Abstract
The central target of spintronics research is to achieve flexible control of highly efficient and spinpolarized electronic currents. Based on firstprinciples calculations and k·p models, we demonstrate that Cu_{2}S/MnSe heterostructures are a novel type of Chern insulators with halfmetallic chiral edge states and a very high Fermi velocity (0.87 × 10^{6} m s^{−1}). The full spinpolarization of the edge states is found to be robust against the tuning of the chemical potential. Unlike the mechanisms reported previously, this heterostructure has quadratic bands with a normal band order, that is, the p/dlike band is below the slike band. Charge transfer between the Cu_{2}S moiety and the substrate results in variation in the occupied bands, which together with spin–orbit coupling, triggers the appearance of the topological state in the system. These results imply that numerous ordinary semiconductors with normal band order may convert into Chern insulators with halfmetallic chiral edge states through this mechanism, providing a strategy to find a rich variety of materials for dissipationless, 100% spinpolarized and highspeed spintronic devices.
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Introduction
Microelectronic developments have given rise to an urgent need for new information storage and transport technologies and materials with very low energy consumption and high response speeds. Over the past two decades, great progress has been made in this regard in the following two areas. The first is based on the approach of spintronics by employing the spin degree of freedom of electrons that usually involves low power consumption. Typically, materials such as magnetic materials, half metals and spingapless semiconductors have been proposed for spintronics.^{1, 2, 3, 4, 5} The second approach has benefited from the discovery of twodimensional topological insulators, for which the edge states are expected to show dissipationless transport of the charge (or spin) current.^{6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17} To date, few studies have combined both of these aspects well, namely, realizing dissipationless transport of a highly spinpolarized current through the chiral edge states of materials such as Chern insulators.^{18} It has been reported that the edge states of Chern insulators are generally not spinpolarized^{19} or are merely partially spinpolarized in the sample plane.^{20, 21, 22} Therefore, it is extremely desirable to realize dissipationless electronic transport through the Chern insulators with halfmetallic chiral edge states, offering great potential for applications of the chiral edge states of Chern insulators in spintronics.
Proposals to produce the Chern insulators generally exploit Dirac electrons^{23} and employ the mechanism of special inverted bands.^{24} These Chern insulators are generally limited to small nontrivial band gaps, show problematic disorder due to magnetic doping, and are found at very low experimental temperatures.^{24} To overcome these drawbacks of Dirac electron systems, one may explore the topological states in numerous ordinary semiconductor materials, for which the dispersions are usually quadratic and show normal band order. Actually, the quantum spin Hall effect has been reported in several nonDirac electron systems.^{25, 26, 27, 28, 29, 30, 31} The appearance of the quantum spin Hall state in these systems is associated with the inverted band order, that is, the p/dlike bands are abnormally above the slike band (Figure 1a),^{25, 26, 27, 28, 29, 30} which has been regarded as an essential and effective strategy for the generation of nontrivial topological states.^{24} Nevertheless, perhaps it is possible to realize nontrivial topology effects in nonDirac electron systems with the normal band order that is found in many semiconductor materials such as Ge, GaAs, silicene, and germanene.^{32, 33} A straightforward approach for achieving a topological state from a nonDirac dispersion with normal band order is illustrated in Figures 1b and c. The bands displayed in Figure 1b give the typical energy dispersion relationship around the Fermi level (E_{F}) of nonDiractype semiconductor materials with normal band order. If the bands shown in Figure 1c can be achieved, a nontrivial band gap may be opened in Figure 1c around the E_{F} after the intrinsic spin–orbit coupling (SOC) is considered. If both branches of the p/dlike bands are in a single spinchannel (spinup or spindown), the edge states generated within the nontrivial band gap may be fully spinpolarized.
In this paper, we report a novel type of Chern insulators with halfmetallic (100% spinpolarized) chiral edge states in the βCu_{2}S films grown on MnSe(111) surfaces. The thin film of βCu_{2}S is an ordinary nonDiractype semiconductor with a normal band order, as shown in Figure 1b, which has been synthesized experimentally and has been found to be stable at room temperature.^{34, 35} The MnSe substrate provides magnetization in the βCu_{2}S bands to break the timereversal symmetry. Based on densityfunctional theory and the Wannier function method, we find that these novel Chern insulators exhibit several prominent advantages: (i) The chiral edge states are fully spinpolarized, and the spin polarization ratio does not change with variation of the chemical potential. (ii) A very high Fermi velocity (0.87 × 10^{6} m s^{−1}) is achieved, comparable to that of graphene. (iii) This type of Chern insulator is achieved in nonDirac electrons with normal band order, implying that many ordinary semiconductor materials may become Chern insulators through this mechanism. (iv) The nontrivial band gap is sizeable (45 meV), supporting the roomtemperature experiments and applications of the Chern insulating behaviors. These discoveries create a new path for the design of Chern insulators with halfmetallic chiral edge states from ordinary semiconductor materials and provide new opportunities to realize highly efficient spintronic devices.
