Quantum valley Hall states and topological transitions in Pt(Ni, Pd)-decorated silicene: A first-principles study Bao Zhao, Jiayong Zhang, Yicheng Wang, and Zhongqin Yang Citation: The Journal of Chemical Physics 141, 244701 (2014); doi: 10.1063/1.4904285 View online: http://dx.doi.org/10.1063/1.4904285 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/24?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Strain-tunable topological quantum phase transition in buckled honeycomb lattices Appl. Phys. Lett. 106, 183107 (2015); 10.1063/1.4919885 Quantum spin/valley Hall effect and topological insulator phase transitions in silicene Appl. Phys. Lett. 102, 162412 (2013); 10.1063/1.4803084 Quantum spin Hall effect induced by electric field in silicene Appl. Phys. Lett. 102, 043113 (2013); 10.1063/1.4790147 Electronic structures of an epitaxial graphene monolayer on SiC(0001) after metal intercalation (metal = Al, Ag, Au, Pt, and Pd): A first-principles study Appl. Phys. Lett. 100, 063115 (2012); 10.1063/1.3682303 First-principles study of metal–graphene interfaces J. Appl. Phys. 108, 123711 (2010); 10.1063/1.3524232
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THE JOURNAL OF CHEMICAL PHYSICS 141, 244701 (2014)
Quantum valley Hall states and topological transitions in Pt(Ni, Pd)-decorated silicene: A first-principles study Bao Zhao,1 Jiayong Zhang,1 Yicheng Wang,1 and Zhongqin Yang1,2,a) 1
State Key Laboratory of Surface Physics and Key Laboratory for Computational Physical Sciences (MOE) and Department of Physics, Fudan University, Shanghai 200433, China 2 Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing, 210093, China
(Received 10 September 2014; accepted 1 December 2014; published online 23 December 2014) The electronic states and topological behaviors of Pt(Ni, Pd)-decorated silicene are investigated by using an ab-initio method. All the three kinds of the adatoms prefer hollow sites of the silicene, guaranteeing the Dirac cones unbroken. The Pt(Ni, Pd)-decorated silicene systems all present quantum valley Hall (QVH) states with the gap opened exactly at the Fermi level. The gaps of the QVH states can be increased substantially by applying a positive electric field. Very fascinating phase transitions from QVH to quantum spin Hall (QSH) and then to QVH again are achieved in the Pt/Ni-decorated silicene when a negative electric field is applied. The QSH state in the Pd case with a negative electric field is, however, quenched because of relatively larger Rashba spin-orbit coupling (SOC) than the intrinsic SOC in the system. Our findings may be useful for the applications of silicene-based devices in valleytronics and spintronics. C 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4904285]
I. INTRODUCTION
Silicene, a monolayer of silicon atoms, has recently attracted much attention due to its exotic electronic states and topological behaviors.1–8 It has been successfully synthesized in experiments.9,10 Different to graphene, silicene possesses a buckled honeycomb structure,1,10 resulting in the electronic behaviors controllable by applying an external electric field to silicene.2,4 The relatively strong intrinsic spin-orbit coupling (SOC) in silicene causes the predicted quantum spin Hall (QSH) effect to be observed in an experimentally accessible low temperature regime.3 When exchange fields are considered, the time-reversal symmetry in the silicene system is broken, which may induce quantum anomalous Hall (QAH) effects and abundant other new topological states,4–8 including valley polarized metals,4 marginal valley polarized metals,4 and the interesting hybrid topological state of QSH and QAH effects.6 Especially, a brand new and very unique topological state of valley-polarized QAH is found in silicene by tuning the strengths of Rashba and intrinsic SOCs in a tight-binding model,8 which may have special applications in silicene spintronics. Currently, valleytronics becomes one of the focuses in the field of condensed matter physics,11 where the electric current is controlled by the valley degree of freedom instead of the traditional charge and spin of electrons in two-dimensional (2D) crystals with a honeycomb lattice. Silicene, graphene, and monolayers of transition metal dichalcogenides MX2 (M = Mo, W; X = S, Se), etc., are all potential materials in valleytronics, with band structures consisting of two degenerate but inequivalent valleys located at the corners of the hexagonal Brillouin zone (BZ). The broken inversion symmetry in these systems, induced by such as substrate potentials, external eleca)Electronic address: [email protected]
0021-9606/2014/141(24)/244701/9/$30.00
tric fields, or intrinsic inequivalent atoms, allows a (quantum) valley Hall effect,12–18 where carriers in different valleys flow to opposite transverse edges when an in-plane electric field is applied.12,19 Some theoretical models of certain valleytronic devices, such as valley filters,19,20 valley valves,19 and devices generating pure valley current,21 are proposed. More charmingly, valley-dependent interactions of electrons with light of different circular polarizations are enabled in the systems. Thus, optical measurements and manipulations of the valleydependent physics become possible.22 It is very meaningful to explore new materials or paths to acquire the valley Hall effects and apply them to valleytronics. In this paper, we intend to achieve quantum valley Hall (QVH) states in silicene by adsorbing Pt, Ni, or Pd atoms onto the silicene plane, from first-principles calculations. We find that all the three kinds of the adatoms prefer hollow sites of the silicene, guaranteeing the silicene Dirac cones unbroken. The systems are nonmagnetic and the band gaps are opened just around the Fermi level (EF ). These behaviors together with the inversion symmetry broken by the adatoms lead to QVH appearance in the systems. How an electric field tunes the QVH states is also investigated. When a positive electric field is applied, the gaps of the QVH states in the three systems are increased drastically. Interesting topologically phase transitions from QVH to QSH and then to QVH again are found in the Pt and Ni cases when a negative electric field is applied. The QSH state is, however, quenched in the Pd case. The mechanisms are understood based on a low-energy effective model. II. METHODS AND MODELS
The electronic structure calculations were performed by using projector augmented wave formalism23 within the framework of density functional theory (DFT),24 as implemented
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in the Vienna ab-initio simulation package (VASP).24 The Perdew, Burke, and Ernzerhof parameterized generalizedgradient approximation was used to treat the exchange and correlation functional.25 To avoid the interactions between the two adjacent silicene layers, we chose a vacuum space of larger than 15 Å. The plane-wave cutoff energy was 500 eV and the convergence criterion for the total energy was set to be 10−5 eV. All atomic positions were relaxed until the Hellmann-Feynman force on each atom was less than 0.01 eV/Å. A 4 × 4 supercell of silicene adsorbed with one adatom was employed, as shown in Fig. 1(a). We considered three typical adsorption sites on silicene: hollow (H), top A (T A), and top B (T B) (Fig. 1(a)). Besides the H, T A, and T B adsorption sites, the metal atoms may be also adsorbed above the bridge site, the middle point between the two silicon atoms. Our calculations, however, showed that the adatom placed initially above the bridge site finally moved to the T A site after the structural optimization,26 indicating the instability of the bridge site. The 8×8×1 gamma central Monkhorst-Pack grids were used to conduct the first BZ integration. The positive direction of the applied electronic field was defined to be along +z, as denoted in Fig. 1(b). Berry curvatures were calculated to identify the QVH effect12,27 after the DFT calculations were completed. According to the Kubo formula,28,29 the Berry curvature Ω(k) can be expressed by Ω(k) = f n Ωn (k), n
~ ⟨ψ nk|v x |ψ mk⟩ ψ mk|v y |ψ nk , Ωn (k) = −2Im (Em − En )2 m,n 2
(1)
where the summation is over all n occupied states, f n is the Fermi-Dirac distribution function, |ψ nk⟩ is the Bloch function with the eigenvalue En , and υ x(y) are the velocity operators. Here, the Berry curvatures were calculated in maximally localized Wannier functions (MLWFs) bases.30–33 The constructions of MLWFs were non-self-consistent calculations on 5 × 5×1 k-point grids based on the previously converged DFT self-
consistent charge potential. We checked that the adopted Wannier functions included sufficient number of unoccupied bands so that the first-principles band structures could be precisely reproduced. III. RESULTS AND DISCUSSION A. Electronic structures and QVH states
Table I gives the favored adsorption site and the corresponding adsorption energy and structural properties of the 4× 4 supercell of silicene adsorbed with the different metal atoms. The adsorption energy is defined as Eb = ES + E M − ES+M , where ES , E M , and ES+M represent the total energies of the bare silicene supercell, the free adatom, and the 4 × 4 silicene adsorbed with one of the metal atoms, respectively. For all of the three kinds of adatoms, the hollow adsorption site is found to be the most stable configuration, ensuring the silicene Dirac cones unbroken. The obtained pretty large adsorption energies (4.2-5.8 eV), in good agreement with the results in Refs. 7 and 26, indicate very strong structural stability of Pt(Ni, Pd)silicene systems. The strong interaction between the adatoms and the substrate can also be seen by obviously local structural distortions of silicene after the metal atoms are adsorbed. As illustrated in Fig. 1(c), the Si atoms around the Pt adatom distinctly sink and deviate much from the initial positions. Thus, the inversion symmetry of silicene is broken, making the QVH appearance possible.12 The z coordinates of the atoms in the fully relaxed structures for the three systems are given in Fig. 1(d). Explicitly, both of the Si atoms in the A and B sublattices, nearest to the adatom, shift downward by at least 0.4 Å. If the bond lengths between the adatom and its nearest Si atoms in A/B sublattices are denoted as d(M-Si A/B), d(M-Si A) is always larger than d(M-Si B) (by about 0.25 Å, Table I) for the three systems, indicating the stronger interaction between M-Si B than that between M-Si A. Namely, the electrostatic potential of the neighboring Si A atoms increases while that of the nearby Si B atoms decreases due to the existence of the adatoms. The influence of the adatoms can, thus, be divided
FIG. 1. (a) Top view of the 4 × 4 supercell of silicene. The three adsorption sites considered for the adatom are labeled as H, T A, and T B . (b) Side view of silicene with one adatom (grey) adsorbed at the H site before the structural relaxation. The A/B sublattices and the direction of the applied positive electric field (E ⊥) are also shown. (c) The relaxed structure of the Pt-decorated silicene. (d) The z coordinates of the adatoms and the Si atoms in the fully optimized structures. The 17th atom in (d) denotes the adatom (Pt, Ni, or Pd). The left and right of the adatom indicate the Si atoms in the A and B sublattices, respectively. Especially, the 14–16th and 18–20th denote the Si atoms nearest to the adatom.
