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# Origin of phase stability in Fe with long-period stacking order as
intermediate phase in cyclic $\gamma$-$\epsilon$ martensitic transformation

Takao Tsumuraya1,2 1International Center for Young Scientists, National
Institute for Materials Science, Tsukuba 305-0044, Japan  2Priority
Organization for Innovation and Excellence, Kumamoto University, Kumamoto
860-8555, Japan  Ikumu Watanabe3 and Takahiro Sawaguchi3 3Research Center for
Structural Materials, National Institute for Materials Science, Tsukuba
305-0047, Japan

###### Abstract

A class of Fe-Mn-Si based alloys was found to exhibit a reversible martensitic
transformation between the $\gamma$ ($fcc$) and $\epsilon$ ($hcp$) phases.
During the deformation-induced $\gamma$-$\epsilon$ transformation, a new phase
was identified that is different from the $\epsilon$-phase, where the electron
diffraction spots are located at the 1/3 position corresponding to the $hcp$
(2H) phase, which suggests long-period stacking order (LPSO). To understand
the actual stacking pattern and discuss the possible realization of a LPSO
phase as the intermediate phase between the $\gamma$ and $\epsilon$ phases,
the phase stability of various structural polytypes of iron was examined using
first-principles calculations with a spin-polarized form of the generalized
gradient approximation in density functional theory. We found that an
antiferromagnetic ordered 6H2 structure is the most stable among candidate
LPSO structures and is energetically close to the $\epsilon$ phase, which
suggests that the observed LPSO-like phase adopts the 6H2 structure. We
determined that the origin of the phase stability can be attributed to the
depth of the deep valley in the density of states close to the Fermi level.

Martensitic transformation, magnetism, Density functional theory, phase
stability

###### pacs:

Valid PACS appear here

##  I Introduction

Austenitic steel is an industrial structural material with a long history.
Ever since wear-resistant Fe-Mn-C steel was invented in the end of the 19th
century, much attention has been paid to its superior mechanical properties.
Among these alloys, those that have 28–32 weight percent (wt.$\%$) Mn and 4–7
wt.$\%$ Si are known to demonstrate a shape memory effect that is governed by
a non-diffusive solid-to-solid phase transformation from the
$\gamma$-austenite with a face-centered cubic ($fcc$) structure to an
$\epsilon$-martensite with a hexagonal close-packed ($hcp$) structure. Sato
_et al._ (1986) The plastic deformation and subsequent shape recovery upon
heating are associated with the forward $\gamma$ $\rightarrow$ $\epsilon$ and
reverse $\epsilon$ $\rightarrow$ $\gamma$ transformations, respectively. Ogawa
and Kajiwara (1993); Kikuchi  _et al._ (1995); Ohtsuka  _et al._ (1995);
Sawaguchi  _et al._ (2006)

Figure 1 illustrates the atomic displacement during the $\gamma$
$\leftrightarrow$ $\epsilon$ transformation. The (111) planes in the $fcc$
structure are parallel to the basal (0001) planes of the $hcp$ lattice. The
formation of the $\epsilon$ phase from the $\gamma$ phase is induced by
stacking faults bounded by Shockley partial dislocations with an
$a$/$\sqrt{6}$ shift on the (111) plane. The partial dislocations occur every
two layers in the pathway from the $fcc$ to the $hcp$ structure. Nishiyama
(1978) This transformation is one of the noticeable plastic deformation modes
in austenite steels. The reversible transformation between the $\gamma$ and
$\epsilon$ phases occurs not only during heating and cooling cycles, but also
during cyclic plastic deformation. In a previous study, one of the present
authors revealed that a class of alloys with modified chemical compositions
exhibits a long fatigue lifetime ($N_{f}$), which is defined as the number of
cycles to fracture.Sawaguchi  _et al._ (2015) The Fe-15Mn-10Cr-8Ni-4Si (in
mass$\%$) alloy that we developed has been implemented in seismic damping
components for architectural construction, Sawaguchi  _et al._ (2016) and has
an almost 10 times longer $N_{f}$ than conventional steels.

Figure 1: Atomic displacement and formation of partial dislocation-stacking
fault units associated with (a) $\gamma$ $\rightarrow$ (b) $\epsilon$
martensitic transformation.

Ever since a dual-phase magnesium based alloy with a long-period stacking
order (LPSO) structure and $\alpha$-Mg ($hcp$) exhibited superior mechanical
properties with a tensile yield strength of $\sim$600 MPa, Kawamura  _et al._
(2001) there has been growing attention to the development of novel alloys
with LPSO structures.Zhang  _et al._ (2020) However, very few reports are
available on the existence of a LPSO phase in steels and Fe based alloys. In
the 1960s, Lysak and Nikolin discovered a new phase that is different from the
$\epsilon$-phase in a disordered Fe-Mn-C alloy subjected to heating and
cooling cycles of 400 $\leftrightarrows$ –196 ∘C. Lysak and Nikolin (1963);
Kononenko  _et al._ (1969); Lysak  _et al._ (1970) The new phase was referred
as to $\epsilon^{\prime}$ phase. Based on the experimental results, the
distance between stacking faults was found to be six atomic layers. Lysak and
Nikolin (1967) As many as different LPSO phases have been discovered in
various Fe-based alloys using TEM. Oka  _et al._ (1972, 1973); Ohtsuka  _et
al._ (1995); Hwang  _et al._ (1991); Chao and Ho (1992); Lee  _et al._ (1995)
Although thermodynamic modeling verifies the presence of the LPSO phase, Wan
_et al._ (2001) actual stacking pattern and phase stability of the LPSO phase
could not be identified from experiments alone.

