The stability, structural, electronic, and optical properties of hydrogenated silicene under hydrostatic pressures: a first-principle study

The structural, electronic, and optical properties of hydrogenated silicene have been studied under different hydrostatic pressures using first-principle calculations. The binding energy and band structure have been calculated for chair (C-) and boat (B-) structures, which are having good stability at 0 GPa, 3 GPa, 6 GPa, 9 GPa, 12 GPa, 15 GPa, and 18 GPa hydrostatic pressures. Stability has been verified using binding energy and phonon calculations. The C- and B-structures have become metallic and unstable at 21 GPa. The optical properties of B-configuration have been studied in the energy range of 0–20 eV. Five optical parameters such as conductivity threshold (σth), dielectric constant ε(0), refractive index n(0), birefringence Δn(0), and plasmon energy (ħωp) have been calculated for the first time under different hydrostatic pressures. The calculated values are in good agreement with the reported values at 0 GPa.


Introduction
In the recent years, substantial scientific efforts have been given on graphene, as it exhibits outstanding properties such as high carrier mobility, current carrying capacity, non-zero berry's phase, unusual quantum Hall effect, and linear dispersion energy bands [1][2][3]. However, it may not be adaptable to the present silicon based industry. Accordingly, researchers investigated the other IVA group elements, such as silicene and germanene. Silicene and germanene are the graphene analogy of silicon and germanium. Silicon is the more abundant element compared to germanium, and also silicene has important properties of linear dispersing energy bands, massless Dirac distribution, [4][5][6], ferromagnetic [7], half metallicity [8], giant magnetoresistance [9], and superconductivity properties [10]. It shows a good spin-orbit gap at Dirac points, spin-orbit coupling [11], and the emergence of valley polarized metal phase for spintronic applications [12,13]. Silicene has zero energy bandgap like graphene and also has less thermal conductivity [14,15]. To introduce the energy band gap many procedures have been used such as chemical functionality [16,17], application of electric field [18], doping [19], substrate effect [20], nanoribbon [21,22], and introducing nanoholes [23,24]. Hydrogenation is a well-known technique to create a bandgap and increases the thermal conductivity of silicene [25,26]. Normally, the hydrogenated silicene is also called as silicane, which is experimentally tested on Ag (111) [27][28][29].
First-principle calculations have used by several workers to study the structural, electronic, magnetic, and thermal properties of hydrogenated silicene [30][31][32][33][34][35]. The thermal properties of silicane have been calculated by Lui et al. [36] who predicted that hydrogenation leads to a large increment of thermal conductivity from 22.5 Wm −1 K −1 to 78.0 Wm −1 K −1 . The influence of hydrogen in the silicene sheet has been studied in which half-silicane shows ferromagnetic behavior [37]. Nguyen et al. [38] have studied the hydrogenated silicene and graphene that confirms silicene has a strong binding with hydrogen compared to graphene. Several workers have deliberated on carrier mobility and electron transport properties of silicene and hydrogenated silicene [39][40][41]. Molecular dynamics simulation reveals the stability of adsorption configurations of hydrogenated silicene [42]. Other researchers have studied the hydrogenated silicene and observed that chair (C-) and boat (B-) structures have good stability [25,26]. The authors [43][44][45][46][47][48] have also studied several properties of hydrogenated graphene, hydrogenated silicene, and hydrogenated germanene using first-principle calculations. So far, the research on the properties of C-and B-structures of hydrogenated silicene under different hydrostatic pressures has been temporarily vacant. Therefore, to bridge up the gap we have studied the structural, electronic, and optical properties of C-and B-conformers of hydrogenated silicene under external pressures.
In this paper, first-principle calculations based on density functional theory is used to optimize the geometry structures and calculate the lattice constants (a, b, and c), bond lengths (d Si-Si and d Si-H ), bond angles (θ Si-Si-Si and θ Si-Si-H ), energy bandgap (E g ), and binding energy (E b ) at 0, 3, 6, 9, 12, 15, 18, and 21 GPa pressures. The stability is studied using the phonon and binding energy data. Five parameters of optical properties, n(0), Δn(0), ε(0), σ th , and ћω p , are reported in in-plane (E⊥c) and out of plane (E||c) polarization for the first time.

Methods
The calculations have been performed using Cambridge Sequential Total Energy Package (CASTEP) simulation package [49]. The generalized gradient approximation (GGA) has been used with the Perdew-Burke-Ernzerhof (PBE) [50] scheme to estimate the structural, electronic and optical properties of hydrogenated silicene. A plane-wave basis set kinetic energy cut-off of 272 eV is used in an ultrasoft pseudopotential representation in the reciprocal crystal lattice [51]. The optimized structure is obtained by applying the Broyden-Fletcher-Goldfarb-Shanno (BFGS) scheme [52]. Throughout the geometry optimization, maximum tolerances of total energy of 0.2 × 10 −4 eV/atom, Fermi energy of 0.27 × 10 −13 eV, and Hellmann-Feynman ionic force convergence of 0.05 eV/Å are used. The maximum stress component tolerances and ionic displacement tolerances are 0.1 GPa and 0.2 × 10 −2 Å, respectively. The calculations can also be performed using the LDA, GW, and HSE approximations, but LDA underestimates the energy gap [53], while GW and HSE overestimate the values 40-50% greater than the original or experimental values [54,55].

