Robust Tunable Large-Gap Quantum Spin Hall States in Monolayer Cu2S on Insulating Substrates.

Quantum spin Hall (QSH) insulators with large band gaps and dissipationless edge states are of both technological and scientific interest. Although numerous two-dimensional (2D) systems have been predicted to host the QSH phase, very few of them harbor large band gaps and retain their nontrivial band topology when they are deposited on substrates. Here, based on a first-principles analysis with hybrid functional calculations, we investigated the electronic and topological properties of inversion-asymmetric monolayer copper sulfide (Cu2S). Interestingly, we found that monolayer Cu2S possesses an intrinsic QSH phase, Rashba spin splitting, and a large band gap of 220 meV that is suitable for room-temperature applications. Most importantly, we constructed heterostructures of a Cu2S film on PtTe2, h-BN, and Cu(111) substrates and found that the topological properties remain preserved upon an interface with these substrates. Our findings suggest Cu2S as a possible platform to realize inversion-asymmetric QSH insulators with potential applications in low-dissipation electronic devices.

Introduction

The seminal discovery of two-dimensional (2D) topological insulators (TIs), also known as quantum spin Hall (QSH) insulators, is the beginning of an era of intensive theoretical and experimental studies on several topological quantum materials. The nontrivial band topology of TIs causes gapless conducting states with an odd number of Dirac cones at the edges of 2D TIs, while the bulk remains insulating.[1−3] The gapless edge states are protected by the constraints of time-reversal symmetry and exhibit linear energy dispersion around the Dirac point. Moreover, these topologically protected edge states are quite robust against surface defects, environmental perturbations, and the backscattering of charges. The edge states are highly spin polarized due to the strong spin–orbit coupling effect that is commonly found in TIs,[4−7] thereby making TIs candidates of significant importance for new applications in low-power-consuming electronic devices.

In comparison to their 3D counterparts, 2D TIs offer more advantages in terms of application due to their flexibility, easier integration into existing electronic devices, and robustness against backscattering.[5,8,9] Up to now over hundreds of 2D materials have been predicted to be QSH insulators,[10−12] including the graphene-like materials germanene,[13,14] silicene,[13,14] stanene,[15] and III–V buckled honeycombs,[16,17] Janus materials,[18] and several transition-metal dichalcogenides in 1T and 1T′ phases.[19−22] In addition, a QSH to quantum anomalous Hall insulator phase transition has been demonstrated in GaBi by doping.[23] However, the QSH effect has been experimentally verified in only a few 2D materials.[24] The majority of the predicted 2D TIs are pristine or are functionalized in freestanding form with small band gaps. However, 2D materials need to be transferred on a substrate in practical applications and later be integrated into the semiconductor industry. Oftentimes, the interaction between 2D TIs and substrates can affect the crystal structure and electronic properties of freestanding films, resulting in the destruction of the QSH state. Most of the studies on 2D TIs on substrates are constrained to quantum well systems that possess small band gaps and exhibit 2D TI states at unrealistically low temperatures.[24−38] Therefore, it remains a challenge to find new 2D TIs that can keep their QSH states and band gap on substrates at room temperature, and thus more theoretical studies on material selections are needed regarding the interaction between 2D TIs and substrates.

Copper sulfide, Cu2S, which is known as a liquidlike thermoelectric material,[39] is gaining considerable attention and has been intensely studied in recent years due to its high thermoelectric figure of merit and promising optoelectronic properties.[40] Cu2S is an earth-abundant, low-cost, and nontoxic p-type semiconductor with a narrow band gap.[41] Recently, a β-Cu2S monolayer[42] has been synthesized and found to be stable at room temperature.[43] However, the topological properties of Cu2S have not yet been fully explored.

Here, on the basis of first-principles calculations, we investigated the electronic and topological properties of the recently synthesized β-Cu2S monolayer. The monolayer Cu2S possesses a buckled structure where the buckling height can be tuned by tensile strain. The freestanding Cu2S film exhibits the QSH phase with a typical band inversion of the s orbital of Cu and the p orbital of S, and strong SOC leads to a large band gap of 220 meV. Our band calculations reveal the coexistence of Rashba-type spin splitting in both the valence and conduction bands of monolayer Cu2S due to its inversion-asymmetric structure. The QSH phase in Cu2S has been further confirmed by calculations of a Z2 topological invariant and topologically protected edge states. We further investigate the electronic and topological properties of a Cu2S film in contact with three different substrates to facilitate practical applications. We construct Cu2S/PtTe2, Cu2S/h-BN, and Cu2S/Cu heterostructures, and the topological properties are found to be preserved with all substrates. Our findings expand the 2D TI family on earth-abundant compounds (Cu2S) that have great applications in low-power and nontoxic electronic devices.