In the present work, transmission electron microscopy (TEM) measurements were
performed for Fe-Mn-Si based alloys subjected to cyclic deformation. Electron
diffraction spots at the 1/3 positions relative to the $\\{$10-11$\\}$ spots
for the $\epsilon$ phase (Fig. 2) suggest the existence of a LPSO structure.
However, for Fe-Mn-Si based alloys under cyclic deformation, the observed LPSO
phase is unstable during room temperature aging, probably due to sensitivity
to temperature variations or electron beam irradiation, and information on the
structural and magnetic properties are still not to be available. Therefore,
we study the structural and magnetic stability of candidate LPSO structures
(structural polytypes) of pure Fe using first-principles calculations based on
density functional theory (DFT).Hohenberg and Kohn (1964); Kohn and Sham
(1965) The relative stability with the $fcc$ and $hcp$ phases is examined by
structural optimization to understand the possible realization of an LPSO
phase as an intermediate phase in the $\gamma$–$\epsilon$ martensitic
transformation. Antiferromagnetic (AFM) order is a key factor in the
stabilization of phases with $fcc$ and $hcp$ structures for Fe-Mn based
alloys. Ohno and Mekata (1971); Endoh and Ishikawa (1971); Kennedy and Hicks
(1987); Bleskov  _et al._ (2016); Reyes-Huamantinco  _et al._ (2012)
Therefore, we assume AFM spin structures with collinear spin order, and
compare the total energies obtained from non-spin-polarized calculations. The
total energies of the candidate LPSO structures are compared with respect to
the more stable structure of $hcp$ at 0 K, and the origin of structural
stabilities is discussed based on the difference in the total density of
states (DOS) near the Fermi level.

For the sake of simplicity, pure Fe is used as a model for Fe-rich Fe-$X$
alloys. The ground state of Fe at ambient pressure is widely known to be
ferromagnetic with a body-centered cubic ($bcc$) structure, and $hcp$-Fe with
a nonmagnetic ground state only appears under high pressure. Taylor  _et al._
(1982); Nicol and Jura (1963) In contrast, various hexagonal Fe-rich Fe-$X$
alloys exhibit a rich variety of magnetic states at both ambient and high
pressures. Fe-rich Fe-Mn alloys with $hcp$ structure ($\epsilon$ phase) shows
the Néel temperature of 230 K.Ohno (1971) $\epsilon$-Fe-$X$ alloys ($X$ = Ru,
and Os) have AFM ground states at ambient pressure with Néel temperatures of
$\sim$100 K. Fe2Ta alloy exhibits a paramagnetic state in which either an
excess of Fe or Ta can induce ferromagnetic ordering at low temperatures
($\sim$150 K).Kai  _et al._ (1970); Arapan  _et al._ (2020) Iron hydride FeH
with a double $hcp$ ($dhcp$) structure exhibit a ferromagnetic ground state
under pressure.V. E. Antonov  _et al._ (1980); Antonov  _et al._ (1998);
Vočadlo  _et al._ (2000) On the other hand, no internal magnetic field has
been detected for $hcp$ Fe-Ni alloy, even at 11 K under 21 GPa. In most of
these alloys, the amount of alloying element ($X$) is relatively low (less
than 25 atom$\%$), and the Fe content is large. Therefore, the study of
polytypism for pure Fe is not limited to the possible realization of an LPSO
phase during cyclic $\gamma$-$\epsilon$ transformation, but is also valuable
for investigating the possible realization of a metastable phase with multiple
partial dislocations in such Fe-rich alloys, and a new $\epsilon$ phase found
in pure-Fe under extremely high pressures.Saxena  _et al._ (1995); Yoo  _et
al._ (1996)

The paper is organized as follows. The details of the first-principles DFT
method are given in Sec. II. Experimental observations of LPSO-like phases in
Fe-Mn-Si based alloys are presented in Sec. III. Section IV.1 describes the
calculated structural models for nonmagnetic and AFM states of Fe polytypes.
The calculated structural and magnetic stabilities are discussed using the
equation of state (EoS) and convex hull diagram in Sec. IV.2. Section IV.3
presents the analysis of the DOS for AFM states of LSPO structures to discuss
the electronic origin of phase stabilities, followed by conclusions.

Figure 2:  TEM image of (a) the $\epsilon$ ($hcp$) phase and (b) an LPSO phase
observed in Fe-33Mn-4Si alloy after being subjected to cyclic tensile-
compressive loading at room temperature and a constant strain amplitude of
0.01 to fatigue failure. The $\epsilon$ and LPSO phases are identified from
the selected area electron diffraction (SAED) patterns shown in (c) and (d),
respectively.

##  II Calculation methods

First-principles DFT calculations were performed using the all-electron full-
potential linearized augmented plane wave (FLAPW) method, which was
implemented in the QMD-FLAPW12 code. Wimmer  _et al._ (1981); Weinert (1981);
Koellng and Arbman (1975) This method is known as the most accurate among the
first-principles methods. The exchange-correlation functional used was a spin-
polarized form of the generalized gradient approximation (GGA) proposed by
Perdew, Burke, and Ernzerhof (PBE). Perdew  _et al._ (1996) For nonmagnetic
phases, integration of the Brillouin-zone was performed using $k$-point grids
of 16$\times$16$\times$8 for $hcp$-Fe, 16$\times$16$\times$4 for $dhcp$-Fe,
and 16$\times$16$\times$2 for 6H and 10H structures. The $k$-point mesh used
for AFM phases of $hcp$, 6H1 and 6H2 was 10$\times$6$\times$4\. To obtain the
electronic structure and the EoS accurately, the cut-off energies for the LAPW
basis functions in the interstitial were set to be 36 and 276 Ry for the plane
waves and potentials, respectively. The common muffin-tin (MT) sphere radius
of Fe was set to 1.16 Å for all the structural polytypes of Fe. The angular
momentum expansion inside the MT sphere was truncated at $l$ = 8 for Fe atoms.
The method of explicit orthogonalization was used in the present study.
Weinert  _et al._ (2009)

##  III Experiment

Figures 2(a) and 2(b) show TEM images of the $\epsilon$ and LPSO phases in
Fe-33Mn-4Si alloy, respectively. Two different phases were identified from the
SAED patterns shown in Fig. 2(c) and 2(d). The electron diffraction spots for
the LPSO phase are located at 1/3 positions relative to the $\\{$10-11$\\}$
spots for the $\epsilon$ phase. The experimental evidence suggests that the
LPSO-like structure was present in the Fe-Mn-Si alloys after being subjected
to a low-cycle fatigue failure in a series of studies in the search for a
novel seismic damping alloy.

The observed LPSO like phase appears below the temperature at which the $fcc$
phase starts transforming to $hcp$ phase ($M_{s}$). At Fe-rich and low Mn
content, the $M_{s}$ stay higher than the Néel temperature ($T_{N}$) of the
$fcc$-phase. Cotes  _et al._ (1995); Dong  _et al._ (2006) The Néel
temperatures of the $fcc$ phase are 338 K for Fe-33Mn-4Si alloy and 118 K for
Fe-15Mn-10Cr-8Ni-4Si alloy.Sawaguchi  _et al._ (2015) In fact, LPSO phase was
observed at room temperature in the former alloy system, and at much lower
temperatures below the Néel temperature in the latter alloy system.