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TABLE I. The results for 4 × 4 silicene adsorbed with Pt, Ni, or Pd atoms. The properties listed contain the favored adsorption site and the corresponding adsorption energy (E b , eV), distance between the adatom to its nearest Si atoms in A/B sublattice (d(M-Si A/B ), Å), magnetic moment per unit cell (M, µ B ), electron transfer from the silicene to the adatom (dQ, e), and the band gaps without (E g1, meV) and with (E g2, meV) the SOC. The intrinsic (λ SO , meV) and Rashba (λ R , meV) SOC interactions around the zero UT are also evaluated. Atom Pt Ni Pd
Site
Eb
d(M-Si A)
d(M-Si B )
M
dQ
E g1
E g2
λ SO
λR
H H H
5.78 4.75 4.26
2.68 2.57 2.71
2.42 2.32 2.46
0.0 0.0 0.0
0.88 0.13 0.36
27.0 4.6 17.5
20.2 2.3 16.5
2.15 0.87 ∼0
1.32 ∼0 0.77
into two aspects: the local staggered AB-sublattice potential and the local structural distortions around the adatoms, both of which will trigger trivial band gaps in the systems. Among Pt, Ni, and Pd adatoms, the bond length between Ni-Si A/B is the smallest (Table I), due to the smallest radius of the Ni atoms. The largest bond length found in the Pd case can be ascribed to the relatively large radius and strong stability of the Pd atoms (4d 10, closed-shell). The trend is consistent with the height sequence of the adatoms in Fig. 1(d). The Pt(Ni, Pd)-decorated silicene systems are all nonmagnetic (Table I), different from the silicene adsorbed with many other 3d-transition metal atoms (Sc-Co).7 The non-magnetism can be rationalized by densities of states (DOSs) given in Fig. 2. For the Pt (Ni) adatom, the energy of the 5d (3d) shell is lowered due to interactions with the Si substrate, causing the one 6s (two 4s) electron(s) transferring to the 5d (3d) orbital. Hence, a closed 5d (3d) shell is formed, illustrated well by the
FIG. 2. The calculated total and partial (the right scale) DOSs of silicene adsorbed with different metal atoms. The SOC is not considered. For clarity, the partial DOSs of s orbitals are multiplied by a factor of 5.
almost all occupied Pt 5d (Ni 3d) states in the partial DOS of Fig. 2. The closed shells of Pt 5d and Ni 3d lead to the vanishing magnetic moment in the systems. For the Pd case, the Pd atom has initially a closed 4d shell without any 5s electrons, also giving rise to a non-magnetism ground state. The Bader analysis, actually, shows there are some electrons transferring from the silicene to the adatoms (Table I), especially for the Pt system. These gained electrons from the silicene may occupy also the Pt 5d orbitals. Besides, they can occupy the Pt 6s orbitals as well, seen by plenty of the occupied states in the Pt 6s DOS in Fig. 2(a). The band structures of the three adsorbed silicene systems are illustrated in Fig. 3. The Pt-decorated silicene presents rather “perfect” Dirac cones around the EF , without (Fig. 3(a)) or with (Fig. 3(b)) the SOC interaction, very like the bands in the pristine silicene. This characteristic benefits from the most stable hollow sites in the adsorption systems. Amazingly, the Dirac cones in Figs. 3(a) and 3(b) are exactly located at the EF in spite of slight electron transfer between the Pt adatom and the substrate. The status is the same for the Ni and Pd cases. This behavior is a very desirable property to observe Dirac-cone effects owing to no gate voltage needed to tune the chemical potential in experiments.34 Magnifying the bands around the EF in Fig. 3(a), we find a band gap (27 meV) opened at the EF (Fig. 3(c)). In comparison with the gapless band of pure silicene,1 this trivial band gap is opened due to the local structural distortion and the staggered AB-sublattice potential, induced by the Pt adatoms. Relatively large spin splits are aroused by the SOC, in the valence (6.7 meV) and conduction (10.4 meV) bands near the K and K′ points, which is clearly shown in the magnified bands in Fig. 3(d). Because of these spin splits, the direct band gaps at the two inequivalent K and K′ points become a little bit smaller (20.2 meV). These spin splits around the valence-band maximum (VBM) and conductionband minimum (CBM) with opposite spin moments at the two valleys come from the time-reversal symmetry and the broken inversion symmetry in the Pt-decorated silicene. For the Ni and Pd systems, the bands around the EF (−0.25 to 0.25 eV) are almost the same as Figs. 3(a) and 3(b), except different values of the band gaps and spin splits. Different from the large spin splits in the Pt system, the spin splits induced by the SOC in Ni and Pd systems are very small (Figs. 3(e) and 3(f)). Therefore, the SOC interactions in the latter two systems are very small, ascribed to the light Ni atoms and the closed 4d shell of Pd atoms. The trivial gap (without the SOC) opened in the Ptsilicene is the maximum (Table I), due to the largest net effect of the local structural distortions (Fig. 1(d)) and the staggered
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FIG. 3. (a) and (b) Band structures of silicene adsorbed with Pt atoms without and with the SOC, respectively. (c) and (d) Magnifications of (a) and (b) around the E F , set at energy zero, respectively. The red and black curves in (d) denote the spin-up and spin-down states, respectively. (e) and (f) The same as (d) except for Ni- and Pd-silicene systems, respectively.