Computational Methods

First-principles calculations based on density functional theory (DFT)[44] were performed under the Perdew–Burke–Ernzerhof (PBE) generalized gradient approximation (GGA)[45] and projector augmented-wave method[46] using the Vienna ab Initio Simulation Package (VASP).[47,48] The nonlocal optB86b density functional was used to account for the van der Waals interactions between the films and the substrates.[49,50] The kinetic cutoff energy was set to 400 eV for freestanding Cu2S and 500 eV for Cu2S on substrates. The relaxation of all studied systems was conducted in the absence of SOC until the residual forces per atom were less than 10–3 eV/Å. The self-consistent electronic convergence criterion for electronic structure calculations was set to 10–6 eV. To avoid interactions between layers due to periodicity, a vacuum space of 15 Å was added. The hybrid functional HSE06[51] was used to overcome the underestimation of band gaps under the GGA functional and to accurately predict the topological phases of the Cu2S monolayer. The 2D Brillouin zone was sampled using γ-centered 21 × 21 × 1 and 9 × 9 × 1 Monkhorst–Pack grids[52] for GGA and HSE06 calculations, respectively. The topology of the studied structures was identified by calculating the Z2 topological invariant[1] via the Wannier-derived Z2 topological invariant method[53] using Z2Pack[54] and WannierTools.[55] The systems with Z2 = 1 were characterized as TIs while those with Z2 = 0 were taken as ordinary insulators. The edge states were calculated using an iterative Green function as implemented in the WannierTools package.[55]

Results and Discussion

The unit cell of the Cu2S monolayer is formed by two Cu atoms and one S atom. It can be regarded as a honeycomb formed by the first Cu and one S, while the other Cu is in the center of the honeycomb. Figure a displays the top view of these atomic structures where each S atom is surrounded by six Cu atoms while each Cu atom is bonded with three Cu and three S atoms. Three types of Cu2S honeycomb-like structures, namely planar, two-sided buckled, and one-sided buckled, were constructed and further relaxed as illustrated in Figure a–d, and the associated Brillouin zone (BZ) of the hexagonal lattice is shown in Figure e. These three structures have a similar top view. The planar structure possesses space group P6/mmm (No. 191), which is the same that for as other typical planar M2X materials, while the one-sided and two-sided buckled structures possess space group P6mm (No. 183) and P3m1 (No. 156), respectively, as the high buckling reduced the symmetry of the atomic structure.

Figure 1

(a) Top view of the crystal structure of monolayer Cu2S honeycomb. Side views of (b) two-sided buckled, (c) planar, and (d) one-sided buckled structures. (e) First Brillouin zone with high-symmetry points. (f) Energies and (g) vertical buckling heights as a function of strain.

The total energy and buckling distance of the monolayer Cu2S as a function of the biaxial strain and lattice constant are shown in Figure f,g. The magnitude of biaxial strain (ε) and the variational ratio of buckling height (Δh) are defined as ε ≡ (l – lo)/l × 100% and Δh ≡ (h – Δb)/h, respectively, where lo and Δb represent the equilibrium lattice constant and buckling height, respectively. The buckling height is defined as the vertical distance between the Cu and S atoms in two-sided and one-sided buckled structures (see Figure b,d). The energy–strain curves of all three structures have a single minimum that indicates each of their optimized lattice constants of 4.105, 4.097, and 4.102 Å for planar, one-sided, and two-sided buckled atomic structures, respectively. These three lattice constants differ by less than 0.5%. Their total energies are relatively close as well. Among the three structures, the two-sided buckled structure is the most stable structure, which is lower in energy by only 8.6 and 10 meV/atom than the one-sided buckled (Figure d) and planar structures (Figure b), respectively.