We also found that the observed LPSO phase in these Fe based alloys do not
seem to have ordering of the solute atoms, and it originates solely from the
periodic order of the stacking faults. This is different from the Mg based
alloy with the LPSO structure, where solute atoms of Zn and Y are enriched in
a few layers and periodic partial dislocations occur on the planes where the
solute atoms are arranged. Abe  _et al._ (2002); Egusa and Abe (2012)

A sample of Fe-33Mn-4Si alloy (mass$\%$) was prepared by induction furnace
melting into a 9.5 kg ingot. The ingot was hot forged and rolled at 1273 K. It
was then annealed for solidification and homogenization at 1273 K for 1 h
under an argon flow, followed by water quenching. Dog-bone shaped low-cycle
fatigue (LCF) test specimens with a gauge diameter of 8 mm were machined from
the annealed bar. Axial strain-controlled LCF tests were conducted under
symmetric push-pull loading at a total strain amplitude of 0.01 with a
triangular wave at a strain rate of 0.004 s-1. The cross-section of the
fatigue failure specimen ($N_{f}$ = 3,162 cycles), approximately 5 mm from the
fracture surface, was cut into a disc several hundred micrometers thick.
Further thinning was performed by mechanical and electrolytic polishing to
obtain a damage-free TEM foil. TEM (JEOL JEM-4010) observations were performed
with a double-tilt stage at an acceleration voltage of 400 kV.

##  IV Calculation results

###  IV.1 Structural models of LPSO structures

The crystal structures of LPSO phases were developed using periodic boundary
conditions to determine the structural stabilities. Table. 1 shows the
candidate LPSO structures and the optimized lattice parameters within the
nonmagnetic calculations. A previous report on martensitic transformation
described the presence of the $\epsilon^{\prime}$ phase, and suggested two
possible patterns of triple-hexagonal structures of 6H1 and 6H2 as candidates.
Nishiyama (1978) In nonmagnetic cases, the $hcp$, $dhcp$, and 6H1 structures
of Fe belong to the same space group ($P$63/$mmc$); however, the stacking
sequence along the $z$-axis of the hexagonal lattice is different. The
stacking sequence means that the lattice constant $c$ in the $hcp$ structure
is approximately half that in the $dhcp$ structure; therefore, the unit cell
of the $dhcp$ structure contains four Fe atoms and the 6H structures contain
six Fe atoms, while the $hcp$-Fe structure has two. Due to the limitation of
symmetry operations, the 10H structure has an orthorhombic cell with a space
group $Cmcm$, and it contains 20 Fe atoms. As illustrated in Fig. 3, the
sequences of the $hcp$, $dhcp$, 6H1 structures are ABAB, ABCA, and ABACBC
stacking on Fe triangular layers, respectively. The 10H structure was also
investigated, which is an LPSO structure with a longer period than the 6H
structures. Among these structures, only the 6H2 structure, which has ABCBCB
stacking, belongs to the noncentrosymmetric $P\bar{6}m2$ space group. The ABC
notation is not a unique expression; therefore, a configurational notation
called hc notation (= hk, k, from the German word kubik for cubicChristian and
Vitek (1970)), was introduced and is summarized in Table. 1. The symbol h
represents a local set of three layers in a hexagonal pattern (layers with
identical neighbors, $i.e.$ ABA or ACA), and the symbol c represents a set of
three layers in an $fcc$-like pattern (layers with different neighbors, $i.e.$
ABC or BCA).

Table 1: Various structural polytypes of Fe. Stacking sequences are
characterized by three different notations; Ramsdell, ABC, and hc notations.
The listed lattice parameters are relaxed structures for nonmagnetic states.
The axis ratio of $c$/$a$ is scaled by the 2H ($hcp$) structure. The
equilibrium lattice volume $V_{0}$, is given per Fe atom. $B_{0}$ is the bulk
modulus. $\Delta E$ is the relative total energy with respect to that of the
nonmagnetic state in $hcp$-Fe.

Type | ABC |  hc Christian and Vitek (1970) | Space group | $a$ | $b$ | $c$ |  $c$/$a$ | $V_{0}$ | $B_{0}$ | Wyckoff | $x$ | $y$ | $z$ | $\Delta E$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  | [Å] | [Å] | [Å] |  | [Bohr3] | [GPa] |  |  |  |  | [meV]
2H ($hcp$) | AB | hh | $P6_{3}/mmc$ | 2.46 |  | 3.89 | 1.58 | 68.81 | 138 | $2c$ | 1/3 | 2/3 | 1/4 | 0
4H ($dhcp$) | ABAC | hchc | $P6_{3}/mmc$ | 2.45 |  | 7.89 | 1.61 | 69.09 | 143 | $2a$ | 0 | 0 | 0 | 56
|  |  |  |  |  |  |  |  |  | $2c$ | 1/3 | 2/3 | 1/4 |
6H1 | ABACBC | chcchc | $P6_{3}/mmc$ | 2.45 |  | 11.85 | 1.62 | 69.21 | 139 | $2b$ | 0 | 0 | 1/4 | 54
|  |  |  |  |  |  |  |  |  | $4f$ | 1/3 | 2/3 | –0.0863 |
6H2 | ABCBCB | hchhhc | $P\bar{6}m2$ | 2.45 |  | 11.78 | 1.60 | 69.02 | 141 | $1c$ | 1/3 | 2/3 | 0 | 31
|  |  |  |  |  |  |  |  |  | $2i$ | 2/3 | 1/3 | 0.166 |
|  |  |  |  |  |  |  |  |  | $2g$ | 0 | 0 | 1/3 |
|  |  |  |  |  |  |  |  |  | $1f$ | 2/3 | 1/3 | 1/2 |
10H | ABCBCBACAC | cchhhcchhh | $Cmcm$ | 2.45 | 4.25 | 19.61 | 1.60 | 68.91 | 140 | $4c$ | 0 | 0.6666 | 1/4 | 33
|  |  |  |  |  |  |  |  |  | $8f$ | 0 | 0 | 0.8489 |
|  |  |  |  |  |  |  |  |  | $8f$ | 0 | 0.3330 | –0.0515 |
3C ($fcc$) | ABC | ccc | $Fm\bar{3}m$ | 3.45 |  |  |  | 69.35 | 139 | 4$a$ | 0 | 0 | 0 | 83

