Reversible giant out-of-plane Rashba effect in two-dimensional GaXY (X= Se, Te; Y= Cl, Br, I) compounds for persistent spin helix

Abstract

The coexistence of ferroelectricity and spin-orbit coupling (SOC) in noncentrosymmetric systems may allow for a nonvolatile control of spin degrees of freedom by switching the ferroelectric polarization through the well-known ferroelectric Rashba effect (FRE). Although the FER has been widely observed for bulk ferroelectric systems, its existence in two-dimensional (2D) ferroelectric systems is still very rarely discovered. Based on first-principles calculations, supplemented with k⃗⋅p⃗\vec{k}\cdot\vec{p} analysis, we report the emergence of the FRE in the GaXY (X= Se, Te; Y= Cl, Br, I) monolayer compounds, a new class of 2D materials having in-plane ferroelectricity. Due to the large in-plane ferroelectric polarization, a giant out-of-plane Rashba effect is observed in the topmost valence band, producing unidirectional out-of-plane spin textures in the momentum space. Importantly, such out-of-plane spin textures, which can host a long-lived helical spin mode known as a persistent spin helix, can be fully reversed by switching the direction of the in-plane ferroelectric polarization. Thus, our findings can open avenues for interplay between the unidirectional out-of-plane Rashba effect and the in-plane ferroelectricity in 2D materials, which is useful for efficient and non-volatile spintronic devices.

Full text

Reversible giant out-of-plane Rashba effect in two-dimensional GaXY (X= Se, Te; Y=
Cl, Br, I) compounds for persistent spin helix
Siti Amalia Sasmito, Muhammad Anshory, Ibnu Jihad, and Moh. Adhib Ulil Absor∗
Department of Physics, Universitas Gadjah Mada, Sekip Utara, BLS 21 Yogyakarta Indonesia.
(Dated: June 22, 2021)
The coexistence of ferroelectricity and spin-orbit coupling (SOC) in noncentrosymmetric systems
may allow for a nonvolatile control of spin degrees of freedom by switching the ferroelectric polariza-
tion through the well-known ferroelectric Rashba effect (FRE). Although the FER has been widely
observed for bulk ferroelectric systems, its existence in two-dimensional (2D) ferroelectric systems is
still very rarely discovered. Based on first-principles calculations, supplemented with ~
k·~p analysis,
we report the emergence of the FRE in the GaXY (X= Se, Te; Y= Cl, Br, I) monolayer compounds,
a new class of 2D materials having in-plane ferroelectricity. Due to the large in-plane ferroelectric
polarization, a giant out-of-plane Rashba effect is observed in the topmost valence band, producing
unidirectional out-of-plane spin textures in the momentum space. Importantly, such out-of-plane
spin textures, which can host a long-lived helical spin mode known as a persistent spin helix, can
be fully reversed by switching the direction of the in-plane ferroelectric polarization. Thus, our
findings can open avenues for interplay between the unidirectional out-of-plane Rashba effect and
the in-plane ferroelectricity in 2D materials, which is useful for efficient and non-volatile spintronic
devices.
I. INTRODUCTION
During the last decade, spin-orbit coupling (SOC) has
attracted increasing interest in various fields, including
spintronics, quantum computing, topological matter, and
cold atom systems1,2. In particular, the SOC links the
spin degree of freedom to the orbital motion of electrons
in a solid without additional external magnetic field,
thus playing an important role in semiconductor-based
spintronics1–3. For a system with a lack of inversion
symmetry, the SOC induces an effective magnetic field,
which results in spin-splitting bands and non-trivial spin
textures in the momentum space, known as the Rashba4
and Dresselhaus5effects. The Rashba effect has been
widely observed on a system having structural inversion
asymmetry such as semiconductor quantum well6,7, sur-
face heavy metal8,9, and several two-dimensional (2D)
layered compounds10–14, while the Dresselhaus effect oc-
curs on a system hold bulk inversion asymmetries such as
bulk zincblende5and wurtzite semiconductors15. Re-
cently, ferroelectric materials have witnessed a surge of
interest in the field of spintronics since they enable in-
tegration of the SOC and ferroelectricity through the
well-known ferroelectric Rashba effect (FRE)16. In such
functionality, the spin textures of the spin-splitting bands
can be fully reversed in a non-volatile way by switching
the direction of the ferroelectric polarization. As such,
the FER is very promising for spintronic devices imple-
menting, for instant, tunneling anomalous and spin Hall
effects17,18. The FER was first predicted theoretically in
bulk GeTe19 and experimentally confirmed in GeTe thin-
film20,21. After that, numerous candidates for FRE ma-
terials have been recently proposed, which mainly comes
from the bulk metal-organic halide perovskite, includ-
ing (FA)SnI322,23, hexagonal semiconductors (LiZnSb)24,
and oxides (KTaO325, HfO226, BiAlO327 ).
While the FRE has been widely studied for the bulk
ferroelectric materials16,19,22–27, due to favorable spin-
tronic applications in nanoscale devices28,29, ultrathin
two-dimensional (2D) materials supporting the FRE
would be more desirable. However, the small thickness
in the 2D materials may lose the FRE functionality since
the ferroelectric polarization is suppressed by an enor-
mous depolarizing field30,31. Recently, a new class of 2D
materials exhibiting robust ferroelasticity and ferroelec-
tricity has been reported, which comes from GaXY (X=
Se, Te; Y= Cl, Br, I) monolayer (ML) compounds32,33.
These compounds are stable under room temperature
exhibiting the large in-plane ferroelectricity32–35. More-
over, due to the strong SOC in these materials, the large
band splitting with tunable spin polarization in the con-
duction band minimum have recently been predicted35.
In addition to the observed large spin splitting in the
GaXY ML compounds, the SOC induces doubly degen-
erate nodal loops featuring an hourglass type dispersion
has also been reported34 . Considering the fact that the
GaXY ML exhibits large in-plane ferroelectricity and
strong SOC, it is expected that achieving the FRE in
these materials is highly plausible, which is expected to
be useful for spintronic applications.