Although the lattice constants and total energies are very similar, there are stark differences between the structures in terms of the location of Cu atoms with respect to S atoms as well as the overall buckling structure of the Cu2S monolayer. These results reflect the liquidlike nature of Cu2S, as the Cu atoms have more freedom to move around within the rigid S structure.

It can be seen in Figure g that the buckling height decreases gradually until it reaches a plateau at approximately 4.22 Å, which indicates that both the one-sided and two-sided buckled structures transform into the planar honeycomb structure with larger tensile strain. Furthermore, the two-sided buckled Cu2S (Figure c), which is energetically more favorable in a freestanding state in comparison to the other two morphologies, will transform into the one-sided buckled phase when it is optimized upon the presence of a substrate, which will be elaborated further in the latter part of the discussion section. Since the one-sided buckled phase is more stable when it is placed on a substrate and has been synthesized recently,[43] we will focus on the electronic and topological properties of the one-sided buckled structure of Cu2S.

The electronic band structure of one-sided buckled monolayer Cu2S obtained from the HSE06 functional calculation is shown in Figure , while the band structures of the other two structures are shown in Figure S1 in the Supporting Information. In the absence of spin–orbit coupling (SOC), monolayer Cu2S is gapless with the conduction band minimum (CBM) and heavy-hole valence band maximum (VBM) degenerate at the Γ point, as shown in Figure a. This degeneracy occurs exactly at the Fermi level and disperses linearly over a substantial energy range of −2.0 to +2.4 eV.

Figure 2

Hybrid-functional-calculated band structure of the one-sided buckled monolayer Cu2S (a, c) without (left column) and (b, d) with SOC (right column). The red/blue dots represent s/p orbitals.

When SOC is included in the computations (Figure b), the gapless feature in monolayer Cu2S transforms into a gapped insulating state with a large direct band gap of 220 meV. The significant band gap opening in Cu2S can be attributed to the heavier elements. In addition to the band-opening feature due to SOC, a Rashba spin splitting (RSS) is pronounced at the topmost valence and the lowest conduction bands around the Γ point, as shown in Figure d. The origin of the Rashba splitting is the induced effective electric field by the breaking of inversion symmetry in the Cu2S crystal structure. The strength of RSS is determined by the Rashba energy (ER), the corresponding k-space shift, or the momentum offset (kR) and the Rashba constant (αR).[56] For monolayer Cu2S, the ER, kR, and αR values of the valence band maximum (VBM) and conduction band minimum (CBM) are summarized in Table .

Table 1

Rashba Energy (ER), k-Space Shift (kR), and the Rashba Constant (αR) of the Valence Band Maximum (VBM) and Conduction Band Minimum (CBM) of Monolayer Cu2S

	VBM	CBM
ER (meV)	0.3	2.457
kR (Å–1)	0.0024	0.0138
αR (eV Å)	0.5	0.348

Alongside Rashba spin splitting, a band inversion occurs between the conduction and valence bands. To reveal the band inversion mechanism, we projected the band structures with the orbital contribution of Cu and S atoms in Figure c,d. As the CBM and VBM cross only at the Γ point, it is expected that the CBM and VBM must be inverted at Γ. Thus, only the contribution of the orbitals in the vicinity of Γ around the Fermi level has been focused on. Figure c shows that, without SOC, the CBM and VBM are mainly derived from the s orbital of Cu and the p and p orbitals of S atoms, respectively. Notably, the p orbitals of Cu and p orbital of S do not contribute to states near the Fermi level. If the SOC is taken into consideration, the CBM and VBM are switched and now the valence band edge is comprised of the s orbital of Cu and the conduction band edge is comprised of p and p orbitals of S, which is opposite to the band alignment in the case without SOC (Figure d). This clearly demonstrates that the band structure of monolayer Cu2S has been intrinsically inverted, which is a characteristic of QSH insulators. It is noteworthy that the band inversion occurred due to the crystal field effect and persists even without SOC, while SOC only participates to open a large band gap.

To further confirm the topological property of monolayer Cu2S, we calculated the Z2 topological invariant. The studies on 2D TIs reveal that, in the presence of time-reversal symmetry, 2D TIs can be characterized by Z2 topological invariants. As inversion symmetry is absent in the Cu2S monolayer, we used the Wannier charge center method to determine the Z2 invariant and we obtained Z2 = 1, which verifies that monolayer Cu2S is a QSH insulator. The nontrivial band gap of monolayer Cu2S is large enough to realize the QSH state at room temperature and for device implementation.