AB-sublattice potential, induced in the system. The Hubbard U effect on the bands around the Dirac point is very weak due to the primary Si 3p states, as also reported in Ref. 5. To identify whether the three studied systems present the QVH effect, the Berry curvatures Ω(k) were calculated and shown in Fig. 4. In Fig. 4(a), two sharp peaks with opposite signs of Ω(k) are obtained around the K and K′ points. The distribution of the Berry curvatures in 2D momentum space plotted in Fig. 4(b) more clearly shows that Ω(k) is only finite and has opposite signs at the two valleys, indicating valley Hall effect12,17 appearing in the Pt-silicene system. Obviously, 1 2 integrating the Ω(k) over the first BZ (C = 2π BZ d kΩ n ), n
zero Chern number (C = 0) is obtained, leading to zero Hall conductivity originating from the time-reversal symmetry in the system. While if the integral of Ω(k) is performed only around the K or K′ points, a finite valley Chern number Cv = CK − CK ′ can be obtained, which is usually employed to characterize the QVH effect.8 In the Pt-silicene, the calculated CK = −CK ′ = 1. Thus, Cv = 2, displaying realization of QVH state in the Pt-silicene. The distributions of Ω(k) for the Nisilicene and Pd-silicene are shown in Figs. 4(c) and 4(d), respectively. The obtained Cv = 2 proves the two systems also present QVH ground states. Our results provide new routes to design the valleytronics in silicene. B. Topological transitions under an electric field
Now we investigate whether the gap of the QVH state obtained can be further manipulated and how the QVH states
evolve, under an external electric field, which has been shown to tune flexibly the band gap of silicene.2 The electric field (E⊥) is applied vertically to the silicene plane (Fig. 1(b)), where positive direction of the electric fields is also defined. The band gaps as a function of E⊥ for the three systems are presented in Figs. 5(a)–5(c), where very rich topological phase diagrams emergent. For all the Pt-, Ni-, and Pd-silicene systems, the band gaps of the QVH states can be increased monotonically by applying a positive electric field. For example, the QVH gap in the Pt-silicene increases from 20.2 meV (with SOC) without E⊥ to 40 meV with E⊥ = 0.3 V/Å, accessible strength of electric fields in experiments. When a negative electric field is applied (E⊥ < 0), the status becomes various. For the Pt and Ni cases, the gap with the SOC decreases first to zero, then increases to a finite value, decreases to zero again, and finally increases with the negative electric field (Figs. 5(a) and 5(b)), while for the Pd-silicene, the gap just closes once during the process. Before we analyze concretely the mechanism of the gap variations, the factors associated with the trivial-gap opening are discussed. When a vertical electric field is applied, the E⊥ will induce a uniformly staggered AB-sublattice potential (UE ). With a positive electronic field (E⊥ > 0, Fig. 1(b)), the electrostatic potential of the A sublattice decreases while that of the B sublattice increases. Its effect is opposite to the local staggered AB-sublattice potential (US ) induced by the adatoms (such as Pt), as discussed above. Thus, the US effect from the adatoms can be regarded as a local negative electric field applied to the system. Besides, the adatoms
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FIG. 4. (a) The Berry curvatures (in atomic units (a.u.)) for the whole valence bands around the K and K′ points for the Pt-silicene system. (b) The 2D distribution of the Berry curvatures (in a.u.) in the momentum space for the Pt-silicene. (c) and (d) The same as (a) except for the Ni- and Pd-silicene systems, respectively. The insets in (c) and (d) are the corresponding 2D distributions of the Berry curvatures (in a.u.).
also induce the local structural distortion around the adatoms, whose corresponding potential is denoted as UD . Since the six Si atoms nearest to the adatom all move along the −z direction (Fig. 1(d)), the surrounding electrons also shift along the −z direction. Hence, the UD effect is equivalent to a local positive electric field. Thus, the total electrostatic potential induced externally in the system contains three terms: UT = UE + US + UD , where the latter two terms coming from the adatoms are approximately not varied with the E⊥, while the UE is determined by the E⊥ and its sign is opposite when a
minus E⊥ is applied. Namely, the strength of this total potential (UT ) is tunable by the E⊥. For the Pt-silicene (Fig. 5(a)), when a positive E⊥ is applied, the trivial gap (the gap without the SOC) increases, indicating the UT increase and |U D | > |US |. If a negative E⊥ is exerted, the UT becomes small due to the minus UE . At E⊥ = −0.36 V/Å, the trivial gap reduces to 4.28 meV. When the SOC is considered at this E⊥, the spin splitting of the valence and conductance bands nearest to the EF happens, giving rise to the zero band gap in the system (Fig. 5(a)). Thus, for E⊥ >
FIG. 5. Topological phase diagrams for silicene adsorbed with Pt (a), Ni (b), or Pd (c) atoms. The QSH state is quenched in (c). (d) shows the energy band gap dependent on the relative SOC strength (λ/λ0) for the Pt-silicene system at E ⊥ = −0.42 V/Å. λ0 is the real SOC strength in the material.
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−0.36 V/Å (the right cyan region in Fig. 5(a)), the effect of UT is stronger than the SOC and the system is still in QVH state as that at E⊥ = 0. The total potential UT is reduced to zero at E⊥ = −0.43 V/Å and then the UT changes the sign, meaning the average electrostatic potential of the A sublattice becomes higher than that of the B sublattice. This property leads to the effect of UT is symmetric about the point of E⊥ = −0.43 V/Å. That is, when the field E⊥ = −0.50 V/Å, the effect of UT is increased to the same as that of the SOC and the band gap with the SOC becomes zero again as the status of E⊥ = −0.36 V/Å. If E⊥ < −0.50 V/Å, the situation becomes the same as E⊥ > −0.36 V/Å and the system is QVH state again. Now, we focus on the case when the effect of the total potential UT is weaker than the SOC in the range −0.50 V/Å < E⊥ < −0.36 V/Å. Fig. 5(d) illustrates the band gap as a function of the relative SOC strength at E⊥ = −0.42 V/Å in the Pt-silicene system. The band gap first decreases to zero then increases again with the continuous evolution of λ from 0 to λ0, the real SOC strength in the material. This gap-closing at K and K′ Dirac points in the inversion-asymmetric 2D system is generally accompanied with the change of Z2 topological number.35 Here, it reflects topological transitions from QVH to QSH (at −0.36 V/Å) and then to QVH again (at −0.50 V/Å). The nontrivial topological behavior of the yellow region in Fig. 5(a) will be illustrated intuitively by the edge states calculated. The maximum nontrivial gap obtained in the QSH region is 2.3 meV, larger than the 1.5 meV in pristine silicene.3 This increase can be ascribed to the hybridization of Pt 5d electrons with the silicene Dirac cones. It is interesting to note that for all the three systems, the gap closing occurs at negative electric fields (Fig. 5). This trend contradicts to the results of silicene depositing on GaS (Ref. 36) or MoS2 (Ref. 37) substrates, in which the gap closing happens at positive electric fields. (Note that the direction definition of the electric field there may be different.) To understand this problem, some further calculations were performed, in which all the Si atoms were fixed and only the adatoms were allowed to relax to a local energy minimum. With such relaxed structures, the obtained gaps as a function of E⊥ for the three systems are given in Fig. 6. Interestingly, the gap closing occurs now at positive electric fields for all the three systems. Here, since the UD = 0, UT = UE +US . The gap closing at a positive electric field means US effect equivalent to a negative electric
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field, consistent with the above analysis. Therefore, it is the local structural distortions (UD ) induced by the adatoms that cause the gap closing occurring at negative electric fields in these systems. Fig. 5(b) shows the results of the Ni-silicene, the QSH state occurs at the range −0.11 V/Å < E⊥ < −0.04 V/Å. And the largest nontrivial gap is about 1.75 meV at E⊥ = −0.07 V/Å. The Ni-silicene case is very similar to the Ptsilicene. Here, we find the electric field can not only increase the gap of the QVH states but also lead to the amazing topological phase transitions. As observed in Fig. 5(c), there is only one gap-closing point at E⊥ = −0.22 V/Å in the Pd-silicene, revealing the disappearance of the QSH state. In experiments, silicene is generally deposited on a substrate, such as metallic Ag(111) (Ref. 9) or Ir(111) (Ref. 10). To maintain the exotic Dirac-electron behaviors in silicene, some insulating substrates, including hexagonal BN,38,39 SiC (0001),38 and MgBr2(0001),40 have been proposed to support the silicene. The lattice mismatch between the silicene and these substrates is usually small ( λSO). The Rashba interaction is, however, neglected in their work. In Fig. 7, when λ R increases, such as λ R = 0.3 and 0.5λSO, the two gap-closing points gradually approach each other, accompanied by the decrease of the energy gap at the U = 0. It is valuable to note that the QSH survives as λ R is less than 2/3λSO, certified by the stable existence of gapless edge states as shown in Fig. 8. Thus, the topological transitions in the Pt/Ni-decorated silicene are identical to above cases. Particularly, this model also gives that
FIG. 8. Evolution of edge states of semi-infinite silicenes at the marked points in Fig. 7; QSH1 (a), QSH2 (b), M (c), and QVH (d), where the energies are in units of t (t = 10λ SO ).
