Hydrogenated group-IV binary monolayers: a new family of inversion-asymmetric topological insulators

Abstract

Band topology and Rashba spin splitting (RSS) are two extensively explored exotic properties in condensed matter physics. However, the coexistence has rarely been reported in simplest stoichiometric films so far. Here, by using first-principles calculations, we demonstrate that a series of inversion-asymmetric group-IV XYH₂ monolayers (X, Y = Si, Ge, Sn, Pb) allow for the simultaneous presence of topological order and large RSS that derives from their peculiarly atomic structure. The topological bulk gaps and RSS energies of PbSnH₂, PbGeH₂, and PbSiH₂ are tunable over a wide range of strains (−8 to 8%), even the maximum value can be enhanced to 0.68 eV and 0.24 eV under achievable strain, but another three configurations transform from trivial to nontrivial phases under tensile strain. Furthermore, we find that the Te(111)-terminated BaTe surface is an ideal substrate for the growth of these monolayers, without destroying their intrinsic band topology. Our findings provide a possible route to future applications of inversion-asymmetric topological insulators in spintronic devices.

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The discovery of topological insulators (TIs) has drawn intense attention due to the intriguing properties of this new type of quantum matter.^1–3 These novel materials are insulating in the bulk but support gapless edge states for two-dimensional (2D) TIs and gapless surface states for three-dimensional (3D) TIs, which are protected by time reversal invariance. Although a number of 3D TIs have been synthesized experimentally, the 2D TIs, also known as quantum spin hall (QSH) insulators, are better suited for applications in spintronics and quantum computing in view of the remarkable robustness of their edge states against backscattering. The experimental demonstration of the existence of topological spin transport channels, however, is only limited to HgTe/CdTe^4,5 and InAs/GaSb^6,7 quantum wells with band gap too small for room-temperature applications. Therefore, the search for new classes of 2D TIs with sizable band gaps has become a challenge of urgent importance.

Currently, intensive efforts have been devoted to explore 2D groups-IV and V films, including carbon,⁸ silicon,⁹ germanium,¹⁰ tin¹¹ and bismuth,¹² which can harbor 2D topological nontrivial phase in the honeycomb structure. More importantly, following the experimental advancement of graphene,¹³ chemical functionalization on honeycomb structure has been considered as an efficient way to synthesis suitable 2D nanomaterials. In this respect, the effects of hydrogenation and halogenation on electronic properties of a variety of 2D thin films have been explored.^14–20 Recently, the organic molecule functionalization is found to be an effective way to stabilize 2D films and then enhance nontrivial bulk gaps in group-IV films.^21–25 All make these 2D TI materials very promising for future device applications.

On the other hand, the Rashba spin splitting (RSS), which originates from the inversion-symmetry breaking along the perpendicular direction to 2D film, leads to spin-polarized band dispersion curves with in-plane opposite helical spin texture,²⁶ allowing the control of spin direction through an electric field. Compared with 2D inversion-symmetric (IS) TIs,^14–25 these systems are called inversion-asymmetric TIs (IASTIs), which may be more promising due to their perfect performance in realizing new topological phenomena, such as crystalline-surface dependent topological electronic states,²⁷ pyroelectricity,²⁸ natural topological p–n junctions,²⁹ and so on. All these characters lead to a great potential of IASTIs in device paradigms for spintronics and quantum information processing. Recently, some excellent works has been reported, e.g. functionalized III-Bi bilayers³³ and functionalized GeSn.³⁴ Additionally, concerning the group-IV element based 2D TIs, functionalized Ge³⁵ and functionalized Bi and Pb³⁶ has been realized, indicating their potentials in practical applications. Despite the importance of IASTIs, up to now, few materials have been found to have large gap and large RSS.^30–36 which gives rise to the fundamental questions: is it possible to realize the IASTIs in hexagonal group-IV films with a tunable RSS, and if so, could they be observed at room temperature?

In this work, we report on a series of 2D IASTIs in hydrogenated group-IV binary films, named as XYH₂ monolayers, where A and B are group-IV elements (Si, Ge, Sn, Pb), as illustrated in Fig. 1(a). XYH₂ monolayer has the simplest stoichiometric layered structure, which is distinctive from convention IASTIs such as BiTeI film. Together with the topologically nontrivial feature, we demonstrate that these IASTIs can produce remarkable RSS as large as 0.17 eV, deriving from inversion-asymmetry induced strong polar field. The nontrivial band gap and RSS in PbSnH₂, PbGeH₂, PbSiH₂ are effectively modulated over a wide range of strains (−8 to 8%), even with the maximum band gap being improved to 0.68 and 0.24 eV under achievable stain. It is known that the experimental realization of compressed lattice of 2D monolayer directly is challengeable, while it is possible to grow these 2D monolayers on the appropriate substrates with smaller lattice constant by employing molecular beam epitaxial (MBE) method. And the main goal here is to predict the influence of compressive strain to QSH effect. We believe our findings provide a promising platform for development of IASTIs and may enable topological quantum computing in spintronics.