The existence of the time-reversal symmetry-protected helical edge states is a unique feature of QSH insulators. Therefore, to demonstrate the nontrivial topology of monolayer Cu2S, we calculated the edge states of Cu2S using VASP and Wannier90,[57] as shown in Figure . The edge states of Cu-terminated (Cu-T), S-terminated (S-T), and armchair-terminated (AC-T) monolayer Cu2S are shown in Figure a–c, respectively. The edge states are manifested as a single pair of linearly dispersed Dirac-cone-like features inside the bulk band gap. For the armchair termination, both edges are the same and thus have the same edge state dispersion. In contrast, the Dirac point of the Cu-terminated edge state is slightly downward toward the conduction band and the S-terminated Dirac point is slightly upward due to electric polarization caused by the zigzag termination of edges. The misalignment in Dirac points at Cu- and S-terminated edges facilitate the easy tunability of the dominant current channel among two (Cu and S) edges by the chemical potential which is commonly seen in 2D TIs.[16,17,58]

Figure 3

Band structures of different edges of one-sided buckled monolayer Cu2S with SOC: (a) Cu-terminated (Cu-T) and (b) S-terminated (S-T) edge states, respectively; (c) armchair (AC-T) edge states.

The growth of thin films on substrates generally results in the effect of the strain. To examine the robustness of the TI phase against strain, we analyzed the effect of biaxial strain on all three phases of monolayer Cu2S. All of the phases sustain their QSH phase over a wide range of strains, as shown in Figure a). As was mentioned before, the band inversion occurs in monolayer Cu2S due to the crystal field effect instead of SOC; therefore, any change in Cu and S bonds or buckling height by strain causes corresponding changes in the orbital ordering and crystal field splitting, thus resulting in a change in the band gap and topology. The band inversion strength (BIS) here is defined as the direct band gap between VBM and CBM at Γ, and it provides an estimate of how far a system is from the topological critical state. BIS is negative when no band inversion occurs, and the system is topologically trivial while it is positive otherwise.

Figure 4

(a) Band inversion strength (BIS) as a function of strain for planar and one-sided and two-sided buckled structures. The positive and negative BIS values represent the quantum spin Hall insulator and trivial insulator states, respectively. The bottom row (b–e) displays the SOC band structures of four selected strains for the one-sided buckled phase.

We compared the BIS and topological phase of Cu2S without strain (0%) in Figure a by examining the three structure variants of Cu2S to have the same lattice constants.[59] We found that the buckling of the Cu atoms affects their topological phase transition. The band inversion strength at 0% strain in Figure a decreases as the planar structure with a nontrivial phase becomes more buckled, until it becomes negative in the two-sided buckled structure and transforms to a trivial phase. The planar and one-sided buckled band structures both exhibit a band closing without SOC and band opening with SOC, whereas the two-sided buckled case is already gapped without SOC and remains a trivial insulator with SOC (see Figure S1 in the Supporting Information).

Applying tensile and compressive strains to each of the three structures results in another topological phase transition in Figure a. We look more closely at the band structures of four selected strains for the one-sided buckled phase. The one-sided buckled phase behaves like a normal semiconductor at −3% compressive strain (3.980 Å) with a direct band gap of 246 meV (Figure b). Reducing the compressive strain causes a decrease in band gap, and it reaches 170 meV at −2%. At −1.5%, the gap between CBM and VBM closes completely at Γ, indicating a topological critical point (Figure c). A further reduction in compressive strain led to a band inversion between CBM and VBM at Γ with a direct band gap opening of 161 meV, extending over the entire BZ. The evolution of BIS is shown over a wide range of strains in Figure a. The inverted band gap attains a value of 288 meV at 3% tensile strain, and then the band gap starts to decrease gradually with a further increase in tensile strain, keeping its nontrivial nature. Notably, we find that the band gap decreases with a decrease in buckling height from −3% to −1.5% strain and vanishes at buckling height of 0.301 Å in the trivial phase. A further reduction in buckling height induces band inversion at Γ in the one-sided buckling phase of monolayer Cu2S. The buckling height reaches zero at 4% tensile strain, and a structural transition from the one-sided buckling phase to the planar phase occurs while the nontrivial topology of the system is preserved. The same trend is seen in the two-sided buckled phase of monolayer Cu2S as well. This indicates that the band inversion in monolayer Cu2S mainly depends upon the buckling height rather than the bond length between Cu and S atoms.