In this paper, through first-principles density-
functional theory (DFT) calculations, complemented
with ~
k·~p analysis, we predict the emergence of the FRE in
the 2D GaXY ML compounds. We find that due to the
large in-plane ferroelectric polarization in the GaX Y ML,
a giant out-of-plane Rashba effect is observed in the top-
most valence band, exhibiting unidirectional out-of-plane
spin textures in the momentum space. Importantly, such
out-of-plane spin textures, which can host a long-lived he-
lical spin mode known as a persistent spin helix (PSH),
can be fully reversed by switching the direction of the
in-plane ferroelectric polarization. Moreover, the physi-
cal mechanism of the FRE found in the present system is
well analyzed within the framework of the ~
k·~p Hamilto-
arXiv:2106.10596v1 [cond-mat.str-el] 20 Jun 2021

2
FIG. 1. Atomic structure of the GaXY ML compounds cor-
responding to their symmetry operations viewed in the x−y
and x−zplanes, respectively, is presented. The unit cell of
the crystal is indicated by the black lines and characterized
by aand blattice parameters in the xand ydirections. The
crystal is characterized by the glide reflection ¯
Mxy consisted of
reflection about z= 0 plane followed by a/2 translation along
the xaxis and b/2 translation along the yaxis, the twofold
screw rotation ¯
C2yconsisted of π/2 rotation around y=b/2
line followed by a/2 translation along the xaxis, and the mir-
ror reflection Myz around the x= 0 plane. ~rGa−X(X0)and
~rY(Y0)−Ga are the vectors connected the Ga atom to chalco-
gen X(X0) atom and the halogen Y(Y0) atom to Ga atom,
respectively in the unit cell. These vectors determine the dis-
tortion vector as indicated by ~r0. (b) first Brillouin zone of
the GaXY ML compounds is shown, where high symmetry ~
k
points (Γ, X,Y, and M) are indicated.
nian model incorporating the in-plane ferroelectricity and
point group symmetry of the crystal. Finally, a possible
implication of the reversible spin textures of the present
system for spintronics will be discussed.
II. COMPUTATIONAL DETAILS
We have performed first-principles DFT calculations
using the OpenMX code36–39 , based on norm-conserving
pseudo-potentials and optimized pseudo-atomic localized
basis functions. The exchange-correlation functional was
treated within generalized gradient approximation by
Perdew, Burke, and Ernzerhof (GGA-PBE)40,41. The ba-
sis functions were expanded by linear combination of mul-
tiple pseudo atomic orbitals generated using a confine-
ment scheme36,38, where two s-, two p-, two d-character
numerical pseudo atomic orbitals were used. The accu-
racy of the basis functions as well as pseudo-potentials
we used were carefully bench-marked by the delta gauge
method42.
We applied a periodic slab to model the GaXY ML,
where a sufficiently large vacuum layer (20 ˚
A) was ap-
plied in order to avoid the spurious interaction between
slabs [Fig. 1(a)]. The 12 ×10 ×1k-point mesh was used
to discretize the first Brillouin zone (FBZ) [Fig. 1(b)].
We adopted the modern theory of polarization based
on the Berry phase (BP) method43 implemented in the
OpenMX code to calculate the ferroelectric polarization.
During the structural relaxation, the energy convergence
criterion was set to 10−9eV. The lattice and positions of
the atoms were optimized until the Hellmann-Feynman
force components acting on each atom was less than 1
meV/˚
A.
The SOC was included self consistently in all calcu-
lations by using j-dependent pseudo potentials44. We
calculated the spin textures by deducing the spin vector
components (Sx,Sy,Sz) in the reciprocal lattice vector ~
k
from the spin density matrix45. The spin density matrix,
Pσσ0(~
k, µ), were calculated using the following relation,
Pσσ0(~
k, µ) = ZΨσ
µ(~r,~
k)Ψσ0
µ(~r,~
k)d~r, (1)
where Ψσ
µ(~r,~
k) is the spinor Bloch wave function. This
methods has been successfully applied on our recent stud-
ies on various 2D materials35,46–50.
III. RESULTS AND DISCUSSION
A. Atomic structure, symmetry, and
ferroelectricity
First, we characterize the optimized structural pa-
rameters, symmetry of the crystal, and ferroelectric-
ity of the GaXY ML compounds, where the atomic
structure is displayed in Fig. 1(a). The crystal struc-
ture of the GaXY ML is noncentrosymmetric having a
black-phosphorene-type structure belonging to P mn21
space group32–35,51. For the convenience ofdiscussion, we
choose the x(y) axis to be along the zigzag (armchair)
direction in the real space so that the reciprocal space is
characterized by the FBZ as shown in Fig. 1(b). There
are six atoms in the unit cell consisted of two Ga atoms,
two chalcogen atoms (labeled by Xand X0), and two
halogen atoms (labeled by Yand Y0). These atoms are
invariant under the following symmetry operations: (i)
identity operation E, (ii) the glide reflection ¯
Mxy con-
sisted of reflection about z= 0 plane followed by a/2
translation along the xaxis and b/2 translation along
the yaxis, where aand bis the lattice parameters of the
crystal, (iii) the twofold screw rotation ¯
C2ydefined as

3
TABLE I. The optimized structural-related parameters and ferroelectric polarization are displayed. Here, the lattice parameters
[a(in ˚
A), b(in ˚
A)], the bondlength between the Ga and chalcogen X(X0) atoms [|~r|Ga−X(X0)(in ˚
A)], the bondlength between
the halogen Y(Y0) and Ga atoms [|~r|Y(Y0)−Ga (in ˚
A)], the magnitude of the distortion vector |~r0|(˚
A), and the in-plane electric
polarization P(in pC/m) obtained for each GaXY ML compounds are shown.
GaXY ML a(˚
A) b(˚
A) |~r|Ga−X(X0)(˚
A) |~r|Y(Y0)−Ga (˚
A) |~r0|(˚
A) P(pC/m)
GaSeCl 3.87 5.53 2.47 2.23 0.23 478.9
GaSeBr 3.95 5.63 2.47 2.37 0.15 459.1
GaSeI 4.17 5.93 2.49 2.60 0.08 352.5
GaTeCl 4.17 5.93 2.70 2.26 0.33 542.6
GaTeBr 4.26 6.08 2.71 2.37 0.28 530.1
GaTeI 4.41 6.33 2.73 2.61 0.25 519.8
π/2 rotation around y=b/2 line followed by a/2 trans-
lation along the xaxis, and (iv) the mirror reflection
Myz around the x= 0 plane. The optimized lattice pa-
rameters (a,b) for each GaXY ML compound are listed
in Table 1. We find that due to the difference value be-
tween the aand bparameters, the crystal geometry of the
GaXY ML is anisotropic, implying that these materials
have different mechanical responses being subjected to
uniaxial strain along the x- and y-direction similar to that
observed on various group IV monochalcogenide46,52,53.
The atomic structure of the GaXY ML can be viewed
as GaX(X0) ML surface functionalized by halogen Y(Y0)
atoms bonded to the Ga atoms forming a sandwiched
structure with Y-GaX(X0)-Y0sequence [see Fig. 1(a)].