Figure 3: Types of close-packed structures associated with different local
stacking sequences of (001)hcp/(111)fcc planes; (a) 2H ($hcp$), (b) 3C
($fcc$), (c) 4H ($dhcp$), (d) 6H1, and (e) 6H2 in Ramsdell notation. Schematic
ball and stick models represent the local (only for nearest neighbor)
environment of each layer, either (a) $hcp$ or (b) $fcc$-like. Nishiyama
(1978)

For the spin-polarized state, magnetic supercells were constructed to realize
AFM ordered states. The AFM structures proposed for 6H1, 6H2, and $hcp$ (2H)
are orthorhombic structures with the space groups $Pmcm$, $Pm2m$, and $Pmma$,
respectively. The orthorhombic cell for the $hcp$ structure is equivalent to
the previously reported structure, which is called AFM (II). The lattice
constants of these orthorhombic cells are taken as $a_{orth}$ = $a_{h}$,
$b_{orth}$ = $\sqrt{3}$$a_{h}$, and $c_{orth}$ = $c_{h}$, as illustrated in
Fig. 4. The common transformation matrix $m_{i,j}$, between hexagonal and
orthorhombic lattices can be written as (1, 1, 0), (–1, 1, 0), and (0, 0, 1).
AFM unit cells for all the structural polytypes of Fe are constructed with
this matrix to study their structural and magnetic stabilities. Among these,
the AFM states of $fcc$, 6H1, 6H2, and $hcp$ are determined to be stable. For
the $fcc$ structure, two possible AFM states have been reported previously
using body-centered tetragonal ($bct$) lattices. The results that an AFM
double layer (AFM-D) with a $q$-vector of [001] is more energetically
favorable than that of single-layer (AFM-S) were verified.Herper  _et al._
(1999) The magnetic unit cells of AFM-S and AFM-D belong to the space groups,
$P4/mmm$ and $Pmm2$, respectively. The $c$/$a$ ratio for the $bct$ cell for
AFM-S and AFM-D are $\sqrt{2}$ and $2\sqrt{2}$, respectively, which
corresponds to the $fcc$ structure.

Figure 4: Orthorhombic unit cell for AFM ordered states. $a_{h}$ and
$a_{orth}$ are the $a$ lattice constants for the hexagonal and orthorhombic
lattices, respectively. $b_{orth}$ is the $b$ lattice constant for the
orthorhombic cell. $c_{orth}$ is equal to $c_{h}$.

For both spin-polarized and unpolarized calculations, structural optimizations
for lattice parameters and internal atomic coordinates were performed by
maintaining the space group symmetry and stacking sequences. To determine the
equilibrium lattice parameters in the ground state at ambient pressure, the
calculated total energies at different volumes were fitted using the Birch-
Murnaghan EoS. The calculated equilibrium volumes and bulk modulus for
nonmagnetic and AFM states are listed in Tables. 1 and 2, respectively. For
the structural optimizations, the equilibrium volume was first determined by
fixing the axial ratio ($c$/$a$) at a constant value, and the $c$/$a$ ratio
was then optimized by maintaining the volume at equilibrium using the fourth
order of the numerical fitting. The optimized structural parameters are
summarized in Table. 2. The details of the structural and magnetic properties
at the ground state are discussed in the next subsection.

Figure 5: Schematic illustrations of the spin arrangement in the AFM states for (a) 2H ($hcp$), (b) 6H1, and (c) 6H2 with the direction of spins (red arrows). The spin structures are mapped on each hexagonal structure, although the spin-polarized calculations were performed with an orthorhombic cell. Black spheres with up-arrows represent Fe atoms with up-spin, and blue spheres with down-arrows indicate those with down-spin. The direction and magnitude of spin moments ($m_{spin}$) on the A, B, and C layers are illustrated with yellow, blue, and green arrows, respectively. Wyckoff positions with spin in the orthorhombic cells are shown.  Table 2: Crystallographic data and spin magnetic moments for various structural polytypes of Fe with AFM ordered states together with optimized lattice constant ratios ($c$/$a$) in a hexagonal cell. Wyckoff positions and their internal atomic coordinates ($x$, $y$, $z$) are shown. The equilibrium lattice volume $V_{0}$ is given per Fe atom. $B_{0}$ is the bulk modulus. $\Delta E$ is the relative total energy with respect to that of the AFM (II) state in $hcp$-Fe. Type | Space group | $a$ | $b$ | $c$ |  $c$/$a$ | $V_{0}$ | $B_{0}$ | Wyckoff | $x$ | $y$ | $z$ | $m_{spin}$ | $\Delta E$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  | [Å] | [Å] | [Å] |  |  [Bohr3] | [GPa] |  |  |  |  |  [$\mu_{B}$] | [meV]
2H ($hcp$) | $Pmcm$ | 2.47 | 4.28 | 3.98 | 1.616 | 70.96 | 100 | $2c$ | 0 | 0.3333 | 1/4 | 1.17 | 0
|  |  |  |  |  |  |  | $2c$ | 1/2 | 0.8333 | 1/4 | –1.17 |
6H1 | $Pmnm$ | 2.47 | 4.29 | 12.02 | 1.620 | 71.73 | 87 | $2b$ | 3/4 | –0.2485 | 1/4 | –0.91 | 53
|  |  |  |  |  |  |  | $4f$ | 1/4 | 0.0888 | –0.0863 | –1.46 |
|  |  |  |  |  |  |  | $2a$ | 1/4 | –0.2520 | 1/4 | 0.91 |
|  |  |  |  |  |  |  | $4f$ | 1/4 | 0.4116 | 0.0862 | 1.46 |
6H2 | $Pm2m$ | 2.47 | 4.29 | 11.97 | 1.611 | 71.49 | 100 |  1$a$ | 0 | –0.3508 | 0 | –1.38 | 15
|  |  |  |  |  |  |  |  1$c$ | 0 | 0.3343 | 1/2 | –1.09 |
|  |  |  |  |  |  |  |  2$e$ | 0 | 0.0096 | –0.3359 | –1.27 |
|  |  |  |  |  |  |  |  2$e$ | 0 | 0.3320 | –0.1671 | –1.37 |
|  |  |  |  |  |  |  |  1$b$ | 1/2 | 0.1492 | 0 | 1.38 |
|  |  |  |  |  |  |  |  1$d$ | 1/2 | –0.1657 | 1/2 | 1.09 |
|  |  |  |  |  |  |  |  2$f$ | 1/2 | 0.510 | –0.3359 | 1.27 |
|  |  |  |  |  |  |  |  2$f$ | 1/2 | –0.1681 | –0.1671 | 1.37 |
3C AFM-S | $P4/mmm$ | 2.47 | 2.47 | 3.49 |  | 71.85 | 100 |  1$a$ | 0 | 0 | 0 | 1.36 | 63
($fcc$) |  |  |  |  |  |  |  |  1$d$ | 1/2 | 1/2 | 1/2 | –1.36 |
3C AFM-D | $Pmm2$ | 2.50 | 2.50 | 3.54 |  | 74.76 | 63 |  1$a$ | 0 | 0 | 0 | 1.48 | 56
($fcc$) |  |  |  |  |  |  |  |  1$d$ | 1/2 | 1/2 | –0.25 | 1.48 |
|  |  |  |  |  |  |  |  1$a$ | 0 | 0 | 0.5 | –1.48 |
|  |  |  |  |  |  |  |  1$d$ | 1/2 | 1/2 | 0.25 | –1.48 |

