Strain Induced Phase Transitions in Silicene Bilayers: a First Principles and Tight-Binding Study Chao Lian and Jun Ni

Strain induced phase transitions in silicene bilayers: a first principles and tight-binding study Chao Lian and Jun Ni

Citation: AIP Advances 3, 052102 (2013); doi: 10.1063/1.4804246 View online: http://dx.doi.org/10.1063/1.4804246 View Table of Contents: http://scitation.aip.org/content/aip/journal/adva/3/5?ver=pdfcov Published by the AIP Publishing

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Strain induced phase transitions in silicene bilayers: a ﬁrst principles and tight-binding study Chao Lian and Jun Nia Department of Physics and State Key Laboratory of Low-Dimensional Quantum Physics, Tsinghua University, Beijing 100084, P.R.China (Received 5 February 2013; accepted 23 April 2013; published online 2 May 2013)

Using ﬁrst principles and tight-binding calculations, we have investigated the structures of silicene bilayers under the isotropic tensile strain. We ﬁnd that (i) the strain induce several barrierless phase transitions. (ii) After the phase transitions, the bilayer structures become planar, similar with the AA-stacking graphene bilayers, but combined with the strong covalent interlayer bonds. The tight-binding results demonstrate that this silicene bilayer is characterized by intralayer sp2 hybridization and the interlayer sp1 hybridization. (iii) The electronic properties of the silicene bilayers change from semiconducting to metallic with the increase of strain. C 2013 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License.[http://dx.doi.org/10.1063/1.4804246]

I. INTRODUCTION Silicene, the two-dimensional silicon nanosheet, has attracted extensive research interests in recent years.1–26 Silicene shares many important properties with graphene, especially the most fas- cinating one: the Dirac-type dispersions at the Fermi surface, which is shown by both first-principles calculations7–14 and the measurements from different experiments.1, 3, 4 Other than the similarity with graphene, silicene has its own advantages. First, the highly developed silicon microelectronic industry provide a strong technologic basis for the applications of silicene. Second, the π electrons in silicene are much more active than that of graphene.24 The active π electrons in silicene lead to a different structure from graphene. The silicene monolayer has low-buckled structure instead of a planar structure.7 The bilayer structures also show different configurations. The most stable graphene bilayer is in the AB stacking and the interlayer bonds in graphene bilayer are the weak Van der Waals interactions. However, if the two silicene monolayers are manipulated into the AB stacking silicene bilayer, the strong covalent bonds will form between different layers. The original low buckled silicene monolayer will become a complete sp3 Si(111) sheet. Morishita et.al. have demonstrated that, because of the unpaired pz electrons, this kind of AB stacking silicene bilayer(SSBL) is only metastable.25, 26 Instead, a reconstruction will occur in the SSBL with lower energy. This reconstructed silicene bilayer (RSBL) is the most stable structure. The structures of the silicene monolayers and bilayers show that the π bonds in silicene is not pure sp2 hybridization, but a mixed sp2-sp3 hybridization. Recently, Wu et.al. have demonstrated that the strain will turn several wurtzite materials into stable graphitic thin films.27 However, even under very large strain, the silicene does not become planar but remain buckled. It indicates that the planar monolayer structure is difficult to be obtained because of the active π electrons. In silicene bilayers, the π electrons is partly saturated when the interlayer bonds are formed. It may lead to a possible planar bilayer structure tuned by strain. Thus, it is important to discuss the changes of the silicene bilayer structures under strain. In this paper, we study the structures of silicene bilayers under isotropic tensile strains. Both SSBL and RSBL are considered. We find that the phase transitions occurs under certain tensile strains for SSBL and RSBL, both with no energy barrier.

aElectronic mail: [email protected]

2158-3226/2013/3(5)/052102/10 3, 052102-1 C Author(s) 2013

After the transition, the SSBL and RSBL turn into a graphene-like silicene bilayer (GBSL) structure. Different from the AB stacking graphene bilayer with the Van der Waals interlayer interactions, the GBSL is combined in the AA stacking with the strong covalent bonds. Using the tight-binding model, we demonstrate that the hybridization of the electrons in the GSBL is the sp2 − sp1 hybridization without any sp3 part. Meanwhile, with the increase of strain, the electronic property of the silicene bilayer also turn from semiconducting to metallic. This article is organized as follows: Section I is the background information. Section II and Section III are the ﬁrst principles results of the structural transitions and the electronic phase transitions induced by the strain. Section IV is the tight-binding model of the GSBL. Section V is the summary.