We then introduce a distortion vector, ~r0, defined as
~r0=~rGa−X+~rY−Ga +~rGa−X0+~rY0−Ga,(2)
where ~rGa−X(X0)and ~rY(Y0)−Ga are the vectors con-
nected the Ga atom to chalcogen X(X0) atom and the
halogen Y(Y0) atom to Ga atom, respectively, in the unit
cell [see left side in Fig. 1(a)]. Here, the magnitude
|~r|Ga−X(X0)and |~r|Y(Y0)−Ga represent the Ga-X(X0) and
Y(Y0)-Ga bond lengths, respectively. Due to the Myz
mirror symmetry operation along the y−zplane, we
obtain that ~r0·ˆx= 0, while the screw operation ¯
C2yim-
plies that ~r0·ˆz= 0. Accordingly, ~r0should be parallel
to the in-plane ydirection and induces intrinsic spon-
taneous polarization along the ydirection. Generally,
the optimized structures of the GaXY ML compounds
show that the |~r|Ga−X(X0)bond lengths are larger than
the |~r|Y(Y0)−Ga bond lengths [see Table I]. However, the
|~r|Y(Y0)−Ga bond lengths substantially increases for the
compounds with the same chalcogen (X) atoms but have
the heavier halogen (Y) atoms, thus decreasing the mag-
nitude of the distortion vector, |~r0|. Therefore, the de-
creased in magnitude of the in-plane electric polarization
is expected, which is in fact confirmed by our BP cal-
culation results shown in Table I. The existence of the
in-plane ferroelectricity allows us to maintain the FRE
in the GaXY ML compounds, which is expected to be
observed due to the large SOC.
In the next section, we will show how the in-plane fer-
roelectricity plays an important role in the SOC and elec-
tronic properties of the GaXY ML compounds.
B. Spin-orbit coupled ferroelectric and spin
textures
We will start our analysis by deriving the general SOC
Hamiltonian in the 2D systems having in-plane ferroelec-
tricity. The SOC Hamiltonian is further analyzed for the
GaXY ML compounds within the framework of the ~
k·~p
Hamiltonian model using the method of invariant54. Fi-
nally, we discuss the important implication of the derived
SOC Hamiltonian in terms of the spin splitting and spin
textures involving to the in-plane ferroelectricity.
The SOC occurs in solid-state materials when an elec-
tron moving at velocity ~v through an electric field ~
Eex-
periences an effective magnetic field due to the relativistic
transformation of electromagnetic fields. A general form
of the SOC Hamiltonian HSO can be expressed as:
HSO =~
Ω·~σ, (3)
where ~σ = (σx, σy, σz) are the Pauli matrices and ~
Ω is
a wave-vector dependent spin-orbit field (SOF) that is
simply written as
~
Ω(~
k) = αˆ
E×~
k, (4)
where αis the strength of the SOC that is proportional
to the magnitude of the electric field, |~
E|,ˆ
Edenotes the
electric filed direction, and ~
kis the wave vector repre-
senting the momentum electron. The HS O is invariant
under time reversal symmetry operations, T, so that the
following relation holds, T HSO T−1=−~
Ω(−~
k)·~σ =HSO .
Accordingly, the SOF is a odd in wave vector ~
k, i.e.
~
Ω(−~
k) = −~
Ω(~
k), which also depends on the crystal sym-
metry of the system.
Lets us consider the general 2D systems having in-
plane ferroelectricity, where we assumed that the sponta-
neous in-plane electric polarization being oriented along
the in-plane y-direction. In this case, an effective elec-
tric field is induced, which is also oriented along the y-
directions, ~
E=Eˆx. Due to the 2D nature of the systems,
we have ~
k=kxˆx+kyˆyfor the wave vector ~
k, and by us-
ing the explicit form of the effective electric field ~
E, we
find that the SOF ~
Ω in Eq. (4), can be expressed as
~
Ω = αkxˆz. (5)

4
We can see that for the 2D systems having in-plane fer-
roelectricity, the SOF is enforced to be unidirectional in
the out-of-plane direction. Inserting the Eq. (4) to the
Eq. (3), we find that
HSO =αkxσz.(6)
The Eq. (6) clearly shows that the HSO is characterized
only by one component of the wave vector kxand the
out-of-plane spin vector σz, yielding a unidirectional out-
of-plane Rashba effect.
TABLE II. Transformation rules for the wave vector ~
kand
spin vector ~σ under the considered point-group symmetry op-
erations. Time-reversal symmetry, implying a reversal of both
spin and momentum, is defined as T=iσyK, where Kis the
complex conjugation, while the point-group operations are
defined as ˆ
C2y=iσy,ˆ
Myz =iσx, and ˆ
Mxy =iσz. The last
column shows the invarian terms, where the underlined term
are invariant under all symmetry operations.
Symmetry (kx, ky) (σx, σy, σz) Invariants
Operations
ˆ
T=iσyK(−kx,−ky) (−σx,−σy,−σz)kiσj
(i, j =x, y, z)
ˆ
C2y=iσy(−kx, ky) (−σx, σy,−σz)kxσx,kxσz,kyσy
ˆ
Myz =iσx(−kx, ky) (σx,−σy,−σz)kxσy,kxσz,kyσx
ˆ
Mxy =iσz(kx, ky) (−σx,−σy, σz)kxσz,kyσz
The HSO in Eq. (6) is also obtained by considering
the wave-vector symmetry group at the high symmetry
points in the first-Brillouin zone. Here, we assumed that
only linear terms with respect to the wave vector ~
kcon-
tribute to the HSO . For the case of the GaXY ML com-
pounds, the wave-vector symmetry group of the P mn21
space group at the high symmetry points such as Γ, X,
and Ypoints, belongs to C2vpoint group35 , which has
two mirror reflections about the x−yplane (Mxy) and the
y−zplane (Myz ) as well as twofold rotation C2yaround
the y-axis. The transformation rules for the wave vec-
tor ~
kand spin vector ~σ under the considered point-group
symmetry operations are listed in Table II. By applying
the methods of invariant54, we list all the invariant term
of the HSO in the form of product between the ~
kand ~σ
components (see the right column in Table II) and select
those specific terms which are invariant under all sym-
metry operations as indicated by the underlined terms in
the right column in Table II. We find that only kxσzterm
of the HSO is invariant under all symmetry operations of
the C2v, which is identical to the HSO shown in Eq. (6).
Next, we characterize low energy properties of the
present system involving the HSO term of Eq. (6). The
effective ~
k·~p Hamiltonian Hincluding the kinetic and
HSO terms can be expressed as
H=~2k2
2m∗+αkxσz.(7)
Solving eigenvalue problem involving the Hamiltonian of
FIG. 2. (a) Schematic view of band dispersion showing an
anisotropic splitting around ~
k= (0,0,0) and (b) the corre-
sponding Fermi line are presented. The red and blue arrows
indicate Szand −Szspin orientation in the momentum space,
respectively. In Fig. (b), the Fermi line has the shifted two
pair loops characterized by the shifting wave vector ~
Qalong
the kxdirection where the spins are persisting in the fully out-
of-plane direction, resulting in the formation of the persistent
spin textures (PST). (c) Schematic view of the persistent spin
helix (PSH) mode emerge under the PST formation with the
wavelength of lP SH = 2π/|~
Q|. (d) Schematic correlation be-
tween the in-plane ferroelectricity and the spin textures is
shown. The two stable ferroelectric phases of the 2D ma-
terials having opposite in-plane ferroelectric polarization are
indicated. The switching spin textures are expected by re-
versing the in-plane ferroleectric polarization.