###  IV.2 Calculated structural and magnetic stability

####  IV.2.1 Nonmagnetic states

We first discuss the nonmagnetic phase stability of LPSO structures together
with $hcp$ and $fcc$ structures. Table 1 shows the equilibrium structural
parameters and total energy differences among different LPSO phases. The
equilibrium lattice volumes and bulk moduli for all the structural polytypes
are quite small, 69 Bohr3 per atom and 140 GPa, respectively. However, those
for AFM states are slightly different to each other as listed in Table 2. The
differences are expected to come from the magneto-volume effect and spin-spin
interactions on the anisotropic triangular lattice. Within the nonmagnetic
calculations, $hcp$-Fe has the lowest energy at the lowest temperature of 0 K.
Therefore, the energy difference is compared with respect to the $hcp$ phase.
Among these candidates, the 6H2 and 10H structures are energetically close to
each other, which is similar to the results of DFT studies for pure-Mg. Iikubo
_et al._ (2012) On the other hand, the 6H1 structure is energetically more
unfavorable than 6H2, but energetically close to $fcc$ and $dhcp$-Fe.

####  IV.2.2 Stability of AFM states

FLAPW calculations were also performed with the spin-polarized form of GGA-
PBE, and stable AFM spin structures were found for the 6H1, 6H2, $hcp$, and
$fcc$ structures. These magnetic states are energetically lower than those of
the nonmagnetic phases. Several possible ordering patterns were also assumed
for the $dhcp$ (4H) and 10H structures, including ferromagnetic and
ferrimagnetic states; however, no magnetic orderings were stabilized. Figure 5
illustrates the direction and magnitude of the calculated spin moments
($m_{spin}$) for LPSO structure mapping on the hexagonal lattice. For
$hcp$-Fe, the calculated $m_{spin}$ values are close to those reported in
Refs. Steinle-Neumann  _et al._ (1999, 2004a), as shown in Fig. 5(a) and Table
2. The magnetic ground state AFM (II) has been reported to be stable up to
approximately 50 GPa, and the calculated EoS for the AFM (II) configuration
shows better agreement with the experimental EoS and elasticity measurements
than those obtained from nonmagnetic calculations. Steinle-Neumann  _et al._
(1999, 2004a, 2004b); Lizárraga  _et al._ (2008) Therefore, we only show the
volume dependence of total energies that consider AFM ordering in Fig. 6,
together with those for $fcc$ and $hcp$-Fe. With AFM order taken into account,
the differences of the total energies between different structures become more
distinct. Here we find that the 6H2 structure becomes more energetically close
to the most stable structure of $hcp$. On the other hand, the 6H1 structure is
also stabilized by AFM ordering; however, it is as unstable as $fcc$ with
AFM-D ordering. Therefore, we suggest that the LPSO-like phase observed by TEM
measurements adopts the 6H2 structure.

Next, we discuss the difference in the magnetic properties between the 6H1 and
6H2 structures, focusing on the site dependence of spin moments in each layer.
The equilibrium volume and $c$/$a$ ratio for the two structures with AFM
states are almost the same, as plotted in Fig. 6. As schematically shown in
Fig. 5(b), the absolute values of $m_{spin}$ for the inequivalent Fe atoms in
the 6H1 structure at the B and C layers are 0.91 and 1.46 $\mu_{B}$/atom,
respectively. The $z$ position for the 2$a$ and 2$b$ sites are constrained by
the space group symmetry. A layers (2$a$ sites) are sandwiched between two
adjacent Fe layers with 4$f$ sites. The most interesting feature in the 6H1
structure is that there is no symmetry constraint on the $z$-coordinate of the
4$f$ position, $z$(4$f$). This small displacement of $z$(4$f$) affects the
magnetic properties of 6H1-Fe. The ideal $z$ position for the 4$f$ site is
0.0833; however, the equilibrium $z$ positions determined by structural
relaxation under AFM order is –0.0863 and 0.0862. A similar $z$-position
dependence of the magnetic properties in ferromagnetic $dhcp$-FeH has been
discussed by one of the present authors.Tsumuraya  _et al._ (2012) Fig. 5(c)
shows the spin structure for 6H2-Fe. In this spin structure, the inequivalent
Fe atoms result in four different values of $m_{spin}$, while 6H1 has two.
Thus, 6H2 has more degrees of freedom. The $z$ positions for the 2$e$ and 2$f$
sites that belong to the B and C layers do not have symmetry constraints on
the $z$ axis. Only the $m_{spin}$ values at the 1$d$ and 1$c$ sites that
belong to the B layer are relatively small (1 $\mu_{B}$/atom), and those of
the other three layers are larger than those for the B layer.