Structural and electronic properties
The hydrogenation of silicene leads to the disruption of π-bond of adjacent p z orbitals of silicon atoms. The hexagonal sp 2 silicene lattice changes to tetrahedral sp 3 hybridized silicane (hydrogenated silicene) with strong σ-bonding. As a consequence, the perpendicular distance between sub-lattices increases, which increases the bond lengths and lattice constant. The hydrogenation of silicene forms many structures such as chair, boat, tricycle, twistedboat, and wash board, in which chair (C-) and boat (B-) structures have good stability [25,26]. Figure 1a shows the stable configuration of C-structure of silicane, which belongs to a space group of P-3M1, having a trigonal structure with lattice constants of 3.857 Å, 3.857 Å, and 4.59 Å, respectively, for a, b, and c. Figure 1b shows the structure of B-silicane that belongs to the orthorhombic structure with space group of PMMN. The optimized lattice parameters of B-silicane are 6.271 Å, 3.714 Å, and 6.015 Å, respectively, for a, b, and c.
The calculated values of bond lengths (d Si-Si and d Si-H ) and bond angles (θ Si-Si-Si and θ Si-Si-H ) are listed in Tables 1  and 2, respectively, for B-and C-configurations. There are two pairs of Si-Si bond lengths in B-configuration, one is parallel to the plane, and the other is perpendicular to the plane of the silicene sheet. The first bond length has a higher value than the second due to H-H repulsion bonded with adjacent silicon atoms [56]. The bond length (d Si-Si ) of two configurations of hydrogenated silicene shows a higher value than pristine silicene because of the depopulation of bonding orbitals of silicon atoms. This depopulation is due to the electronegativity difference between hydrogen and silicon atoms. Tables 1 and 2 show the calculated structural parameters are in good agreement with the available reported values. The two structures have been optimized for 100 iterations, and the estimated total energies for B-and C-conformers are − 494.45 eV and − 247.30 eV, respectively. The corresponding Pseudo atomic total energies of Si and H are − 165.0639 eV and − 12.3526 eV. The equation used for the calculation of binding energy has been taken from Sahin et al. [57,58] is the total energies of hydrogen, E Sil T is the total energy of silicene, E SiH T is the total energy of hydrogenated silicene, n H is the number of hydrogen atoms, and n is the total number of atoms per unit cell. The calculated value of E b shows that the B-and C-configurations have good stability. The stability has also been verified using phonon calculations, in which positive phonon dispersion (Fig. 2a) confirms the stability of hydrogenated silicene [44].
The nature of the bond has been verified using the bond population [59]. The high value of the bond population confirms the nature of the covalent bond; the medium is semiionic and the low value approves ionic bond. The value of spilling parameter σ should be low, to verify the calculated values are reliable or not. It has been calculated using the relation = 1 is the number of PW states, (k) is the Eigen states obtained  for given wave vector k, p(k) is the projection operator of Bloch functions with wave vector k. In B-configuration, the charge transfer from silicon to hydrogen is 0.08, and the bond population for S-H is 0.86. The value of the spilling parameter is 1.39%. In C-silicane, the calculated bond population of Si-H is 0.82, which conveys the covalent bond, and charge transfer from silicon to hydrogen is 0.08. The value of the spilling parameter is 1.46%. The band structure of hydrogenated silicene has been calculated along with the high symmetry points for two structures and shown in Fig. 3a and b. The C-silicane shows an indirect bandgap while the B-configuration has a direct bandgap. The direct bandgap of the B-structure is applicable for solar cells and photodetectors [60]. The calculated values of the energy gap are listed in Tables 1 and 2, respectively, for B-and C-configurations along with the known values. Reasonably good agreements are obtained between them.
The density of states (DOS) gives information about the character of orbital in the formation of energy bands and states of hybridization. The total density of states (TDOS) and partial density of states (PDOS) are calculated and shown in Fig. 4a and b, respectively, for B-and C-silicane. For B-configuration, the valence band consists of three parts. The first part is mainly dominated by Si-s orbital in the energy range of − 11.25 eV to − 8.12 eV. The second part is from − 8.13 eV to − 5.27 eV, which is predominantly influenced by Si-s and H-1 s orbitals, and the last part from − 5.28 eV to Fermi level, is dominated by Si-p and H-s states. However, the conduction band is the combination of H-s, Si-s, and Si-p orbitals. This analysis shows that, the creation of an energy band gap between the valence band and conduction band is mainly due to the hybridization of H-1 s and S-3p orbitals. The explanation for the density of states of C-configuration has been given in previous publication [44].