Eq. (7) leads to the eigenstates
Ψ~
k↑=ei~
k↑·~r 1
0(8)
and
Ψ~
k↓=ei~
k↓·~r 1
0,(9)
corresponding to the energy dispersion,
E↑↓ =~2k2
2m∗±αkx.(10)

5
This dispersion indicates that a strongly anisotropic spin
splitting occurs around the ~
k= (0,0,0) point, i.e., the
energy bands are lifted along kxdirection but are degen-
erated along the kydirection [Fig. 2(a)]. Importantly,
this dispersion is characterized by the shifting property,
E↓(~
k) = E↑(~
k+~
Q), where the ~
Qis the shifting wave
vector given by
~
Q=2m∗α
~2[1,0,0].(11)
The Eqs. (10) and (11) implies that a constant-energy
cut shows two Fermi loops whose centers are displaced
from their original point by ±~
Qas schematically shown
in Fig. 2(b).
The spin texture, which is ~
k-dependent spin configu-
ration, is determined from the expectation values of the
spin operators, i.e., ~
S= (~/2)hψ~
k|~σ|ψ~
ki, where ψ~
kis the
electron’s eigenstates. By using ψ~
kgiven in Eqs. (8) and
(9), we find that
~
S±=±~
2[0,0,1].(12)
This shows that the spin configuration in the k-space is
locked being oriented in the out-of-plane directions as
schematically shown in Fig. 2(b). Such a typical spin
configuration forms a persistent spin textures (PST) sim-
ilar to that observed for [110] Dresselhauss model55. Pre-
viously, it has been reported that the PST is known to
host a long-lived helical spin mode known as a persistent
spin helix (PSH)55–57, enabling long-range spin transport
without dissipation55–60, and hence very promising for an
efficient spintronic devices.
The PSH arises when the SOF is unidirectional, pre-
serving a unidirectional spin configuration in the k-space.
When an electron moving in the real space is accompa-
nied by the spin precession around the SOF, a spatially
periodic mode of the spin polarization is generated. Ac-
cording to Eq. (5), the magnitude of the effective mag-
netic field can be expressed as B= 2|~
Ω|/γ~, where γ
is the gyromagnetic ratio. Therefore, the angular fre-
quency of the precession motion, ω, can be calculated
using the relation, ω=−γB = 2αkx/~. The spin pre-
cession angle, θ, around the yaxis at time t, is obtained
by θ=ωt = 2αkxt/~. At the same time, the traveling
distance of the electron is given by l=vt =~kxt/m∗,
where vis the electron velocity. By eliminating t, we
find that θ= 2αm∗l/~. When θ= 2π, we then obtain
the wavelength of the PSH, lP SH 55,
lP SH =π~2/(m∗α).(13)
Furthermore, in term of the shifting wave vector ~
Qde-
fined in Eq. (11), we can write the lP SH as
lP SH = 2π/|~
Q|.(14)
A schematic picture of the PSH mode enforced by
the unidirectional SOF is displayed in Fig. 2(c),
where a spatially periodic mode of the spin polariza-
tion with the wavelength lP SH is shown. Such spin-
wave mode protects the spins of electrons from decoher-
ence through suppressing the Dyakonov-Perel spin relax-
ation mechanism61 and renders an extremely long spin
lifetime55–60.
Finally, we study the correlation between spin textures
and ferroelectricity. Here, an important property called
reversible spin textures holds, i.e., the direction of the
spin textures is locked and switchable by reversing the
direction of the spontaneous electric polarization. Fig.
2(d) shows a schematic view of the spin textured fer-
roelectric in the GaXY ML compounds showing fully
reversible out-of-plane spin textures. It is shown that
switching the direction of the in-plane ferroelectric po-
larization from ~
Pto −~
Pleads to reversing the direction
of the out-of-plane spin textures from z- to −z-direction.
From the symmetry point of view, switching the elec-
tric polarization direction ~
Pis equivalent to the space
inversion symmetry operation I, which changes the wave
vector from ~
kto −~
k, but preserves the spin vector ~
S62.
Now, suppose that |ψ~p (~
k)iis the Bloch wave function of
the crystal with electric polarization ~
P. The inversion
symmetry operation Ion the Bloch wave function hold
the following relation, I|ψ~
P(~
k)i=|ψ−~
P(−~
k)i. Applying
the time-reversal symmetry Tbrings −~
kback to ~
kbut
flip the spin vector ~
S, thus T I|ψ~
P(~
k)i=|ψ−~
P(~
k)i. The
expectation values of spin operator hSican be further
expressed in term of ~
Pand ~
kvectors as
h~
Si−~
P ,~
k=hψ−~
P(~
k)|~
S|ψ−~
P(~
k)i
=hψ~
P(~
k)|I−1T−1~
ST I |ψ~
P(~
k)i
=hψ~
P(~
k)|(−~
S)|ψ~
P(~
k)i
=h−~
Si~
P ,~
k,
(15)
which clearly shows that the spin directions is fully re-
versed by switching the direction of the electric polariza-
tion ~
P.
In the next section, we implement these general de-
scription of the spin-orbit coupled ferroelectric to discuss
our results from the first-principles DFT calculations on
various GaXY ML compounds.
C. First-principles DFT analyses
Figure 3 shows the electronic band structure of the
GaXY ML compounds calculated along the selected ~
k
paths in the FBZ corresponding to the density of states
(DOS) projected to the atomic orbitals. It is found that
the GaXY ML compounds are semiconductors with di-
rect or indirect band gaps depending on the chalcogen
(X) atoms. In the case of the GaSeYMLs, the electronic
band structure shows a direct bandgap where the valence
band maximum (VBM) and conduction band minimum

6
FIG. 3. Electronic band structures calculated with (purple lines) and without (black lines) including the SOC for various
GaXY ML compounds: (a) GaSeCl, (b) GaSeBr, (c) GaSeI, (d) GaTeCl, (e) GaTeBr, and (d) GaTeI. Partial density of states
(PDOS) projected to the atomic orbitals is also shown. In the PDOS, the black, blue, yellow, red, green, and pink lines indicate
the Ga-s,X(Se, Te)-s, Ga-p,X(Se, Te)-p, and Y(Cl, Br, I)-porbitals, respectively.
(CBM) is located at the Γ point [Figs. 3(a)-3(c)]. The
VBM at the Γ point retains for the case of the GaTeY
MLs but the CBM shifts to the kpoint along the Γ −Y
line, resulting in an indirect bandgap [Figs. 3(d)-3(f )].
We find that the band gap significantly decreases for the
compounds with the same chalcogen Xatoms but has the
larger Znumber of the halogen (Y) atoms. For an in-
stant, the calculated bandgap for the GaTeCl ML is 2.17
eV under GGA level, which is much larger than that for
the GaTeI ML (1.10 eV). Our calculated DOS projected
to the atomic orbitals confirmed that the VBM is mostly
dominated by the contribution of the chalcogen X-por-
bital with a small admixture of the Ga-pand halogen Y-
porbitals, while the CBM is mainly originated from the
Ga-sorbital with a small contribution of Ga-p, chalcogen
X-pand halogen Y-porbitals [Figs. 3(a)-(f)].