II. STRUCTURAL TRANSITIONS The first principles calculations are performed with the Vienna Ab initio Simulation Pack- age (VASP).28–30 We use the projector augmented-waves method31 and Perdew-Burke-Ernzerhof exchange-correlation.32 The plane-wave cutoff energy is set to be 250 eV. Using the conjugate gra- dient method, the positions of atoms are optimized until the convergence of the force on each atoms is less than 0.005 eV/Å. The vacuum space is larger than 15 Å and sufficient to make the system isolated. The Monkhorst-Pack scheme33 is used to sample the Brillouin zone. The k mesh of 8 × 8 × 1 including the point is used in the structure optimization, and a larger 12 × 12 × 1kmesh is used in the total energy calculations. There is a 2 × 1 reconstruction in the RSBL, thus a 2 × 2 supercell is chosen to reproduce the RSBL structure. The atom positions in the RSBL are set to be thesameasinRefs.25 and 26, and then relax to ensure it become the most stable configuration. The supercell used in the calculations are the 2 × 2 silicene unit cell, containing 16 silicon atoms. Spin-polarized calculations are performed with both ferromagnetic and antiferromagnetic initial states in all the configurations. No magnetic order emerges in the final state. We apply isotropic tensile strains on the initial structures, including the SSBLs and RSBLs. The RSBL is the most stable structure which will be essentially studied. We also choose the SSBL for comparison. The SSBL is the most important metastable structure which possibly exists in the beginning of the experimental growth procedure at low temperature. Other metastable configurations are also reported in the Ref. 25. However, these configurations can not be synthesized at low temperature and will turn into the RSBL at high temperature. Thus we consider the RSBL and the SSBL as the initial structures. The strain is expressed as = (a − a0)/a0, where a and a0 are the lattice constants with and without strain, respectively.√ The strain is applied by extending the basis vec- = , , = 1 , 3 , = + , , = 1 + , tors√ of the cell a1 (a0 0 0), a2 ( 2 a0 2 a0 0) to a1 (a0(1 ) 0 0), a2 ( 2 a0(1 ) 3 + , = α + β = α + 2 a0(1 ) 0). Any direction in the silicene plane a a1 a2 becomes a (1 )a1 + β(1 + )a2 = (1 + )a, which indicates that the strain is equal along every direction and thus isotropic. The coordinates of each atom r = xa1 + ya2 + za3 become r = x(1 + )a1 + y(1 + )a2 + za3 and then relax to the most stable positions. The SSBL is relaxed from the bilayer slice of Si(111) and the RSBL is obtained similarly with the previous work.25, 26 As shown in Fig. 1,the phase I stands for the SSBL and the phase II stands for the RSBL. The mixed sp2-sp3 hybridizations are observed for both SSBL and RSBL. For the SSBL, the T1 atoms in Fig. 1(a) form a nearly regular tetrahedrons with the nearest neighbors, which is a typical feature of the sp3 hybridizations. As for the RSBL, the T2 atoms in Fig. 1(b) protrude and form the 2 × 2 reconstruction, which indicates that there is partial sp3 hybridization.25 Zhao has demonstrated that this sp2-sp3 hybridization is the determining factor of the stability in silicene system.34 The variation of the weight of the sp3 part in the sp2-sp3 hybridizations is important to analyze the stability of the silicene bilayer structures. The 3 2 3 weight of the sp part in the sp -sp hybridizations is measured by the buckle lengths of T1 atoms and T2 atoms, where the T1(T2) atom is the outmost atom in the SSBL(RSBL). Thus we will analyze the variation of the buckle lengths with the increase of strain. For the SSBLs, as shown in Fig. 2(a), the buckle length of T1 atoms in the SSBL (phase I) slowly decrease with the increases of strain. Under the strain of 15.8%, the buckle length reaches its yield point and a barrierless phase transition occurs. Then the buckle length decreases to zero, indicating

FIG. 1. The structures of the (a) SSBL (phase I), (b) RSBL (phase II), (c) GSBL (phase III) and (d) ISBL (phase IV). The arrows represent the changes of the phases in transitions. T1 atoms and T2 atoms represent the outmost atoms in phase I and phase II, respectively.

the bilayer structure becomes totally planar. As shown in Fig. 1(c), this silicene bilayer is arranged in the AC stacking and the silicon atoms in the top layer form a tilted bonds with the atoms in the bottom layer. We name this phase (phase IV) the intermediate silicene bilayer phase(ISBL) because it only exists under the strain ranging from 15.8% to 16.8%. At the strain = 16.8%, another phase transition occurs. A relative slide between the two layers in this transition makes the ISBL into the AA stacking. Because this phase (phase III) is similar as the AA stacking graphene bilayer and thus named as graphene-like silicene bilayer (GSBL). However, the interlayer distance is ∼2.46 Å and barely changes with the strain, indicating that the two silicene layers form strong covalent bonds between corresponding atoms, completely different from the Van der Waals interlayer interactions in graphene.