Fig. 1 Top view and side view of 2D PbSnH₂ monolayer (a), phonon band dispersions of (b) denote the PbSnH₂ film. (c) Total energies of PbSnH₂, PbGeH₂, PbSiH₂, SnGeH₂, SnSiH₂ and GeSiH₂ as a function of lattice constant, respectively. (d) 2D Brillouin zones with specific symmetry points.

All calculations were carried out using the plane wave basis Vienna ab initio simulation pack (VASP) code^37,38 implementing density functional theory (DFT). The projector-augmented wave (PAW) method³⁹ was used to describe electron-ion potential. The exchange-correlation potential was approximated by generalized gradient approximation (GGA) in Perdew–Burke–Ernzerhof (PBE) form.⁴⁰ Considering the possible underestimation of GGA method, the Heyd–Scuseria–Ernzerhof (HSE06) hybrid functional was employed to check the band topology.⁴¹ We used an energy cutoff of 500 eV and maximum residual force less than 0.001 eV Å⁻¹. Periodic boundary conditions are employed to simulate these 2D systems, and the Brillouin zone was sampled by using a 17 × 17 × 1 gamma-centered Monkhorst–Pack grid. Moreover, SOC effect was included in self-consistent electronic structure calculations. The phonon spectra were calculated using a super cell approach within the PHONON code.⁴²

To identify the nontrivially topological character of these films, we introduced the evolution of Wannier Center of Charges (WCCs)⁴³ to calculate the Z₂ invariant, in which the Wannier functions (WFs) related with lattice vector R can be written as:

Here, a WCC x̄_n can be defined as the mean value of 〈0n|X̂|0n〉, where the X̂ is the position operator and |0n〉 is the state corresponding to a WF in the cell with R = 0. Then we can obtained

Assuming that with S = I or II, where summation in α represents the occupied states and A is the Berry connection. So we have the format of Z₂ invariant:

The Z₂ invariant can be obtained by counting the even or odd number of crossings of any arbitrary horizontal reference line.

Fig. 1(a) displays the buckled honeycomb structure of XYH₂ (A, B = Si, Ge, Sn, Pb) with two group-IV and two hydrogen atoms, accompanied with its Brillouin zone in Fig. 1(d). It is iso-structural to stannane with C_3v symmetry (P3m1, space group no. 156) with group-IV atoms sp³ hybridized. The total energy per unit cell is plotted in Fig. 1 (c) as a function of lattice constant a, where a is varied to identify structural ground state. We find a double-well energy curve in all cases, with two local ground states named as high-buckled (HB) and low-buckled (LB) states, respectively. In the HB state, the vertical layer distance is in the range of 3.216–4.076 Å, while the distance in LB state is within 0.685–0.800 Å. Such double-well energy curve is also found in III–V films.⁴⁴ The LB structure here, however, is more stable by at least 1.12 eV per unit cell than HB structure, as compared with those of III-Bi films.⁴⁴ The high structural stability can be verified by the formation energy expressed as

ΔE_f = E(XYH₂) − E(XY) − E(H₂)

where E(XYH₂) and E(XY) are, respectively, the total energies of hydrogenated and pristine AB films, while E(H₂) is the chemical potential of hydrogen atoms. The results are found to be in the range of −3.04 to −5.46 eV per atom, indicating no phase separation in these systems. As further evidence, we also calculate phonon dispersion curves of all films, and no imaginary phonon modes are observable, as shown in Fig. 1(b), confirming their dynamical stability. Detailed structure parameters of these six monolayer structures are shown in Table 1.

