off Switching of Silicene Superlattice - The Journal of

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Bandgap on/off Switching of Silicene Superlattice Tian-Tian Jia, Mengmeng Zheng, Xin-Yu Fan, Yan Su, ShuJuan Li, Hai-ying Liu, Gang Chen, and Yoshiyuki Kawazoe J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.5b06626 • Publication Date (Web): 17 Aug 2015 Downloaded from http://pubs.acs.org on August 23, 2015

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The Journal of Physical Chemistry

Bandgap on/off Switching of Silicene Superlattice Tian-Tian Jia1, Meng-Meng Zheng2, Xin-Yu Fan1, Yan Su1, Shu-Juan Li1, Hai-Ying Liu1, Gang Chen1,*, and Yoshiyuki Kawazoe3,4 1

Laboratory of Advanced Materials Physics and Nanodevices, School of Physics and Technology, University of Jinan, Jinan, Shandong 250022, P. R. China 2 Shandong Provincial Key Laboratory of Laser Polarization and Information Technology and Department of Physics, Qufu Normal University, Qufu, Shandong 273165, P. R. China 3 New Industry Creation Hatchery Center, Tohoku University, Sendai, Miyagi 980-8577, Japan 4 Kutateladze Institute of Thermophysics, Siberian Branch of Russian Academy of Sciences, Novosibirsk 630090, Russia

Abstract On the basis of density functional theory calculations with generalized gradient approximation, we have investigated in detail the cooperative effects of uniaxial strain and degenerate perturbation on manipulating bandgap in silicene. The uniaxial strain would split π bands into πa and πz bands making Dirac cone to move. Then, the hexagonal antidot would split πa (πz) bands into πa1 and πa2 (πz1 and πz2) bands accounting for the bandgap opening in the superlattices with Dirac cone being folded to Γ point, which is a different mechanism as compared to the sublattice equivalence breaking. The energy interval between the split πa and πz bands could be tuned to switch bandgap on/off, suggesting a reversible switch between the high charge carrier velocity properties of massless Fermions attributed to linear energy dispersion relation around Dirac point and the high on/off properties associated with sizable bandgap. Even more, the gap width could be continuously tuned by manipulating strain, showing fascinating application potentials.

Keywords: Silicene, Electronic Properties, Bandgap Tuning, Degenerate Perturbation

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1. INTRODUCTION Recently, the two dimensional (2D) material silicene has gained extraordinary research interests. Unlike its carbon counterpart graphene, it is found to be unstable to form a perfectly planar honeycomb structure due to its weaker sp2 hybridization. As early as 1994, Takeda and Shiraishi1 reported the sheet structure with dimples rather than perfect planar structure of 2D silicon material based on quantum mechanical ab initio calculations. The notion of silicene was firstly used by Guzmán-Verri and Lew Yan Voon2 in 2007, who also demonstrated the presence of Dirac cone based on tight-binding calculations. The low-buckled silicene has been verified to be stable due to the mixing of sp2 and sp3 hybridization. The slightly buckled honeycomb lattice consists of two sublattices with one of them being displaced vertically of about half angstrom with respect to the other. Besides the experimental progresses of synthesizing silicene on substrate3-12, the suspended silicene sheet has also been fabricated by using chemical exfoliation method, etc, which paves the way for exploring its unique properties and prospective applications13,14. Furthermore, recent theoretical studies suggest the presence of quasi free-standing silicene15-17. The hexagonal BN and hydrogen-processed Si(111) are found to be good substrates to support quasi free-standing silicene, whose binding energies between them are comparable to or twice as that of graphite. Also, the quasi freestanding silicene shows fascinating electronic properties as those observed in graphene. For example, the massless charge carriers also have high group velocity attributed to the linear energy dispersion relation around Dirac point3,18,19. For the extraordinarily attractive electronic properties, silicene is regarded as superiorly advanced material for nanotechnology, which would be most likely favored over graphene in integrating into Si-based nanoelectronics. As for its digital electronic applications in the high performance nanoelectronics, a sizable bandgap needs to be opened to gain high on/off ratio, which can be realized by substitution doping20, substrate effects8,15,21, adsorption