FIG. 2. The energies (solid lines) and buckle lengthes hb (dash lines) of (a) SSBL (b) RSBL and (c) GSBL with strains. The arrows represent the variations of the strain.

For the RSBL, the variation of the buckle length is nearly the same as the SSBL. As shown in Fig. 2(b), the buckle lengths of T2 atoms in the RSBL (phase II) decrease to 0.08 Å under the strain of 10.8%. After that, the RSBL undergoes a phase transition. Different from the SSBL, the RSBL turns directly into the GSBL without the intermediate state ISBL. The phase transition can be more directly illustrated by the changes of the total energy with the strain, as shown in Fig. 2. As the strain increases, the total energy of the SSBL increases substantially. When applied with a strain of 15.8%, the SSBL becomes unstable, leading to a phase transition. The decrease in the total energy in this transition is 0.24eV/atom. After the transition, the SSBL (phase I) turns into the ISBL (phase IV). When the strain continues to increase from 15.8% to 16.8%, the total energy of the ISBL increases within a small range. When the strain reaches 16.8%, another transition occurs and the GSBL (phase III) state emerges, accompanied with a decrease of 0.11 eV/atom in the total energy. For the phase transition from the RSBL (phase II) to the GSBL (phase III), the process is similar, as shown in Fig. 2(b). However, the values of the transition strains differ a lot. The RSBL-GSBL transition occurs at the strain of 10.3%, with a decrease of 0.074 eV/atom in the

total energy. Because the decreases in the total energy are substantial, the bilayer structures become much more stable after the phase transition. Although the GSBL emerges under the strain over 10% from the barrierless phase transitions, the GSBL has already become the most stable structure when the strain reaches 6.5%, as shown in Fig. 2(c). As we gradually decrease the strain from 20% to 0% on the GSBL, the total energy decreases to the minimum of −4.96 eV/atom at the strain of 8.3% and increases after that. Before the strain decreases to 6.5%, the GSBL is always the most stable structure. The SSBL and RSBL will eventually turn into the GSBL when the strain is larger than 6.5%, but with energy barriers. The extra strain is needed for the SSBL and RSBL to overcome the barriers and the barrierless phase transitions occur at = 16.8% and = 10.3%, respectively. The barrier also exists in the transition from the GSBL back to the RSBL when the strain decreases, as shown in Fig. 2(c). Although the total energy of the GSBL is higher than that of the RSBL with the strain smaller than 6.5%, the GSBL does not spontaneously become the RSBL until the strain reaches 2.9%. At the strain of 2.9%, another barrierless phase transition occurs, from the GSBL into the RSBL, with a decrease of 0.15eV/atom in total energy.

III. ELECTRONIC PHASE TRANSITION In addition to the structural transition, the strain also change the electronic properties of the silicene bilayers by altering the bond lengths. The intralayer bonds are elongated with the increase of strain for each phase, which deduces the hopping between adjacent atoms. As mentioned above, in the four phases, only the RSBL (phase II) and GSBL (phase III) can be the most stable states under a given strain. Therefore, we focus on the variations of the electronic properties of these two phases. The RSBL exists under the strain from 0% to 10.3%, being the most stable state from 0% to 6.5%. We ﬁnd that it is a semiconductor with a small gap of 0.23eV without strain, as shown in Fig. 3(a) and 3(b), consistent with the previous conclusions.25 When the strain increases, the RSBL becomes metallic with the band gap decreased to zero. As shown in Fig. 3(a), the bands near the Fermi level, named as band 1 and band 2, are isolated without strain. With the strain increases, the band 1 shifts up and the band 2 shifts down, becoming across near the M point. The density of states (DOS) of the RSBL at the Fermi level is zero under the strain = 0% but become nonzero under the strain = 3.8%. Under the strain larger than = 6.5%, the GSBL is the most stable structure. We choose the GSBL under = 5.3% and = 7.3% to demonstrate the variation of the electronic properties, as shown in Fig. 3(c). Though the hopping integrals decrease with the increase of strain, the symmetries of the GSBL remain unchanged. Thus, with the increase of strain from 5.3% to 7.3%, the energies of the corresponding bands increase but the main feature of dispersion relations of each band remains.

