Abstract
For potential applications in spintronics and quantum computing, it is desirable to place a quantum spin Hall insulator [i.e., a 2D topological insulator (TI)] on a substrate while maintaining a large energy gap. Here, we demonstrate a unique approach to create the largegap 2D TI state on a semiconductor surface, based on firstprinciples calculations and effective Hamiltonian analysis. We show that when heavy elements with strong spin orbit coupling (SOC) such as Bi and Pb atoms are deposited on a patterned HSi(111) surface into a hexagonal lattice, they exhibit a 2D TI state with a large energy gap of ≥0.5 eV. The TI state arises from an intriguing substrate orbital filtering effect that selects a suitable orbital composition around the Fermi level, so that the system can be matched onto a fourband effective model Hamiltonian. Furthermore, it is found that within this model, the SOC gap does not increase monotonically with the increasing strength of SOC. These interesting results may shed new light in future design and fabrication of largegap topological quantum states.
Introduction
Recently there has been a surge in the investigation of topological insulators (TIs)^{1,2,3}. TIs are characterized by topologically protected metallic surface or edge states with helical spin polarization residing inside an insulating bulk gap. These states have negligible elastic scattering and Anderson localization^{4,5}, which may provide ideal dissipationless spin current for future electronic devices with low power consumption. To realize their potential applications, it is desirable for the TIs to have an energy gap as large as possible^{6}, i.e., for room temperature applications. As for 2D TIs [i.e., quantum spin Hall (QSH) insulators], they also need to be grown or placed onto a substrate^{7,8,9,10} or formed as an interface^{12,13}, while maintaining a large gap. One desired approach is to directly fabricate largegap TIs on semiconductor surfaces, which may avoid problems like transfer or interfacing a 2D layer over a foreign substrate^{11}. So far, however, this goal remains allusive.
The HgTe quantum well, as the first theoretically predicted^{12} and experimentally confirmed^{13} QSH insulator, has a small gap of 40 meV with topological edge states only detectable at very low temperature (<10 K)^{13}. Recent studies pertaining to Bi/Sb(111) films^{17,14,15,16}, Sn films^{18}, metaldecorated graphene^{21,19,20}, silicene/germanene^{22} and 2D organometallic frameworks^{26,23,24,25} have largely enriched the family of 2D TIs and some of them have a large gap^{14,18,21}. However, a critical drawback with most previous theoretical studies of 2D TIs is their reliance on the electronic and topological properties of freestanding films, whose existence can be in doubt. Even a freestanding film does exist, its properties are expected to be influenced by the underlying substrate in real applications^{8,9,10}.
Here, we demonstrate a unique approach of creating QSH state on a conventional semiconductor surface via depositing heavy elements with strong spin orbit coupling (SOC) onto a patterned Si(111) surface into a hexagonal lattice, which exhibit TI state with a large energy gap of ~0.5 eV. Here, the substrate plays a ‘positive’ role acting as an orbital filter to critically select the orbital composition around the Fermi level to realize nontrivial largegap topological state^{27}. Specifically, the surface system can be matched onto an effective fourband model Hamiltonian which captures the underlying physics. We depict a unified picture of energy gap as a function of SOC to achieve largegap QSH state. Importantly, we found that it is not necessarily true to have a large gap with a heavier atom of larger SOC, a noteworthy point for future design of TIs.
We have performed density functional theory (DFT) based firstprinciples calculations of band structure and band topology of 2D hexagonal lattices of various metal atoms, including Bi, Pb, Sb, Sn, Ga, In and Tl, grown on a patterned Hsaturated Si(111) surface. The detailed methodologies are presented in the Supplementary Information. We will first discuss in detail the results of Bi and Pb, as representative examples and leave the results of other metals for later discussion. Atomically flat HSi(111) surface has been prepared for decades and is a widelyused substrate for epitaxial growth of ordered overlayers^{28,29}. The surface dangling bonds are passivated by H to avoid surface reconstruction. In order to form a hexagonal metal overlayer lattice, we propose a twostep fabrication process, as shown in Fig. 1. First, to create a desirable surface template pattern for metal growth, H atoms are selectively removed in hexagonal symmetry using scanning tunneling microscopy, as discussed in Refs. 30, 31; Second, heavy metal atoms with large SOC can be deposited to grow or selfassemble into the exposed Si sites, as already demonstrated for other systems^{32,33,34}.