Introducing the SOC, however, strongly modifies the
electronic band structures of the GaXY ML compounds.
Here, we observed a significant band splitting produced
by the SOC due to the lack of the inversion symmetry,
which is mainly visible at the kbands along the Γ−X−M
symmetry lines [Figs. 3(a)-(f)]. However, along the Γ−Y
line in which the wave vector ~
kis parallel to the effective
electric field associated with the in-plane ferroelectric po-
larization, the bands are double degenerated. Since the
electronic states near the Fermi level are important for
transport carriers, we then focused our attention on the
bands near the VBM. Fig. 4(a) shows the calculated
band structure along the Y−Γ−Xline around the VBM
for GaTeCl ML as a representative example of the GaXY
ML compounds. At the Γ point, the electronic states at
the Γ point are double degenerated due to time reversibil-
ity. This doublet splits into singlet when considering the
bands ~
kalong the Γ −Xline. However, the doublet re-
mains for the ~
kalong the Γ −Yline, which is protected
by the ¯
C2yscrew rotation and the ¯
Mxy glide mirror re-
flection. Accordingly, a strongly anisotropic splitting is
clearly observed around the Γ point as highlighted by the
red line in Fig. 4(a), which is in good agreement with the
energy dispersion shown in Eq. (10) as well as Fig. 2(a).
We noted here that the remaining band degeneracy
at the wave vector ~
kalong the Γ −Yline can be ex-
plained in term of the symmetry analysis. Since the
wave vector ~
kat the Γ −Yline is invariant under
¯
C2yand ¯
Mxy symmetry operations, the folowing rela-
tion holds, ¯
Mxy ¯
C2y=−e−ikx¯
C2y¯
Mxy, where the minus
sign comes from the fact that both ¯
C2yand ¯
Mxy oper-
ators are anti-commutative, ¯
C2y,¯
Mxy= 0 due to the
anti-commutation between σyand σzspin rotation oper-
ators, σy, σz= 0. Supposed that 
ψgis an eigenvector
of ¯
Mxy operator with the eigenvalue of g, we obtain that
¯
Mxy(¯
C2y
ψg) = −g(¯
C2y
ψg). This evident shows that
both 
ψgand ¯
C2y
ψg) states are distinct states degen-

7
FIG. 4. (a) Electronic band structures calculated with the SOC around the VBM along Y−Γ−Xline for the GaTeCl ML as
representative example of the GaXY ML compounds. The spin-split bands at the VBM around the Γ point is highlighted. (b)
Momentum-resolved map of the spin-splitting energy calculated along the entire of the first Brillouin zone is shown. The color
bars in Fig. 4(b) shows the spin-splitting energy ∆E=|E(~
k↑)−E(~
k↓)|, where E(~
k↑) and E(~
k↓) are the energy for the
~
kbands with spin up and spin down, respectively. (c) Spin textures projected to the k-space for the upper and lower bands
at the VBM around the Γ point are shown. Here, color bars represent expectation values of the out-of-plane spin component
hSzi. (d) hSziprojected to the Fermi line calculated at constant energy cut of 1 meV below the degenerate state at the VBM
around the the Γ point.
erated at the same energy, thus ensuring the double de-
generacy of the states at the wave vector ~
kalong the Γ−Y
line. To further clarify the observed anisotropic splitting
around the Γ point, we show in Fig. 4(b) momentum-
resolved map of the spin-splitting energy calculated along
the entire of the FBZ. Consistent with the band struc-
tures, we identify the non-zero spin-splitting energy ex-
cept for the bands ~
kalong the Γ −Yline. Here, the
largest splitting is observed at the ~
kclosed to the Γ point
at along the Γ −Xline, where the splitting energy up to
0.25 eV is achieved. Such value is comparable with the
splitting energy observed on various 2D transition metal
dichalcogenides MX2(M= Mo, W; X= S, Se, Te) MLs
[0.15 eV - 0.46 eV]13,14,63–65. The large splitting energy
observed in the present system is certainly sufficient to
ensure proper function of spintronic devices operating at
room temperature66 .
The nature of the anisotropic splitting around the Γ
point at the VBM is further analyzed by identifying the
spin textures of the spin-split bands. As shown in Fig.
4(c), it is found that a uniform pattern of the spin tex-
tures is observed around the Γ point, which is mostly
characterized by fully out-of-plane spin components Sz
rather than the in-plane spin components (Sx, Sy). These
spin textures are switched from Szto −Szwhen crossing
at kx=0 along the Γ−Yline. Although we identified large
in-plane spin components (Sx, Sy) in the Γ −Yline, the
net in-plane spin polarization vanishes, which is due to
the equal population of the opposite in-plane spin polar-
ization between the outer and inner branches of the spin
split bands [see black arrows in Fig. 4(c)]. Such a pat-
tern of the spin textures, which is similar to that observed
on several 2D ferroelectric materials such as WO2Cl269
and various group IV monochalcogenide MLs46,47,50,67, is
strongly different from the in-plane Rashba-like spin tex-
tures reported on the widely studied 2D materials12–14,64.
Moreover, the fully out-of-plane spin texture becomes

8
TABLE III. Several selected PST systems in 2D materials and parameters characterizing the strength of the SOC (α, in eV˚
A)
and the wavelength of the PSH mode (lP SH , in nm).
2D materials α(eV˚
A) lP SH (nm) Reference
GaXY compounds
GaSeCl 1.2 2.89 This work
GaSeBr 0.85 4.09 This work
GaSeI 0.53 6.57 This work
GaTeCl 2.65 1.20 This work
GaTeBr 2.40 1.45 This work
GaTeI 1.90 1.83 This work
Group IV Monochalcogenide
(Sn,Ge)X(X= S, Se, Te) 0.07 - 1.67 8.9×102- 1.82 Ref.47
GeXY (X , Y = S, Se, Te) 3.10 - 3.93 6.53 - 8.52 Ref.50
Layeted SnTe 1.28 - 2.85 8.80 - 18.3 Ref.67
Strained SnSe 0.76 - 1.15 Ref.46
SnSe-X(X= Cl, Br, I) 1.60 - 1.76 1.27 - 1.41 Ref.48
Defective transition metal dichalcogenides
line defect in PtSe20.20 - 1.14 6.33 - 28.19 Ref.49
line defect in (Mo,W)(S,Se)20.14 - 0.26 8.56 - 10.18 Ref.68
Other 2D ML
WO2Cl20.90 Ref.69
clearly visible when measured at the constant energy
cut of 1 meV below the degenerated states at the VBM
around the Γ point [Fig. 4(d)]. Here, two circular loops
of the Fermi lines with the opposite Szspin components
are observed, which are shifted along the Γ −X(kx) di-
rection. The observed spin textures, as well as Fermi
lines, are all consistent well with our ~
k·~p Hamiltonian
model presented in Eq. (7) and the schematic pictures
shown in Figs. 2(a)-(b). Remarkably, the observed unidi-
rectional out-of-plane spin textures in the present system
lead to the PST55,57, which can host a long-lived helical
spin-wave mode through the PSH mechanism55–60.