Materials and methods
The constructed heterostructure of the βCu_{2}S monolayer on MnSe(111) surfaces is displayed in Figures 2a and b. Very thin βCu_{2}S films have been fabricated experimentally.^{34, 35} One exotic characteristic of this kind of Cu_{2}S films (Figure 2c) is that the structure usually exists in a solid–liquid hybrid phase with a framework of graphenelike fixed S layers and mobile (liquidlike) Cu atoms within the framework.^{34, 35, 36} For the substrate material, the bulk MnSe compound has a cubic rocksalt structure in which the Mn atoms in the (111) plane form a ferromagnetic (FM) triangular lattice, and the neighboring two Mn(111) planes, containing one Se plane between them, form an antiferromagnetic structure.^{37, 38} The Cu_{2}S monolayer can be magnetized by the topmost FM Mn plane through the proximity effect.^{13, 14} The lattice of the βCu_{2}S monolayer^{35} (Figure 2c) matches well with that of the MnSe (111) plane (with a small mismatch of 2%). The structural details and different contact configurations considered for the Cu_{2}S and MnSe interfaces can be found in the Supporting Information. The structure displayed in Figures 2a and b is found to be the most stable configuration (top site), and its electronic structures and topology will be discussed in detail below. A relatively strong chemical bonding exists between the bottom Cu atoms and the topmost Mn atoms, confirmed by the large value of the binding energy (2 eV per unit cell). This tendency is consistent with the small distance d (d=1.39 Å) between the bottom Cu plane and the topmost Mn layer in the substrate after the geometry optimization (see Figure 2a). The structural stability is also further confirmed by molecular dynamics simulations (see Supporting Information).
The geometric optimization and electronic structure calculations of the Cu_{2}S monolayer grown on the MnSe substrate are performed with the projector augmented wave method based on ab initio densityfunctional theory.^{39, 40} The Perdew–Burke–Ernzerhof generalized gradient approximation (GGA) is used to describe the exchange and correlation functional.^{41} The Mn 3d orbitals are treated with the GGA plus Hubbard U (GGA+U) method,^{42} in which the effective onsite Coulomb interaction U and exchange interaction J are set to 5.0 and 1.0 eV,^{43, 44} respectively. The planewave cutoff energy is set to 500 eV, and a vacuum space larger than 15 Å is adopted to avoid the interaction between the two adjacent slabs. The 9 × 9 × 1 gamma central Monkhorst–Pack grids are employed to perform the first Brillouin zone (BZ) integral. MD simulations are carried out to study the structural stability using Born–Oppenheimer ground states with 4 fs as the time step. A thermostat is used to control the temperature to remain at room temperature (300 K). The structure has been simulated for 2.4 ps. The edge state and its spintexture are calculated by using the surface Green’s function method.^{45}
Results and discussion
We now explore the topological electronic structures of the Cu_{2}S/MnSe heterostructure shown in Figures 2a and b. The band structures without and with consideration of SOC are given in Figures 3a and b, respectively. As shown in Figure 3a, when the Cu_{2}S monolayer is placed on the (111) surface of the MnSe substrate, an obvious spin splitting appears in the Cu_{2}S bands. The Cu_{2}S is magnetized well by the topmost FM Mn layer. Our calculations also show that the magnetization of the heterostructure along the caxis is 4 meV more stable than that lying in the ab plane, enabling improvements in the experimental observation of the quantum anomalous Hall (QAH) effect. Without the SOC, the Cu_{2}S/MnSe system is gapless, with the top of the valence bands and the bottom of the conduction bands degenerate at the Г point in the spinup channel, as displayed in Figure 3a. The degeneracy is primarily composed of Cu d_{x}^{2}_{−y}^{2} and d_{xy} orbitals and arises from the C_{3v} symmetry of the heterostructure. However, the SOC splits the double degeneracy at the Г point, resulting in a global and sizeable band gap of approximately 45 meV opening around the E_{F} (Figure 3b), which is large enough to support the corresponding experimental work and possible applications of the heterostructure at room temperature. To identify the topological nature of the insulating phase, the Chern number is calculated by integrating the Berry curvatures over the first BZ.^{46, 47, 48, 49, 50} The red curve in Figure 3c gives the obtained Berry curvatures along the highsymmetry lines in the first BZ. The obtained Chern number of C=−1 demonstrates the nontrivial topological features of the band gap opened by the SOC.