We found a very strong binding between the deposited metal atoms and the exposed Si atoms in the HSi(111) surface, as evidenced by the calculated adsorption length [d, see Fig. 1(c)] of 2.68 Å and 2.75 Å for Bi and Pb, respectively. The high structural stability is also indicated by a large adsorption energy (E_{ad}), defined as E_{ad} = E_{M}_{@H−Si(111)} − (2E_{M} + E_{H−Si}_{(111)}) + E_{H}_{2}, where E_{M}_{@H−Si(111)}, E_{M}, E_{H−Si}_{(111)} and E_{H}_{2} denote the energy of Bi/Pb@HSi(111), single metal atom, pristine HSi(111) surface and H_{2} gas molecule, respectively. The adsorption energy is found to be 3.88 eV and 3.92 eV for Bi and Pb, respectively, which are much larger than the binding energies of bulk Bi (2.18 eV) and Pb (2.03 eV) in the crystalline solid form, indicating high thermodynamic stability of the surface systems.
To examine the band topology of such surface structures, we first purposely exclude SOC from calculation. The resulting band structures for Bi and Pb@HSi(111) are shown in Figs. 2 (a–b), respectively. For Bi@HSi(111), there are two Dirac bands residing inside the bulk gap of Si with a Dirac point at K point, which locates nearby the Fermi level [Fig. 2(a)]. Analysis of band composition further showed that the two Dirac bands consist mainly of p_{x} orbitals of Bi atoms. Another dispersive band, consisting of p_{y} orbitals of Bi, sits below the bulk conduction band edge of Si and touches the upper Dirac band of p_{x} orbitals at Γ point. The band structure of Pb@HSi(111) is similar to the case of Bi, represented by two Dirac bands inside the Si gap; but the Dirac point sits ~0.8 eV above Fermi level and the upper Dirac band largely overlaps with the conduction band of Si [Fig. 2(b)]. Such different behaviors originate from the different valance electron configuration of Pb and Bi, e.g., [Xe].4f^{14}.5d^{10}.6s^{2}.6p^{2} for Pb and [Xe].4f^{14}.5d^{10}.6s^{2}.6p^{3} for Bi. With two electrons less in Pb@HSi(111) per unit cell, the lower Dirac band becomes almost unoccupied compared to that of Bi@HSi(111).
Next, we include SOC in calculation and the resulting band structures of Bi and Pb@HSi(111) are shown in Figs. 2(c) and (d), respectively. One sees that for Bi@HSi(111), two Dirac bands are split apart; one large energy gap of ~0.7 eV opens at K point and another gap of ~0.5 eV opens at Γ point, which is the global gap. The p_{y} and the upper p_{x} bands are also separated by SOC, with an indirect gap ~ 0.45 eV. We note that the spin degeneracy of these bands is lifted with most noticeable splitting at K point, which is due to the Rashba effect^{35} induced by the broken spatial inversion symmetry. Although Rashba effect has shown to be detrimental to QSH phase^{19}, in our systems such extrinsic spin splitting is relatively small compared to the intrinsic SOC induced band gap [see Fig. 2(c)], suggesting the QSH state is robust against Rashba effect in our surface systems. It should also be noted that the SOC strength of Si is orders of magnitude smaller that Bi, making the SOC of Bi a decisive factor in opening the energy gap. Similarly, the SOC opens a gap at K point for Pb@HSi(111), with the upper branch of Dirac bands moving completely into the Si conduction band, as shown in Fig. 2(d). Nevertheless, the energy splitting between the two Dirac bands is found to be around 0.65 eV at K point and an effective SOC gap of ~0.54 eV could be counted by the energy difference between the Si conduction band minimum and the top of the Pb p_{x} band. However, to truly make the Pb@HSi(111) a 2D TI, ntype doping is needed to shift the Fermi level into the SOC gap, such as by Si substrate doping or electric gating.