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of functional groups22-26 and electric field27. However, an issue needs to be addressed. The high group velocity of massless Dirac fermions would be sacrificed along with the bandgap opening due to the breaking of the linear dispersion relation, especially for the case with wide bandgap. Hence, it would be interesting if the bandgap could be switched on/off for a specific silicene-based nanomaterial toward different application demands. Recently, many experimental studies reported the synthesis of 2D nanomeshes by using nanotechnologies28-31. Bai et al.28 fabricated the nanomesh by using the block copolymer lithography, in which the antidots were periodically arranged. Liang et al.29 combined the nanoimprint lithography and the block-copolymer self-assembly for high-resolution nanoimprint template patterning in preparing the nanomesh. Kim et al.30 also successfully nanoperforated the nanomesh by using the cylinder-forming diblock copolymer templates. A method combining the e-beam lithography and the oxygen reactive ion etching was proved efficiently in making tunable periodic superlattices31. These studies shed light on the feasibility in producing superlattice structures with nice precision in fabricating regular patterns. The experimental advances also show the possibilities in patterning silicene into superlattice whose Born-von Karman boundary conditions would be modulated32-34. For the fundamental importance and as a contribution to the field of bandgap engineering, we have performed detailed studies on the bandgap engineering of silicene superlattice. In comparison with the well-known sublatticeequivalence breaking, the mechanism of bandgap opening has been discussed in detail for the silicene superlattice. For our studied superlattices, a kind of superlattices would have Γ as four-fold degenerate Dirac point according to the energy band-folding analysis. Then, the inversion symmetry preserved antidot used to shed light on the circularly shaped nanomesh hole synthesized in experiment could open a sizable bandgap. Furthermore, the opened bandgap could be reversibly switched on/off and the gap width could also be continuously tuned by applying uniaxial strain.



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2. COMPUTATIONAL DETAILS Our spin-polarized first principles calculations were carried out based on the density functional theory (DFT) as implemented in the Vienna ab initio simulation package (VASP)35. The projector augmented-wave (PAW) method36 and the generalized gradient approximation (GGA) were used. The exchange-correlation energy was described by the Perdew, Burke and Ernzerhof (PBE)37 parameterized functional. The solution of Kohn-Sham (KS) equation was solved by an efficient matrix diagonalization technique based on a sequential band-by-band residual minimization method35. The planewave basis set was adopted. The corresponding cutoff energy is 250 eV for dealing with the silicene and antidotpatterned silicene superlattice, while it is 400 eV for studying the silicene superlattice patterned by the regularly arranged (AlP)3 nanoflake patches. Because the lattice constant of 5.45 Å of AlP bulk agrees well with the 5.43 Å of silicon bulk, the (AlP)3 nanoflake is adopt to reduce the structural distortion after substitution doping. The 2D material was modeled in the supercell. The sheet material was placed in the XY plane which was separated by about 15 Å vacumm in Z direction to eliminate the interaction with its neighboring images. In the structural optimization, all the atoms were fully relaxed until the force acting on each atom converged to 0.01 eV/Å. The k-sampling in reciprocal lattice for calculating electronic properties was carried out by using the Monkhorst-Pack technique38. For the studying of silicene with the primitive unit cell and the smallest rectangle silicene superlattice, the integrations of electronic structures were carried out by using a 21×21×1 k-mesh. The structural parameters of the primitive unit cell were firstly optimized, which were then used to construct superlattice. The schematic structure of silicene was shown in Fig. 1. In our studies, the in plane lattices of the hexagonal primitive unit cell were calculated to be 3.87 Å and the displacement height between the sublattices of the low-buckled structure was found to be 0.45 Å. For the large-size orthogonal superlattices investigated in our studies, the corresponding


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unit cells are 20.111.615 Å3 and 40.223.215 Å3, respectively. Accordingly, the kmeshes of 5×7×1 and 3×5×1 were used for studying their electronic properties, respectively. Also, as for testing the accuracy of the method, we have studied the lattice constant of diamond silicon, which was calculated to be 5.47 Å falling within 1% to the experimental value of 5.43 Å39.

