructures are energetically close to each other, however, the6H2 structure is the most energetically close to hcp Fe. Thisresult for the NM states is similar to the results of DFT studiesfor pure Mg, in which the energy difference between variousLPSO phases is quite small [64]. On the other hand, the 6H1structure is less energetically favorable than 6H2, althoughenergetically close to dhcp Fe.FLAPW calculations with the spin-polarized form of theGGA-PBE functional were also performed, and stable AFMspin structures were determined for the 6H1, 6H2, hcp, and fccstructures (Table II). Several possible magnetic order patternswere also examined for the 4H and 10H structures, includingferromagnetic and ferrimagnetic states; however, no magneticorderings were stabilized. Therefore, Fig. 9 shows only thevolume dependence of the total energies for 6H1 and 6H2 thatconsider AFM ordering and the NM state, as well as those forfcc and hcp Fe. As shown in Fig. 10, the AFM states of the6H1, 6H2, hcp, and fcc structures are energetically lower thanthose of the NM phase by approximately 30–50 meV/atom.033215-8ORIGIN OF PHASE STABILITY IN Fe WITH … PHYSICAL REVIEW RESEARCH 3, 033215 (2021) 0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1.0Hexagonality �  0 20 40 60 80100120�E (meV/atom)AFM2AFM1AFM-DAFM-S NM state AFM stateFIG. 10. Difference of the total energies for different structuralpolytypes of pure iron as a function of hexagonality, β. �E wascalculated with respect to hcp Fe with the AFM-II phase and arelisted in Table II. Open and solid circles represent NM and AFMstates, respectively. The values of β are listed in Table I.Figure 10 also describes the difference of total energy (�E )with respect to hcp Fe as a function of the hexagonality,β. In both the NM and AFM phases, �E decreases almostproportionally as β increases.The equilibrium volumes and bulk moduli are also listedin Table II. The 6H2 structure with AFM ordering is energeti-cally closest to the ground state structure of hcp Fe at ambientpressure. While the 6H1 structure is also stabilized by AFMordering, the 6H1-AFM1 structure is as unstable as fcc withAFM-D ordering. Nevertheless, the AFM2 state of 6H1 has alower energy than AFM1, and it becomes energetically closerto the AFM phase of the 6H2 structure. There is a notableenergy difference of 13 meV/atom (approximately 150 K)between the 6H1-AFM2 and 6H2-AFM phases. Therefore, wesuggest that the LPSO-like phase observed by TEM measure-ments most likely has the 6H2 structure.Figure 9 shows the volume dependence of the total energydifference of stacking variants with respect to the AFM-IIstate of hcp Fe. The equilibrium lattice volumes for theNM states of all the structural polytypes are quite small(69 bohr3/atom), and their bulk moduli (approximately280 GPa) are generally much higher than the experimentalvalues measured at finite temperatures. The correspondingvalues for the AFM states are slightly different from eachother, which is expected to be due to the magnetovolumeeffect and spin-spin interactions. When AFM order is consid-ered, the differences of bulk moduli between different typesof LPSO become more distinct. As summarized in Table II,hcp Fe with the AFM-II state leads to a better agreement withthe experimental bulk modulus within 10% overestimation[45–47] than NM calculations. The calculated bulk modulusfor bcc Fe is also slightly overestimated from the experimentalmodulus [65,66]. The lattice volume and bulk modulus for theAFM-D state of the fcc (3C) structures are closer to the ex-perimental values (6% underestimation from the experimentalbulk modulus at 293 K [67,68]). While the bulk modulus forthe AFM-S state overestimates the experimental values, theequilibrium lattice volume for AFM-S agrees well with thoseof the hcp, 6H1, and 6H2 structures.Structural optimization for lattice parameters and internalatomic coordinates was performed for both the spin-polarizedand unpolarized calculations. To determine the equilibriumlattice parameters in the ground state at ambient pressure,we first determined the equilibrium volume by specifying theaxis ratio of c/a at a constant value, and the calculated totalenergies at different volumes were fitted using the third-orderBirch-Murnaghan EoS. The c/a ratio is optimized by speci-fying the lattice volume at equilibrium using the fourth-orderof the fitting. In Fig. 9, the energy-volume curves are plottedwith the c/a ratio optimized at ambient pressure.D. Origin of phase stabilitiesIt is also interesting to understand how the stacking se-quence along the c axis changes the electronic structure. TheDOS for metals with an hcp structure is generally character-ized by a deep valley or dip near the Fermi level (EF ), whereEF is at the lowest position (the bottom) of the deep valley[69–71]. In the 1970s, before first-principles calculations wereestablished, Inoue and Yamashita suggested that the depth ofthe valleys in the DOS represents the magnitude of splittingof the main peaks of the DOS between those in the occupiedand unoccupied states [69]. This is a type of energy separationbetween the bonding and antibonding molecular orbitals dueto the significant hybridization between the s and p states[69,72]. From a comparison of the electronic structure ofhcp Be and hcp Mg, they also suggested that the degree ofenergy splitting between the two main peaks (the width of thedeep valley or