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the gap at U = 1λSO without the SOC is equal to the gap at U = 0 with the SOC (Fig. 7). Thus, from the nontrivial gaps at UT = 0 and the gaps without the SOC at the phase transition borders in Figs. 5(a) and 5(b), the Rashba strengths and the intrinsic SOC values in the two systems can be evaluated, listed in Table I. The Rashba SOC in the Ni-silicene is negligible, while that in the Pt-silicene is not. It is exciting that the finite value of Rashba SOC in the Pt-silicene does not destroy the QSH. In Fig. 7, when λ R increases to or greater than 2/3λSO, such as λ R = 2/3 and 1.5λSO, the two gap-closing points merge. Simultaneously, the energy gaps at U = 0 decrease to zero in the process, implying the QSH state is quenched and the QVH states directly connect across the gap-closing point. This exactly corresponds to the case of Pd-silicene system. The Rashba and intrinsic SOC strengths for the Pd-decorated silicene were also evaluated and listed in Table I. From above discussion, we can see that the adatoms and the applied electric field both can break the inversion symmetry in silicene, resulting in an effective staggered potential. The competition between this staggered potential (U), controlled by the external electric field, and the intrinsic and Rashba SOCs in the system finally leads to the rich phase diagram in Fig. 7. To intuitively understand the appearance and absence of QSH states, the evolution of edge modes inside the energy gap is investigated. Fig. 8 plots the edge states of semi-infinite silicenes with zigzag edges at marked points in Fig. 7 via calculating the local density of state (LDOS) based on the iterative Green’s function method.43 In Fig. 8(a), the gapless helical edge states driven from the QSH effect emerge in the bulk gap and directly link the conduction and valence bands. Although nonzero Rashba SOC destroys the spin conservation of electrons, one pair of edge states is still found to connect the conduction and valence bands only if the energy gap is not closed (Fig. 8(b)). When the λ R is equal or greater than 2/3λSO together with zero average staggered potential, the silicenes are always a gapless semiconductor with quadratically dispersion as shown in Fig. 8(c). Based on Fig. 8(c), any perturbations such as the staggered potential may trigger a trivial gap and quench the QSH state, as shown in Fig. 8(d). However, in this case, the interesting QVH state is aroused, as confirmed by the calculations of the Berry curvatures. IV. CONCLUSIONS
We systematically investigated the electronic and topological properties of silicene adsorbed with Pt, Ni, or Pd atoms. All the three kinds of adatoms are found to prefer the hollow adsorption sites of silicene and the systems are nonmagnetic. The QVH states with the gap opened exactly around the EF are achieved in the adsorption systems. An external electronic field is applied to consider the effect of electrical tuning for nanoelectronic applications. The band gaps of the QVH states can be increased effectively by applying a positive electronic field. Topological transitions from QVH states to QSH states and then to QVH states again occur at a suitable range of negative electronic fields in the Pt/Ni-silicene. It is the strong structural distortions induced by the adatoms that cause the transitions happening at negative electric fields. The QSH state vanishes during the transitions in the Pd-silicene due to the relatively
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larger Rashba SOC than the intrinsic SOC in the system. Our results provide new routes to design valleytronics in silicenebased systems and are also helpful to the understanding of the SOC effects in the systems. ACKNOWLEDGMENTS
This work was supported by 973-project under Grant No. 2011CB921803, Natural Science Foundation of Shanghai with Grant No. 14ZR1403400, and Fudan High-end Computing Center. 1S.
Cahangirov, M. Topsakal, E. Aktürk, H. Sahin, ¸ and S. Ciraci, Phys. Rev. Lett. 102, 236804 (2009). 2Z. Y. Ni, Q. H. Liu, K. C. Tang, J. X. Zheng, J. Zhou, R. Qin, Z. X. Gao, D. P. Yu, and J. Lu, Nano Lett. 12, 113 (2012). 3C.-C. Liu, W. Feng, and Y. Yao, Phys. Rev. Lett. 107, 076802 (2011). 4M. Ezawa, Phys. Rev. Lett. 109, 055502 (2012). 5J. Y. Zhang, B. Zhao, and Z. Q. Yang, Phys. Rev. B 88, 165422 (2013). 6M. Ezawa, Phys. Rev. B 87, 155415 (2013). 7X.-L. Zhang, L.-F. Liu, and W.-M. Liu, Sci. Rep. 3, 2908 (2013). 8H. Pan, Z. S. Li, C. C. Liu, G. B. Zhu, Z. H. Qiao, and Y. G. Yao, Phys. Rev. Lett. 112, 106802 (2014). 9P. Vogt, P. De Padova, C. Quaresima, J. Avila, E. Frantzeskakis, M. C. Asensio, A. Resta, B. Ealet, and G. Le Lay, Phys. Rev. Lett. 108, 155501 (2012). 10L. Meng, Y. Wang, L. Zhang, S. Du, R. Wu, L. Li, Y. Zhang, G. Li, H. Zhou, W. A. Hofer, and H.-J. Gao, Nano Lett. 13, 685 (2013). 11K. Behnia, Nat. Nanotechnol. 7, 488 (2012). 12D. Xiao, W. Yao, and Q. Niu, Phys. Rev. Lett. 99, 236809 (2007). 13M. Ezawa, Phys. Rev. B 86, 161407 (2012). 14W. Yao, D. Xiao, and Q. Niu, Phys. Rev. B 77, 235406 (2008). 15J. Ding, Z. H. Qiao, W. X. Feng, Y. G. Yao, and Q. Niu, Phys. Rev. B 84, 195444 (2011). 16M. Tahir, A. Manchon, K. Sabeeh, and U. Schwingenschlögl, Appl. Phys. Lett. 102, 162412 (2013). 17W. X. Feng, Y. G. Yao, W. G. Zhu, J. J. Zhou, W. Yao, and D. Xiao, Phys. Rev. B 86, 165108 (2012). 18K. F. Mak, K. L. McGill, J. Park, and P. L. McEuen, Science 344, 1489 (2014). 19A. Rycerz, J. Tworzydlo, and C. W. J. Beenakker, Nat. Phys. 3, 172 (2007). 20T. Fujita, M. B. A. Jalil, and S. G. Tan, Appl. Phys. Lett. 97, 043508 (2010). 21Y. Jiang, T. Low, K. Chang, M. I. Katsnelson, and F. Guinea, Phys. Rev. Lett. 110, 046601 (2013). 22H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, Nat. Nanotechnol. 7, 490 (2012). 23G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999). 24G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996). 25J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996). 26X. Lin and J. Ni, Phys. Rev. B 86, 075440 (2012). 27D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010). 28D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982). 29Y. G. Yao, L. Kleinman, A. H. MacDonald, J. Sinova, T. Jungwirth, D. S. Wang, E. Wang, and Q. Niu, Phys. Rev. Lett. 92, 037204 (2004). 30N. Marzari and D. Vanderbilt, Phys. Rev. B 56, 12847 (1997). 31I. Souza, N. Marzari, and D. Vanderbilt, Phys. Rev. B 65, 035109 (2001). 32A. A. Mostofi, J. R. Yates, Y.-S. Lee, I. Souza, D. Vanderbilt, and N. Marzari, Comput. Phys. Commun. 178, 685 (2008). 33X. J. Wang, J. R. Yates, I. Souza, and D. Vanderbilt, Phys. Rev. B 74, 195118 (2006). 34C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. C. Zhang, M. H. Guo, K. Li, Y. B. Ou, P. Wei, L.-L. Wang, Z.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. F. Jia, X. Dai, Z. Fang, S. C. Zhang, K. He, Y. Y. Wang, L. Lu, X.-C. Ma, and Q.-K. Xue, Science 340, 167 (2013). 35S. Murakami, S. Iso, Y. Avishai, M. Onoda, and N. Nagaosa, Phys. Rev. B 76, 205304 (2007). 36Y. Ding and Y. L. Wang, Appl. Phys. Lett. 103, 043114 (2013).
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.30.242.61 On: Sun, 31 May 2015 18:51:23
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Zhao et al.
37N. Gao, J. C. Li, and Q. Jiang, Phys. Chem. Chem. Phys. 16, 11673 (2014). 38H.
S. Liu, J. F. Gao, and J. J. Zhao, J. Phys. Chem. C 117, 10353 (2013). 39T. P. Kaloni, M. Tahir, and U. Schwingenschlögl, Sci. Rep. 3, 3192 (2013). 40J. J. Zhu and U. Schwingenschlögl, ACS Appl. Mater. Interfaces 6, 11675 (2014).