3. RESULTS AND DISCUSSION 3.1. Energy Band Folding and Bandgap Opening For the silicene, the electronic properties are studied in Fig. 1c, which have the Dirac cone located at K (K′) point due to the sublattice equivalence. The recent advances of nanotechnologies28-31 show the feasibilities of regularly arranging antidots in silicene sheet, making silicene nanomesh. This would impose new Born-von Karman boundary conditions to form silicene superlattice32-34. Though both hexagonal and orthogonal superlattices could be synthesized in experiment, we would like to use the orthogonal one as a prototype for the convenience to discuss the effects of uniaxial strain on modulating electronic properties, which could be surely extended to the case of hexagonal superlattice. In Fig. 1d, the geometrical structure of our studied superlattice is schematically illustrated. In each unit cell A×B, a high symmetry D6h antidot is presented which could shed light on the circular antidot hole studied in experiment. The antidot preserves the inversion symmetry of silicene. However, as to be discussed later in this paper, such vacancy defect could still open bandgap for a kind of silicene superlattices by a different bandstructure engineering mechanism in comparison with the subliattice-equivalence breaking. In order to facilitate discussion, we would like to define the pseudo superlattice. As illustrated in Fig. 2a, the A1×B1 rectangle unit cell is the smallest orthogonal supercell. The basis lattices of an orthogonal supercell could be written as A=PA1 and B=QB1. So, the notation (P,Q) could be used to account for the supercell A×B. Accordingly, we would like to



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refer the “nanostructure” calculated with the (P,Q) supercell without any defect in it as the pseudo silicene superlattice (PSS). Once a perturbation such as an antidot is introduced in the supercell, the periodic modification of Born-von Karman boundary conditions would then make the real silicene superlattice. In Fig. 2b, besides the hexagonal Brilliouin zone (h-BZ) corresponding to the primitive unit cell, the rectangular Brilliouin zone (r-BZ) for the (1,1) PSS is also shown. It is obvious that the Dirac point K in the h-BZ of primitive cell is now folded to the T1 ( Γ Y ) point in the r-BZ of (1,1) PSS. The corresponding energy

bandstructure of (1,1) PSS is presented in Fig. 2c. In comparison with the bandstructure shown in Fig. 1c, the Dirac point located at T1 point in Fig. 2c confirms the band folding discussion. On the basis of our detailed analysis, the studied PSSes could be sorted into two categories. One is for the superlattices (P,Q) (Q=3q) whose Dirac point would be folded to Γ ΓY) point point in the corresponding r-BZ. All the other PSSes keep the Dirac point at T (

correspondingly. In Figs. 2d-f, the (1,3) PSS is used as an example. The folding of Dirac point to Γ2 point could be seen in both Fig. 2e of the r-BZ and Fig. 2f of the energy bandstructure. Introducing a defect in the unit cell of PSS would make the real superlattice, which may act quite differently in different type silicene superlattices. In this paper, we concentrate on studying the D6h antidot to shed light on the experimentally investigated nanomeshes, which would preserve the inversion symmetry. In our studies, it is found to open bandgap for the superlattice (P,Q) (Q=3q), while it would keep the semimetal conducting properties for all the other studied superlattices. In Figs. 2g-i, we use (5,6), (5,7), and (5,8) superlattices to illustrate the different effects. The corresponding unit cell contains a Si6 antidot (In order to facilitate discussion, we would like to use the notation Sin to account for the antidot formed by removing a n-atom D6h silicon nanoflake from the silicene with the hole edge passivated by hydrogen). Only the (5,6) superlattice, whose corresponding PSS has band closed Dirac


