Tuan V. Vu^{ab},
Nguyen N. Hieu*^{cd},
A. A. Lavrentyev^{e},
O. Y. Khyzhun^{f},
Chu V. Lanh^{g},
A. I. Kartamyshev^{ab},
Huynh V. Phuc^{h} and
Nguyen V. Hieu*^{i}
^{a}Division of Computational Physics, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam. E-mail: vuvantuan@tdtu.edu.vn
^{b}Faculty of Electrical & Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam
^{c}Institute of Research and Development, Duy Tan University, Da Nang, Vietnam. E-mail: hieunn@duytan.edu.vn
^{d}Faculty of Natural Sciences, Duy Tan University, Da Nang, Vietnam
^{e}Department of Electrical Engineering and Electronics, Don State Technical University, 1 Gagarin Square, 344010 Rostov-on-Don, Russian Federation
^{f}Frantsevych Institute for Problems of Materials Science, National Academy of Sciences of Ukraine, 3 Krzhyzhanivsky Street, 03142 Kyiv, Ukraine
^{g}Department of Physics, Vinh University, 182 Le Duan, Vinh City, Vietnam
^{h}Division of Theoretical Physics, Dong Thap University, Dong Thap, Vietnam
^{i}Physics Department, The University of Danang – University of Science and Education, Da Nang, Vietnam. E-mail: nvhieu@ued.udn.vn
First published on 11th March 2022
In this paper, the structural, electronic, and transport properties of Janus GaInX_{3} (X = S, Se, Te) single-layers are investigated by a first-principles calculations. All three structures of GaInX_{3} are examined to be stable based on the analysis of their phonon dispersions, cohesive energy, and Born's criteria for mechanical stability. At the ground state, The Janus GaInX_{3} is a semiconductor in which its bandgap decreases as the chalcogen element X moves from S to Te. Due to the vertical asymmetric structure, a difference in the vacuum level between the two surfaces of GaInX_{3} is found, leading to work functions on the two sides being different. The Janus GaInX_{3} exhibit high directional isotropic transport characteristics. Particularly, GaInX_{3} single-layers have high electron mobility, which could make them potential materials for applications in electronic nanodevices.
Along with monochalcogenides and dichalcogenides, trichalcogenides have also received a lot of attention recently. In_{2}Se_{3} nanosheets have also been successfully synthesized experimentally.^{6} The group-III trichalcogenides are the 2D quintuple-layer atomic materials and their crystal structure belongs to R3m space group. Due to vertical asymmetric structure, the group-III trichalcogenide single-layers possess novel physical properties that can not exist in the monochalcogenides or dichalcogenides. The group-III trichalcogenide single-layers are predicted to be stability.^{7} Zhao and co-workers have revealed that the group-III trichalcogenides In_{2}X_{3} (X = S, Se, Te) are indirect semiconductors and they are can be served as photocatalysts in the photolytic field.^{8} The other group III trichalcogenide single-layers such as Al_{2}X_{3} or Ga_{2}X_{3} are also reported to have promising applications in water splitting and their solar-to-hydrogen efficiency is predicted to be high.^{7} An intrinsic 2D out-of-plane ferroelectricity has been experimentally observed in atomically thin crystal In_{2}Se_{3} due to the locking between in-plane lattice asymmetry and out-of-plane dipoles.^{9} Based on the density functional theory, Lu and Huang have predicted that Ga_{2}X_{3} and In_{2}X_{3} single-layers possess excellent out-of-plane and in-plane second harmonic generation and piezoelectricity properties.^{10} Recently, quintuple-layer atomic structure Ga_{2}O_{3} has been reported to be high carrier mobility and low lattice thermal conductivity, which is suitable for applications in thermoelectric devices.^{11} The vertical asymmetric structure leads to the appearance of an intrinsic electric field and dipole in the trichalcogenide and trioxide single-layers. Consequently, the adsorption energy, as well as the ability to perform water splitting applications at different surfaces of single-layers, can be different.^{7,12} In parallel with the study of single-layers, van der Waals heterostructures based on the group-III trichalcogenides have also been of great interest. Ding and co-workers have previously considered the electronic and ferroelectric characteristics of In_{2}Se_{3}/graphene and In_{2}Se_{3}/WSe_{2} heterostructures. It was found that the Schottky barrier in van der Waals bilayer heterostructures based on In_{2}Se_{3} can be modulated by switching the orientation of electric dipole in In_{2}Se_{3} layer and they are suitable for a wide range of applications in nanoelectronics.^{13}
Recently, the Janus structures of the group-III trichalcogenide single-layers have begun to be studied.^{14,15} The Janus In_{2}X_{2}Y (X/Y = S, Se, Te) single-layers were confirmed to be strong solar absorption and superior out-of-plane and in-plane piezoelectric response.^{14} Particularly, the electron mobility and solar-to-hydrogen efficiency of the Janus In_{2}X_{2}Y can be higher than that in In_{2}X_{3} single-layers.^{7,14} Excited by the successful synthesis of the group-III trichalcogenide nanosheets and the great recent achievements in theoretical studies on Janus group-III trichalcogenide single-layers, we here investigate the structural, electronic, and transport properties of novel Janus GaInX_{3} (X = S, Se, Te) single-layer using the density functional theory (DFT). The obtained results not only give deep insight into the electronic and transport properties of the novel Janus GaInX_{3} single-layers, but also provide an impetus for further both experimental and theoretical studies of this exciting family of materials.
