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Indones. J. Chem., xxxx, xx (x), xx - xx
Ibnu Jihad et al.
1
Prediction of Rashba Effect on Two-dimensional MX Monochalcogenides (M = Ge, Sn
and X = S, Se, Te) with Buckled Square Lattice
Ibnu Jihad*, Juhri Hendrawan, Adam Sukma Putra, Kuwat Triyana, and Moh. Adhib Ulil Absor
Department of Physics, Faculty of Mathematics and Natural Science, Universitas Gadjah Mada,
Sekip Utara BLS 21, Yogyakarta 55281, Indonesia
* Corresponding author:
tel: +62-895364829488
email: ibnu.jihad@ugm.ac.id
Received: September 5, 2019
Accepted: January 7, 2020
DOI: 10.22146/ijc.49331
Abstract
: The Rashba splitting are found in the buckled square lattice. Here, by applying
fully relativistic density
-functional theory (DFT) calculation, we confirm the existence of
the Rashba splitting in the conduction band minimum of various two
-dimensional MX
monochalcogenides (
M = Ge, Sn and X =
S, Se, Te) exhibiting a pair inplane Rashba
rotation of the spin textures.
A strong correlation has also been found between the size of
the Rashba parameter and the atomic number of chalcogen
atom for Γ and M point in
the first Brillouin zone. Our investigation clarifies that the buckled square lattice are
promising for inducing
the substantial Rashba splitting suggesting that the present system
is promising for spintronics device.
Keywords:
Ge monochalcogenides; Sn monochalcogenides; DFT method; spintronics;
square lattice; Rashba effect; spin textures
■ INTRODUCTION
Spintronics is a combinational word from spin
transport electronics or spin electronics, which is the next
generation of electronics. Spintronics explores the spin
properties of electrons rather than the charge properties
in electronics device, in which give us more degree of
freedom [1]. This fact makes spintronics device have
higher information density and also energetically efficient
because the smaller movement is needed for changing
(reading and writing) the spin structure. One branch of
the spintronics research field is spin-orbitronics, which is
focused on the exploitation of non-equilibrium material
properties using spin-orbit coupling (SOC). In the spin-
orbitronics, the investigation is aiming for searching
materials that have large enough SOC energy splitting,
especially the Rashba effect type to build spin field-effect
transistor (SFET) [2].
The Rashba splitting in the electronics band
structures is caused by the lack of inversion symmetry of
the materials [3], which occurred in a two-dimensional
(2D) structures of materials. Graphene as the first
example of the 2D materials having hexagonal structure
have been considered for this purpose. However, due to
the weak SOC, this material is not suitable for spin-
orbitronics [4-6]. Another example of 2D materials
having hexagonal structure is coming from the
transition metal dichalcogenides (TMDs) family [7].
Here, the breaking of inversion symmetry in the
hexagonal crystal, together with the strong SOC of
transition metal atom leads to the strong SOC splitting.
Recently, the 2D Group-IV monochalcogenides in
square lattice with black phosphorus structure have been
recently studied and proved as a suitable semiconductor
with sizeable Rashba splitting [9].
Although the SOC has been widely studied on the
2D materials with the hexagonal structure and black
phosphorus structure, this effect should apparently
appear on the other non-centrosymmetric 2D materials.
One of the candidates is 2D material which posses a
square lattice structure. The square lattice structure with
buckling has also been experimentally observed on
Bismuth (Bi) [10], making the realization of this
structure is plausible. Computational research on
buckled square lattice of lead chalcogenides showing
significant first-order Rashba effect in this structure [8].
Previously, a preliminary study on the electronic
structure of the MX monochalcogenides (M = Ge, Sn
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Indones. J. Chem., xxxx, xx (x), xx - xx
Ibnu Jihad et al.
2
and X = S, Se) has been conducted with buckled squared
lattice and also the calculation of Rashba splitting size up
to the-third order correction that was showing a sizeable
Rashba splitting in these materials [9].