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point at Γ, opens a bandgap. In comparison with the sublattice-equivalence breaking, the fact of bandgap opening in Q=3q superlattice shows the mechanism by removing the four-fold degeneracy, which could be attributed to the bond symmetry breaking. As depicted previously for the 3q superlattice40, the inversion symmetry preserved perturbation could in fact break the bond symmetry, which as a result opens the bandgap due to the different transfer integrals of two bond groups. Besides of the above discussed orthogonal superlattices, we have also examined the superlattices patterned by the hexagonally arranged antidots. Actually, no matter what kind of repeated units to be adopted, the corresponding PSSes could be classified into two groups: (1) the PSSes with Dirac cone being folded to Γ point and (2) the other PSSes. Then, the inversion symmetry preserved defect such as the hexagonal antidot would open bandgap at Dirac point of the group one superlattices while it does not perturb the bandgap closing properties of the group two superlatices. However, if the antidot does not hold inversion symmetry, it would open bandgaps for all the superlattices. Considering the recent progresses in synthesizing graphene nanomeshes with the advanced nanotechnologies by regularly arranging circularly shaped antidots, we would like to expect the silicene nanomesh to be prepared in experiment. The bandgap opening phenomena of a kind of silicene superlattices by the bond symmetry breaking mechanism should be interesting to call for experimental studies.

3.2. Splitting Between πa and πz Bands It is interesting that the hexagonal antidot only opens bandgap of (P,Q) (Q=3q) superlattice. So, one may wonder what effects the studied antidot would have providing the Dirac point to be slightly shifted away Γ point in such superlattice. The Dirac point could be moved along ΓY by applying uniaxial strain. The lattices A and B of our studied superlattice are along the armchair and zigzag edges, respectively. The effects of uniaxial strain could be investigated by applying stretching strains along the A (σa) and B (σz) directions. After



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applying the strain along a lattice, the other lattice of the studied unit cell has been fully optimized in our calculations. The studied results of the (1,1) PSS are presented in Fig. 3. For silicene, the analogy to graphene, its electronic structure could be regarded to have a mixture of sp2 and sp3 hybridizations. Therefore, we would like to use the notion of π-type bands to discuss the electronic properties shown in Fig. 3. It is clear that the strain σa could move the Dirac point along ΓY axis toward Y-type point, while the strain σz would move it toward Γtype point. In fact, we have also calculated the deformation potentials of both (3,3) PSS and Si6-antidot patterned (3,3) superlattice (see Figure S1 in the Supporting Information), which show almost same evolutions along with strain enhancing. The maximum deformation potentials are only ~30 meV per atom induced by the strains of σz=5% and σa=5%, suggesting the possibilities for the strains to be manipulated in experiment. As for the (P,Q) (Q=3q) PSS, the uniaxial strain would make the Dirac point to leave Γ forming two Dirac cones around it. In Fig. 4a, the bandstructure of the (3,3) PSS without strain is shown, which confirms the fact of Dirac cone at Γ point. The corresponding threedimensional (3D) plotting of this Dirac cone is shown in Fig. 4b. By applying 5% σz strain, the band closing point is now split resulting into two separated Dirac points adjacent to Γ (see Fig. 4c). The corresponding 3D Dirac cones are shown in Fig. 4d. Also, one could find two separated band crossing points located at Γ point above and below the Fermi level as marked with πz and πa in Fig. 4c, respectively. By projecting wavefunctions, the band-decomposed charge densities of them are studied in Figs. 4e and 4f, respectively. The πa (πz) bands are found to correspond to the bonds along armchair (zigzag) edge. Interesting it is that the density distributions of πa and πz bands show anisotropic characters. However, due to the fact that they would cross with each other at Fermi level in the neighborhood of Γ point, the anisotropic characters could not be observed in the conducting property studies. But, as to be discussed, the peculiar phenomena may be obtained in the strain engineered defect-patterned


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suplattice. In fact, the bonds along zigzag edge could be continuously elongated by increasing the uniaxial strain σz (see Figure S2 in the Supporting Information) while the ones along armchair edge would keep almost unchaged. Under the σz=5% strain, the Si-Si bonds along zigzag edge are calculated to be ~2% elongated while the ones along armchair edge are only slightly shortened (~0.2%). These suggest the decrease and increase of the corresponding transfer integrals41, respectively, which may account for the splitting between the πa and πz bands. In order to facilitate the forthcoming discussion, we would like to use Estrain to account for the energy interval between πz and πa bands at Γ point in Fig. 4c. The crossings of the πa and πz bands at Fermi level form two Dirac cones, accounting for the Dirac cone moving. As to be discussed later, the uniaxial strain itself indeed cannot open bandgap to alter the conducting properties of both the semiconducting superlattices (P,Q) with Q=3q and the other semimetallic ones, which is in the line with the previous studies42-44. However, the uniaxial strain could combine with the bond symmetry breaking mechanism to show cooperative effects in tuning bandgap for the semiconducting Q=3q superlattices.