The obtained edge state (red curve) of the heterostructure is displayed in Figure 3d and is located at the edge of the semiinfinite plane of the slab system. In particular, the calculated spin texture of the edge state (Figure 3e) shows that the edge state is 100% spinpolarized in the zdirection (along the caxis), indicating the halfmetallic behavior of the chiral edge state (see the inset of Figure 3d). The halfmetallic behavior obtained here is distinct from the traditional halfmetallic property^{3} for which the spinpolarization may be highly suppressed upon consideration of the SOC.^{51} Importantly, the constant value of S_{z}/S in Figure 3e demonstrates that the full spinpolarization is robust against the tuning of the chemical potential, unlike the case in Zhang et al.^{22}; this is beneficial for the applications of the edge states in spintronics. The onedimensional (1D) boundary of the heterostructures slab is a closed manifold in real space, and the electrons in the chiral edge state will acquire a quantized π Berry phase after they evolve one cycle along the boundary. Thus, the full spinpolarization of the edge state can be recognized as the 1D realspace topological states.^{20} The halfmetallic behavior of the edge state can be associated with the bulk nontrivial band gap introduced by both bands in the spinup channel (Figure 3a) without SOC. The conservation of the S_{z} of the edge states (Figures 3d and e) indicates that the Rashba SOC plays a negligible role in the bandgap opening. Namely, the band gap is primarily opened by atomic SOC. The sharp slope of the almost linear dispersion of the edge state implies the very high Fermi velocity v_{F}(~0.87 × 10^{6} m s^{−1}) of the electrons, as evaluated by (). This Fermi velocity is even comparable to that of graphene (1 × 10^{6} m s^{−1}).
These results prove that the Cu_{2}S/MnSe heterostructure is a novel Chern insulator with exotic halfmetallic chiral edge states. Because of these edge states, the Cu_{2}S/MnSe heterostructure has great potential for the fabrication of a dissipationless, 100% spinpolarized, and highspeed spintronic devices and is superior to the Chern insulators with no spin polarization or partially spinpolarized edge states.^{19, 21, 22} It is also important to note that the E_{F} in this system lies in the nontrivial band gap, enabling improvements in the experimental observations.
To understand the origin of the Chern insulators, the band evolution of the heterostructure with different contact distances is explored. Since the influence of the MnSe substrate on the Cu_{2}S sample can be tuned by changing the distance (d) between the bottom Cu plane and the topmost Mn plane (Figure 2a), how the substrate varies the bands of the Cu_{2}S film can be examined by tuning the distance d. The projected band structures with different contact distances d are displayed in Figures 4a–c, where the blue squares and red circles represent the Cu d_{x}^{2}_{−y}^{2} /d_{xy} and Mn d_{x}^{2}_{−y}^{2} /d_{xy}/d_{z}^{2} orbitals, respectively. SOC is not considered in this figure. When imposing a 180% tensile strain (defined as (d−d_{0})/d_{0}, where d stands for the distance between the bottom Cu plane and the topmost Mn plane with the applied strain and d_{0} stands for the relaxed distance without the strain (1.39 Å)), the Cu bands (blue curves) cross the Mn bands (red curves). Importantly, no spin splitting is found for the Cu bands (blue curves) in Figure 4a because of the very weak interaction from the topmost FM Mn plane in the substrate (d=3.89 Å).
Upon decreasing the tensile strain from 180% to 20%, the Cu bands hybridize strongly with the Mn bands around the E_{F}, as illustrated in Figure 4b. The spin polarization of the Cu d bands is induced by the Mn plane due to the chemical bonding formed between the interface Cu and Mn atoms. In particular, the Cu bands move downward in energy while the Mn bands show the opposite effect. Thus, electron transfer occurs from the Mn atoms in the substrate to the Cu atoms in the Cu_{2}S sample, which is likely due to the higher electronegativity of the Cu atoms relative to that of the Mn atoms. With the tensile strain further decreasing to zero (Figure 4c), more Mn d electrons transfer to the Cu d states. Since the stretched strain occurs only in the c direction (Figure 2a), the C_{3v} symmetry is not broken in the process. Hence, the double degeneracy of the Cu d orbitals (d_{x}^{2}_{−y}^{2}, d_{xy}) at the Γ point is preserved in the process of the tensile strain variation (Figures 4a and b). A gapless semiconductor is finally achieved in Figure 4c without consideration of SOC, with the E_{F} crossing exactly the degenerate point of Cu d_{x}^{2}_{−y}^{2} and d_{xy}.