Pressure effect on structural and electronic properties of silicane
The behavior of structural and electronic properties under different pressures has been studied for B-and C-configurations. The estimated values of lattice constants, bond lengths, bond angles, total energy, unit cell volume, binding energy, and energy bandgap are presented in Tables 1  and 2  shows that B-and C-configurations become unstable at 21 GPa external pressure. The above statement is verified by the phonon calculation in which imaginary frequencies (ω 2 (k, j < 0)), shown as negative values below the zero level. These negative values are called soft modes [61]. The soft modes that occur at K-point confirm instability (Fig. 2b). Table 1 shows the direct bandgap of B-conformer decreases for every 3 GPa rise in pressure and becomes zero at 21 GPa, whereas for C-configuration in Table 2, the indirect bandgap decreases with the increase of pressure and becomes zero at 21 GPa. The behavior of energy bandgap and binding energy under different pressures is shown in Fig. 5a and b.

Optical properties
Optical parameters such as refractive index, dielectric constant, birefringence, conductivity threshold, complex dielectric function, and plasmon energy of B-configuration hydrogenated silicene under 0 GPa, 6 GPa, and 12 GPa external pressures have been analyzed. The imaginary part dielectric function ε 2 (ω) conveys the transitions from occupied electronic states to unoccupied conduction states. These transitions can be divided into direct and indirect transitions. The direct transitions are further divided into intra-band and inter-band transitions. The intra-band transitions affect the infrared part of the spectrum and calculated using Drude's contribution with a damping factor δ = 0.5 eV. The interband transitions are calculated using momentum matrix elements between the valence and conduction band wave functions. The ε 2 (ω) and momentum matrix elements are derived using the relation given by Momida et al. [62].
The C-configuration shows an indirect bandgap, which is not suitable for optoelectronic applications. The optical properties of B-configuration have been calculated along the in-plane (E⊥c) and out-of-plane polarization (E||c). The behavior of 2 ( ) of B-configuration is plotted in Fig. 6a and b for E⊥c and E||c planes, respectively, in the energy range of 0-20 eV. The conduction starts at threshold energy (E th ) of 2.205 eV in the in-plane polarization and rises rapidly due   Table 3 which decreases with an increase of external pressure. The peak value increases with pressure due to an increase in the range of band structure. In out-of-plane polarization, the conduction starts at 3.646 eV and reaches a maximum value at 6.883 eV. The optical band gap also decreases with the increase of pressure from 0 to 6 GPa and 6 GPa to 12 GPa.
The real part of the dielectric function has been calculated using the Kramer-Kronig relation [63] and shown in Fig. 7a and b. The calculated dielectric constant in in-plane and out-of-plane polarizations has been listed in Table 3. The estimated static dielectric function of ε(0) shows the hydrogenation increases the mobility of electrons in pristine silicene. The ε(0) increases with the increase of external pressure, which is due to a decrease in the optical bandgap of the material. Using the Penn model [64], the dielectric constant is inversely proportional to bandgap energy, i.e., (0) = 1 + The polarizabilility also increases for every 6 GPa rise in hydrostatic pressures. The refractive index n(ω) has been derived using the relation given by Sahin et al. [65]. The calculated n(ω) has been shown in Fig. 8a and b for E⊥c and E||c for 0 GPa, 6 GPa, and 12 GPa, respectively. The static refractive index in in-plane and out-of-plane polarization is   Table 3, which increases with the increase of external pressures.
The birefringence is the difference between the refractive index in different directions and the calculated birefringence shown in Fig. 9. In the mid UV region, hydrogenated silicene shows large birefringence and the phase-matching condition is no longer satisfied. The electron energy loss function L(ω) has been calculated in both directions using the relation Sahin et al. [65] and shown in Fig. 10a and b. The hydrogenated silicene shows high Plasmon energies compared to pristine silicene due to the Π to σ bond translation, which causes transitions from the lowest occupied valence band to the highest unoccupied conduction band (σ- * plasmon).

Conclusions
The B-configuration shows a direct bandgap behavior, and C-shows an indirect bandgap. The energy bandgap of both the structures decreases with an increase in external pressure and becomes zero at 21 GPa. The stability decreases with the increase of pressure and becomes unstable at 21 GPa. The optical bandgap of the B-structure decreases with the increase of external pressure in both in-plane and out-ofplane polarization. The hydrogenation increases the mobility of electrons and shows high plasmon energy compared   to pristine silicene. The value of ε(0) increases with the increase of external pressure. The values of E th , ε(0), n(0), Δn(0), ћω p , E g , and E b under different pressures are reported for the first time. The knowledge of these parameters is very important in designing various optoelectronic devices and integrated circuits.