APPLIED PHYSICS LETTERS 107, 213105 (2015)

Theoretical discovery of stable structures of group III-V monolayers:
The materials for semiconductor devices
Tatsuo Suzuki (鈴木 達夫)a)
Tokyo Metropolitan College of Industrial Technology, 8-17-1, Minami-Senju, Arakawa-ku, Tokyo 116-8523,
Japan(Received 24 September 2015; accepted 9 November 2015; published online 25 November 2015)
Group III-V compounds are very important as the materials of semiconductor devices. Stable structures of the monolayers of group III-V binary compounds have been discovered by using firstprinciples calculations. The primitive unit cell of the discovered structures is a rectangle, which
includes four group-III atoms and four group-V atoms. A group-III atom and its three nearestneighbor group-V atoms are placed on the same plane; however, these connections are not the sp2
hybridization. The bond angles around the group-V atoms are less than the bond angle of sp3 hybridization. The discovered structure of GaP is an indirect transition semiconductor, while the discovered
structures of GaAs, InP, and InAs are direct transition semiconductors. Therefore, the discovered
structures of these compounds have the potential of the materials for semiconductor devices, forexample, water splitting photocatalysts. The discovered structures may become the most stable strucC 2015 AIP Publishing LLC.
tures of monolayers which consist of other materials. V[http://dx.doi.org/10.1063/1.4936553]

Graphene is a monolayer which consists of carbon
atoms.1 Graphene has the potential of the material for various devices. However, graphene cannot be applied to semiconductor devices because the energy gap of graphene iszero. If the monolayers which consist of other materials havethe finite energy gaps, these monolayers have the great
potential of the materials for semiconductor devices.
Graphene has a hexagonal planar (PL) structure (Fig.
1(a)); and silicene which is the monolayer of silicon atoms
has a hexagonal buckled (BK) structure (Fig. 1(b)).2,3
Although carbon and silicon belong to the same group-IV
elements, graphene and silicene have different structures.
The structures of monolayers are very important, because thestructures influence the electronic and optical properties ofthe monolayers. Bulk group III-V compounds such as GaN,
GaAs, InP, etc., are very important as the materials of semiconductor devices, especially optoelectronic devices.
Therefore, we consider the most stable structures of themonolayers of group III-V binary compounds, which include
B, Al, Ga, or In atoms as group-III elements, and N, P, or As
atoms as group-V elements. Previous researches4–8 reported
that the most stable structures of BN, BP, BAs, AlN, GaN,
and InN monolayers were the PL structures like a graphene.
Zhuang et al.7 reported that AlP and AlAs monolayers
became tetragonal (TETRA) structures (Fig. 1(c)). Sahin
et al.4 reported that GaP, GaAs, InP, and InAs monolayers
became the BK structures like silicene, while Tong et al.8
reported that these monolayers became hexagonal zigzag(ZIG) structures (Fig. 1(d)) recently. However, it has been
discovered9 that the most stable structures of GaP, GaAs,
InP, and InAs monolayers become unknown structures (Fig.
1(e)), which will be called hexagonal armchair (ARM) structures after the side view on yz-coordinates. Therefore, thispaper reports these discovered stable structures in detail.
a)Electronic mail: dr.tatsuosuzuki@gmail.com

0003-6951/2015/107(21)/213105/4/$30.00

We discuss the most stable structures of monolayers ofgroup III-V binary compounds based on the first-principle
calculations. The structural relaxations and energy calculations are performed by using two different methods withhigh accuracy.
One method is a density functional theory (DFT) using aplane wave basis set. The projector augmented wave (PAW)
potentials10,11 and the generalized gradient approximation
given by Perdew-Burke-Enzerhof (PBE)12 are adopted in thesame way as in previous works.6–8 The full Heyd-ScuseriaErnzerhof (HSE06) functionals13,14 are not used because oflong calculation time. Cutoff energies for the plane wave basis set are 500 eV for compounds including N atoms and
400 eV for other compounds. The k-point sampling uses
Monkhorst-Pack scheme, which employs a 20  19  1 meshfor the ARM structures, and a 32  32  1 mesh for otherstructures. First, approximate structures are searched by thefull relaxations. Next, the local minimum energies aresearched by the cell shape conserving relaxations under theconditions that the z distance between layers of the supercells is fixed at 18 Å, and that each lattice constant ischanged at 0.01 Å intervals. The convergence criteria ofelectronic self-consistent calculations and ionic relaxations
are 106 and 105 eV, respectively. These calculations are
performed by using VASP 5.3.3 package.15,16
Another method is a DFT using a Gaussian type orbital(GTO) basis set under the periodic boundary conditions.
This method does not need the super-cells and does not use
pseudo-potential because of all-electron calculations. HSE06
functionals and 6–311G(d,p) basis set are adopted. InP and
InAs cannot be calculated, because 6–311G(d,p) basis set
can apply to elements only from hydrogen (H) to krypton(Kr) in the periodic table. About 2000 k-points are requested
in a unit cell. The convergences of electronic self-consistentcalculations are that the maximum and the root mean square(RMS) of the variations of the density matrix are less than107, 213105-1