dip) appears as a difference in the enthalpy offormation (or cohesive energies) between the metals [73]. Inthe early 1970s, it would have been difficult to quantitativelycalculate the difference in the heats of formation.Later, Andersen derived the force theorem, which de-scribes how the change in the total energy of an electronsystem can be calculated to the first order in a virtual displace-ment [74]. With this theorem, the energy difference can besimply calculated with DFT as the differences of appropriatesums of the one-electron eigenvalue energies [75,76]. There-fore, the phase stability analysis based on DOS near the EFis effective to understand the energy change associated withsmall displacement based on the force theorem.Figures 11(a)–11(d) compare the total DOS for the hcp,6H2, and 6H1 structures with AFM ordering. A typical deepvalley in the DOS observed in hcp metal is evident inFig. 11(a). The total DOS for the 6H2 structure also has asimilar DOS valley near EF [Figs. 11(b)], and several smallpeaks appear near the bottom of these DOS, compared withthose for hcp Fe. This is evident in the ABCBCBA stacking ofthe 6H2 structure, where the A layer appears every six layers,and the other layers are stacked in a BCBCB pattern.The DOS for the 6H1 structure with the AFM2 configura-tion is similar to that of the 6H2 phase; however, the EF islocated at slightly higher energy than the bottom of the valleyof the DOS [Figs. 11(c)]. We consider this difference in DOSto be the origin of the energy difference between the 6H1 and6H2 structures. A comparison of the spin structures of hcp and6H2 indicates that the local spin arrangement and the stacking033215-9TSUMURAYA, WATANABE, AND SAWAGUCHI PHYSICAL REVIEW RESEARCH 3, 033215 (2021)Energy (eV)010−5−10 0 52030Energy (eV)0510−5−10 0 5(a) (b)(c)Energy (eV)010−5−10 0 52030Total DOS (/eV)0102030Energy (eV)−5−10 0 5(d)Total DOS (/eV)Total DOS (/eV)Total DOS (/eV)FIG. 11. The total DOS for the AFM state of the (a) 2H (hcp),(b) 6H2, (c) 6H1 with AFM2, and (d) 6H1 with AFM1 structures ofpure iron at ambient pressure. Dashed lines represent the Fermi level.sequence agree well with those of the ABAB pattern in the hcpstructure. Nevertheless, the total DOS for the unstable 6H1structure with AFM1 ordering has many peaks near EF , andthe valley of the DOS is obscured [Fig. 11(d)]. We have shownthat this deviation of the DOS from the hcp structure is themicroscopic origin of the structural stability of the candidateLPSO structures. Based on this analysis, one of the presentauthors studied the origin of phase stability in an Mg-Zn-Yalloy with LPSO, in which solute elements of Zn and Y wereembedded in the Mg matrix near stacking faults [77]. Theresults will be reported elsewhere shortly.V. SUMMARYThe structural and magnetic properties of long-periodstacking order structures (polytypism) in pure iron were stud-ied by first-principles DFT calculations. During deformation-induced martensitic transformation from γ -austenite toε-martensite, a phase (different from the ε phase) was discov-ered in Fe-Mn-Si–based alloys. In this phase, the additionaldiffraction spots are located at the 1/3 positions that corre-spond to the {0002} plane of the ε (hcp) phase with the 2Hstructure, which suggests a 6H structure. However, the actualstacking pattern of the 6H phase is unknown. Therefore, weproposed several structural models for the LPSO structure ofpure iron, including 4H, 6H1, 6H2, and 10H structures, andstructural optimization was performed using first-principlesDFT calculations. From a search among the stable magneticphases, stable AFM states were identified in the 6H1 and6H2 structures. An AFM state of 6H2 was also revealed asenergetically closest to the hcp structure, and the observedLPSO-like phase has a high probability of adopting the 6H2structure. Due to the probably coherent nature between thepossible 6H structures, a negligibly low elastic contributionmay not affect the highest probability of the appearance of the6H2 structure. The electronic origin of the phase stability isattributed to the depth of the valley in the DOS near the Fermilevel; the energy splitting between the two peaks in occupiedand unoccupied states is large, which maximizes the phasestability. The relationship between the electronic structure andthe phase stability was quantitatively verified for the LPSOand hcp phases in Fe, which was proposed for hcp metals inthe 1970s.ACKNOWLEDGMENTSThe authors thank A. Singh, T. Oguchi, W.-T. Geng, andD. S. Shih for stimulating discussions. We also acknowledgeD. Drozdenko, I. Tkach, D. Korotin, and Z. Pchelkina forcollecting and translating old Russian papers by Lysak et al .This research was funded by a Grant-in-Aid for ScientificResearch (Grants No. JP19K04988, No. JP20H00312, No.JP21H01220, and No. JP21H01659) from the Japan Societyfor the Promotion of Science (JSPS) and a CREST Grant(No. JPMJCR2094) from the Japan Science and TechnologyAgency (JST). This work was performed under the GIMRTProgram of the Institute for Materials Research (IMR), To-hoku University and the Cooperative Research Program ofthe Network Joint Research Center for Materials and De-vices. T.T. was supported in part by the Leading Initiativefor Excellent Young Researchers (LEADER), a Ministry ofEducation, Culture, Sports, Science and Technology (MEXT)program, Japan. 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