It should be noted that determining the ground state for both $hcp$ and $fcc$
Fe from first-principles calculations has proven to be a difficult task
because the energy difference between different magnetic ordering patterns is
in a narrow energy window. Asada and Terakura (1992); Jiang and Carter (2003);
Lebert  _et al._ (2019) Spin frustration occurs in the alternate stacking of
the triangular lattice, where magnetic spins are expected to be frustrated
with respect to AFM ordering, in that these lattices cannot be ordered
antiferromagnetically, so that all neighbors have opposite spins. This leads
to incommensurate spin waves,Thakor  _et al._ (2003) non-collinear spin
order,Lizárraga  _et al._ (2008) spin nematic states,Lebert  _et al._ (2019)
and quantum spin liquid states. In this paper, we do not aim to obtain perfect
relative energies between LPSO structures and $hcp$/$fcc$-Fe. In stead we
intend to determine which LPSO structures are energetically close to the
$hcp$-phase. The evaluation of magnetic contributions for Gibbs free energy
are also a further problem to be investigated.Körmann  _et al._ (2008)

Figure 6: Calculated total energies for AFM ordered states in the $fcc$, 6H1,
6H2, and $hcp$ structures within the GGA-PBE functional. Total energy per atom
as a function of the lattice volume per atom. Solid and open triangles
represent the 6H1 and 6H2 structures, respectively. Open squares show the
$fcc$ structure with the AFM-D ordered state. Solid circles show the $hcp$
structure with the AFM (II) state.

The phase stabilities with AFM states are discussed next using the convex hull
diagram. The polytypes are characterized by their hexagonality, Raffy  _et
al._ (2002) which is defined as the ratio of the number of hexagonal layers
($n_{h}$) to the total number of layers per unit cell, $\alpha$ =
$n_{h}$/($n_{h}$ \+ $n_{c}$). The $\alpha$ parameters for the end member of
the $fcc$ and $hcp$ (2H) polytypes are 0 and 1, respectively, while those of
the intermediate polytypes, $dhcp$ (4H), 6H1 and 6H2, are 0.5, 0.333, and
0.666, respectively. $n_{h}$ and $n_{c}$ are the respective numbers of locally
hexagonal and cubic layers in each polytype, which are obtained using hc
notation for each unit cell shown in Table. 1. Figure 7 shows the convex hull
diagram: the total energies for the stable AFM phases of the $fcc$, $hcp$,
6H1, and 6H2 structures as a function of the hexagonality. The total energies
for the $fcc$ and $hcp$ structures are taken as the reference energies, and
the convex hull is calculated as $\Delta$$E$ = $E$($Y$) – [$E$(${fcc}$) +
$E$(${hcp}$)]/2, where $Y$ represents the 6H1 and 6H2 structures in the
present case. The calculated convex hull for the four close-packed structures
indicates that the 6H2 structure with AFM ordering is on the hull, while the
6H1 structure is above the hull with a relatively large hull distance. This
result clearly suggests the possible observation of the 6H2 structure between
the $\gamma$ and $\epsilon$ phases.

Figure 7: Convex hull diagram for Fe as a function of hexagonality $\alpha$.
Solid circles are the differences of the total energies with respect to those
of the $fcc$ and $hcp$ structures; $\Delta$$E$ = $E$($Y$) – [$E$(${fcc}$) +
$E$(${hcp}$)]/2, where $Y$ represents 6H1 and 6H2. The total energy $E$, for
each phase is that for the DFT-optimized structure with the most stable AFM
spin arrangement pattern.

###  IV.3 Origin of phase stabilities

It is also interesting to understand how the electronic structure is changed
by the stacking sequence along the $c$-axis. The DOS for metals with a $hcp$
structure is generally characterized by the presence of a deep valley or dip
near the Fermi level ($E_{F}$), where $E_{F}$ is at the lowest position (the
bottom) of the deep valley.Inoue and Yamashita (1973); Paxton  _et al._
(1997); Blaha  _et al._ (1988) In the 1970s, before first-principles
calculations were established, Yamashita $et$ $al.$ suggested that the depth
of the valleys in the DOS represents the magnitude of splitting of the main
peaks of the DOS between those in the occupied and unoccupied states. This is
a kind of energy separation between the bonding and antibonding molecular
orbitals due to the significant hybridization between the $s$ and $p$
states.Slater (1965); Inoue and Yamashita (1973) By comparing the electronic
structure of $hcp$-Be and $hcp$-Mg, they also suggested that the degree of
energy splitting between the two main peaks (the width of the deep valley or
the dip) appears as a difference in the enthalpy of formation (or cohesive
energies) between the different metals.Yamashita (1973) At that time, it would
have been difficult to quantitatively calculate the difference of the heats of
formation; however, these quantities can now be calculated from first
principles.

Figures 8(a)–8(c) compare the calculated total DOS for the $hcp$, 6H2, and 6H1
structures with AFM ordering. A typical deep valley in the DOS observed in
$hcp$ metal is evident in Fig. 8(a). The total DOS of the 6H2 structure also
has a similar DOS valley near $E_{F}$ [Fig. 8(b)], and we find several small
peaks appear near the bottom of the DOS compared with those for $hcp$-Fe. This
can be understood from the ABCBCA stacking of the 6H2 structure; the A layer
appears every six layers, and layers other than the A layers are stacked in a
BCBC pattern. A comparison of the spin structures of $hcp$ and 6H2 shown
schematically in Figs. 5(a) and 5(b) indicates that the local spin arrangement
and the stacking sequence agree well with those of the ABAB pattern in the
$hcp$ structure. On the other hand, the total DOS for the 6H1 structure has
many peaks near $E_{F}$ and the valley of the DOS is obscured, as shown in
Fig. 8(c). We have shown that this deviation of the DOS from the $hcp$
structure is the microscopic origin of the structural stability of the
candidate LPSO structures.

Figure 8: Total DOS for the AFM ordered state of the (a) 2H ($hcp$), (b) 6H2,
and (c) 6H1 structures of Fe at ambient pressure. Dashed lines represent the
Fermi levels.