The SOCinduced gap opening at the Dirac point in Bi@HSi(111) and Pb@HSi(111) indicates possible existence of 2D TI state. To check this, we calculated the topological edge states of Bi@HSi(111) by the Wannier90 package^{36}. Using DFT bands as input, we construct the maximally localized Wannier functions and fit a tightbinding Hamiltonian with these functions. Figure 3(a) shows the DFT and fitted band structures, which are in very good agreement. Then, the edge Green's function of a semiinfinite Bi@HSi(111) is constructed and the local density of state (DOS) of Bi zigzag edge is calculated, as shown in Fig. 3(b). Clearly, one sees gapless edge states that connect the upper and lower band edge of the bulk gap, forming a 1D Dirac cone at the center of Brillouin zone (Γ point). This indicates that the Bi@HSi(111) is a 2D TI with a large gap of ~0.5 eV.
To further confirm the above topological edgestate results, we also calculated Z_{2} topology number. As the spatial inversion symmetry is broken in these systems, we used the method developed by Xiao et al.^{22,37}. In this method, Z_{2} is calculated by considering the Berry gauge potential and Berry curvature associated with the Bloch wave functions, which does not require any specific pointgroup symmetry. Indeed, we found that Z_{2} = 1 for Bi@HSi(111) (Fig. S1 in Supplementary Information), confirming the existence of QSH state in this surface. Assuming a shift of Fermi level above the lower branch of Dirac band, we also found Z_{2} = 1 for Pb@HSi(111) (Fig. S2).
The physical origin of QSH state in Bi@HSi(111) and Pb@HSi(111) can be understood by a substrate orbital filtering effect as discussed recently in a related system of Bi on halogenated Si surfaces^{27}. Figure 3(c) shows the partial density of states (DOS) of Bi@HSi(111) around the Fermi level. It is seen that the p_{z} orbital of Bi hybridizes strongly with the dangling bond of the exposed surface Si atom overlapping in the same energy range, which effectively removes the p_{z} bands away from the Fermi level, leaving only the p_{x} and p_{y} orbitals. We also analyzed the charge density redistribution [see upper panel of Fig. 3(d)], which clearly shows that charge redistribution induced by Si surface mainly happens to the p_{z} orbital of Bi, in a similar way to the saturation of Bi p_{z} orbital by using hydrogen [lower panel of Fig. 3(d)]. It has been shown that the freestanding planar hexagonal lattice of Bi is a topologically trivial insulator with Z_{2} = 0 (see Fig. S3). When it is placed onto the HSi(111) surface or adsorbed with H, it becomes topologically nontrivial (Figs. S1 and S4). This originates from the intriguing orbital filtering effect imposed by the substrate or H saturation, which selectively remove the p_{z} orbitals from the Bi lattice to realize the largegap QSH phase.
Specifically, we can describe the Bi@HSi(111) using a simplified (p_{x}, p_{y}) fourband model Hamiltonian in a hexagonal lattice as^{24,38},
in which , , , a_{1}, a_{2} is the lattice vector, V_{ppσ} (V_{ppπ}) is the SlaterKoster parameter^{39} and σ_{Z} = ±1 is the spin eigenvalue.
Diagonalization of Eq. (1) in reciprocal space gives the band structures shown in Fig. 4, which shows typical four bands as a function of SOC strength. One sees that without SOC, this Hamiltonian produces two flat bands and two Dirac bands with a Dirac point formed at K point and two quadratic points at Γ point [Fig. 4(a)]. Inclusion of a small SOC (λ = 0.2t) opens one energy gap (ΔE_{1}) at K point and two energy gaps (ΔE_{2}) at Γ point [Fig. 4(b)], with both gaps topologically nontrivial^{24}. With the increasing SOC strength, both ΔE_{1} and ΔE_{2} increase [Fig. 4(c)], which eventually leads to the formation of a different energy gap (ΔE_{3}) between the upper and lower Dirac bands at Γ point when ΔE_{3} becomes smaller than both ΔE_{1} and ΔE_{2} [Fig. 4(d)]. As such, for sufficiently large SOC, ΔE_{3} replaces ΔE_{1} to be the global gap and correspondingly the global gap shifts from K to Γ point. Further increase of SOC will tend to decrease ΔE_{3}, indicating that for sufficiently large SOC the band gap decreases with increasing SOC.