This work showed that the Rashba splitting is also
observed in the MTe monochalcogenides (M = Ge, Sn)
with buckled squared lattice as additional materials to MX
monochalcogenides (M = Ge, Sn and X = S, Se) buckled
square lattice. It is found that the substantial Rashba
splitting is achieved in the conduction band minimum of
various 2D MX monochalcogenides showing clockwise-
anticlockwise rotation of the spin in the momentum
space. From these various TMDs materials, the size of the
Rashba parameter is strongly affected by the atomic
number of chalcogen atoms for different high symmetry
points in the first Brillouin zone.
■ COMPUTATIONAL METHOD
This work was started by modeling the two-
dimensional monolayer buckled square lattice crystal
(Fig. 1). The distance between slabs is set to 30 Å to
prevent interaction between slabs. The coordinate axis is
x = a, y = b, and c = z. The geometry is relaxed until the
force of each atom less than 1 meV/Å.
The calculation was performed using a density
functional theory (DFT) approach implemented on
OPENMX code [10] within the generalized gradient
approximation (GGA) for the exchange-correlation
energy [11]. The norm-conserving pseudopotentials [12]
was set to energy cut off 300 rydberg for charge density.
The optimum k-point sampling for the calculation is 8 ×
8 × 1. The wave function was expanded by a confinement
scheme of a linear combination of multiple pseudoatomic
orbitals [13-14]. The numerical pseudoatomic orbitals
used for M atom was two-s, two-p, three-d, and one-f,
and for X atom was three-s, three-p, two-d, and one-f.
The j-dependent pseudopotentials were applied for SOC
[15]. After gaining the stable structure of the materials,
the process continue to calculate the electronics band
structures with the corresponding density of states
(DOS) projected to the atomic orbitals.
■ RESULTS AND DISCUSSION
The optimized structure and the related
parameters, are summarized in Table 1 and have good
agreement with previous report [9]. The PbS and PbSe
calculations have been presented to confirm the
computational results with previously reported data [8].
The result of the electronic band structures is
displayed in Fig. 2. It is observed that in the calculation
without including SOC (blue lines in band structure),
the valence band maximum (VBM) located in Y (and X’)
point for all MX. While In conduction band, the
extremum valley or the conduction band minimum
occurred along the Y-M line, for all MX. This result has
a similarity with the previous result of MX with black
phosphorene lattice structure. However, in black
phosphorene, the different band structure was observed
[16]. Our calculated DOS projected to the atomic
orbitals confirmed that the VBM is dominated by a
mixture of X-pz and M-pz. The CBM is contributed by
the coupling of M-pz and M-py. The band gap energy of
all MX buckled square lattice monolayer has an indirect
type. These calculated values are different from black
phosphorene structure [9]. However, both structures
have typical pattern in the band gap properties. The
band gap are decreased along with the increasing
chalcogen atomic mass in the buckles square lattice, thus
(a)
(b)
(c)
Fig 1. 2D buckled square lattice (drawing using Vesta) (a) from top and (b) side and (c) the first Brillouin zone
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Ibnu Jihad et al.
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Table 1. Calculation result of lattice constant, buckling distance, and buckling angle
Calculation
References
a(Å)
dz(Å)
θ (°)
a(Å)
dz(Å)
θ (°)
Source
PbS
3.76
1.04
21.4
3.76
1.0841
22.17
[8]
PBSe
3.79
1.27
25.4
3.85
1.2087
23.92
[8]
GeO
2.86
0.76
20.6
-
-
-
This work
GeS
3.25
1.14
26.4
3.31
1.10868
25.35
[9]
GeSe
3.42
1.22
26.8
3.46
1.18338
25.83
[16]
GeTe
3.69
1.30
26.5
-
-
-
This work
SnO
3.16
0.76
18.8
-
-
-
This work
SnS
3.54
1.20
25.6
3.55
1.18601
25.32
[16]
SnSe
3.62
1.34
27.6
3.68
1.27148
26.07
[16]
SnTe
3.94
1.41
26.8
-
-
-
This work
Fig 2. The Brillouin zone in k-space of the square lattice structure. Blue and pink line indicating the energy band
without SOC and with SOC, respectively. The dominant DOS is displayed with corresponding orbitals
resulting in a lower band gap [17-19]. The band gap values
are GeS = 0.660 eV, GeSe = 0.085 eV, GeTe = no band gap,
SnSe = 1.097 eV, SnSe = 0.627 eV, and SnTe = no bandgap.
