Silicane nanoribbons: electronic structure and electric field modulation

We present electronic band structure, Gibbs free energy of formation, and electric field modulation calculations for silicane nanoribbons (NRs), i.e., completely hydrogenated or fluorinated silicene NRs, using density functional theory. We find that although the completely hydrogenated silicene (H-silicane) sheet in the chair-like configuration is an indirect-band-gap semiconductor, a direct band gap can be achieved in the zigzag H-silicane NRs by using Brillouin-zone folding. Compared to H-silicane NRs, the band gaps of completely fluorinated silicene (F-silicane) NRs reduce at least by half. For all silicane NRs considered here, the Gibbs free energy of formation is negative but shows different trends by changing the ribbon width for H-silicane NRs and F-silicane NRs. Furthermore, by analyzing the effect of transverse electric fields on the electronic properties of silicane NRs, we show that an external electric field can make the electrons and holes states spatially separated and even render silicane NRs self-doped. The tunable electronic properties of silicane NRs make them suitable for nanotechnology application.


Introduction
Silicene, a two-dimensional (2D) monolayer of silicon (Si) atoms in the honeycomb lattice, has recently aroused tremendous attention in the scientific community, owing to its graphene-like electronic properties [1], such as the presence of a Dirac cone in the vicinity of K points, and its compatibility with current Si-based electronics. Synthesis of silicene on the different substrates, such as Ag(110), Ag(111), and ZrB 2 (0001), has been reported by some groups [2][3][4][5][6]. Being different from the flat geometry of graphene, silicene has a stable, low-buckled honeycomb structure, as a result of which a perpendicular electric field can induce a tunable band gap in silicene by breaking the inversion symmetry [7,8]. Adsorption of a foreign chemical species, such as hydrogen (H), fluorine (F), or chlorine (Cl), is also a promising approach to tailor the electronic properties of silicene [9][10][11][12][13]. Houssa et al [10] demonstrated using density functional theory (DFT) calculations that completely hydrogenated silicene, also called silicane, has a wide band gap. Quhe et al [13] predicted that single-side adsorption of alkali atoms can open a band gap in silicene, and the band gap size is controllable by changing the adsorption coverage, with a maximum band gap up to 0.50 eV. Furthermore, the dimensionality of materials plays a crucial role in modulating their electronic properties. One-dimensional (1D) silicene nanoribbons (NRs) possess distinct electronic and magnetic properties depending on their chirality, width, and edge structure. Ding et al [14] found that the band gap of edge hydrogenated armchair silicene NRs oscillates with the width in a period of three. Fang et al [15] studied the effect of edge hydrogenation and doping on the properties of zigzag silicene NRs, showing that the dihydrogenated edges can effectively stabilize the antiferromagnetic semiconducting ground state of zigzag silicene NRs and nitrogen (or phosphorus) doped monohydrogenated zigzag silicene NR is a magnetic semiconductor. Though the chemical functionalization is considered one of the important methods to tune the properties of silicene, few theoretical works have systematically examined the properties of surface-functionalizated silicene NRs.
In this work, we present a DFT study on the electronic band structure and relative stability of completely hydrogenated and fluorinated silicene NRs. Moreover, we investigate the effect of transverse electric fields on the electronic properties of NRs, showing that an external electric field can make the electrons and holes states localized at the different sides of an NR and even render the NR self-doped.

Method
Our calculations were performed using DFT with the norm-conserving pseudopotential method as implemented in the Siesta [16,17] code. The Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) [18] was used for the exchange-correlation functional. Doublezeta plus polarization numerical atomic orbitals were used for all atoms. A cutoff of 250 Ry for the grid integration was employed to guarantee a good description of the charge density. One primitive unit cell was used in all calculations. A vacuum region larger than 10 Å was used to avoid interaction between the neighboring images of NRs. Brillouin-zone (BZ) integration was performed on a × × (30 1 1) and × × (1 18 1) Monkhorst-Pack K POINTS [19] meshs for the atomic relaxation of zigzag NRs and armchair NRs, respectively. All geometries were relaxed until the maximum atomic force was smaller than 0.03 eVÅ −1 . The uniform transverse electric field was modeled by a sawtooth-like potential.