The results of strain calculations indicate the robustness of the topological phase of monolayer Cu2S against strain, offering a wider range of potential substrates for experimental realization and device fabrication. However, it is essential to find a suitable substrate that can also preserve the topological properties of the system. We have found that monolayer PtTe2 and h-BN can be possible insulating substrates to grow monolayer Cu2S, since the lattice constant of a 1 × 1 unit cell of monolayer PtTe2 (4.019 Å) and √3 × √3 supercell of h-BN (4.337 Å) closely match the lattice constant of a 1 × 1 unit cell of the Cu2S film. Additionally, Chin et al. used Cu(111) as a substrate to fabricate the Cu2S thin film; therefore, we also simulate the Cu2S monolayer on a √3 × √3 supercell of the Cu (4.336 Å) substrate.[43]

Our calculations show that monolayer Cu2S on Cu(111), PdTe2, and h-BN substrates retains its nontrivial phase. However, when Cu2S is on Cu(111), it becomes metallic. Furthermore, Cu2S/h-BN is a QSH insulator. The h-BN substrate has already been proposed as a substrate for supporting a few QSH materials[32−35] and has no significant effect on the band structure in general. The results with the h-BN and Cu substrates can be found in Figures S2 and S3 in the Supporting Information.

Next, we focus our discussion on the Cu2S/PtTe2 heterostructure. Figure a,b displays the top and side views of the optimized energy structure of Cu2S film on monolayer PtTe2, respectively. The buckling height of Cu2S on PtTe2 is 0.155 Å, and the interlayer distance between monolayer Cu2S and PtTe2 is 3.26 Å. To see the effect of the PtTe2 substrate on the topological properties of monolayer Cu2S, we calculated the band structure of the Cu2S/PtTe2 heterostructure under the HSE06 functional, as shown in Figure c,d. The bands located in the gapped region of PtTe2 entirely originate from Cu2S. Without consideration of the SOC, the Cu2S/PtTe2 heterostructure shows semimetallic behavior with a Dirac-cone-like crossing at the Γ point. Moreover, the CBM is completely dominated by s orbitals of Cu atoms while the VBM is dominated by p and p orbitals. The second-highest valence band gets some contribution from the s orbital of Cu atoms as well. In the presence of SOC, the Dirac-cone-like crossing opens and a band gap of 141 meV appears at Γ, leading to a band inversion between the s orbitals of Cu and the p and p orbitals of S. In addition to the reduction of the band gap, the Cu2S/PtTe2 heterostructure preserved all the prominent electronic features of the freestanding monolayer Cu2S, which indicates that the Cu2S film perfectly maintains its topological properties on the PtTe2 substrate. The reduction in the band gap can be attributed to the narrow band gap of the PtTe2 substrate. To verify the presence of a QSH phase in Cu2S/PtTe2, we calculate the Z2 invariant and obtain an Z2 index of 1, which confirms that Cu2S/PtTe2 possesses a QSH phase.

Figure 5

Crystal structure of PtTe2-supported one-sided buckled monolayer Cu2S: (a) top view; (b) side view. The orbital-resolved band structure of Cu2S on a PtTe2 substrate under HSE06 functional (c) without and (d) with SOC. Red and blue circles represent the copper s and sulfur p orbital contributions, respectively.

Conclusion

In conclusion, we demonstrated that monolayer Cu2S is a large band gap QSH insulator where an s–p type band inversion is triggered by the crystal field effect and SOC facilitates the opening of the large band gap. The nontrivial phase was confirmed via the presence of topological edge states. The Dirac-like edge states are tunable via chemical potential due to the electric polarization effect of zigzag edges. Moreover, we demonstrated that the nontrivial phase of Cu2S films remains preserved upon an interface with Cu(111), h-BN, and PtTe2 substrates. The tunability of the band gap size and the preservation of the nontrivial topology on insulating and metallic substrates are attractive features for potential applications of the Cu2S monolayer in low-power electronic devices.