J. Chem. Phys. 141, 244701 (2014) 41H.
Jiang, Z. Qiao, H. Liu, J. Shi, and Q. Niu, Phys. Rev. Lett. 109, 116803 (2012). 42X.-F. Wang, Y. B. Hu, and H. Guo, Phys. Rev. B 85, 241402(R) (2012). 43M. P. L. Sancho, J. M. L. Sancho, J. M. L. Sancho, and J. Rubio, J. Phys. F: Met. Phys. 15, 851 (1985).
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THE JOURNAL OF CHEMICAL PHYSICS 141, 244701 (2014)
Quantum valley Hall states and topological transitions in Pt(Ni, Pd)-decorated silicene: A first-principles study Bao Zhao,1 Jiayong Zhang,1 Yicheng Wang,1 and Zhongqin Yang1,2,a) 1
State Key Laboratory of Surface Physics and Key Laboratory for Computational Physical Sciences (MOE) and Department of Physics, Fudan University, Shanghai 200433, China 2 Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing, 210093, China
(Received 10 September 2014; accepted 1 December 2014; published online 23 December 2014) The electronic states and topological behaviors of Pt(Ni, Pd)-decorated silicene are investigated by using an ab-initio method. All the three kinds of the adatoms prefer hollow sites of the silicene, guaranteeing the Dirac cones unbroken. The Pt(Ni, Pd)-decorated silicene systems all present quantum valley Hall (QVH) states with the gap opened exactly at the Fermi level. The gaps of the QVH states can be increased substantially by applying a positive electric field. Very fascinating phase transitions from QVH to quantum spin Hall (QSH) and then to QVH again are achieved in the Pt/Ni-decorated silicene when a negative electric field is applied. The QSH state in the Pd case with a negative electric field is, however, quenched because of relatively larger Rashba spin-orbit coupling (SOC) than the intrinsic SOC in the system. Our findings may be useful for the applications of silicene-based devices in valleytronics and spintronics. C 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4904285]
I. INTRODUCTION
Silicene, a monolayer of silicon atoms, has recently attracted much attention due to its exotic electronic states and topological behaviors.1–8 It has been successfully synthesized in experiments.9,10 Different to graphene, silicene possesses a buckled honeycomb structure,1,10 resulting in the electronic behaviors controllable by applying an external electric field to silicene.2,4 The relatively strong intrinsic spin-orbit coupling (SOC) in silicene causes the predicted quantum spin Hall (QSH) effect to be observed in an experimentally accessible low temperature regime.3 When exchange fields are considered, the time-reversal symmetry in the silicene system is broken, which may induce quantum anomalous Hall (QAH) effects and abundant other new topological states,4–8 including valley polarized metals,4 marginal valley polarized metals,4 and the interesting hybrid topological state of QSH and QAH effects.6 Especially, a brand new and very unique topological state of valley-polarized QAH is found in silicene by tuning the strengths of Rashba and intrinsic SOCs in a tight-binding model,8 which may have special applications in silicene spintronics. Currently, valleytronics becomes one of the focuses in the field of condensed matter physics,11 where the electric current is controlled by the valley degree of freedom instead of the traditional charge and spin of electrons in two-dimensional (2D) crystals with a honeycomb lattice. Silicene, graphene, and monolayers of transition metal dichalcogenides MX2 (M = Mo, W; X = S, Se), etc., are all potential materials in valleytronics, with band structures consisting of two degenerate but inequivalent valleys located at the corners of the hexagonal Brillouin zone (BZ). The broken inversion symmetry in these systems, induced by such as substrate potentials, external eleca)Electronic address: [email protected]
0021-9606/2014/141(24)/244701/9/$30.00
tric fields, or intrinsic inequivalent atoms, allows a (quantum) valley Hall effect,12–18 where carriers in different valleys flow to opposite transverse edges when an in-plane electric field is applied.12,19 Some theoretical models of certain valleytronic devices, such as valley filters,19,20 valley valves,19 and devices generating pure valley current,21 are proposed. More charmingly, valley-dependent interactions of electrons with light of different circular polarizations are enabled in the systems. Thus, optical measurements and manipulations of the valleydependent physics become possible.22 It is very meaningful to explore new materials or paths to acquire the valley Hall effects and apply them to valleytronics. In this paper, we intend to achieve quantum valley Hall (QVH) states in silicene by adsorbing Pt, Ni, or Pd atoms onto the silicene plane, from first-principles calculations. We find that all the three kinds of the adatoms prefer hollow sites of the silicene, guaranteeing the silicene Dirac cones unbroken. The systems are nonmagnetic and the band gaps are opened just around the Fermi level (EF ). These behaviors together with the inversion symmetry broken by the adatoms lead to QVH appearance in the systems. How an electric field tunes the QVH states is also investigated. When a positive electric field is applied, the gaps of the QVH states in the three systems are increased drastically. Interesting topologically phase transitions from QVH to QSH and then to QVH again are found in the Pt and Ni cases when a negative electric field is applied. The QSH state is, however, quenched in the Pd case. The mechanisms are understood based on a low-energy effective model. II. METHODS AND MODELS
The electronic structure calculations were performed by using projector augmented wave formalism23 within the framework of density functional theory (DFT),24 as implemented
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in the Vienna ab-initio simulation package (VASP).24 The Perdew, Burke, and Ernzerhof parameterized generalizedgradient approximation was used to treat the exchange and correlation functional.25 To avoid the interactions between the two adjacent silicene layers, we chose a vacuum space of larger than 15 Å. The plane-wave cutoff energy was 500 eV and the convergence criterion for the total energy was set to be 10−5 eV. All atomic positions were relaxed until the Hellmann-Feynman force on each atom was less than 0.01 eV/Å. A 4 × 4 supercell of silicene adsorbed with one adatom was employed, as shown in Fig. 1(a). We considered three typical adsorption sites on silicene: hollow (H), top A (T A), and top B (T B) (Fig. 1(a)). Besides the H, T A, and T B adsorption sites, the metal atoms may be also adsorbed above the bridge site, the middle point between the two silicon atoms. Our calculations, however, showed that the adatom placed initially above the bridge site finally moved to the T A site after the structural optimization,26 indicating the instability of the bridge site. The 8×8×1 gamma central Monkhorst-Pack grids were used to conduct the first BZ integration. The positive direction of the applied electronic field was defined to be along +z, as denoted in Fig. 1(b). Berry curvatures were calculated to identify the QVH effect12,27 after the DFT calculations were completed. According to the Kubo formula,28,29 the Berry curvature Ω(k) can be expressed by Ω(k) = f n Ωn (k), n
~ ⟨ψ nk|v x |ψ mk⟩ ψ mk|v y |ψ nk , Ωn (k) = −2Im (Em − En )2 m,n 2
(1)
where the summation is over all n occupied states, f n is the Fermi-Dirac distribution function, |ψ nk⟩ is the Bloch function with the eigenvalue En , and υ x(y) are the velocity operators. Here, the Berry curvatures were calculated in maximally localized Wannier functions (MLWFs) bases.30–33 The constructions of MLWFs were non-self-consistent calculations on 5 × 5×1 k-point grids based on the previously converged DFT self-
consistent charge potential. We checked that the adopted Wannier functions included sufficient number of unoccupied bands so that the first-principles band structures could be precisely reproduced. III. RESULTS AND DISCUSSION A. Electronic structures and QVH states
Table I gives the favored adsorption site and the corresponding adsorption energy and structural properties of the 4× 4 supercell of silicene adsorbed with the different metal atoms. The adsorption energy is defined as Eb = ES + E M − ES+M , where ES , E M , and ES+M represent the total energies of the bare silicene supercell, the free adatom, and the 4 × 4 silicene adsorbed with one of the metal atoms, respectively. For all of the three kinds of adatoms, the hollow adsorption site is found to be the most stable configuration, ensuring the silicene Dirac cones unbroken. The obtained pretty large adsorption energies (4.2-5.8 eV), in good agreement with the results in Refs. 7 and 26, indicate very strong structural stability of Pt(Ni, Pd)silicene systems. The strong interaction between the adatoms and the substrate can also be seen by obviously local structural distortions of silicene after the metal atoms are adsorbed. As illustrated in Fig. 1(c), the Si atoms around the Pt adatom distinctly sink and deviate much from the initial positions. Thus, the inversion symmetry of silicene is broken, making the QVH appearance possible.12 The z coordinates of the atoms in the fully relaxed structures for the three systems are given in Fig. 1(d). Explicitly, both of the Si atoms in the A and B sublattices, nearest to the adatom, shift downward by at least 0.4 Å. If the bond lengths between the adatom and its nearest Si atoms in A/B sublattices are denoted as d(M-Si A/B), d(M-Si A) is always larger than d(M-Si B) (by about 0.25 Å, Table I) for the three systems, indicating the stronger interaction between M-Si B than that between M-Si A. Namely, the electrostatic potential of the neighboring Si A atoms increases while that of the nearby Si B atoms decreases due to the existence of the adatoms. The influence of the adatoms can, thus, be divided
FIG. 1. (a) Top view of the 4 × 4 supercell of silicene. The three adsorption sites considered for the adatom are labeled as H, T A, and T B . (b) Side view of silicene with one adatom (grey) adsorbed at the H site before the structural relaxation. The A/B sublattices and the direction of the applied positive electric field (E ⊥) are also shown. (c) The relaxed structure of the Pt-decorated silicene. (d) The z coordinates of the adatoms and the Si atoms in the fully optimized structures. The 17th atom in (d) denotes the adatom (Pt, Ni, or Pd). The left and right of the adatom indicate the Si atoms in the A and B sublattices, respectively. Especially, the 14–16th and 18–20th denote the Si atoms nearest to the adatom.