##  V Summary

The structural and magnetic properties of long-period stacking order
structures (polytypism) in pure-Fe were studied by first-principles DFT
calculations. During deformation-induced martensitic transformation from
$\gamma$-austenite to $\epsilon$-martensite, a new phase different from the
$\epsilon$ phase was discovered in Fe-Mn-Si based alloys. To clarify the
actual stacking sequence of the LPSO-like phase, we proposed several
structural models for the LPSO structure of pure Fe, such as 4H, 6H1, 6H2, and
10H structures, and structural optimization was performed using first-
principles DFT calculations. From a search among the stable magnetic phases,
stable AFM states were identified in the 6H1 and 6H2 structures. An
nonmagnetic and AFM state of 6H2 was also revealed as the most energetically
close to the $hcp$ structure, and the observed LPSO-like phase has a high
probability to adopt the 6H2 structure. We show that the magnetic contribution
can stabilize the LPSO phases. The electronic origin of the phase stability is
attributed to the depth of the valley in the DOS near the Fermi level, where
the energy splitting between two peaks in occupied and unoccupied states is
large, which leads to maximization of the phase stability. The relationship
between the electronic structure and the phase stability was quantitatively
verified for LPSO and $hcp$ phases in Fe, which was proposed for $hcp$-metals
in the 1970s.

##  VI ACKNOWLEDGEMENTS

One of the authors (T.T.) thank T. Oguchi, W-T. Geng, and D. Drozdenko for
stimulating discussions. This research was funded by a Grant-in-Aid for
Scientific Research (No. 19K04988) from the Japan Society for the Promotion of
Science (JSPS). T. T. is supported in part by the Leading Initiative for
Excellent Young Researchers (LEADER) program of the Ministry of Education,
Culture, Sports, Science and Technology, Japan (MEXT), and the Cooperative
Research Program of Network Joint Research Center for Materials and Devices.
Computations were mainly conducted using the computer facilities of ITO at
Kyushu University and MASAMUNE at IMR, Tohoku University.