C 2015 AIP Publishing LLC
V

213105-2

Tatsuo Suzuki

Appl. Phys. Lett. 107, 213105 (2015)

FIG. 1. The various structures of group
III-V monolayers. The upper left is the
top view on xy-coordinates. The lower
left and upper right are the x-z and y-z
side views, respectively. A green
parallelogram is a primitive unit cell.
Blue and red circles are group-III andgroup-V atoms, respectively. In orderto understand the structures clearly,
different marks are used according to thepositions in the primitive unit cell, even
if the atoms are the same element.
a; b; and c are the bond lengths.
a; b; and c are the bond angles aroundthe group-III atoms. /, h, and w are thebond angles around the group-V atoms.
Tx ; Ty ; x3 ; y3 ; z3 ; y5 ; and z5 are the structural parameters.

105 and 107, respectively, and that the variation of thetotal-energy is less than 105 Hartree. The convergences ofionic relaxations are that the maximum and RMS of force
are less than 2  106 and 106 Hartree/Bohr, respectively,
and that the maximum and RMS of displacement are lessthan 6  106 and 4  106 Bohr, respectively. These calculations are performed by using Gaussian 09 package.17
Calculations are performed about all binary compoundsof B, Al, Ga, or In atoms as group-III elements, and N, P, or
As atoms as group-V elements. First, we consider BN, BP,
BAs, AlN, GaN, and InN. The BK, ZIG, and ARM structures of these compounds changed into the PL structuresduring the structural relaxations. Table I shows the totalenergy differences per atom DE of the TETRA structures ofthese compounds. DE is defined as the difference from thetotal-energy per atom of the PL structure of the same compound calculated by the same calculation method. The
smaller the DE is, the more stable the structure is. DE of the
TETRA structures are positive; therefore, the most stablestructures of BN, BP, BAs, AlN, GaN, and InN monolayers
are the PL structures. These results agree with S
ahin’s conclusion that “all III-V compounds containing first row elements B or N have planar stable structures,”4 and agree withother previous researches.5–8
Next, we consider AlP and AlAs. Table I shows DE of
the BK, TETRA, ZIG, and ARM structures of these compounds. The TETRA structure of each compound have thesmallest DE in all structures by either calculation method;
therefore, the most stable structures of AlP and AlAs monolayers are the TETRA structures. These results also agreewith previous researches.7,8Finally, we consider GaP, GaAs, InP, and InAs. In
Table I, the ARM structure of each compound has the smallest DE in all structures. Therefore, the most stable structuresof GaP, GaAs, InP, and InAs monolayers are the ARM structures. In order to examine the dynamical stability of the
ARM structures, phonon spectra were calculated by the finite
displacement method in a plain wave basis set with PBE
approximation. A 2  2  1 supercell is used, and the z distance between layers is 14 Å. Cutoff energies are 320 eV,

TABLE I. Total-energy differences per atom DE of various structures of
all compounds calculated with PBE approximation and with HSE06
functional.
DE (meV/atom)
Material

Structure

PBE

HSE06

BN
BP
BAs
AlN
GaN
InN
AlP

TETRA
TETRA
TETRA
TETRA
TETRA
TETRA
BK
TETRA
ZIG
ARM
BK
TETRA
ZIG
ARM
BK
TETRA
ZIG
ARM
BK
TETRA
ZIG
ARM
BK
TETRA
ZIG
ARM
BK
TETRA
ZIG
ARM

1109
406
219
73
416
209
PLb
213
56
61
14
296
TETRAdTETRAd
13
13
76
83
62
142
154
157
24
4
102
110
70
138
173
179

NCa
NCa
NCa
NCa
NCa

AlAs

GaP

GaAs

InP

InAs

a

Non-convergence.
Changed into the PL structure.
c
Calculated by using 6-31g(d) basis set.
dChanged into the TETRA structure.
b

PLb
147c
26
34
6
334c
92
99
2
NCa
66
74
31
NCa
129
135

213105-3

Tatsuo Suzuki

Appl. Phys. Lett. 107, 213105 (2015)

