Energy bands of atomic monolayers of various materials:
Possibility of energy gap engineering
Tatsuo Suzuki∗,a,b , Yushi Yokomizoba

Tokyo Metropolitan College of Industrial Technology, 8-17-1, Minami-Senju,
Arakawa-ku, Tokyo, 116-0003, Japanb
Tokyo Metropolitan College of Aeronautical Engineering, 8-17-1, Minami-Senju,
Arakawa-ku, Tokyo, 116-0003, Japan

Published in Physica E 42 (2010) 2820-2825, doi:10.1016/j.physe.2010.04.031
This paper was first presented at the 14th International Conference on Modulated Semiconductor Structures (MSS14) held in Kobe, Japan, on 23 July2009.
(Received 31 August 2009; received in revised form 1 April 2010; accepted
12 April 2010; available online 8 June 2010)
Abstract
The mobility of graphene is very high because the quantum Hall eﬀects
can be observed even at room temperature. Graphene has the potential ofthe material for novel devices because of this high mobility. But the energy
gap of graphene is zero, so graphene can not be applied to semiconductor
devices such as transistors, LEDs, etc. In order to control the energy gaps, wepropose atomic monolayers which consist of various materials besides carbonatoms.
To examine the energy dispersions of atomic monolayers of various materials, we calculated the electronic states of these atomic monolayers using
density functional theory with structural optimizations. The quantum chemical calculation software Gaussian 03 was used under periodic boundary conditions. The calculation method is LSDA/6-311G(d,p), B3LYP/6-31G(d),∗Corresponding author
Email address: tatsuo@acp.metro-cit.ac.jp (Tatsuo Suzuki)

or B3LYP/6-311G(d,p).
The calculated materials are C (graphene), Si (silicene), Ge, SiC, GeC,GeSi, BN, BP, BAs, AlP, AlAs, GaP, and GaAs. These atomic monolayers
can exist in the flat honeycomb shapes. The energy gaps of these atomic
monolayers take various values. Ge is a semimetal; AlP, AlAs, GaP, andGaAs are indirect semiconductors; and others are direct semiconductors.
We also calculated the change of energy dispersions accompanied by thesubstitution of the atoms. Our results suggest that the substitution of impurity atoms for monolayer materials can control the energy gaps of the atomicmonolayers.
We conclude that atomic monolayers of various materials have the potential for novel devices.
1. Introduction
Graphene [1] is an atomic monolayer which consists of carbon atoms
arranged in a honeycomb lattice. The mobility of graphene is very highbecause the quantum Hall eﬀects can be observed even at room temperature[2]. Graphene has the potential of the material for novel devices because ofthis high mobility. But the energy gap of graphene is zero [1], so graphene can
not be applied to semiconductor devices such as transistors, LEDs, etc. Inorder to control the energy gaps, we propose atomic monolayers which consistof various materials besides carbon atoms. Theoretical works show that theenergy gaps of silicene which is the atomic monolayer of silicon atoms [3, 4, 5]
and the atomic monolayer of germanium atoms [3] are zeros, but a pioneeringexperimental work about atomic monolayers of boron nitride formed on themetal surfaces [6] shows the finite energy gaps. So we hope that other atomic
monolayers also have the finite energy gaps and have the potential for noveldevices. To examine the energy dispersions of atomic monolayers of various
materials, we calculated the electronic states of these atomic monolayers
using density functional theory with structural optimizations. A honeycomb
lattice was assumed as a crystal structure (Fig.1). The atoms were alternately
placed like Fig.1(b) in the case of a binary compound.
2. Computational Method
All calculations were performed using Gaussian 03 package [7]. Gaussian
is a very famous quantum chemical calculation software, and has been used2