## References

  * Sato  _et al._ (1986) A. Sato, Y. Yamaji, and T. Mori, Acta Metall. 34, 287 (1986).
  * Ogawa and Kajiwara (1993) K. Ogawa and S. Kajiwara, Mater. Trans. 34, 1169 (1993).
  * Kikuchi  _et al._ (1995) T. Kikuchi, S. Kajiwara, and Y. Tomota, Mater. Trans. 36, 719 (1995).
  * Ohtsuka  _et al._ (1995) H. Ohtsuka, S. Kajiwara, T. Kikuchi, T. Ishihara, and K. Nagai, J. de physique IV 5, 451 (1995).
  * Sawaguchi  _et al._ (2006) T. Sawaguchi, P. Sahu, T. Kikuchi, K. Ogawa, S. Kajiwara, A. Kushibe, M. Higashino, and T. Ogawa, Scripta Mater 54, 1885 (2006).
  * Nishiyama (1978) Z. Nishiyama,  _Martensitic transformation_ (London Academic Press, 1978).
  * Sawaguchi  _et al._ (2015) T. Sawaguchi, I. Nikulin, K. Ogawa, K. Sekido, S. Takamori, T. Maruyama, Y. Chiba, A. Kushibe, Y. Inoue, and K. Tsuzaki, Scripta Mater 99, 49 (2015).
  * Sawaguchi  _et al._ (2016) T. Sawaguchi, T. Maruyama, H. Otsuka, A. Kushibe, Y. Inoue, and K. Tsuzaki, Mater. Trans. 57, 283 (2016).
  * Kawamura  _et al._ (2001) Y. Kawamura, K. Hayashi, A. Inoue, and T. Masumoto, Mater. Trans. 42, 1172 (2001).
  * Zhang  _et al._ (2020) X. Zhang, C. Li, Q. Wang, M. Zheng, Z. Ye, and J. Gu, Mater. Res. Lett. 8, 454 (2020).
  * Lysak and Nikolin (1963) L. I. Lysak and B. I. Nikolin, Dokl. Akad. Nauk SSSR 153, 812 (1963).
  * Kononenko  _et al._ (1969) V. L. Kononenko, L. N. Larikov, L. I. Lysak, B. I. Nikolin, and Y. F. Yurchenko, Fiz. metal. metalloved. 28, 889 (1969).
  * Lysak  _et al._ (1970) L. I. Lysak, Y. N. Makogon, and B. Nikolin, Fiz. metal. metalloved. 29, 512 (1970).
  * Lysak and Nikolin (1967) L. I. Lysak and B. I. Nikolin, Fiz. metal. metalloved. 23, 93 (1967).
  * Oka  _et al._ (1972) M. Oka, Y. Tanaka, and K. Shimizu, Jpn. J. Appl. Phys. 11, 1073 (1972).
  * Oka  _et al._ (1973) M. Oka, Y. Tanaka, and K. Shimizu, Mater. Trans. 14, 148 (1973).
  * Hwang  _et al._ (1991) K. H. Hwang, W. S. Yang, T. B. Wu, C. M. Wan, and J. G. Byrne, Acta Metallurgica et Materialia 39, 825 (1991).
  * Chao and Ho (1992) C.-H. Chao and N.-J. Ho, Scr Metall Mater 26, 1863 (1992).
  * Lee  _et al._ (1995) W. Lee, F.-R. Chen, S. Chen, G. Olson, and C. Wan, Acta Metallurgica et Materialia 43, 21 (1995).
  * Wan  _et al._ (2001) J. Wan, S. Chen, and T. Y. Hsu, Calphad 25, 355 (2001).
  * Hohenberg and Kohn (1964) P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).
  * Kohn and Sham (1965) W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).
  * Ohno and Mekata (1971) H. Ohno and M. Mekata, J. Phys. Soc. Jpn 31, 102 (1971).
  * Endoh and Ishikawa (1971) Y. Endoh and Y. Ishikawa, J. Phys. Soc. Jpn 30, 1614 (1971).
  * Kennedy and Hicks (1987) S. J. Kennedy and T. J. Hicks, J. Phys. F: Met. Phys. 17, 1599 (1987).
  * Bleskov  _et al._ (2016) I. Bleskov, T. Hickel, J. Neugebauer, and A. Ruban, Phys. Rev. B 93, 214115 (2016).
  * Reyes-Huamantinco  _et al._ (2012) A. Reyes-Huamantinco, P. Puschnig, C. Ambrosch-Draxl, O. E. Peil, and A. V. Ruban, Phys. Rev. B 86, 060201(R) (2012).
  * Taylor  _et al._ (1982) R. D. Taylor, G. Cort, and J. O. Willis, J. Appl. Phys. 53, 2064 (1982).
  * Nicol and Jura (1963) M. Nicol and G. Jura, Science 141, 1035 (1963).
  * Ohno (1971) H. Ohno, J. Phys. Soc. Jpn 31, 92 (1971).
  * Kai  _et al._ (1970) K. Kai, T. Nakamichi, and M. Yamamoto, J. Phys. Soc. Jpn 29, 1094 (1970).
  * Arapan  _et al._ (2020) S. Arapan, P. Nieves, H. C. Herper, and D. Legut, Phys. Rev. B 101, 014426 (2020).
  * V. E. Antonov  _et al._ (1980) V. E. V. E. Antonov, I. T. Belash, V. F. Degtyareva, E. G. Ponyatovskii, and V. I. Shiraev, Sov. Phys. Dokl. 25, 490 (1980).
  * Antonov  _et al._ (1998) V. Antonov, K. Cornell, V. Fedotov, A. Kolesnikov, E. Ponyatovsky, V. Shiryaev, and H. Wipf, J. Alloys Compd. 264, 214 (1998).
  * Vočadlo  _et al._ (2000) L. Vočadlo, J. Brodholt, D. Alfè, M. J. Gillan, and G. D. Price, Phys. Earth Planet. Inter 117, 123 (2000).
  * Saxena  _et al._ (1995) S. K. Saxena, L. S. Dubrovinsky, P. Häggkvist, Y. Cerenius, G. Shen, and H. K. Mao, Science 269, 1703 (1995).
  * Yoo  _et al._ (1996) C. Yoo, P. Söderlind, J. Moriarty, and A. Cambell, Phys. Lett. A 214, 65 (1996).
  * Wimmer  _et al._ (1981) E. Wimmer, H. Krakauer, M. Weinert, and A. J. Freeman, Phys. Rev. B 24, 864 (1981).
  * Weinert (1981) M. Weinert, J. Math. Phys. 22, 2433 (1981).
  * Koellng and Arbman (1975) D. D. Koellng and G. O. Arbman, J. Phys. F: Metal Phys. 5, 2041 (1975).
  * Perdew  _et al._ (1996) J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
  * Weinert  _et al._ (2009) M. Weinert, G. Schneider, R. Podloucky, and J. Redinger, J. Phys.: Condens. Matter 21, 084201 (2009).
  * Cotes  _et al._ (1995) S. Cotes, M. Sade, and A. F. Guillermet, Metall. Mater. Trans. A 26, 1957 (1995).
  * Dong  _et al._ (2006) Z. Dong, T. Sawaguchi, T. Kikuchi, F. Yin, K. Ogawa, P. Sahu, and S. Kajiwara, Mater. Sci. Eng. A 442, 404 (2006).
  * Abe  _et al._ (2002) E. Abe, Y. Kawamura, K. Hayashi, and A. Inoue, Acta Metall. 50, 3845 (2002).
  * Egusa and Abe (2012) D. Egusa and E. Abe, Acta Metall. 60, 166 (2012).
  * Christian and Vitek (1970) J. Christian and V. Vitek, Reports on Progress in Physics 33, 307 (1970).
  * Herper  _et al._ (1999) H. C. Herper, E. Hoffmann, and P. Entel, Phys. Rev. B 60, 3839 (1999).
  * Iikubo  _et al._ (2012) S. Iikubo, K. Matsuda, and H. Ohtani, Phys. Rev. B 86, 054105 (2012).
  * Steinle-Neumann  _et al._ (1999) G. Steinle-Neumann, L. Stixrude, and R. E. Cohen, Phys. Rev. B 60, 791 (1999).
  * Steinle-Neumann  _et al._ (2004a) G. Steinle-Neumann, L. Stixrude, and R. E. Cohen, Phys. Rev. B 69, 219903(E) (2004a).
  * Steinle-Neumann  _et al._ (2004b) G. Steinle-Neumann, L. Stixrude, and R. E. Cohen, Proc. Nat. Acac. Sci. 101, 33 (2004b).
  * Lizárraga  _et al._ (2008) R. Lizárraga, L. Nordström, O. Eriksson, and J. Wills, Phys. Rev. B 78, 064410 (2008).
  * Tsumuraya  _et al._ (2012) T. Tsumuraya, Y. Matsuura, T. Shishidou, and T. Oguchi, J. Phys. Soc. Jpn 81, 064707 (2012).
  * Asada and Terakura (1992) T. Asada and K. Terakura, Phys. Rev. B 46, 13599 (1992).
  * Jiang and Carter (2003) D. E. Jiang and E. A. Carter, Phys. Rev. B 67, 214103 (2003).
  * Lebert  _et al._ (2019) B. W. Lebert, T. Gorni, M. Casula, S. Klotz, F. Baudelet, J. M. Ablett, T. C. Hansen, A. Juhin, A. Polian, P. Munsch, G. Le Marchand, Z. Zhang, J.-P. Rueff, and M. d’Astuto, Proc. Nat. Acac. Sci. 116, 20280 (2019).
  * Thakor  _et al._ (2003) V. Thakor, J. B. Staunton, J. Poulter, S. Ostanin, B. Ginatempo, and E. Bruno, Phys. Rev. B 67, 180405(R) (2003).
  * Körmann  _et al._ (2008) F. Körmann, A. Dick, B. Grabowski, B. Hallstedt, T. Hickel, and J. Neugebauer, Phys. Rev. B 78, 033102 (2008).
  * Raffy  _et al._ (2002) C. Raffy, J. Furthmüller, and F. Bechstedt, Phys. Rev. B 66, 075201 (2002).
  * Inoue and Yamashita (1973) S. Inoue and J. Yamashita, J. Phys. Soc. Jpn 35, 677 (1973).
  * Paxton  _et al._ (1997) A. T. Paxton, M. Methfessel, and D. G. Pettifor, Proc. R. Soc. Lond. A 453, 1493 (1997).
  * Blaha  _et al._ (1988) P. Blaha, K. Schwarz, and P. H. Dederichs, Phys. Rev. B 38, 9368 (1988).
  * Slater (1965) J. C. Slater,  _Quantum Theory of Molecules and Solids II_ (McGraw-Hill, 1965).
  * Yamashita (1973) J. Yamashita,  _Kotai-densiron (in Japanese)_ (Asakura Publishing Co., Ltd., 1973).