Table 1 Calculated equilibrium lattice parameters a (Å), the buckled height h (Å), band gaps E_g (eV), band gaps with SOC E_g-SOC (eV), the band gap located at Γ point E_Γ (eV), gap at Γ with SOC E_Γ-SOC (eV) and topological invariants (Z₂) for six films

Structure	a (Å)	h (Å)	Eg (eV)	EΓ (eV)	Eg-SOC (eV)	EΓ-SOC (eV)	Z2
PbSnH2	4.89	0.80	0	0	0.500	0.691	1
PbGeH2	4.60	0.75	0	0	0.338	0.507	1
PbSiH2	4.49	0.76	0	0	0.103	0.103	1
SnGeH2	4.46	0.76	0.463	0.463	0.327	0.327	0
SnSiH2	4.38	0.75	1.074	1.074	0.981	0.981	0
GeSiH2	4.08	0.68	1.386	1.386	1.334	1.334	0

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Now, we check the electronic band structure of XYH₂ films with respect to external strain, which indicates interesting topological phase transitions. For convenience, the band inversion strength (BIS) here, which is defined as energy difference of s and p_xy at Γ point, provides a measure of that how far these systems are from a topological critical point. BIS is positive when the band is inverted and negative otherwise. Fig. 2(a) gives BIS values as a function of strain, in which BIS increases with the increase of atomic number. One can see that SnGeH₂, SnSiH₂ and GeSiH₂ films are all in trivial phases at zero strain but transform into a nontrivial phase under a strain of 4, 8, and 10%, respectively, while for PbSnH₂, PbGeH₂ and PbSiH₂ films, they are in nontrivial phases at equilibrium state. By comparing with each other in the latter cases, however, PbSiH₂ transforms into a trivial phase under negative strain of −2%, whereas both PbSnH₂ and PbGeH₂ are nontrivially topological over a wide range of strains. To confirm the TI character, we display the evolution lines of WCCs for PbSnH₂ as an example in Fig. 4(a). As expected, the WCCs evolution curves cross any arbitrary reference lines odd times, yielding Z₂ = 1.

Fig. 2 (a) Variation of band inversion strength (BIS) for all systems and (b) nontrivial band gap and RSS for PbSnH₂, with respect to external strain.

To demonstrate the character of inverted electronic bands, we highlight the band structures for PbSnH₂, Sn₂H₂ and Pb₂H₂ films as representative examples, these being three exemplar cases in which both hydrogenation and SOC strength play a different role in TI nature. As illustrated in Fig. 3, the s-orbital of H atom is strongly hybridized with p_z state of group-IV atoms, resulting in downshift and up shift of unsaturated p_z orbitals near the Fermi level, so that the states near the Fermi level become dominated by the s and p orbitals of group-IV atoms. Accordingly, the TI character is determined by s–p band inversion from group-IV atoms. By comparing Fig. 3(a) and (b) with Fig. 3(c) and (d), which possess inversion symmetry in honeycomb structure, their bands are all degenerated even with SOC. The band topology in Sn₂H₂ is trivial with Z₂ = 0, while Pb₂H₂ becomes nontrivial with Z₂ = 1.¹⁶ For PbSnH₂, however, whose inversion symmetry is broken due to the hybridization/non-equivalence of Sn and Pb atoms, the resulting bands in Fig. 3(e) and (f) are different significantly, exhibiting RSS near Γ point which derives from their peculiarly polar atomic structure, as illustrated in Fig. 4(e). In addition, the band topologies are robust over a wide range of strains, due to the stronger SOC from Pb atoms. Noticeably, PbSnH₂ has a large RSS of 0.24 eV and supports a band gap as large as 0.68 eV under achievable strain, as shown in Fig. 2(b), which exceeds the thermal energy at room temperature.

Fig. 3 Orbitals-resolved band structures with and without SOC for (a) and (b) Sn₂H₂, (c) and (d) Pb₂H₂, as well as (e) and (f) PbSnH₂ monolayer.

Fig. 4 (a) Evolutions of Wannier centers along k_y. The evolution lines cross the arbitrary reference line (red dash line) parallel to k_y odd times, yielding Z₂ = 1. Total (left panel) and spin (right panel) edge density of states for (b) and (d) PbSnH₂. In the spin edge plot, blue/red lines denote the spin up/down polarization. (c) Enlarger view of the bands around Fermi level at the Γ point for the PbSnH₂ structure. (e) Real-space charge distribution for PbSnH₂. (f) Spin texture in the highest valence bands for PbSnH₂ monolayer. Arrows refer to the in-plane orientation of spin, and the color background denotes the z component of the spin.