widely to calculate the electronic states of isolated molecules. Gaussian wasimproved in 2003, and Gaussian 03 has been able to calculate the electronicstates using periodic boundary conditions. It can calculate the electronicstates of not only three-dimensional crystals, but also two-dimensional orone-dimensional materials.
We can specify the combination of method and basis set in Gaussian
03 package. We want to calculate the electronic states of the Kohn-Shamequations of density functional theory, using the hybrid B3LYP exchange
and correlation functionals combined with 6-311G(d,p) basis set (B3LYP/6311G(d,p)). But unfortunately most calculations using B3LYP/6-311G(d,p)
did not converge. So we used local spin density approximation (LSDA) combined with 6-311G(d,p) basis set (LSDA/6-311G(d,p)) or B3LYP combinedwith 6-31G(d) basis set (B3LYP/6-31G(d)) instead. The adopted combinations of method and basis set are noted in Table 1.
We carried out the calculations of all electrons, so the contributions fromcore electrons are included.
Furthermore the merits of our calculations are structural optimizations,and the translation vectors are also changed after every self-consistent calculation.
3. Results and discussion
3.1. A IV group element
We show the energy dispersions of atomic monolayers which consist of a
IV group element in the periodic table in Fig.2. We plot the lowest conduction bands and the highest valence bands from other calculations by coloredlines.
Fig.2(a) is a result of graphene which consists of carbon atoms. The
conduction band and the valence band of graphene contacts at K points,
which is famous as Dirac points. The highest valence band of our calculationagrees very well with other calculations [8, 9, 10, 11], but the energy of theconduction band of our calculation at Γ point is rather higher than the resultsof other calculations. We cannot understand this reason at present. This is
our problem to be solved.
Fig.2(b) is a result of silicene which consist of silicon atoms. The conduction band and the valence band of silicene also contacts at K points as Diracpoints. Our result of silicene agrees very well with the solution of the local
density approximation (LDA) with the projected augumented wave (PAW)
3method [3] and the solution of LDA in a plane wave basis set with Vanderbilt
ultrasoft pseudopotentials [4]. The bond length of our calculation is 2.23 Å,
the bond length of LDA with the PAW method is 2.23 Å [3], and the bondlength of LDA with Vanderbilt ultrasoft pseudopotentials is 2.22 Å [4]. Thesebond lengths also agree very well. On the other hand, our result of silicene
does not agree with the solutions of tight-binding model [5].
Fig.2(c) is a result of an atomic monolayer which consists of germanium
atoms. The conduction band and the valence band also contacts at K points,
but the bottom of conduction band at Γ point is a little lower than the Fermi
energy. So the atomic monolayer of germanium is a semimetal. This result
agrees very well with the solution of LDA with the PAW method [3]. The
bond length of our calculation is 2.31 Å and the bond length of LDA with
the PAW method is 2.32 Å [3]. These bond lengths also agree very well.
3.2. Binary compounds located in IV group elements
We show the energy dispersions of atomic monolayers which consist ofbinary compounds located in IV group elements in the periodic table in Fig.
3.
The energy dispersions of SiC (Fig.3(a)) and GeC (Fig.3(b)) resemble
each other, and the conduction band and the valence band does not contact
at K points. The energy gap of SiC is 3.526 eV and the energy gap of GeC
is 3.160 eV. These values correspond to the wave lengths of ultraviolet rays.
But the conduction band and the valence band of GeSi (Fig.3(c)) almost
contact at K points. This energy dispersion resembles the energy dispersionof silicene (Fig.2(b)).
At present we cannot explain the physical reason why the conduction
band and the valence band do not contact at K points for SiC and GeC
while both bands contact for GeSi.
3.3. Binary compounds located between III and V group elements
We show the energy dispersions of atomic monolayers which consist ofbinary compounds located between III and V group elements in the periodic