Fig. 1 Top and side views of optimized atomic structures of the Janus GaInX_{3} (X = S, Se, Te) single-layers. The unit cell is indicated by a rhombus. |
a (Å) | d_{1} (Å) | d_{2} (Å) | d_{3} (Å) | d_{4} (Å) | φ_{∠In–X–Ga} (deg) | φ_{∠X–In–X} (deg) | φ_{∠X–Ga–X} (deg) | Δh (Å) | E_{c} (eV) | |
---|---|---|---|---|---|---|---|---|---|---|
GaInS_{3} | 3.74 | 2.38 | 2.21 | 2.73 | 2.53 | 127.68 | 95.47 | 103.38 | 6.20 | 4.35 |
GaInSe_{3} | 3.90 | 2.52 | 2.34 | 2.84 | 2.66 | 127.53 | 94.40 | 101.92 | 6.60 | 3.95 |
GaInTe_{3} | 4.19 | 2.71 | 2.55 | 3.04 | 2.87 | 127.30 | 93.71 | 101.23 | 7.17 | 3.49 |
To investigate the stabilities of the studied systems, we firstly examine the strength of chemical bonds via the analysis of their cohesive energy. In principle, GaInX_{3} can be constructed from Ga_{2}X_{3}, which has successfully been fabricated by experiment.^{6} Also, previous study suggested that Ga_{2}X_{3} and In_{2}X_{3} single layers are energetically favorable.^{10} The cohesive energy E_{c} of the Janus GaInX_{3} single-layers can be evaluated by:
(1) |
The obtained calculations for E_{c} of GaInX_{3} single-layers are summarized in Table 1. It is found that the cohesive energy of GaInX_{3} single-layers is quite high, from 3.49 to 4.35 eV per atom and these three structures of GaInX_{3} are confirmed to be energetically favorable. The cohesive energy E_{c} decreases as the atomic size of X element increasing.
Next, we calculate the phonon spectra to evaluate the dynamical stability of GaInX_{3} single-layers. The phonon spectra of GaInX_{3} single-layers, which are calculated by the DFPT method,^{21} are presented in Fig. 2. Note that the phonon splitting corrections^{23} are not included in the present calculations. There are five atoms in the primitive cell of GaInX_{3}, therefore, its phonon dispersions contain 15 normal vibrational modes at the center of the Brillouin zone (the Γ point), including three acoustic modes in the low-frequency region and twelve optical modes in the higher frequency regions. It is known that the larger the mass atomic mass of elements, the softer the vibrations.^{24} Then, vibration frequencies in the phonon dispersions of GaInX_{3} are downshifted as the chalcogen element X changes from S to Te. As shown in Fig. 2, we can see that the phonon energies of GaInS_{3} single-layers are lower than those of GaInSe_{3} and GaInTe_{3} single-layers at the same point in the Brillouin zone. More importantly, there are no negative frequencies in the phonon dispersions of all three Janus GaInX_{3}. The dynamic stabilities of materials were confirmed when the evaluated phonon dispersions contain only positive frequencies in the whole Brillouin zone. If the imaginary frequencies are available, the restoring force, which opposes the atom displacement of the atoms, will no longer exist. Obtained calculations for the phonon spectra, as depicted in Fig. 2, indicate that the Janus GaInX_{3} single-layers are dynamically stable.