From this band gap, all of MX is semiconductor except
GeTe and SnTe, which is metallic.
When the calculation includes the SOC term, all of
the bands are split due to the lack of inversion symmetry,
which is occurs in all MX. The clarification of the
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Ibnu Jihad et al.
4
occurrence of the spin-split states is by considering the
characteristic of the atomic orbitals of the band. From
atomic point of view, coupling between atomic orbitals
will contribute to the non-zero SOC matrix element
through the relation L
S
,, where is the angular-
momentum-resolved atomic SOC strength, with l =
(s, p, d), L
and S
are the orbital angular momentum and
Pauli spin operator, respectively, for the (u, v) atomic
orbitals. It is revealed that the dominant contribution in
CBM is coming from coupling between p orbitals of M
atom and p orbitals of chalcogen atom for all MX. This p-
p coupling is enhanced by increasing the atomic number
of chalcogen atom. This finding is consistent with the
enhancement of the spin splitting size around the Γ point.
The more detailed discussion about the quantitative size
of the splitting will be presented later.
Next, the spin-splitting near Γ and M-points of the
CBM is claimed as Rashba-type splitting because the spin
orientation is rotational [3], and we will show the
calculation of this spin orientation (which is known as
spin textures). It is well-known that there are three kinds
of SOC splitting in non-centrosymmetric and non-
magnetic materials, namely Rashba, Dresselhaus, and
Zeeman-like splitting [20], where the spin splitting and
their corresponding spin textures are displayed in Fig. 3.
The figure shows that the Rashba and Dresselhaus
splitting is very similar except for the spin textures.
Next, the calculation of the size of the Rashba
splitting in the buckled MX square lattice materials and
the spin textures is conducted to clarify the claim that
this is an actual Rashba splitting. To calculate the size of
Rashba splitting (Rashba parameter) for higher-order,
first is to use symmetry analysis to derive the effective
Hamiltonian via k.p perturbation approximation. This
approximation allows us to describe the electronic
properties of our 2D MX monochalcogenides and can be
used to analyze the properties of the band structure, such
as spin splitting and spin texture for points around VBM
and CBM. This method has already successfully used in
various 2D material [7,21-25]. The symmetry group of
the 2D square lattice structure is isomorphic to the C2v
or 2 mm two-dimensional space group [26], and from
Fig 3. Three kinds of SOC splitting with the spin orientation
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Ibnu Jihad et al.
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previous analysis using group theory [24], by deriving the
general Hamiltonian using the direct product of
irreducible representations. The elements of the group are
denoted by E: {x, y, z}, C:{x, y, z}, T:{x, y, z}, and
T:{x, y, z}. This group has four one-dimensional
irreducible representations, A, A, B, and B with the
character table is written in Table 2.
The corresponding component of the irreducible
representation is set to the polar-vector k and the axial-
vector for the first order combination that corresponds
to the A irreducible representation is B: k, , B: k, ,
and A:. For the second-order, A: k
, k
, k
, and
A: kk, which is this second-order implies the form of
kinetic energy. Finally, the third-order term is the possible
product of B: k
, kk
, and B: k
, kk
with . Then a
new total Hamiltonian, that leaves the old one invariant,
is constructed to get the new Hamiltonian related to SOC
up to third-order as:
H(k) = E(k) +
()k+
()k+
()k
+
()k
k+
()kk
+
()k
(1)
where E(k) = (
+
) is the nearly free-electron
energy, () is the i-th coefficient of the j-order, , ,
is the Pauli spin matrices, (k, k) is the position in k-
space and k = k
+ k
. The eigenvalues problems for Eq.