table in Fig. 4. We calculated AlN and GaN also, but the calculations ofAlN and GaN did not converge.
The energy gap of BN (Fig.4(a)) is very large (6.377 eV). There areexperimental data of the valence band of atomic monolayers of BN formedon the metal surfaces [6]. Our result agrees well with this experimental bandstructures.
4The energy dispersions of BP (Fig.4(b)) and BAs (Fig.4(c)) resemble
each other. The energy gap of BP is 1.912 eV, which corresponds to thewave length of an orange light. The energy gap of BAs is 1.594 eV, whichcorresponds to the wave length of an infrared ray.
The energy dispersions of AlP (Fig.4(d)), AlAs (Fig.4(e)), GaP (Fig.4(f))
and GaAs (Fig.4(g)) resemble one another. The bottoms of conduction bands
are located at Γ point, so these are indirect semiconductors.
We cannnot understand the physical factor which determines the transition type (direct or indirect) at present.
3.4. Possibility of energy gap engineering
We show that the change of energy dispersions accompanied by the substitution of the atoms in Fig.5. In the case of graphene (Fig.5(a)) and silicene(Fig.5(e)), the conduction bands and the valence bands make Dirac points at
K points, so the energy gaps are zeros. But when the 12.5% of carbon atoms
are replaced with Si atoms (Fig.5(b)), the energy gap becomes open. This
fact suggests that the substitution of impurity atoms for monolayer materials
can control the energy gaps of the atomic monolayers. These atomic monolayers are direct semiconductors, and the energy gap of SiC corresponds tothe wave length of an ultraviolet ray. So these atomic monolayers can beapplied to novel optical devices.
We wanted to show the relation between the gap energy and the concentration of the substituted atoms. But by the diﬃculty of convergence, the calculation method is not unified. So we cannot estimate the gap-concentration
relation quantitatively.
3.5. Flatness
In our all converged calculations, all atoms are located on the same plane
which is formed by the two translation vectors. So the atomic monolayers ofvarious materials can exist in the flat honeycomb shapes.
4. Conclusion
We calculated the electronic states of atomic monolayers which consist ofvarious materials using density functional theory with structural optimizations. The calculated materials are C (graphene), Si (silicene), Ge, SiC, GeC,GeSi, BN, BP, BAs, AlP, AlAs, GaP, and GaAs. These atomic monolayers
can exist in the flat honeycomb shapes. The energy gaps of these atomic
5monolayers take various values. Ge is a semimetal; AlP, AlAs, GaP, andGaAs are indirect semiconductors; and others are direct semiconductors.
We also calculated the change of energy dispersions accompanied by thesubstitution of the atoms. Our results suggest that the substitution of impurity atoms for monolayer materials can control the energy gaps of the atomicmonolayers. So we conclude that atomic monolayers of various materials
have the potential for novel devices.
Our important problem is to establish synthetic methods of these atomicmonolayers.
References[1] K. Novoselov, A. Geim, S. Morozov, D. Jiang, Y. Zhang, S. Dubonos,
I. Grigorieva, A. Firsov, Science 306 (2004) 666.
[2] K. Novoselov, Z. Jiang, Y. Zhang, S. Morozov, H. Stormer, U. Zeitler,
J. Maan, G. Boebinger, P. Kim, A. Geim, Science 315 (2007) 1379.
[3] S. Lebègue, O. Eriksson, Phys. Rev. B 79 (2009) 115409.
[4] X. Yang, J. Ni, Phys. Rev. B 72 (2005) 195426.
[5] G. Guzmán-Verri, L. Lew Yan Voon, Phys. Rev. B 76 (2007) 075131.
[6] A. Nagashima, N. Tejima, Y. Gamou, T. Kawai, C. Oshima, Phys. Rev.
B 51 (1995) 4606; A. Nagashima, N. Tejima, Y. Gamou, T. Kawai,C. Oshima, Phys. Rev. Lett. 75 (1995) 3918.
[7] M. Frisch, et al., Gaussian 03, Revision D.01, Gaussian, Inc., Wallingford
CT (2004).
[8] T. Miyake, S. Saito, Phys. Rev. B 68 (2003) 155424.
[9] P. Trevisanutto, C. Giorgetti, L. Reining, M. Ladisa, V. Olevano, Phys.
Rev. Lett. 101 (2008) 226405.
[10] M. Klintenberg, S. Lebègue, C. Ortiz, B. Sanyal, J. Fransson, O. Eriksson, J. Phys. Condens. Matter 21 (2009) 335502.
[11] S. Latil, L. Henrard, Phys. Rev. Lett. 97 (2006) 036803.