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TABLE I. The results for 4 × 4 silicene adsorbed with Pt, Ni, or Pd atoms. The properties listed contain the favored adsorption site and the corresponding adsorption energy (E b , eV), distance between the adatom to its nearest Si atoms in A/B sublattice (d(M-Si A/B ), Å), magnetic moment per unit cell (M, µ B ), electron transfer from the silicene to the adatom (dQ, e), and the band gaps without (E g1, meV) and with (E g2, meV) the SOC. The intrinsic (λ SO , meV) and Rashba (λ R , meV) SOC interactions around the zero UT are also evaluated. Atom Pt Ni Pd
Site
Eb
d(M-Si A)
d(M-Si B )
M
dQ
E g1
E g2
λ SO
λR
H H H
5.78 4.75 4.26
2.68 2.57 2.71
2.42 2.32 2.46
0.0 0.0 0.0
0.88 0.13 0.36
27.0 4.6 17.5
20.2 2.3 16.5
2.15 0.87 ∼0
1.32 ∼0 0.77
into two aspects: the local staggered AB-sublattice potential and the local structural distortions around the adatoms, both of which will trigger trivial band gaps in the systems. Among Pt, Ni, and Pd adatoms, the bond length between Ni-Si A/B is the smallest (Table I), due to the smallest radius of the Ni atoms. The largest bond length found in the Pd case can be ascribed to the relatively large radius and strong stability of the Pd atoms (4d 10, closed-shell). The trend is consistent with the height sequence of the adatoms in Fig. 1(d). The Pt(Ni, Pd)-decorated silicene systems are all nonmagnetic (Table I), different from the silicene adsorbed with many other 3d-transition metal atoms (Sc-Co).7 The non-magnetism can be rationalized by densities of states (DOSs) given in Fig. 2. For the Pt (Ni) adatom, the energy of the 5d (3d) shell is lowered due to interactions with the Si substrate, causing the one 6s (two 4s) electron(s) transferring to the 5d (3d) orbital. Hence, a closed 5d (3d) shell is formed, illustrated well by the
FIG. 2. The calculated total and partial (the right scale) DOSs of silicene adsorbed with different metal atoms. The SOC is not considered. For clarity, the partial DOSs of s orbitals are multiplied by a factor of 5.
almost all occupied Pt 5d (Ni 3d) states in the partial DOS of Fig. 2. The closed shells of Pt 5d and Ni 3d lead to the vanishing magnetic moment in the systems. For the Pd case, the Pd atom has initially a closed 4d shell without any 5s electrons, also giving rise to a non-magnetism ground state. The Bader analysis, actually, shows there are some electrons transferring from the silicene to the adatoms (Table I), especially for the Pt system. These gained electrons from the silicene may occupy also the Pt 5d orbitals. Besides, they can occupy the Pt 6s orbitals as well, seen by plenty of the occupied states in the Pt 6s DOS in Fig. 2(a). The band structures of the three adsorbed silicene systems are illustrated in Fig. 3. The Pt-decorated silicene presents rather “perfect” Dirac cones around the EF , without (Fig. 3(a)) or with (Fig. 3(b)) the SOC interaction, very like the bands in the pristine silicene. This characteristic benefits from the most stable hollow sites in the adsorption systems. Amazingly, the Dirac cones in Figs. 3(a) and 3(b) are exactly located at the EF in spite of slight electron transfer between the Pt adatom and the substrate. The status is the same for the Ni and Pd cases. This behavior is a very desirable property to observe Dirac-cone effects owing to no gate voltage needed to tune the chemical potential in experiments.34 Magnifying the bands around the EF in Fig. 3(a), we find a band gap (27 meV) opened at the EF (Fig. 3(c)). In comparison with the gapless band of pure silicene,1 this trivial band gap is opened due to the local structural distortion and the staggered AB-sublattice potential, induced by the Pt adatoms. Relatively large spin splits are aroused by the SOC, in the valence (6.7 meV) and conduction (10.4 meV) bands near the K and K′ points, which is clearly shown in the magnified bands in Fig. 3(d). Because of these spin splits, the direct band gaps at the two inequivalent K and K′ points become a little bit smaller (20.2 meV). These spin splits around the valence-band maximum (VBM) and conductionband minimum (CBM) with opposite spin moments at the two valleys come from the time-reversal symmetry and the broken inversion symmetry in the Pt-decorated silicene. For the Ni and Pd systems, the bands around the EF (−0.25 to 0.25 eV) are almost the same as Figs. 3(a) and 3(b), except different values of the band gaps and spin splits. Different from the large spin splits in the Pt system, the spin splits induced by the SOC in Ni and Pd systems are very small (Figs. 3(e) and 3(f)). Therefore, the SOC interactions in the latter two systems are very small, ascribed to the light Ni atoms and the closed 4d shell of Pd atoms. The trivial gap (without the SOC) opened in the Ptsilicene is the maximum (Table I), due to the largest net effect of the local structural distortions (Fig. 1(d)) and the staggered
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FIG. 3. (a) and (b) Band structures of silicene adsorbed with Pt atoms without and with the SOC, respectively. (c) and (d) Magnifications of (a) and (b) around the E F , set at energy zero, respectively. The red and black curves in (d) denote the spin-up and spin-down states, respectively. (e) and (f) The same as (d) except for Ni- and Pd-silicene systems, respectively.