FIG. 3. (Lower panel) bond lengths and (upper panel) bond angles of ARM
structures. Black and red were calculated with PBE approximation and with
HSE06 functional, respectively. Triangles and pentagons are the bond angles
around the group-III atoms and the group-V atoms, respectively. Bluedashed lines indicate the bond angles of sp2 and sp3 hybridizations.
FIG. 2. Phonon spectra of ARM structures. Red lines of GaAs are calculated
by using a finer FFT-mesh. The inset is the first Brillouin zone of the ARM
structure.

is a rectangle, which include four group-III atoms and fourgroup-V atoms. These atoms are connected in honeycomblike structure. The positions of atoms are expressed by sevenstructural parameters Tx ; Ty ; x3 ; y3 ; z3 ; y5 ; and z5 in Cartesian
coordinates (Table II). The positions of group-III atoms areðx3 ; y3 ; z3 Þ, ðx3 þ Tx =2; y3 ; z3 Þ, ðx3 ; y3 þ Ty =2; z3 Þ, andðx3 þ Tx =2; y3 þ Ty =2;  z3 Þ. The positions of group-V
atoms are ð0; y5 ; z5 Þ, ðTx =2; y5 ; z5 Þ, ð0; y5 þ Ty =2; z5 Þ, andðTx =2; y5 þ Ty =2; z5 Þ. Uneven monolayers like the ARM
structure belong to the 80 crystallographic layer group types
which are three-dimensional groups with two-dimensionaltranslations. The ARM structure has no point group symmetries except the identity, and has two glide reflection and one
screw rotation. In Fig. 1(e), the glide reflections are a reflection in x-z plane with glide vector ðTx =2; 0; 0Þ, and a reflectionin x-y plane with glide vector ð0; Ty =2; 0Þ. The screw rotation
is a rotation of 180 around the axis y ¼ Ty =4; z ¼ 0 withscrew vector ðTx =2; 0; 0Þ. Therefore, the ARM structure
belongs to layer group ðpb21 aÞ.19 The ARM structure has
three kinds of the bond lengths (a; b; c); three kinds of thebond angles (a; b; c) around the group-III atoms; and threekinds of the bond angles (/; h; w) around the group-V atoms.

and the k-point sampling employs a 10  10  1 mesh. The
calculations are performed by using Phonopy 1.8.4.1 package18 and VASP 5.3.3 package. The results are shown by
black lines in Fig. 2. Acoustic phonons near C points slightly
have imaginary frequencies for all compounds. It is known
that in phonon calculations, imaginary acoustic phonons
appear near C points by computational errors because thetranslational invariance is violated in approximated calculations. The main violation comes from the discreteness of the
FFT-mesh.4 In order to examine the influence of the FFTmesh, the additional phonon calculation of GaAs was performed by using a finer FFT-mesh that contains 8 times
more points. In Fig. 2(b), to use finer FFT-mesh, the regionof the imaginary phonon was narrowed. Hence, the imaginary phonons in Fig. 2 are the artifacts of the mesh size. All
spectra of Fig. 2 have real frequencies; therefore, the dynamical stability of the ARM structures is confirmed.
Now, we examine the characteristics of the ARM structures (Fig. 1(e)). The primitive unit cell of the ARM structure

TABLE II. Structural parameters, energy gaps Eg, and transition types of ARM structures calculated with PBE approximation and with HSE06 functional.
Structural parameters (Å)
Material

Method

Tx

Ty

x3

y3

z3

y5

z5

Eg (eV)

Transition

GaP

PBE
HSE06
PBE
HSE06
PBE
PBE

6.43
6.42
6.58
6.58
6.89
7.00

6.98
6.99
6.97
7.03
7.49
7.46

1.04
1.04
1.06
1.05
1.10
1.10

0.89
0.88
0.89
0.89
0.94
0.94

0.27
0.27
0.32
0.32
0.32
0.37

1.03
1.04
1.10
1.10
1.15
1.23

1.06
1.03
1.27
1.22
1.21
1.42

2.11
2.96
1.76
2.56
2.09
1.78

Indirect
Indirect
DirectDirectDirectDirect

GaAs
InP
InAs

213105-4

Tatsuo Suzuki

Appl. Phys. Lett. 107, 213105 (2015)