6

Table 1: The results of calculations.

Materials

C (graphene)
Si (silicene)
Ge
SiC
GeC
GeSi
BNBP
BAs
AlP
AlAs
GaP
GaAs

The
topofvalence
bandK
KKK
KKKKKKKKKThe
bottomofconduction
bandKK
Γ
KK
K
K
KK
ΓΓΓΓ

(a) Element.

Transition

Energy Bondgap length
(eV)
(Å)

Methodandbasis set

DirectDirectSemimetal
DirectDirectDirectDirectDirectDirectIndirect
Indirect
Indirect
Indirect

0.036
0.064
-0.444
3.526
3.160
0.275
6.377
1.912
1.594
3.453
2.938
3.054
2.475

LSDA/6-311G(d,p)
LSDA/6-311G(d,p)
LSDA/6-311G(d,p)
B3LYP/6-31G(d)B3LYP/6-31G(d)B3LYP/6-31G(d)B3LYP/6-311G(d,p)
B3LYP/6-31G(d)B3LYP/6-31G(d)B3LYP/6-31G(d)B3LYP/6-31G(d)B3LYP/6-31G(d)B3LYP/6-31G(d)

1.41
2.23
2.31
1.79
1.83
2.28
1.45
1.87
1.93
2.28
2.34
2.23
2.29

(b) Binary compound.

Figure 1: A honeycomb lattice is assumed as a crystal lattice. Blue lines indicate an unitcell.

7151010105550

Energy (eV)15

Energy (eV)Energy (eV)1500-5-5-5-10-10-10-15K

Γ

-15M

(a) C (Graphene)

KKΓ

-15M

(b) Si (Silicene)

KKΓ

MK(c) Ge

Figure 2: Energy dispersions of atomic monolayers which consist of a IV group elementin the periodic table. The origin of energy is the Fermi energy. Black solid lines areour calculations, whose calcutation method is LSDA/6-311G(d,p). In Fig. 2(a), both
red solid lines [8] and red broken lines [9] are the local density approximation (LDA)
with the GW approximation, green solid lines [10] are LDA with a full potential linear
muﬃn orbital (FP-LMTO) method, and green broken lines [11] are LDA with a normconserving pseudopotential. In Fig. 2(b), red solid lines [3] are LDA with the projectedaugumented wave (PAW) method, red broken lines [4] are LDA in a plane wave basis setwith Vanderbilt ultrasoft pseudopotentials, green solid lines [5] are a first-nearest neighbor(1NN) tight-binding model, and green broken lines [5] are a second-nearest neighbor (2NN)
tight-binding model. In Fig. 2(c), red solid lines [3] are LDA with the PAW method.

8151010105550

Energy (eV)15

Energy (eV)Energy (eV)1500-5-5-5-10-10-10-15KΓ(a) SiC-15
M

KK

Γ

(b) GeC-15
M

KK

Γ

MK(c) GeSi

Figure 3: Energy dispersions of atomic monolayers which consist of binary compounds
located in IV group elements in the periodic table. The origin of energy is the Fermi
energy. The calcutation method is B3LYP/6-31G(d).

9151010105550Energy (eV)15

Energy (eV)Energy (eV)1500-5-5-5-10-10-10-15K

Γ

-15M

KK-15M

KK(b) BP15151010550-5-10-10KΓ

-15M

KK

15151010550-5-10-10Γ(f) GaP

ΓMKM

K0-5KK(e) AlAs

Energy (eV)Energy (eV)(d) AlP-15

M

0-5-15Γ(c) BAs

Energy (eV)Energy (eV)(a) BNΓ

-15
M

KK

Γ

(g) GaAs

Figure 4: Energy dispersions of atomic monolayers which consist of binary compounds
located between III and V group elements in the periodic table. The origin of energy isthe Fermi energy. The calcutation method of (a) is B3LYP/6-311G(d,p), and the methodof others is B3LYP/6-31G(d).

10151010105550

Energy (eV)15

Energy (eV)Energy (eV)1500-5-5-5-10-10-10-15
K

Γ

-15
M

KK

15101055
Energy (eV)Energy (eV)150-5-10-10

Γ

-15
M

(d) Si0.875C0.125

KKK

Γ

M

K

(c) SiC

0-5K-15
M(b) Si0.125C0.875(a) C (Graphene)

-15

Γ

K

Γ

M

K

(e) Si (Silicene)

Figure 5: (Top) The positions of atoms. Black balls are carbon atoms, and red balls aresilicon atoms. Blue lines indicates an unit cell. (Bottom) Energy dispersions of atomic
monolayers which consist of carbon atoms and silicon atoms. The origin of energy isthe Fermi energy. The calcutation method of (a) and (e) is LSDA/6-311G(d,p), and themethod of others is B3LYP/6-31G(d).

11