AB-sublattice potential, induced in the system. The Hubbard U effect on the bands around the Dirac point is very weak due to the primary Si 3p states, as also reported in Ref. 5. To identify whether the three studied systems present the QVH effect, the Berry curvatures Ω(k) were calculated and shown in Fig. 4. In Fig. 4(a), two sharp peaks with opposite signs of Ω(k) are obtained around the K and K′ points. The distribution of the Berry curvatures in 2D momentum space plotted in Fig. 4(b) more clearly shows that Ω(k) is only finite and has opposite signs at the two valleys, indicating valley Hall effect12,17 appearing in the Pt-silicene system. Obviously, 1 2 integrating the Ω(k) over the first BZ (C = 2π BZ d kΩ n ), n
zero Chern number (C = 0) is obtained, leading to zero Hall conductivity originating from the time-reversal symmetry in the system. While if the integral of Ω(k) is performed only around the K or K′ points, a finite valley Chern number Cv = CK − CK ′ can be obtained, which is usually employed to characterize the QVH effect.8 In the Pt-silicene, the calculated CK = −CK ′ = 1. Thus, Cv = 2, displaying realization of QVH state in the Pt-silicene. The distributions of Ω(k) for the Nisilicene and Pd-silicene are shown in Figs. 4(c) and 4(d), respectively. The obtained Cv = 2 proves the two systems also present QVH ground states. Our results provide new routes to design the valleytronics in silicene. B. Topological transitions under an electric field
Now we investigate whether the gap of the QVH state obtained can be further manipulated and how the QVH states
evolve, under an external electric field, which has been shown to tune flexibly the band gap of silicene.2 The electric field (E⊥) is applied vertically to the silicene plane (Fig. 1(b)), where positive direction of the electric fields is also defined. The band gaps as a function of E⊥ for the three systems are presented in Figs. 5(a)–5(c), where very rich topological phase diagrams emergent. For all the Pt-, Ni-, and Pd-silicene systems, the band gaps of the QVH states can be increased monotonically by applying a positive electric field. For example, the QVH gap in the Pt-silicene increases from 20.2 meV (with SOC) without E⊥ to 40 meV with E⊥ = 0.3 V/Å, accessible strength of electric fields in experiments. When a negative electric field is applied (E⊥ < 0), the status becomes various. For the Pt and Ni cases, the gap with the SOC decreases first to zero, then increases to a finite value, decreases to zero again, and finally increases with the negative electric field (Figs. 5(a) and 5(b)), while for the Pd-silicene, the gap just closes once during the process. Before we analyze concretely the mechanism of the gap variations, the factors associated with the trivial-gap opening are discussed. When a vertical electric field is applied, the E⊥ will induce a uniformly staggered AB-sublattice potential (UE ). With a positive electronic field (E⊥ > 0, Fig. 1(b)), the electrostatic potential of the A sublattice decreases while that of the B sublattice increases. Its effect is opposite to the local staggered AB-sublattice potential (US ) induced by the adatoms (such as Pt), as discussed above. Thus, the US effect from the adatoms can be regarded as a local negative electric field applied to the system. Besides, the adatoms
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FIG. 4. (a) The Berry curvatures (in atomic units (a.u.)) for the whole valence bands around the K and K′ points for the Pt-silicene system. (b) The 2D distribution of the Berry curvatures (in a.u.) in the momentum space for the Pt-silicene. (c) and (d) The same as (a) except for the Ni- and Pd-silicene systems, respectively. The insets in (c) and (d) are the corresponding 2D distributions of the Berry curvatures (in a.u.).
also induce the local structural distortion around the adatoms, whose corresponding potential is denoted as UD . Since the six Si atoms nearest to the adatom all move along the −z direction (Fig. 1(d)), the surrounding electrons also shift along the −z direction. Hence, the UD effect is equivalent to a local positive electric field. Thus, the total electrostatic potential induced externally in the system contains three terms: UT = UE + US + UD , where the latter two terms coming from the adatoms are approximately not varied with the E⊥, while the UE is determined by the E⊥ and its sign is opposite when a
minus E⊥ is applied. Namely, the strength of this total potential (UT ) is tunable by the E⊥. For the Pt-silicene (Fig. 5(a)), when a positive E⊥ is applied, the trivial gap (the gap without the SOC) increases, indicating the UT increase and |U D | > |US |. If a negative E⊥ is exerted, the UT becomes small due to the minus UE . At E⊥ = −0.36 V/Å, the trivial gap reduces to 4.28 meV. When the SOC is considered at this E⊥, the spin splitting of the valence and conductance bands nearest to the EF happens, giving rise to the zero band gap in the system (Fig. 5(a)). Thus, for E⊥ >
FIG. 5. Topological phase diagrams for silicene adsorbed with Pt (a), Ni (b), or Pd (c) atoms. The QSH state is quenched in (c). (d) shows the energy band gap dependent on the relative SOC strength (λ/λ0) for the Pt-silicene system at E ⊥ = −0.42 V/Å. λ0 is the real SOC strength in the material.
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−0.36 V/Å (the right cyan region in Fig. 5(a)), the effect of UT is stronger than the SOC and the system is still in QVH state as that at E⊥ = 0. The total potential UT is reduced to zero at E⊥ = −0.43 V/Å and then the UT changes the sign, meaning the average electrostatic potential of the A sublattice becomes higher than that of the B sublattice. This property leads to the effect of UT is symmetric about the point of E⊥ = −0.43 V/Å. That is, when the field E⊥ = −0.50 V/Å, the effect of UT is increased to the same as that of the SOC and the band gap with the SOC becomes zero again as the status of E⊥ = −0.36 V/Å. If E⊥ < −0.50 V/Å, the situation becomes the same as E⊥ > −0.36 V/Å and the system is QVH state again. Now, we focus on the case when the effect of the total potential UT is weaker than the SOC in the range −0.50 V/Å < E⊥ < −0.36 V/Å. Fig. 5(d) illustrates the band gap as a function of the relative SOC strength at E⊥ = −0.42 V/Å in the Pt-silicene system. The band gap first decreases to zero then increases again with the continuous evolution of λ from 0 to λ0, the real SOC strength in the material. This gap-closing at K and K′ Dirac points in the inversion-asymmetric 2D system is generally accompanied with the change of Z2 topological number.35 Here, it reflects topological transitions from QVH to QSH (at −0.36 V/Å) and then to QVH again (at −0.50 V/Å). The nontrivial topological behavior of the yellow region in Fig. 5(a) will be illustrated intuitively by the edge states calculated. The maximum nontrivial gap obtained in the QSH region is 2.3 meV, larger than the 1.5 meV in pristine silicene.3 This increase can be ascribed to the hybridization of Pt 5d electrons with the silicene Dirac cones. It is interesting to note that for all the three systems, the gap closing occurs at negative electric fields (Fig. 5). This trend contradicts to the results of silicene depositing on GaS (Ref. 36) or MoS2 (Ref. 37) substrates, in which the gap closing happens at positive electric fields. (Note that the direction definition of the electric field there may be different.) To understand this problem, some further calculations were performed, in which all the Si atoms were fixed and only the adatoms were allowed to relax to a local energy minimum. With such relaxed structures, the obtained gaps as a function of E⊥ for the three systems are given in Fig. 6. Interestingly, the gap closing occurs now at positive electric fields for all the three systems. Here, since the UD = 0, UT = UE +US . The gap closing at a positive electric field means US effect equivalent to a negative electric
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field, consistent with the above analysis. Therefore, it is the local structural distortions (UD ) induced by the adatoms that cause the gap closing occurring at negative electric fields in these systems. Fig. 5(b) shows the results of the Ni-silicene, the QSH state occurs at the range −0.11 V/Å < E⊥ < −0.04 V/Å. And the largest nontrivial gap is about 1.75 meV at E⊥ = −0.07 V/Å. The Ni-silicene case is very similar to the Ptsilicene. Here, we find the electric field can not only increase the gap of the QVH states but also lead to the amazing topological phase transitions. As observed in Fig. 5(c), there is only one gap-closing point at E⊥ = −0.22 V/Å in the Pd-silicene, revealing the disappearance of the QSH state. In experiments, silicene is generally deposited on a substrate, such as metallic Ag(111) (Ref. 9) or Ir(111) (Ref. 10). To maintain the exotic Dirac-electron behaviors in silicene, some insulating substrates, including hexagonal BN,38,39 SiC (0001),38 and MgBr2(0001),40 have been proposed to support the silicene. The lattice mismatch between the silicene and these substrates is usually small ( λSO). The Rashba interaction is, however, neglected in their work. In Fig. 7, when λ R increases, such as λ R = 0.3 and 0.5λSO, the two gap-closing points gradually approach each other, accompanied by the decrease of the energy gap at the U = 0. It is valuable to note that the QSH survives as λ R is less than 2/3λSO, certified by the stable existence of gapless edge states as shown in Fig. 8. Thus, the topological transitions in the Pt/Ni-decorated silicene are identical to above cases. Particularly, this model also gives that
FIG. 8. Evolution of edge states of semi-infinite silicenes at the marked points in Fig. 7; QSH1 (a), QSH2 (b), M (c), and QVH (d), where the energies are in units of t (t = 10λ SO ).