Ered¼ 4:44 eV þ pH  0:059 eV. The oxidation potenHþ =H2
22
Fig. 4 shows
tial is Eox
O2 =H2 O ¼ 5:67 eV þ pH  0:059 eV.
the redox potentials at pH ¼ 0 and pH ¼ 7. The band edgesof GaP and GaAs calculated with HSE06 functional straddle
the redox potentials. The band edges of InP calculated with
PBE approximation straddle the redox potentials at pH ¼ 7.
The band edges of InAs calculated with PBE approximation
do not straddle the redox potentials by a narrow margin.
However, the band edges of the real ARM structure of InAs
probably straddle the redox potentials, because the PBE
approximation underestimates the energy gaps. Therefore,
the ARM structures of these compounds are the candidatesof water splitting photocatalysts.
In conclusion, the most stable structures of BN, BP,
BAs, AlN, GaN, and InN monolayers are the PL structures;
AlP and AlAs monolayers are the TETRA structures.
However, GaP, GaAs, InP, and InAs monolayers become thediscovered ARM structures. The BK structures and the ZIG
structures are not the most stable structures of group III-V
monolayers any longer. The ARM structure of GaP is anindirect transition semiconductor, while the ARM structuresof GaAs, InP, and InAs are direct transition semiconductors.
Therefore, the ARM structures of these compounds have thepotential of the materials for semiconductor devices, forexample, water splitting photocatalysts. The ARM structures
may become the most stable structures of monolayers which
consist of other materials.
FIG. 4. Energy bands of ARM structures. Black and red were calculatedwith PBE approximation and with HSE06 functional, respectively. Blue andgreen lines indicate the redox potentials for water splitting at pH ¼ 0 and
pH ¼ 7, respectively. The origin of energy is the vacuum level.

This work was partially supported by Tokyo Metropolitan
College of Industrial Technology.
1

In Fig. 3, there are hardly any differences of the structures
between the two calculation methods. The bond lengths ofeach compound are almost the same. Very interestingly, therelationship a þ b þ c ¼ 360 holds. This relationship meansthat a group-III atom and its three nearest-neighbor group-V
atoms are placed on the same plane. However, these connections are not the sp2 hybridization, because the relationshipa ¼ b ¼ c ¼ 120 does not hold. On the other hand, the bondangles /; h; w around the group-V atom are less thancos1 ð1=3Þ  109:5 , which is the bond angle of sp3
hybridization.
Fig. 4 shows the energy bands of the ARM structures.
GaP is an indirect transition semiconductor, while GaAs,
InP, and InAs are direct transition semiconductors. In Table
II, Eg calculated with PBE approximation is smaller than Eg
calculated with HSE06 functional. It is well known that the
PBE approximation in a plain wave basis set underestimate
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energy gaps.21 Therefore, Eg calculated with HSE06 functional are more reliable than Eg calculated with PBE
approximation.
Semiconductors can be used as water splitting photocatalysts if the conduction band minimum (CBm) energy ishigher than the reduction potential for Hþ/H2, and thevalence band maximum (VBM) energy is lower than the oxidation potential for O2/H2O. The reduction potential isK. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V.
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P. E. Blöchl, Phys. Rev. B 50, 17953 (1994).
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G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).
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APPLIED PHYSICS LETTERS 107, 213105 (2015)

Supplementary Material

Theoretical discovery of stable structures of group III-V monolayers: The materials forsemiconductor devices
Tatsuo Suzuki (鈴木 達夫)
The process of the discovery of the ARM structure is explained briefly. When other group IIIV materials were calculated, the author noticed that the group-III atoms preferred planarconnections, and the group-V atoms preferred solid connections. The author thought that thebond angle ߠଷ around the group-III atoms of the BK structure might be different from the bondangle ߠହ around the group-V atoms. Every other group-V atom in Fig. S1(a) was reflected in theplane where the group-III atoms existed. Then, the ZIG structure was obtained after the
structural relaxation. The unit cell of the ZIG structure was doubled in y coordinate, and atoms
within purple dashed rectangles in Fig. S1(b) were reflected in the x-y plane. Then, the ARM
structure was obtained after the structural relaxation. The author searched other stable structuresconsidering solid geometry about various unit cells, and discovered several metastable structures,for example, Fig. S2. The total-energy difference per atom ΔE of the metastable structure issmaller than the BK structure; however, ΔE is larger than the ZIG structure. Structure searchalgorithm did not be used.

z
x

θ5
θ3
(a)

(b)

FIG. S1. (a) Solid line is the x-z side view of BK structure. ߠଷ and ߠହ are the bond angles
around the group-III and group-V atoms, respectively. (b) The top view of ZIG structure. Green
rectangle is the modified primitive unit cell.

FIG. S2. Other metastable structures of group III-V monolayers.