Such interesting phenomenon has also been confirmed by the DFT results. By comparing Bi@HSi(111) and Pb@HSi(111), we see that given the correct Fermi energy, the global gap is located at Γ point for Bi@HSi(111) [Fig. 2(c)], but at K point for Pb@HSi(111) [Fig. 2(d)]. This is because the SOC strength in p orbital of Pb (~0.91 eV) is smaller than that of Bi (~1.25 eV)^{40}. Meanwhile, the energy gap between the two p_{y} Dirac bands induced by SOC is actually larger for Pb@HSi(111) (0.65 eV) than that of Bi@HSi(111) (0.5 eV), suggesting that Pb may be a better choice to achieve largegap QSH states on the substrate. This is in sharp contrast with the KaneMele model in graphene, for which an energy gap is opened at Dirac point that is in proportion to the strength of SOC^{19}.
Besides Bi and Pb, we have also conducted calculations of other heavy elements adsorption on the Si surface, including Sb, Sn, Tl, In and Ga. It is found that Sb and Sn have a similar band structure with Bi and Sn, respectively (see Fig. S5 in Supplementary Information), but with a smaller energy gap resulting from their similar valence electron configurations but weaker SOC. Band structures of Tl, In and Ga@HSi(111) are a bit different. As shown in Fig. 5, the Fermi level now sits further below the lower dispersive band that is mainly made of the heavy atom p_{y} orbital. This is due to the one (two) less valence electron compared to the Pb (Bi) group, i.e., [Xe].4f^{14}.5d^{10}.6s^{2}.6p^{1} for Tl. Clearly, one sees that from Ga to Tl, the SOC gap between the lower Dirac band and dispersive band increases dramatically, from around 0.1 eV (for Ga) to 0.5 eV for (Tl), confirming the dependence of energy gap (ΔE_{2}) on SOC as demonstrated in Fig. 4.
In summary, we demonstrate the possibility of ‘controlled’ growth of largegap topological quantum phases on conventional substrate surfaces such as the important Si surface by a unique approach of substrate orbital filtering process combined with a proper choice of SOC. Its underlying physical principles are general, applicable to deposition of different metal atoms on different substrates^{11,27}. It opens up a new and exciting avenue for future design and fabrication of room temperature topological surface/interface states based on current available epitaxial growth and semiconductor technology, where the metal overlayer is atomically bonded but electronically isolated from the underneath semiconductor substrate^{27}.
Methods
Our electronic structure calculations based on density functional theory were performed by using a plane wave basis set^{41} and the projectoraugmented wave method^{42}, as implemented in the VASP code^{43}. The exchangecorrelation functional was treated with the generalized gradient approximation in PerdewBurkeErnzerhof format^{44}. Calculations of Z_{2} triviality were carried out by using the fullpotential linearized augmented planewave method implemented in the WIEN2K package^{45}. Details for models and computations (Z2 invariant calculation results and band structures of Sb and Sn@HSi(111)) are presented in Supplementary Information.
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Acknowledgements
This research was supported by DOE (Grant No: DEFG0204ER46148); Z. F. Wang and W. Ming additionally thank support from NSFMRSEC (Grant No. DMR1121252). We thank NERSC and the CHPC at University of Utah for providing the computing resources.
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M.Z. carried out the theoretical calculations with the assistance of W.M.M., Z.L., Z.F.W. and Y.G.Y.; F.L. guided the overall project. M.Z. and F.L. wrote the manuscript.
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Zhou, M., Ming, W., Liu, Z. et al. Formation of quantum spin Hall state on Si surface and energy gap scaling with strength of spin orbit coupling. Sci Rep 4, 7102 (2014). https://doi.org/10.1038/srep07102
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