Further, we examine the elastic characteristics to elucidate the mechanical stability of the considered Janus single-layers. The elastic constants can give important information about the mechanical stability of materials. The elastic constants C_{ij} can be evaluated from the variation of energy when the small strains are applied to the equilibrium lattice state. With hexagonal structure as depicted in Fig. 1, only two independent elastic coefficients need to be estimated being C_{11} and C_{12} due to C_{11} = C_{22} and C_{66} = (C_{11} − C_{12})/2. The small range of uniaxial strain between −0.015 and 0.015 is applied to the x and y directions (each step being 0.005). The lattice structures at each value of applied strain are optimized and obtained coefficients are so-called relaxed-ion coefficients. By polynomial fitting the small strain-dependence of energy, we can get the coefficients C_{11} and C_{12}.^{25} In Table 2, we summarized the calculated results for the elastic constants C_{ij} of GaInX_{3} single-layers. It is demonstrated that the elastic constants of GaInX_{3} single-layers meet the Born's criteria for mechanical stability for hexagonal structures: C_{11} > 0 and C_{11} − C_{12} > 0.^{26}
Young's modulus and Poisson's ratio depend greatly on the in-plane symmetric structure of the materials. The direction-dependence of Young's modulus Y_{2D}(φ) and Poisson's ratio ν(φ) can be written as^{27,28}
(2) |
(3) |
The polar diagrams of Y_{2D}(θ) and ν(θ) of GaInX_{3} single-layers are presented in Fig. 3. We can see that the direction-dependence of Y_{2D}(θ) and ν(θ) are perfectly circulars. This suggests that GaInX_{3} single-layers show isotropic elastic characteristics. This is due to the 2D isotropic structures of GaInX_{2} as shown in Fig. 1. The Janus GaInX_{3} has a small Young's modulus, from 25.61 to 90.76 N m^{−1}. Obviously, Young's modulus of GaInX_{3} is smaller than that of other available 2D structures, such as Janus MoSSe (113 N m^{−1})^{29} or MoS_{2} (130 N m^{−1}).^{30} This indicates that the Janus GaInX_{3} single-layers are mechanically flexible materials and their mechanical strain threshold can be large.
Fig. 4 Band structures along the high-symmetry direction Γ–M–K–Γ of Janus GaInX_{3} (X = S, Se, Te) single-layers at the PBE (solid lines) and HSE06 (dashed lines) levels. |
E^{PBE}_{g} | E^{HSE06}_{g} | E_{F} | ΔΦ | Φ_{1} | Φ_{2} | |
---|---|---|---|---|---|---|
GaInS_{3} | 0.93 | 1.64 | −2.08 | 1.65 | 4.88 | 6.53 |
GaInSe_{3} | 0.47 | 1.20 | −2.01 | 1.29 | 4.53 | 5.83 |
GaInTe_{3} | 0.25 | 0.75 | −1.18 | 0.90 | 4.16 | 5.06 |
To get more insights into the nature of the energy bands of the considered structures, we evaluate the weighted band of GaInX_{3} single-layers at the HSE06 level as revealed in Fig. 5. There are similarities in the weighted bands between the Janus GaInX_{3} single-layers. It is demonstrated that the VBM of GaInX_{3} is mainly contributed from the p-orbitals of the chalcogen atom X. Meanwhile, Ga-s orbitals have a largely contribution to the CBM of GaInX_{3}. The contribution of the orbitals of Ga and In atoms to the valence band is much smaller than that of the X-p orbitals. Also, the X-p orbitals also make a important contribution to the conduction band in the high-energy region.
Fig. 5 Weighted bands of GaInS_{3} (a), GaInSe_{3} (b), and GaInTe_{3} (c) single-layers at the HSE06 level. |
One of the more important properties of electrons that we need to investigate is the work function. The work function Φ is calculated based on the formula as: Φ = E_{vac} − E_{F}. Here, E_{vac} and E_{F} are the vacuum and Fermi levels, which can obtain by calculating the electrostatic potential of the material. Due to the vertical asymmetric structure, Janus single-layers possess an intrinsic built-in electric field as previously reported by Fu and co-workers.^{7} Therefore, we include the dipole correction in the present calculations to treat the possible errors induced by the periodic boundary condition.^{31} The electrostatic potential of the Janus GaInX_{3} single-layers with dipole correction are depicted in Fig. 6. As predicted, a distinct vacuum level difference ΔΦ between the two sides has been found in the Janus GaInX_{3} structures. Our calculations reveal that the ΔΦ of GaInX_{3} reduces when the atomic size of X element increases. As summarized in Table 3, the vacuum level difference ΔΦ of GaInX_{3} is from 0.90 to 1.65 eV. As a result, there is a difference between the work functions on the two different sides of GaInX_{3}. The calculated work functions on the XGa-side Φ_{1} and InX-side Φ_{2} of GaInX_{3} single-layers are tabled in Table 3. We can see that, in each single-layer, Φ_{1} is always smaller than Φ_{2}. This suggests that electrons can escape from the XGa-surface more easily than the InX-surface. Besides, the difference in the vacuum level leads to the difference in the photocatalytic performance at the two different surfaces of the Janus structures.