(1) gives us the splitting energy:
E±(k, ) = E(k) ± X+ Y (2)
with
Y =
()ksin+
()kcossin+
()ksin
X =
()kcos+
()kcos+
()ksincos
where k=kcos, k=ksin, and is the angle of k with
x-axis in k-space. The eigenvectors related to the
±(X+ Y) eigenvalues is
±(k) = 0
±()
(3)
The square of the difference of energy eigenvalue
can be calculated using:
(E(k))= 4(
()k+
()k
+
()kk
)+ 4(
()k+
()k
k+
()k
) (4)
This equation is used to fit the plot of (E(k))
along Γ-X and Y-M and get the non-vanishing first-
order coefficient as the Rashba parameter. The fitting
results for the Rashba parameter are displayed in Table 3
with PbS calculation is shown as a confirmation. The table
shows that for Γ-point, the size of the Rashba parameter
increases depending on the atomic number of chalcogen
atom, while on the M-point, the parameter is decreased.
Next, the spin texture of the band structures is
determined as follows. For a given k point, its spin
polarization of each eigenstates ±(k) is define as S(k) =
[S(k), S(k), S(k)] where S(k) =
(k)|(k) is
Table 2. Character table of C2v group
E
C
2
T
x
T
y
A
1
1
1
1
1
A2
1
1
-1
-1
B1
1
-1
-1
1
B2
1
-1
1
-1
Table 3. Calculation result of band gap and Rashba parameter in CBM
MX Band Gap
(eV)
Fitting Result (eVÅ )
Reference (eVÅ )
Source
Γ-point
(
()
)
M-point
(
()
)
Γ-point M-point
PbS
0.747
1.20
3.19
1.03
5.10
[8]
GeS
0.660
0.20
0.68
0.201
0.583
[9]
GeSe
0.085
0.27
0.52
0.320
0.468
[16]
GeTe
Conductor
0.38
0.34
-
-
This work
SnS
1.097
0.42
2.36
0.429
2.354
[16]
SnSe
0.627
0.58
1.48
0.548
1.746
[16]
SnTe
Conductor
0.59
1.05
-
-
This work
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Ibnu Jihad et al.
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Fig 4. (left) Spin texture of SnTe for energy 0.4 eV around Γ-point, (right) Spin texture of SnSe for energy 1.15 eV
around M-point. Both have zero z-component of spin. A similar form also occurred in other MX
the i-direction of spin component. This spin polarization
is calculated in k-space around Γ point by using the spin
density matrix of the spinor wave functions obtained
from the DFT calculation [21-23]. The calculation result
is shown in Fig. 4. It is revealed that constant-energy
surface shows the inner and outer circles with
counterclockwise and clockwise rotation of spin direction
with no z-component of spin. This feature is consistent
with the characteristic of the Rashba effect splitting.
The observed spin texture can be explained based on
our derived SOC Hamiltonian given in Eq. (1). The
expectation value of the spin polarization can be
calculated from this to get S±=±
(), S±=
±
(), and S±= 0. This result shows that the spin
textures are consistent with Fig. 4.
■ CONCLUSION
Investigation of the Rashba effect on two-
dimensional MX Monochalcogenides (M = Ge, Sn and X
= S, Se, Te) with buckled square lattice has shown that this
structure is appropriate to create the Rasbha splitting. We
also confirmed the properties of the spin texture of SOC
splitting with the calculation from our DFT results. We
found that some of MX have semiconductor properties
(except GeTe and SnTe) and have different Rashba
parameter. There is a strong correlation between the size
of the Rashba parameter with the chalcogen atomic
number. In Γ-point, the escalation of the chalcogen
atomic number increases the Rasha parameter, but in M-
point the effect is inverting. These various size of the
Rashba parameter make these materials, and this
buckled square lattice is potential to be developed for
spintronics materials.
■ ACKNOWLEDGMENTS
This research was supported by Research Grant for
Young Lecturer 2019, funded by Universitas Gadjah Mada
(UGM) with contract number 3943/UN1/DITLIT/DIT-
LIT/LT/2019. One of the authors (Absor) would like to
acknowledge the Department of Physics UGM for partially
financial support through BOPTN research grant 2019.
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