IV. TIGHT-BINDING MODEL OF THE GSBL The GSBL structure emerges after the strain-induced phase transitions. The two silicene layers in the GSBL are planar and combined with strong covalent bonds. As shown in Fig. 1(c), the silicon atoms in the GSBL are not three-coordinated as silicene, but four-coordinated like the sp3 silicon materials. However, different from the common sp3 silicon materials, the four nearest neighbors of the silicon atoms in the GSBL do not form a regular tetrahedron. It indicates that the hybridization in GSBL are novel and not similar with neither the sp2 hybridization nor the sp3 hybridization. To reveal the hybridization type of the GSBL, we carry on a tight-binding calculation. We focus on the primary interactions in our tight-binding model, in purpose to reproduce the main feature of the electronic structures. We consider only the nearest neighbor interactions. As shown in Fig. 4(a), The A atom has three intralayer nearest neighbors (B atoms), and one interlayer nearest neighbor

FIG. 3. (a) and (b) The band structures and the DOS of the RSBL under the strain of 0% (solid lines) and 3.8% (dash lines). (c) and (d) The band structures and the DOS of the GSBL under the strain of 5.3% (solid lines) and 7.3% (dash lines).

(C atom). The Hamiltonian can be expressed as:

⎛ ⎞ HAA HAB HAC 0 ⎜ ⎟ ⎜ HBA HBB 0 HBD ⎟ H = ⎝ ⎠ , (1) HCA 0 HCC HCD 0 HDB HDC HDD

= † where Hi, j H j,i is a matrix describing the interaction between atom i and atom j. The symmetry demands HAA = HBB = HCC = HDD and HAB = HCD, HAC = HBD. There are four atomic orbitals per 35, 36 silicon atom:s, px, py and pz. Thus in the Slater-Koster scheme, Hi, j is a 4 × 4 matrix. HAA and

μ TABLE I. The expression of the mα ,nβ in Eq. (4).TheVss, Vspσ , Vppσ and Vppπ are tight binding parameters which are ﬁtted with the ﬁrst principle results, and θ is the angle shown in Fig. 4(a).

B B B B s px py pz

A s Vss Vspσ cos θ Vspσ sin θ 0 A − θ 2 θ + 2 θ θ θ − θ θ px Vspσ cos cos Vppσ sin Vppπ sin cos Vppσ sin cos Vppπ 0 A − θ θ θ − θ θ 2 θ + 2 θ py Vspσ sin sin cos Vppσ sin cos Vppπ sin Vppσ cos Vppπ 0 A pz 00 0Vppπ C C C C s px py pz A s Vss 00Vspσ A px 0 Vppπ 00 A py 00 Vppπ 0 A − pz Vspσ 00Vppσ

TABLE II. The values of the Ep, Es, Vss, Vspσ , Vppσ and Vppπ ﬁtted with the ﬁrst principle results. The unit is eV.

Ep Es Vss Vspσ Vppσ Vppπ

0 −6.53 −1.66 1.9 2.76 −1.08

HAB describe the intralayer interactions: ⎛ ⎞ ⎛ ⎞ ε h A B h A B h A B s 000 s s s px s py 0 ⎜ ε ⎟ ⎜ ⎟ ⎜ 0 p 00⎟ ⎜ h A B h A B h A B 0 ⎟ = ⎜ ⎟ , = ⎜ px s px px px py ⎟ . HAA ε HAB (2) ⎝ 00 p 0 ⎠ ⎝ h A B h A B h A B ⎠ py s py px py px 0 ε 000 p h A B 000pz pz