Fig. 6 Planar electrostatic potential of the Janus GaInX_{3} (X = S, Se, Te) single-layers with dipole correction. ΔΦ is the difference in the vacuum levels between the XGa and InX sides. |
(4) |
To calculate the carrier mobility μ_{2D} as expressed in eqn (4), we first evaluate the effective mass of carriers, which strongly affects the mobilities of carriers. The effective masses of carriers can be attained by fitting parabolic function to the CBM (electron) and VBM (hole) via the formula as follows:
(5) |
The effective masses of carriers of GaInX_{3} single-layers are summarized in Table 4. It is found that the effective mass of both electron and hole is highly directional isotropic. This is due to the in-plane isotropic lattice of the Janus GaInX_{3} single-layers. However, the effective mass of the electron is much smaller than that of hole, particularly in the case of GaInS_{3} single-layer. The smaller the carrier mass, the faster carriers respond to the external field, and as a result, they have high mobility. As listed in Table 4, we can see that GaInTe_{3} has the smallest electron effective mas, 0.15 m_{0}. The electron effective mass of GaInS_{3} and GaInTe_{3} is found to be 0.29 m_{0} and 0.18 m_{0}, respectively.
C_{2D}^{x} | C_{2D}^{y} | E_{d}^{x} | E_{d}^{y} | μ_{x} | μ_{y} | ||||
---|---|---|---|---|---|---|---|---|---|
GaInS_{3} | Electron | 0.29 | 0.29 | 66.57 | 66.58 | −9.12 | −9.12 | 202.87 | 203.08 |
Hole | 2.69 | 2.69 | 66.57 | 66.58 | −7.51 | −7.59 | 3.20 | 3.13 | |
GaInSe_{3} | Electron | 0.18 | 0.18 | 54.84 | 54.86 | −8.01 | −8.01 | 580.82 | 580.55 |
Hole | 0.47 | 0.47 | 54.84 | 54.86 | −7.33 | −7.42 | 22.88 | 22.33 | |
GaInTe_{3} | Electron | 0.15 | 0.15 | 43.38 | 43.43 | −8.70 | −8.69 | 530.03 | 532.18 |
Hole | 0.23 | 0.23 | 43.38 | 43.43 | −7.82 | −7.98 | 117.52 | 113.06 |
Along with the effective mass, the carrier mobility depends also on the elastic modulus C_{2D} and deformation potential constant E_{d} as presented in eqn (4). The elastic modulus C_{2D} is given by
(6) |
The deformation potential constant E_{d} is written in the form:
(7) |
The strain-dependent energy change and band edge positions of all three Janus structures GaInX_{3} are shown in Fig. 7. It is found that there is no significant difference in elastic modulus along the x and y directions in each single-layer. The deformation potential constant of GaInX_{3} is also high directional isotropic. This is also due to the in-plane isotropic lattice of the investigated systems. However, there is a small difference in the E_{d} of electron and hole in all three investigated structures. The calculated results for the C_{2D} and E_{d} are summarized in Table 4.
Fig. 7 Strain-dependent total energy (a) and band-edge positions (b) of GaInX_{3} single-layers along two transport directions x and y. |
In Table 4, we list the calculated results for the carrier mobility of GaInX_{3} single-layers along the two transport directions. It is found that the mobility of both electron and hole of GaInX_{3} is highly directional isotropic. There is no significant difference between carrier mobilities along the x and y directions. Due to the largely difference in the effective mass and also deformation potential constant between electron and hole, it is calculated that the electron mobility is much higher than that the hole mobility. The electron mobility of GaInSe_{3} is calculated to be up to about 580 cm^{2} V^{−1} s^{−1} and all Janus GaInX_{3} structures possess electron mobility in excess of 200 cm^{2} V^{−1} s^{−1}. It is also worth noting that, with electron mobility of about 200 cm^{2} V^{−1} s^{−1}, MoS_{2} was able to perfectly apply to high-performance electronic devices as previously reported by Radisavljevic and co-workers.^{34} This suggests that the Janus GaInX_{3} single-layer could be potential materials for applications in nanoelectronic devices.
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