3.3. Splitting Between πa1 (πz1) and πa2 (πz2) Bands As to the question whether the studied antidot could still open bandgap of the (P,Q) (Q=3q) superlattice providing the Dirac point to be slightly shifted away Γ by uniaxial strain? In Fig. 5, we use the (3,3) superlattice under 5% σz strain as a protype for the convenience of discussion. In Fig. 5a, the studied defect in the unit cell of (3,3) is illustrated. The Si-Si bonds in the highlighted hexagon are slightly shortened (3%) to enhance the corresponding transfer integrals which would modulate the Born-von Karman boundary conditions to pattern the superlattice accordingly. The rest atoms are free to relax. As shown in Fig. 5b for the patterned superlattice, in comparison with the (3,3) PSS presented in Fig. 4c, both πa and πz bands are now split by the defect. The split interval Edefect between πa1 (πz1) and πa2 (πz2) is about 0.05 eV. In order to understand the splitting induced by the defect, we have also studied



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the band-decomposed charge densities of the split π-type bands, which are presented in Fig. 5c. One can see that the armchair (zigzag) bonds are now divided into two groups. Compared to the armchair bonds in the strain engineered (3,3) PSS studied in Fig. 4c, the defect makes the bonds corresponding to the πa1 band to elongate of about 0.4% while the bonds corresponding to the πa2 band contract of about 0.5%. In order to facilitate discussion, we would like to refer the former and latter bonds as πa1 and πa2 bonds, respectively. Similar notation would be applied to the zigzag bonds also. In our studies, the πz2 bonds are found to elongate of about 0.4% while the πz1 bonds contract of about 0.5%. For the (3,3) PSS under 5% σz strain, the average bond lengths are 2.274 and 2.325 Å for πa and πz bonds, respectively, which correspond to the -0.105 and 0.093 eV in band energy as referred to Fermi level. In comparison, in the defected (3,3) superlattice shown in Fig. 5a, the average bond lengths are found to be 2.283, 2.263, 2.315, and 2.338 Å for the split πa1, πa2, πz1, and πz2 bonds, which respectively correspond to the -0.084, -0.131, 0.071, and 0.120 eV in band energy as referred to Fermi level. In the spirit of the tight-bonding methodology which is useful to handle the s-p electronic configuration of silicene, the shorter the bond is, the higher the transfer integral is, and then the lower the band energy is. Accordingly, as shown in Fig. 5b, the πa2 and πz1 bands are lower in energy compared to the corresponding πa1 and πz2 bands, accounting for the splitting of πa and πz bands. Also, in order to see whether the above mentioned phenomena would only hold for the special superlattice (P,Q) with Q=3q, we have also analyzed the (3,4), (3,5), and (4,3) lattices which in fact also show π-type band splitting and the same relationship of the band energy evolution versus the bond length change. So, we would like to say that such phenomena would happen to all the types of the (P,Q) superlattices, which would however only make obvious effects in bandgap engineering for the ones with Dirac cone being folded to Γ point. Here, for the superlattice studied in Fig. 5b, one can also note that the Edefect is smaller than the Estrain. In this case, the πz1 and πa1 bands cross


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at Fermi level, forming two Dirac cones around Γ point. By controlling the applied strain to reduce the Estrain, the bandgap could be opened.