One essential characteristic of 2D IASTIs is the existence of gapless edge states protected by TRS. To demonstrate it explicitly, in terms of maximally localized Wannier functions (MLWFs), we calculate the edge Green's function of the semi-infinite lattice using the recursive method.⁴⁵ Fig. 4(b) displays local density of state (LDOS) of the edges. All the edge bands connect completely the conduction and valence bands and span 2D bulk band gap, yielding a 1D gapless edge states. By identifying the spin-up and spin-down contributions in edge spectral function, as illustrated in Fig. 4(d), the counter-propagating edge states exhibit opposite spin-polarization, which benefits from their robustness against nonmagnetic scattering.

Another prominent character in PbSnH₂ is the existence of RSS in the edge of valence bands. It is known that RSS⁴⁶ is driven by the inversion symmetry breaking via SOC, which leads to a spin splitting in the electronic band dispersion, such as ε_± (k) = ℏ²k²/2m* ± α_R|k| with k = (k_x, k_y), where the m* is the effective mass and α_R = eη_soE is the Rashba constant. The spins of the electronic states ε_± (k) are oppositely aligned within the k_x, k_y plane, and are normal to the wave vector k. As can be seen from Fig. 4(f), the most dramatic effect resulting from SOC is the splitting of the VBM along momentum direction at Γ point, which is indicated by the rectangular box in Fig. 4(c). The size of the splitting is given by the momentum offset Δk_R away from the crossing point and the corresponding energy offset E_R, being estimate to be 0.069 Å⁻¹ and 0.24 eV, respectively, which are similar with the result in InGaAs/GaAs quantum dot (0.08–0.12 eV Å⁻¹),⁴⁷ or InGaAs/InAlAs quantum well (0.07 eV Å⁻¹).⁴⁸ The RSS of electronic bands obtained here provides a chance for spintronic device applications without the magnetic field, for instance, spin field-effect transistor (FET).⁴⁹

On the experimental side, choosing the suitable substrate is a key factor in device application, since a free-standing film must be eventually deposited or grown on a substrate. It is known that the topological nontrivial characters in pristine graphene and germanene^9–12 are easily destroyed by substrates. In contrast, although the TI phase of PbSnH₂ appears in free-standing structure, its nontrivial TI character would be robust when they are on specific substrates. To check these, we select the Te-terminated (111) surface of semiconductor BaTe⁵⁰ as a representative example, as illustrated in Fig. 5. In the equilibrium structure, the bottom (top) Sn atoms in PbSnH₂ are located at top (hcp) sites of the substrate, while the top Pb atoms are located at hcp sites, see Fig. 5(a) and (b). On this Te-terminated surface, Sn atoms bind preferably on top of Te atoms by forming chemical bonds, so that p_z orbital of group-IV atoms and dangling bond of the substrate both get fully saturated. Fig. 5(c) and (d) present the corresponding band structures with and without SOC. One can see that a few bands appear within the bulk gap of substrate around the Fermi level, which are mostly contributed by Sn atoms according to orbital analysis, and thus a type-I energy level alignment is formed between the substrate and PbSnH₂. By projecting Bloch wave functions onto atomic orbitals of PnSnH₂, Bloch states contributed by the s orbital of group-IV atoms are visualized in Fig. 5(c) and (d) by red dots. These states stay above the Fermi level, but shift to valence bands around Γ point, suggesting a clear s–p band order. This SOC-induced band gap is topological nontrivial as explained by the band inversion, thus the supported PnSnH₂ is a robust IASTI on Te-terminated substrate.

Fig. 5 Top view (a) and side view (b) of the schematic illustration of epitaxial growth PbSnH₂ on Te(111)-terminated BaTe surface, as well as the band structure without SOC (c) and with SOC (d) for PbSnH₂.

In summary, we propose a series of large-gap IASTIs in hydrogenated group-IV binary monolayers with a large RSS of 0.17 eV, which derives from their peculiarly polar atomic structure. The topological nontrivial gap and RSS of PbSnH₂, PbGeH₂, and PbSiH₂ are effectively modulated over a wide range of strain (−8 to 8%), even the maximum band gap being enhanced to 0.68 and 0.24 eV under pressure, while the other three transform from trivial to nontrivial phases with appropriate strain engineering. More importantly, these thin films remain topological nontrivial when growing on Te(111) surface of semiconductor BaTe, which makes it suitable for epitaxial growth of these monolayers and potential application in spintronics.

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11274143, 11434006, 61172028, and 11304121), Natural Science Foundation of Shandong Province (Grant No. ZR2013AL004, ZR2013AL002), Technological Development Program in Shandong Province Education Department (Grant No. J14LJ03), Research Fund for the Doctoral Program of University of Jinan (Grant No. XBS1402, XBS1452).

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6ra14140d

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