The observed spin splitting and spin textures can be
quantified by the strength of the SOC, α, which is ob-
tained from the unidirectional out-of-plane Rashba model
given by Eq. (7). Here, we can rewrite the energy dis-
persion of Eq. (10) in the following form:
E(k) = ~2
2m∗|k| ± k0+ER,(16)
where ERand k0are the shifting energy and the wave
vector evaluated from the spin-split bands along the Γ −
X(kx) line as illustrated in Fig. 4(a). Accordingly, the
following relation holds,
α=2ER
k0
.(17)
Both Erand k0are important parameters to stabilize
spin precession and achieve a phase offset for different
spin channels in the spin-field effect transistor device70.
In table III, we summarize the calculated result of the
SOC strength αin Table III, and compare this result
with a few selected PST systems previously reported on
several 2D materials. It is found that the calculated value
of αfor the GaTeCl ML is 2.65 eV˚
A, which is the largest
among the GaXY ML compounds. This value is compa-
rable with that observed on the PST systems reported for
several 2D group IV monochalcogenide including GeXY
(X, Y = S, Se, Te) MLs (3.10 - 3.93 eV˚
A)50, layered SnTe
(1.28 - 2.85 eV˚
A)67. However, the calculated value of α
is much larger than that observed on the PST systems
found in other class of 2D materials such as WO2Cl2ML
(0.90 eV˚
A)69 and transition metal dichalcogenide MX2
MLs with line defect such as PtSe2(0.20 - 1.14 eV˚
A)49
and (Mo,W)X2(X=S, Se) (0.14 - 0.26 eV˚
A)68.
The emergence of the PST with large SOC strength α
predicted in the present system indicates that the forma-
tion of the PSH mode with a substantially small wave-
length lP SH of the spin polarization is achieved. Here,
the wavelength lP SH can be estimated by using Eq. (13)
evaluated from the band dispersion along the Γ−Xline in
the VBM [see the insert of Fig. 4(a)]. The resulting wave-
length lP SH for all members of GaX Y ML compounds
are shown in Table III. In particular, we find a very small
wavelength lP SH of the PSH mode for the GaTeCl ML
(1.20 nm), which is the smallest over of all known 2D ma-
terials so far [see Table III]. Importantly, the small wave-
length of the PSH mode observed in the present system is
typically on the scale of the lithographic dimension used
in the recent semiconductor industry71, which is possible
to access the features down to the nanometers scale with
sub-nanosecond time resolution by using near-field scan-
ning Kerr microscopy. Thus, we concluded that that the
present system is promising for miniaturization spintron-
ics devices.
Before summarizing, we highlighted the interplay be-
tween the in-plane ferroelectricity, spin splitting, and the
spin textures in the GaXY ML compounds. Fig. 5(a)
displayed the in-plane electric polarization as a function

9
FIG. 5. Relation between polarization, spin splitting, and spin textures. (a) In-plane electric polarization ~
Pof the GaTeCl
ML as a function of ferroelectric distortion τis shown. The insert shows the optimized structure of the GaTeCl ML in the
ferroelectric phase with ~
Pand −~
Ppolarization. τis defined as the magnitude of the distortion vector |~r|of the systems given
in Eq. (1) normalized by the magnitude of the distortion vector of the optimized ferroelectric phase, |~r0|. Here τ= 0 represents
the paraelectric phase and τ= 1 shows the optimized ferroelectric phase. (b) Band structure of the GaTeCl ML calculated
along Y−Γ−Xline around the VBM as a function of the ferroelectric distortion τis presented. (c) The SOC strength αof
the GaTeCl ML as a function of the ferroelectric distortion τis presented. Reversible out-of-plane spin orientation in GaTeCl
ML calculated at constant energy cut of 1 meV below the degenerate state at the VBM around the Γ point for the optimized
ferroelectric phase with opposite in-plane electric polarization: (d) −~
Pand (e) ~
P.
of the ferroelectric distortion, τ. Here, τis defined as the
magnitude of the distortion vector |~r|of the systems de-
fined by Eq. (1), which is normalized by the magnitude of
the distortion vector of the optimized ferroelectric phase,
|~r0|. Therefore, τ= 0 represents the paraelectric phase,
while τ= 1 shows the optimized ferroelectric phase as
shown by the insert of Fig. 5(a). We can see that it is
possible to manipulate the in-plane electric polarization
~
Pby distorting the atomic position [see Fig. 1(a)]. The
dependence of the in-plane polarization on the ferroelec-
tric distortion τsensitively affects the spin-split bands
at the VBM around the Γ point as shown in Fig. 5(b).
It is found that the splitting energy and the position of
the VBM around the Γ point strongly depend on the
ferroelectric distortion, i.e., a decrease in τsubstantially
reduces the spin splitting energy while the position of
the VBVM shifts up to be higher in energy around the
Γ point. Accordingly, the significant change of the SOC
strength αis achieved, in which a linear trend of αas
a function of τis observed as shown in Fig. 5(c). Im-
portantly, our results also show that the SOC strength α
changes sign when the direction of the in-plane ferroelec-
tric polarization ~
Pis switched, resulting in a full reversal
of the out-of-plane spin textures shown in Figs. 5(d)-(e).
Such reversible spin textures are agreed well with our
symmetry analysis given by Eq. (15), putting forward
GaXY ML compounds as a candidate of the FER class
of 2D materials exhibiting the PST, which is useful for
efficient and non-volatile spintronic devices.
IV. CONCLUSION
In summary, we have investigated the emergence of
the FRE in GaXY (X= Se, Te; Y= Cl, Br, I) ML
compounds, a new class of 2D materials having in-plane
ferroelectricity, by performing first-principles density-
functional theory calculations supplemented with ~
k·~p
analysis. We found that due to the large in-plane ferro-
electric polarization, a giant unidirectional out-of-plane
Rashba effect is observed in the spin-split bands around
the VBM, exhibiting the unidirectional out-of-plane spin

10
polarization persisting in the entirely FBZ. These per-
sistent spin textures can host a long-lived persistent spin
helix mode55–57, characterized by the large SOC strength
and a substantially small wavelength of the helical spin
polarization. Importantly, we observed fully reversible
spin textures, which are achieved by switching the di-
rection of the in-plane ferroelectric polarization, thus of-
fering a possible application of the present system for
efficient and non-volatile spintronic devices operating at
room temperature.