3.4. Reversible on/off Switching of Bandgap Now, we reach the conclusions that (1) the uniaxial strain would split the π bands into πa and πz with an energy interval Estrain and (2) the defect could make the πa (πz) to split with an energy interval Edefect. Actually, the Estrain could be increased along with the increase of the applied strain and the Edefect could be affected by the perturbation strength associated with the introduced defect. Then, one can compare Estrain and Edefect to judge whether the bandgap could be opened in the (P,Q) (Q=3q) superlattice. On the basis of our detailed analysis, the semimetal conducting nature of (P,Q) (Q=3q) superlattice could be preserved if Edefect is smaller than Estrain, which is the case studied in Fig. 5. The introduced defect by contracting the selected silicon hexagon only gives the splitting interval Edefect of about 0.05 eV, while the strain of σz=5% makes the splitting as large as Estrain=0.2 eV. Considering the fact of the circular vacancy hole in experimentally synthesinzed nanomeshes, we have also investigated the effects of uniaxial strain on the hexagonal-antidotpatterned silicene superlattice. In order to facilitate discussion, the (6,6) superlattice is studied in Fig. 6 to illustrate our results. In Fig. 6a, the Dirac cone is located at Γ point for the case without strain. The splitting energy intervals Estrain are 0.12 and 0.20 eV for the PSSes under 3% and 5% strains, respectively. In Fig. 6b, the Si6 antidot is introduced in the unit cell. Though the studied D6h Si6 antidot does not break the inversion symmetry, the corresponding silicene superlattice opens a bandgap at Γ point for the case without strain, accounting for the π-band splitting induced by the defect. The splitting energy interval Edefect is calculated to be 0.14 eV, which would remain almost unchanged in the process of applying strain. For the antidotpatterned (6,6) superlattice under 3% strain, the strain induced splitting energy Estrain of 0.12 eV is smaller than the Edefect. Here, the Estrain is calculated by E =


−

. The


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bandgap keeps opened while it is decreased from 0.14 eV to ~0.03 eV. An interesting point needs to be addressed. Now, the conduction band minimum (CBM) and the valence band maximum (VBM) consist of only πa- and πz-type bands, respectively, which hint the possibilities in obtaining anisotropic conducting properties. The electron and hole charge carriers may respectively transport along armchair and zigzag directions. By increasing the strain up to 5%, the Estrain gets about 0.06 eV larger than the Edefect. The πa1 and πz1 start to cross to form new Dirac point. This may be interesting for that it shows a method for reversibly tuning the bandgap on/off to switch between the high charge carrier velocity properties and the high on/off ratio properties. Furthermore, the sizable bandgap could also be continuously tuned by the strain (see Figure S3 in the Supporting Information). Besides, we have also examined the bandstructure engineering by applying σa strain, which acts quite similarily as the σz strain (see Figure S4 in the Supporting Information). We have also studied the effects of the homogeneous strain on the superlattices, which however could not obviously affect the bandgap properties. In comparison with the D6h-antidot-patterned silicene superlattice, the perturbation with a (AlP)3 flake patched in the antidot hole is also studied. The (AlP)3 patch breaks the sublattice equivalence of silicene. Then, the bandgap is opened due to the inversion symmetry breaking. No matter whether the strain is applied, the bandgap remains. This bandgap opening owe to the sublattice-equivalence breaking is different with that induced by the Si6 antidot, confirming again the different mechanism–bond symmetry breaking for bandgap engineering. Based on our detailed studies, we would like to conclude that the bandgap tuning by combining the effects of the uniaxial strain and bond symmetry breaking mechanism can work for all the inversion symmetry preserved (P,Q) (Q=3q) superlattices. Besides the circular vacancy hole, the other shaped antidots including the irregularly shaped defects can work also providing they could keep the sublattice equivalence. The size of the vacancy hole itself would not harm the above discussed bandgap tuning