HAC describes the interlayer interactions in GSBL: ⎛ ⎞ hs AsC 00hs A pC ⎜ z ⎟ h A C ⎜ 0 px px 00⎟ HAC = ⎜ ⎟ , (3) ⎝ h A C ⎠ 00py py 0 h A C h A C pz s 00pz pz

where εs and εp are the energies of 3s and 3p orbitals of the silicon atoms, respectively, and −ik·(rα −rβ ) hmα ,nβ = Fs (|rα − rβ |)μmα ,nβ e (4) rα rβ is the hopping integral between the orbit m of the atom α and the orbit n of the atom β, with m, n ∈ {s, px, py, pz} and α = β ∈ {A, B, C, D}. The expression of μmα ,nβ isshowninTableI. Here, we note that, the HAC contains only six nonzero matrix elements: (i) hs AsC which describes the A C A C A C s s h A C p p π h A C p p π - bonds, (ii) px px which describes the x - x bonds, (ii) py py which describes the y - y A C A C h A C p s h A C s p bonds, (iv) pz s which describes the z - bonds, (v) s pz which describes the - z bonds and A C h A C p p σ (vi) pz pz which describes the z - z bonds. Other elements vanish because of the antisymmetric + − | − | 37 = 2/ 2 / parts in the p orbits. Fs ( rα rβ ) is the scaling function in Harrison form: Fs (r) r0 r . We use the parameters determined from the first-principles energy band. No extra parameters are introduced when two planar silicene bilayer(PSML) form the GSBL. Therefore, the parameters are fitted with the first-principles band structure of the PSML, as shown in Fig. 4(b) and Table II. Note that the Slater-Koster scheme is used in the tight-binding calculations, which ignores the overlaps of the orbital wave functions in different sites. This leads to a slight difference with the first-principles results quantitatively, but reproduce the electronic structures of the PSML with the same feature.

FIG. 4. (a) The schematic diagram of the tight-binding model. The left panel shows the top view and the right panel shows the side view. A B C and D represent different sublattices. θ is the direction angle of the vector between the atom A and atom B. (b) and (c) The band structures of the PSML and GSBL. The left panels and the right panels show the ﬁrst-principles and tight-binding results, respectively.

As shown in Fig. 4(c), the band structure of GSBL is achieved with the parameters shown in Table II. The tight-binding results agree well with the ﬁrst-principles results, which means that the primary interactions have been included in our tight-binding model. The formation of the interlayer bonds in the GSBL change the band structures of the PSML. Comparing the band structure of PSML(Fig. 4(b)) with that of GSBL(Fig. 4(c)), we ﬁnd that each band in the PSML split into two bands in the GSBL with the original dispersion relations. The Dirac cone in the PSML also spilt in the GSBL. The energies of the Dirac cones change from E = 0 (the Fermi level) in the PSML to E = 0.94 eV and E =−2.18 eV in the GSBL. This phenomenon can be explained as follows. The dispersions of the sp2 electrons in the different layers are degenerate with the absence of the interlayer interactions. When the interlayer bonds are formed, the degenerate bands separate. One band decreases in energy and creates the bonding state, while the other one increases in energy and creates the antibonding state. Moreover, the interlayer bond is a covalent bond, which makes the energy split up to several eV. This is totally different from the weak Van der Waals interactions in the graphene bilayer, which can be viewed as a perturbation to the band structure of the graphene monolayer. It concludes that the intralayer sp2 hybridization in the GSBL is the same as that in the PSML, since HAA and HAB, which describe the intralayer interaction, are exactly the same. In addition, because of the existence of the interlayer interaction HAC, the properties of the GSBL are different from the two isolated PSMLs. We recall that the HAC describe the hybridization just like the sp1 hybridization in the 1D silicon chains. It demonstrates that the hybridization of the silicon atoms in GSBL is sp2-sp1 hybridization. Unlike the other stable monolayer and bilayer structures, the GBSL contains no sp3 part in its hybridization, which shows its uniqueness.

V. SUMMARY In this paper, we have investigated the strain-induced phase transition in the silicene bilayers. As the strain increases, the SSBL and GSBL undergo barrierless structural transitions. After each structural transition, the formation energies of the SSBL and the RSBL decrease. The RSBL and the SSBL eventually turn into the GSBL structures. With the decrease of strain, the GSBL undergoes another barrierless phase transition into the RSBL. The GSBL is like the AA stacking graphene bilayer but combined with the strong covalent interlayer bonds. The GSBL is the most stable state under the strain larger than 6.5%. The tight-binding calculation shows that the hybridization of the GSBL is not the typical sp3 hybridization, in spite of the four-coordinated silicon atoms. The intralayer sp2 hybridization is kept and a new interlayer sp1 hybridization is formed when two PSMLs combine into a GSBL. The hybridization of the silicon atoms in the GSBL is the sp2 − sp1 hybridization. Besides the structural transition, the RSBL also turn from semiconducting to metallic with the increase of the strain.

ACKNOWLEDGMENT This research was supported by the National Science Foundation of China (Grant Nos. 11174171 and 10721404).

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