The reversible unidirectional out-of-plane Rashba ef-
fect found in the present study is solely enforced by the
in-plane ferroelectricity and the non-symmorphic P nm21
space group symmetry of the crystal. Therefore, it is
expected that this effect can also be achieved on other
2D materials having similar crystal symmetry. Recently,
there are numerous 2D materials that are predicted to
have P nm21space group symmetry such as the 2D ele-
mental group V (As, Sb, and Bi) MLs72,73. Due to the
stronger SOC in these materials, the better resolution of
the unidirectional out-of-plane Rashba effect is expected
to be observed. Therefore, our prediction is expected to
trigger further theoretical and experimental studies in or-
der to find novel 2D ferroelectric systems supporting the
unidirectional out-of-plane Rashba effect, which is useful
for future spintronic applications.
ACKNOWLEDGMENTS
This research was partly supported by RTA pro-
gram (2021) supported by Universitas Gadjah Mada.
Part of this research was supported by PDUPT
(No.1684/UN1/DITLIT/DIT-LIT/PT/2021) and PD
(No.2186/UN1/DITLIT/DIT-LIT/PT/2021) Research
Grants funded by RISTEK-BRIN, Republic of Indonesia.
The computation in this research was performed using
the computer facilities at Universitas Gadjah Mada,
Republic of Indonesia.
∗adib@ugm.ac.id
1A. Manchon, H. C. Koo, J. Nitta, S. M. Frolov, and R. A.
Duine, Nat. Matter 14, 871 (2015).
2J. Varignon, L. Vila, A. Barth´el´emy, and M. Bibes, Nature
Physics 14, 322 (2018).
3S. D. Ganichev and L. E. Golub, physica status solidi (b)
251, 1801 (2014).
4E. I. Rashba, Sov. Phys. Solid State 2, 1224 (1960).
5G. Dresselhaus, Phys. Rev. 100, 580 (1955).
6J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys.
Rev. Lett. 78, 1335 (1997).
7A. D. Caviglia, M. Gabay, S. Gariglio, N. Reyren, C. Can-
cellieri, and J.-M. Triscone, Phys. Rev. Lett. 104, 126803
(2010).
8Y. M. Koroteev, G. Bihlmayer, J. E. Gayone, E. V.
Chulkov, S. Bl¨ugel, P. M. Echenique, and P. Hofmann,
Phys. Rev. Lett. 93, 046403 (2004).
9S. LaShell, B. A. McDougall, and E. Jensen, Phys. Rev.
Lett. 77, 3419 (1996).
10 H. L. Zhuang, V. R. Cooper, H. Xu, P. Ganesh, R. G.
Hennig, and P. R. C. Kent, Phys. Rev. B 92, 115302
(2015).
11 Z. S. Popovi´c, J. M. Kurdestany, and S. Satpathy, Phys.
Rev. B 92, 035135 (2015).
12 M. A. U. Absor, I. Santoso, Harsojo, K. Abraha, H. Ko-
taka, F. Ishii, and M. Saito, Phys. Rev. B 97, 205138
(2018).
13 Y. Affandi and M. A. Ulil Absor, Physica E: Low-
dimensional Systems and Nanostructures 114, 113611
(2019).
14 M. A. U. Absor, I. Santoso, Harsojo, K. Abraha, H. Ko-
taka, F. Ishii, and M. Saito, Journal of Applied Physics
122, 153905 (2017), https://doi.org/10.1063/1.5008475.
15 W.-T. Wang, C. L. Wu, S. F. Tsay, M. H. Gau, I. Lo, H. F.
Kao, D. J. Jang, J.-C. Chiang, M.-E. Lee, Y.-C. Chang,
C.-N. Chen, and H. C. Hsueh, Applied Physics Letters 91,
082110 (2007), https://doi.org/10.1063/1.2775038.
16 S. Picozzi, Frontiers in Physics 2, 10 (2014).
17 A. V. Vedyayev, M. S. Titova, N. V. Ryzhanova, M. Y.
Zhuravlev, and E. Y. Tsymbal, Applied Physics Letters
103, 032406 (2013), https://doi.org/10.1063/1.4815866.
18 A. Matos-Abiague and J. Fabian, Phys. Rev. Lett. 115,
056602 (2015).
19 D. Di Sante, P. Barone, R. Bertacco, and S. Picozzi, Ad-
vanced Materials 25, 509 (2012).
20 M. Liebmann, C. Rinaldi, D. Di Sante, J. Kellner, C. Pauly,
R. N. Wang, J. E. Boschker, A. Giussani, S. Bertoli,
M. Cantoni, L. Baldrati, M. Asa, I. Vobornik, G. Panac-
cione, D. Marchenko, J. S´anchez-Barriga, O. Rader,
R. Calarco, S. Picozzi, R. Bertacco, and M. Morgenstern,
Advanced Materials 28, 560 (2016).
21 C. Rinaldi, S. Varotto, M. Asa, J. S lawi´nska, J. Fujii,
G. Vinai, S. Cecchi, D. Di Sante, R. Calarco, I. Vobornik,
G. Panaccione, S. Picozzi, and R. Bertacco, Nano Letters
18, 2751 (2018).
22 A. Stroppa, D. Di Sante, P. Barone, M. Bokdam,
G. Kresse, C. Franchini, M.-H. Whangbo, and S. Picozzi,
Nature Communications 5, 5900 (2014).
23 M. Kepenekian, R. Robles, C. Katan, D. Sapori,
L. Pedesseau, and J. Even, ACS Nano 9, 11557 (2015).
24 A. Narayan, Phys. Rev. B 92, 220101 (2015).
25 L. L. Tao and J. Wang, Journal of Applied Physics 120,
234101 (2016), https://doi.org/10.1063/1.4972198.
26 L. L. Tao, T. R. Paudel, A. A. Kovalev, and E. Y. Tsym-
bal, Phys. Rev. B 95, 245141 (2017).
27 L. G. D. da Silveira, P. Barone, and S. Picozzi, Phys. Rev.
B93, 245159 (2016).
28 E. C. Ahn, npj 2D Materials and Applications 4, 17 (2020).
29 W. Han, APL Materials 4, 032401 (2016),
https://doi.org/10.1063/1.4941712.
30 J. Junquera and P. Ghosez, Nature 422, 506 (2003).
31 P. Gao, Z. Zhang, M. Li, R. Ishikawa, B. Feng, H.-J. Liu,
Y.-L. Huang, N. Shibata, X. Ma, S. Chen, J. Zhang, K. Liu,
E.-G. Wang, D. Yu, L. Liao, Y.-H. Chu, and Y. Ikuhara,
Nature Communications 8, 15549 (2017).
32 W. Zhou, S. Guo, S. Zhang, Z. Zhu, X. Song, T. Niu,

11
K. Zhang, X. Liu, Y. Zou, and H. Zeng, Nanoscale 10,
3350 (2018).
33 S.-H. Zhang and B.-G. Liu, Nanoscale 10, 5990 (2018).
34 W. Wu, Y. Jiao, S. Li, X.-L. Sheng, Z.-M. Yu, and S. A.
Yang, Phys. Rev. Materials 3, 054203 (2019).