Results and discussion
Following the convention of graphene and graphane [20], the hydrogenated and fluorinated silicene are named as H-silicane and F-silicane, respectively. Here, we consider only the chairlike configurations of H-silicane and F-silicane, as shown in figure 1 (a), the stabilities of which have been verified recently [9][10][11]. The optimized lattice constants are a = 3.87 Å, 3.88 Å, and 3.93 Å for silicene, H-silicane, and F-silicane sheets, respectively. Our calculations show that H-silicane sheet has an indirect gap of 2.282 eV with the valence band maximum (VBM) at the Γ point and the conduction band minimum (CBM) at the M point, while the F-silicane sheet has a direct gap of 0.887 eV with both the VBM and CBM at the Γ point, which are in good agreement with the previous calculations [9][10][11]. We then construct the 1D zigzag and armchair silicane NRs by cutting the 2D chair-like silicane sheets. The zigzag H-silicane NR with n zigzag chains is named as H-nZSNR, while the armchair H-silicane NR with n dimer lines is named as H-nASNR. The same definitions are used for the F-silicane NRs. The edge Si atoms are all saturated with H or F atoms to avoid the effects of dangling bonds. Li et al [21] have shown that the bare edges of zigzag silicane NRs can induce magnetic moments, whereas the passivated edges have no magnetism. We define the width of a silicane NR as the distance between the Si atoms at two edges, as shown in figures 1 (b) and (c). The width of ZSNRs and ASNRs is dependent on the parameter n and given by ( 1) A with the silicane lattice constant a.

Electronic structures of silicane NRs
We now study the electronic band structures of zigzag and armchair silicane NRs.  figure 4. For silicane NRs, the band gap decreases as the NR width increases, and the armchair NRs have a slightly larger band

Relative stability of silicane NRs
Next, we study the relative stability of silicane NRs. As these NRs have different chemical compositions, the cohesive energy per atom does not offer an appropriate criterion to compare their relative stability. Therefore, we adopt the approach in references [22,23] and define a zero-temperature molar (per atom) Gibbs free energy of formation δG for a silicane NR as where E c is the cohesive energy per atom of a silicane NR, x i is the molar fraction of atom i (i = H or F) in the NR. We choose μ i as the binding energy per atom of i 2 molecule and μ Si as the cohesive energy per atom of silicene sheet. Using Si silicene atom , we obtain another formula for the δG of a silicane NR as follows where E tot is the total energy of a silicane NR, E i 2 is the total energy of an i 2 molecule, and E silicene is the energy per atom of silicene sheet. The number of Si (i) atoms in a silicane NR is n Si (n i ). Figures 5 (a) and (b) show the δG versus the width of H-silicane NRs and F-silicane NRs, respectively. The Gibbs free energy of formation decreases monotonically with increasing NR width for H-silicane NRs but increases monotonically as the NR width increases for F-silicane NRs, which implies that wider NRs are energetically favorable for H-silicane NRs, but narrower NRs are more likely to form for F-silicane NRs. We note that the calculated Si-H bond energy (BE) in H-silicane, Si-Si BE in silicene, and Si-F BE in F-silicane are −3.021 eV, −4.581 eV,  electric field makes the energy of the electronic states at the low potential region increase, and the energy of the electronic states at the high potential region decrease, so that the VBM on one side of the ribbon is leveled to the CBM on the opposite side when a threshold electric field is reached (see figures 7(a)-(c)). Moreover, the band gap of the ribbon on the two sides remains unchanged, leading to the possibility of the generation of electrons and holes via tunneling [24] across the ribbon, which could induce novel optical and electric properties. When we increase the electric field to 0.2 V Å −1 , the VBM at the low potential region stays above the Fermi energy, and the CBM at the high potential region below the Fermi energy (see figures 7(d)-(f)), indicating that the electric field renders the NR self-doped; p-type on one side and n-type on the opposite side.
Finally, we analyze the dependence of electric field modulation on the width of silicane NRs. Figure 8 plots the band gap of silicane NRs as a function of electric field. The band gap of all silicane NRs decreases as the strength of the electric field increases and reaches almost zero at the threshold electric field. The wider the NR is, the smaller the threshold electric field. For NRs of similar width, the threshold electric field for F-ZSNR is about half of that for H-ZSNR and the armchair NRs have a larger threshold electric field than the zigzag NRs. Although the band gap underestimation by the PBE functional makes the threshold electric field smaller than the actual value, the qualitative trend is usually right.

Conclusion
On the basis of density functional theory calculations, we have studied the electronic band structure and relative stability of silicane nanoribbons. The zigzag H-silicane NRs exhibit a direct band gap, differing from the indirect band gap of H-silicane sheet, whereas the armchair H-silicane NRs are still indirect-band-gap semiconductors. The band gaps of F-silicane NRs become about half of those of H-silicane NRs. For all silicane NRs considered here, the Gibbs free energy of formation is negative and decreases monotonically with the ribbon width for Hsilicane NRs but increases monotonically with the ribbon width for F-silicane NRs. Furthermore, the electric field modulation calculations show that the band gap of silicane NRs decreases as the strength of the electric field increases, and the external electric field increases can even render silicane NRs self-doped. Our results are helpful for understanding the electronic properties of silicane NRs and open up the possibility of using them in nanodevices.