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though it can affect the maximum gap width (the case without strain). However, the distance of vacancy hole with its periodic images, namely the neck width of superlattice, cannot be too small. Too narrow neck width such as only one or two atomic silicon chains would induce large distortion to the corresponding bandstructure, which would make the geometrical structure of the superlattice unstable. Besides the patterning of silicene superlattice by regularly arranging antidots, the substrates as suggested in previous experimental studies may also make the silicene into periodic superlattice structures. Especially, the Dirac cone would be folded onto Γ point of the experimentally fabricated √3 × √3 superlattice in which the bond symmetry breaking bandgap opening mechanism and the gap width tuning by combining the unaxial strain may work. In the process of applying uniaxial stretching strain, the electronic properties would be modified. Due to the linear dispersion relation around Dirac cone, the group velocity of massless Fermions could be calculated from the gradient of band energy ( = ∇ E⁄ℏ)41. The k and E(k) are the wave vector and dispersion relation, respectively. In our studies, the calculated mobilities of the charge carrier electron and hole are almost same. To facilitate discussion, the average velocities of the PSSes studied in Fig. 6a are calculated and presented in Table 1, which are around 5.30, 5.25 and 5.19×105 m/s for the silicene under 0%, 3% and 5% strains. These to some sense agree with the experimentally measured Fermi velocity of ~106 m/s for silicene18. Due to the splitting between πa and πz bands, the velocity has been slightly decreased along with the increase of strain. For the semiconducting silicene superlattices, the linear dispersion relation is destroyed. The dispersion relation could be described by parabola function around the band extrmum. Thus, the corresponding effective mass could be obtained by41 '

$ % " = ℏ # & $



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In Table 1, we present the calculated average values of the hole and electron. Starting from the free-standing Si6-antidot-patterned (6,6) superlattice, the opened bandgap would be reduced by applying strain, and the corresponding effective mass would be decreased also. According to the simple relationship ( = )*⁄" between carrier mobility (µ) and effective mass (m) (τ is the scattering time of charge carrier)41, the smaller the bandgap is, the higher the charge carrier mobility would be. By using the rough estimation of 10-13 s for scattering time, we have also estimated the carrier mobilities of the semiconducting superlattices to be ~103 cm2V-1s-1. Once the bandgap starts to close, the band energy dispersion would gain linear relationship again, resulting in high charge carrier mobility. For the case studied in Fig. 6b under σz=5% strain, the group velocities of massless Fermions have been calculated to be around 3.12×105 m/s.

4. CONCLUSIONS In conclusion, the sueprlattice patterned with hexagonal antidots has been investigated in detail. According to the energy band folding analysis, for the free-standing (P,Q) PSS, the point depending on whether Q is integer Dirac cone would be folded to Γ point or ΓY

multiple of 3. Then, the hexagonal antidot can open bandgap of the Dirac cone at Γ point by the bandgap opening mechanism of four-fold degeneracy breaking, while it cannot affect the band closing properties at

ΓY point. By applying uniaxial strain on (P,Q) PSS, the

equivalence between the Si-Si bonds along armchair edge and those along zigzag edge would be broken. Accordingly, the π bands would be split into πa and πz bands with the energy axis. For the suplerlattice interval Estrain, which results in the moving of Dirac cone along ΓY with Q=3q under uniaxial strain, the crossings between the split πa and πz bands form new Dirac cones adjacent to Γ point. By introducing hexagonal defect in the unit cell, the πa (πz) bands could be further split into πa1 and πa2 (πz1 and πz2) bands with the energy interval Edefect. For a given D6h-defect-patterned superlattice, along with the increase of the applied strain, the 14 ACS Paragon Plus Environment

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Edefect would remain almost unchanged while the Estrain would increase continuously. Thus, starting from the free-standing (P,Q) (Q=3q) superlattice, the bandgap opened by the D6h defect would be continuously reduced in the process of enhancing uniaxial strain until the Estrain=Edefect. Then, the πa1 and πz1 start to cross to form Dirac cone. This suggests a revisable method for switching on/off bandgap by controlling strain. So, one could switch to gain high charge carrier velocity properties due to the linear dispersion relation around Dirac point or high on/off ratio due to the sizable bandgap. Furthermore, the sizable bandgap could be continuously tuned. Besides, the defects to break the sublattice equivalence such as the (AlP)3 patch are found to act quite differently, which always keep the bandgap opening due to the inversion symmetry breaking.