35 M. A. U. Absor and F. Ishii, Phys. Rev. B 103, 045119
(2021).
36 T. Ozaki, Phys. Rev. B 67, 155108 (2003).
37 T. Ozaki, H. Kino, J. Yu, M. J. Han, N. Kobayashi, M. Oh-
futi, F. Ishii, T. Ohwaki, H. Weng, and K. Terakura,
http://www.openmx-square.org/ (2009).
38 T. Ozaki and H. Kino, Phys. Rev. B 69, 195113 (2004).
39 T. Ozaki and H. Kino, Phys. Rev. B 72, 045121 (2005).
40 J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.
Lett. 77, 3865 (1996).
41 W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).
42 K. Lejaeghere, G. Bihlmayer, T. Bj¨orkman, P. Blaha,
S. Bl¨ugel, V. Blum, D. Caliste, I. E. Castelli, S. J. Clark,
A. Dal Corso, S. de Gironcoli, T. Deutsch, J. K. Dewhurst,
I. Di Marco, C. Draxl, M. Du lak, O. Eriksson, J. A.
Flores-Livas, K. F. Garrity, L. Genovese, P. Giannozzi,
M. Giantomassi, S. Goedecker, X. Gonze, O. Gr˚an¨as,
E. K. U. Gross, A. Gulans, F. Gygi, D. R. Hamann,
P. J. Hasnip, N. A. W. Holzwarth, D. Iu¸san, D. B.
Jochym, F. Jollet, D. Jones, G. Kresse, K. Koepernik,
E. K¨u¸c¨ukbenli, Y. O. Kvashnin, I. L. M. Locht, S. Lubeck,
M. Marsman, N. Marzari, U. Nitzsche, L. Nordstr¨om,
T. Ozaki, L. Paulatto, C. J. Pickard, W. Poelmans,
M. I. J. Probert, K. Refson, M. Richter, G.-M. Rignanese,
S. Saha, M. Scheffler, M. Schlipf, K. Schwarz, S. Sharma,
F. Tavazza, P. Thunstr¨om, A. Tkatchenko, M. Torrent,
D. Vanderbilt, M. J. van Setten, V. Van Speybroeck, J. M.
Wills, J. R. Yates, G.-X. Zhang, and S. Cottenier, Science
351 (2016), 10.1126/science.aad3000.
43 R. D. King-Smith and D. Vanderbilt, Phys. Rev. B 47,
1651 (1993).
44 G. Theurich and N. A. Hill, Phys. Rev. B 64, 073106
(2001).
45 H. Kotaka, F. Ishii, and M. Saito, Japanese Journal of
Applied Physics 52, 035204 (2013).
46 M. Anshory and M. A. U. Absor, Physica E: Low-
dimensional Systems and Nanostructures 124, 114372
(2020).
47 M. A. U. Absor and F. Ishii, Phys. Rev. B 100, 115104
(2019).
48 M. A. U. Absor and F. Ishii, Phys. Rev. B 99, 075136
(2019).
49 M. A. U. Absor, I. Santoso, N. Yamaguchi, and F. Ishii,
Phys. Rev. B 101, 155410 (2020).
50 M. A. U. Absor, Y. Faishal, M. Anshory, I. Santoso, and
F. Ishii, Journal of Physics: Condensed Matter 33, 305501
(2021).
51 R. Kniep, A. Wilms, and H. J. Beister, Materials Research
Bulletin 18, 615 (1983).
52 X. Kong, J. Deng, L. Li, Y. Liu, X. Ding, J. Sun, and
J. Z. Liu, Phys. Rev. B 98, 184104 (2018).
53 B. Liu, M. Niu, J. Fu, Z. Xi, M. Lei, and R. Quhe, Phys.
Rev. Materials 3, 054002 (2019).
54 R. Winkler, S. Papadakis, E. De Poortere, and
M. Shayegan, Spin-Orbit Coupling in Two-Dimensional
Electron and Hole Systems (Springer, Berlin, 2003).
55 B. A. Bernevig, J. Orenstein, and S.-C. Zhang, Phys. Rev.
Lett. 97, 236601 (2006).
56 P. Altmann, M. P. Walser, C. Reichl, W. Wegscheider, and
G. Salis, Phys. Rev. B 90, 201306 (2014).
57 J. Schliemann, Rev. Mod. Phys. 89, 011001 (2017).
58 M. Kohda, V. Lechner, Y. Kunihashi, T. Dollinger, P. Ol-
brich, C. Sch¨onhuber, I. Caspers, V. Bel’kov, L. Golub,
D. Weiss, et al., Physical Review B 86, 081306 (2012).
59 M. Walser, C. Reichl, W. Wegscheider, and G. Salis, Na-
ture Physics 8, 757 (2012).
60 J. D. Koralek, C. P. Weber, J. Orenstein, B. A. Bernevig,
S.-C. Zhang, S. Mack, and D. Awschalom, Nature 458,
610 (2009).
61 M. I. Dyakonov and V. I. Perel, Sov. Phys. Solid State 13,
3023 (1972).
62 M. Kim, J. Im, A. J. Freeman, J. Ihm, and H. Jin, Pro-
ceedings of the National Academy of Sciences 111, 6900
(2014).
63 Z. Y. Zhu, Y. C. Cheng, and U. Schwingenschl¨ogl, Phys.
Rev. B 84, 153402 (2011).
64 Q.-F. Yao, J. Cai, W.-Y. Tong, S.-J. Gong, J.-Q. Wang,
X. Wan, C.-G. Duan, and J. H. Chu, Phys. Rev. B 95,
165401 (2017).
65 M. A. U. Absor, H. Kotaka, F. Ishii, and M. Saito, Phys.
Rev. B 94, 115131 (2016).
66 K. Yaji, Y. Ohtsubo, S. Hatta, H. Okuyama, K. Miyamoto,
T. Okuda, A. Kimura, H. Namatame, M. Taniguchi, and
T. Aruga, Nature Communications 1, 17 (2010).
67 H. Lee, J. Im, and H. Jin, Applied Physics Letters 116,
022411 (2020).
68 X. Li, S. Zhang, H. Huang, L. Hu, F. Liu, and Q. Wang,
Nano Letters 19, 6005 (2019).
69 H. Ai, X. Ma, X. Shao, W. Li, and M. Zhao, Phys. Rev.
Materials 3, 054407 (2019).
70 S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990).
71 G. Fiori, F. Bonaccorso, G. Iannaccone, T. Palacios,
D. Neumaier, A. Seabaugh, S. K. Banerjee, and
L. Colombo, Nature Nanotechnology 9, 768 (2014).
72 Y. Pan and J. Zhou, Phys. Rev. Applied 14, 014024 (2020).
73 C. Xiao, F. Wang, S. A. Yang, Y. Lu, Y. Feng, and
S. Zhang, Advanced Functional Materials 28, 1707383
(2018).