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the gap at U = 1λSO without the SOC is equal to the gap at U = 0 with the SOC (Fig. 7). Thus, from the nontrivial gaps at UT = 0 and the gaps without the SOC at the phase transition borders in Figs. 5(a) and 5(b), the Rashba strengths and the intrinsic SOC values in the two systems can be evaluated, listed in Table I. The Rashba SOC in the Ni-silicene is negligible, while that in the Pt-silicene is not. It is exciting that the finite value of Rashba SOC in the Pt-silicene does not destroy the QSH. In Fig. 7, when λ R increases to or greater than 2/3λSO, such as λ R = 2/3 and 1.5λSO, the two gap-closing points merge. Simultaneously, the energy gaps at U = 0 decrease to zero in the process, implying the QSH state is quenched and the QVH states directly connect across the gap-closing point. This exactly corresponds to the case of Pd-silicene system. The Rashba and intrinsic SOC strengths for the Pd-decorated silicene were also evaluated and listed in Table I. From above discussion, we can see that the adatoms and the applied electric field both can break the inversion symmetry in silicene, resulting in an effective staggered potential. The competition between this staggered potential (U), controlled by the external electric field, and the intrinsic and Rashba SOCs in the system finally leads to the rich phase diagram in Fig. 7. To intuitively understand the appearance and absence of QSH states, the evolution of edge modes inside the energy gap is investigated. Fig. 8 plots the edge states of semi-infinite silicenes with zigzag edges at marked points in Fig. 7 via calculating the local density of state (LDOS) based on the iterative Green’s function method.43 In Fig. 8(a), the gapless helical edge states driven from the QSH effect emerge in the bulk gap and directly link the conduction and valence bands. Although nonzero Rashba SOC destroys the spin conservation of electrons, one pair of edge states is still found to connect the conduction and valence bands only if the energy gap is not closed (Fig. 8(b)). When the λ R is equal or greater than 2/3λSO together with zero average staggered potential, the silicenes are always a gapless semiconductor with quadratically dispersion as shown in Fig. 8(c). Based on Fig. 8(c), any perturbations such as the staggered potential may trigger a trivial gap and quench the QSH state, as shown in Fig. 8(d). However, in this case, the interesting QVH state is aroused, as confirmed by the calculations of the Berry curvatures. IV. CONCLUSIONS
We systematically investigated the electronic and topological properties of silicene adsorbed with Pt, Ni, or Pd atoms. All the three kinds of adatoms are found to prefer the hollow adsorption sites of silicene and the systems are nonmagnetic. The QVH states with the gap opened exactly around the EF are achieved in the adsorption systems. An external electronic field is applied to consider the effect of electrical tuning for nanoelectronic applications. The band gaps of the QVH states can be increased effectively by applying a positive electronic field. Topological transitions from QVH states to QSH states and then to QVH states again occur at a suitable range of negative electronic fields in the Pt/Ni-silicene. It is the strong structural distortions induced by the adatoms that cause the transitions happening at negative electric fields. The QSH state vanishes during the transitions in the Pd-silicene due to the relatively
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larger Rashba SOC than the intrinsic SOC in the system. Our results provide new routes to design valleytronics in silicenebased systems and are also helpful to the understanding of the SOC effects in the systems. ACKNOWLEDGMENTS
This work was supported by 973-project under Grant No. 2011CB921803, Natural Science Foundation of Shanghai with Grant No. 14ZR1403400, and Fudan High-end Computing Center. 1S.
Cahangirov, M. Topsakal, E. Aktürk, H. Sahin, ¸ and S. Ciraci, Phys. Rev. Lett. 102, 236804 (2009). 2Z. Y. Ni, Q. H. Liu, K. C. Tang, J. X. Zheng, J. Zhou, R. Qin, Z. X. Gao, D. P. Yu, and J. Lu, Nano Lett. 12, 113 (2012). 3C.-C. Liu, W. Feng, and Y. Yao, Phys. Rev. Lett. 107, 076802 (2011). 4M. Ezawa, Phys. Rev. Lett. 109, 055502 (2012). 5J. Y. Zhang, B. Zhao, and Z. Q. Yang, Phys. Rev. B 88, 165422 (2013). 6M. Ezawa, Phys. Rev. B 87, 155415 (2013). 7X.-L. Zhang, L.-F. Liu, and W.-M. Liu, Sci. Rep. 3, 2908 (2013). 8H. Pan, Z. S. Li, C. C. Liu, G. B. Zhu, Z. H. Qiao, and Y. G. Yao, Phys. Rev. Lett. 112, 106802 (2014). 9P. Vogt, P. De Padova, C. Quaresima, J. Avila, E. Frantzeskakis, M. C. Asensio, A. Resta, B. Ealet, and G. Le Lay, Phys. Rev. Lett. 108, 155501 (2012). 10L. Meng, Y. Wang, L. Zhang, S. Du, R. Wu, L. Li, Y. Zhang, G. Li, H. Zhou, W. A. Hofer, and H.-J. Gao, Nano Lett. 13, 685 (2013). 11K. Behnia, Nat. Nanotechnol. 7, 488 (2012). 12D. Xiao, W. Yao, and Q. Niu, Phys. Rev. Lett. 99, 236809 (2007). 13M. Ezawa, Phys. Rev. B 86, 161407 (2012). 14W. Yao, D. Xiao, and Q. Niu, Phys. Rev. B 77, 235406 (2008). 15J. Ding, Z. H. Qiao, W. X. Feng, Y. G. Yao, and Q. Niu, Phys. Rev. B 84, 195444 (2011). 16M. Tahir, A. Manchon, K. Sabeeh, and U. Schwingenschlögl, Appl. Phys. Lett. 102, 162412 (2013). 17W. X. Feng, Y. G. Yao, W. G. Zhu, J. J. Zhou, W. Yao, and D. Xiao, Phys. Rev. B 86, 165108 (2012). 18K. F. Mak, K. L. McGill, J. Park, and P. L. McEuen, Science 344, 1489 (2014). 19A. Rycerz, J. Tworzydlo, and C. W. J. Beenakker, Nat. Phys. 3, 172 (2007). 20T. Fujita, M. B. A. Jalil, and S. G. Tan, Appl. Phys. Lett. 97, 043508 (2010). 21Y. Jiang, T. Low, K. Chang, M. I. Katsnelson, and F. Guinea, Phys. Rev. Lett. 110, 046601 (2013). 22H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, Nat. Nanotechnol. 7, 490 (2012). 23G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999). 24G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996). 25J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996). 26X. Lin and J. Ni, Phys. Rev. B 86, 075440 (2012). 27D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010). 28D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982). 29Y. G. Yao, L. Kleinman, A. H. MacDonald, J. Sinova, T. Jungwirth, D. S. Wang, E. Wang, and Q. Niu, Phys. Rev. Lett. 92, 037204 (2004). 30N. Marzari and D. Vanderbilt, Phys. Rev. B 56, 12847 (1997). 31I. Souza, N. Marzari, and D. Vanderbilt, Phys. Rev. B 65, 035109 (2001). 32A. A. Mostofi, J. R. Yates, Y.-S. Lee, I. Souza, D. Vanderbilt, and N. Marzari, Comput. Phys. Commun. 178, 685 (2008). 33X. J. Wang, J. R. Yates, I. Souza, and D. Vanderbilt, Phys. Rev. B 74, 195118 (2006). 34C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. C. Zhang, M. H. Guo, K. Li, Y. B. Ou, P. Wei, L.-L. Wang, Z.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. F. Jia, X. Dai, Z. Fang, S. C. Zhang, K. He, Y. Y. Wang, L. Lu, X.-C. Ma, and Q.-K. Xue, Science 340, 167 (2013). 35S. Murakami, S. Iso, Y. Avishai, M. Onoda, and N. Nagaosa, Phys. Rev. B 76, 205304 (2007). 36Y. Ding and Y. L. Wang, Appl. Phys. Lett. 103, 043114 (2013).
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.30.242.61 On: Sun, 31 May 2015 18:51:23
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Zhao et al.
37N. Gao, J. C. Li, and Q. Jiang, Phys. Chem. Chem. Phys. 16, 11673 (2014). 38H.
S. Liu, J. F. Gao, and J. J. Zhao, J. Phys. Chem. C 117, 10353 (2013). 39T. P. Kaloni, M. Tahir, and U. Schwingenschlögl, Sci. Rep. 3, 3192 (2013). 40J. J. Zhu and U. Schwingenschlögl, ACS Appl. Mater. Interfaces 6, 11675 (2014).
J. Chem. Phys. 141, 244701 (2014) 41H.
Jiang, Z. Qiao, H. Liu, J. Shi, and Q. Niu, Phys. Rev. Lett. 109, 116803 (2012). 42X.-F. Wang, Y. B. Hu, and H. Guo, Phys. Rev. B 85, 241402(R) (2012). 43M. P. L. Sancho, J. M. L. Sancho, J. M. L. Sancho, and J. Rubio, J. Phys. F: Met. Phys. 15, 851 (1985).
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.30.242.61 On: Sun, 31 May 2015 18:51:23