ACKNOWLEDGMENTS The authors gratefully acknowledge the SR16000 supercomputing resources from the Center for Computational Materials Science of the Institute for Materials Research, Tohoku University, and the computing resources from the University of Jinan. This work was jointly supported by the funds from Shandong Province (Grant Nos. TSHW20101004 and ZR2012AL10) and the National Natural Science Foundation of China (Grant Nos. 11374128 and 21303072).



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ASSOCIATED CONTENT Supporting Information The evolutions of deformation potential versus the applied strain for both (3,3) PSS and Si6-antidot patterned (3,3) superlattice have been studied. The effects of the uniaxial strains σa and σz are respectively investigated (Figure S1). The elongation of Si-Si bonds along zigzag edge as referred to the ones without strain versus the applied strain σz has been calculated (Figure S2). The evolutions of bandgap and Edefect–Estrain versus the applied uniaxial strain for the Si6-antidot patterned (6,6) superlattice have been schematically shown (Figure S3). The calculated energy bandstructures for the (6,6) PSS and Si6-antidot patterned (6,6) silicene superlattice under 0%, 3%, and 5% σa strains have also been studied, respectively (Figure S4). This material is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION Corresponding Author *E-mail: [email protected] (G. Chen)


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Table 1. The average group velocities of massless Fermions (v, in the unit of 105 m/s) or the average effective masses of charge carriers (m, in the unit of free electron mass m0) of the (6,6) PSS and Si6-antidot patterned superlattice. The corresponding energy bandstructures are studied in Fig. 6. The stretching uniaxial strain σz is applied along the zigzag edge. Material

σz

v

m

PSS

0 3% 5% 0 3% 5%

5.30 5.25 5.19 3.12

0.109 0.099 -

Si6



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Figure Captions Figure 1 The top (a) and side views (b) of the geometrical structure of silicene, and the energy bandstructure (c) calculated by using the primitive unit cell. An anitdot-patterned orthogonal superlattice defined by A×B unit is schematically illustrated in (d). The hole edge of antidot is passivated by hydrogen.

Figure 2 The schematic structure, Brillouin zone r-BZ, and energy bandstructure for the smallest orthogonal (1,1) PSS are shown in (a), (b), and (c), respectively. Those for the (1,3) PSS are presented in (d), (e), and (f), respectively. The Brillouin zone h-BZ corresponding to the primitive unit cell of silicene is also shown in (b) and (e) for comparison. The (g), (h), and (i) are for the energy bandstructures of the antidot-patterned (5,6), (5,7), and (5,8) silicene superlattices.

Figure 3 The shifts of Dirac point along ΓY of the reciprocal lattice for the silicene under 5% uniaxial stretching strain applied along armchair (a) and zigzag (b) edges. The insets show the deviation of Dirac point referred to its position in free-standing silicene as a function of the applied strain. Only π bands those cross to form Dirac point are shown for clarity. The notations of k-points Γ1, T1, and Y1 are illustrated in Fig. 2b.

Figure 4 The band-folded energy bandstructure along Y′-Γ-Y path in reciprocal space (a) and the three-dimensional plotting of the corresponding Dirac cone (b) for the free-standing (3,3) PSS. Those for the (3,3) PSS under 5% σz strain are shown in (c) and (d), respectively. The band-decomposed charge densities at isovalue of ~0.01 e/Å3 for the split πa and πz bands are presented in (e) and (f), respectively. Both top and side views are shown.


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Figure 5 The D6h defect formed by contracting the Si-Si bonds by ~3% of the grey (yellow online) hexagon (a) and the corresponding bandstructure along the Y′-Γ-Y path (b). The corresponding band-decomposed charge densities at isovalue of ~0.01 e/Å3 for the split πa1, πa2, πz1, and πz2 bands are shown in (c).

Figure 6 The calculated energy bandstructures for the silicene-based nanostructures under the 3% and 5% σz strains and the ones for the free-standing materials. The (a), (b), and (c) are for the (6,6) PSS, Si6-patterned superlattice, and (AlP)3-patterned superlattice, respectively.



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Table of Contents


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