The schematic diagram of the above band evolution is illustrated in Figure 4d, where the solid and dashed curves represent the spinup and spindown bands, respectively. The green, blue, and red curves denote the Cu_{2}S slike bands, dlike bands and MnSe dlike bands, respectively. As seen in stage (i), when the distance between the monolayer Cu_{2}S and the substrate is relatively large, no exchange field is induced in the Cu_{2}S bands, which has a typical semiconducting band structure with the normal band order. With a decrease in the distance between the Cu_{2}S film and the substrate, the exchange field, chemical bonding, and charge transfer (indicated by the black arrows) finally give rise to a gapless semiconductor (from stage (ii) to (iii)). After SOC is included, the quadratic band touching at the Γ point is no longer protected by the C_{3v} symmetry according to the double group representations of C_{3v} symmetry.^{52} Thus, a sizable nontrivial band gap is acquired, and a QAH insulator is obtained in the Cu_{2}S/MnSe heterostructure (stage (iv)).
It is important to emphasize that the chemical bonding and charge transfer at the interface of the sample and the substrate result in band deformation (Figures 1b and c) and lead to the QAH state in the heterostructure. The band evolution in Figure 4d clearly illustrates that the formation of the topological states does not at all demand the band inversion of the slike band below the p/dlike band, differing from many previously reported cases in nonDirac electron systems.^{25, 26, 27, 28, 29, 30} Although the sband in Figures 1b, c and 4 is not essential for the appearance of the topological state in the Cu_{2}S/MnSe heterostructure, it is displayed for a good comparison with the traditional inverted band mechanism of the topological states proposed in numerous nonDirac systems (Figure 1a). Furthermore, many semiconductor materials show normal band ordering around the Γ point, that is, the p/dlike band is below the slike band. Hence, our work offers more accessible materials for experimental observation and applications of the Chern insulators. Since no band inversion is required, the exchange interaction does not need to be very large to generate the QAH effect, as seen from the last two stages in Figure 4d. Moreover, the nontrivial gap is opened by the intrinsic atomic SOC rather than by the external Rashba SOC, resulting in a rather large band gap (that is, approaching the atomic SOC strength). Our results show that many nonDiractype semiconductor materials with normal band order may produce the QAH effect based on the mechanism proposed here, providing a new route for discovery of the Chern insulators.
To deeply understand the topological mechanism, an effective Hamiltonian was then constructed in terms of the k·p model for lowenergy physics and based on the invariant theory.^{31, 53} Since the bands near the E_{F} around the Г point are dominated by d_{x}^{2}_{−y}^{2} and d_{xy} orbitals of the Cu atoms in the Cu_{2}S monolayer, it is reasonable to adopt these two orbitals as the basis. For convenience, the two orbitals are transformed to the basis of {d_{+,2↑}〉, d_{−,2↑}〉, d_{+,2↓}〉, d_{−,2↓}〉} with d_{±,2}〉 =(d_{x}^{2}_{−y}^{2}〉 ± id_{xy}〉) and ↑(↓) for the spinup (down) state. The Hamiltonian takes the form:^{31}
In the absence of SOC and magnetism, the system contains timereversal symmetry (T), mirror symmetry (M_{l}) along the laxis() direction, and threefold rotation (C_{3}) along the caxis. Owing to the T symmetry, f(k) should be an even function of k. By taking the C_{3} symmetry into consideration, one can obtain because the angular momentum of the d_{x}^{2}_{−y}^{2} and d_{xy} orbitals equals 2. Thus, by inspecting the two constraints of T and C_{3}, we see that f(k) must take the form of . Moreover, we have , . The FM exchange term as (σ_{z} is the Pauli matrix) can be added to the Hamiltonian (Equation (2)). Such a term can separate the energy bands for spinup and spindown states, as found by the densityfunctional theory calculations. Because the Rashba SOC is trivial compared to the exchange interaction, we can neglect the coupling between the spinup and spindown states and proceed to the discussion of the spinup channel, that is, on the basis of {d_{+,2↑}〉, d_{−,2↑}〉}.
For the atomic SOC effect, H_{so}=λ_{so}L⋅S is diagonal in the selected basis. Thus, the k·p model under {d_{c+,2↑}〉, d_{−,2↑}〉} around the Γ point can be constructed up to the second order of k according to:
To explore the topology and pseudospin texture, we neglect the constant term M⊗1, which does not modify the eigenstates of the system. Thus, the Hamiltonian H is rewritten as:
with the . The calculated pseudospin texture for the total Hamiltonian is shown in Supplementary Figure S2a. An obviously very special twomeron structure with double vorticities is observed in the pseudospin texture.^{6, 31} By calculating the pseudospin Chern number as , the Chern number of −1 is acquired, in agreement with the result obtained from the densityfunctional theory calculations described above and with the result acquired by integrating the following Berry curvature:
derived from .^{23} It is interesting to find that the systems with nonDirac band dispersions composed of other orbitals such as p_{x, y} or d_{xz, yz}, have Chern numbers with opposite sign and inverse edge states for the same spin channel due to their different angular momentum. The corresponding pseudospin textures also have the opposite chirality (Supplementary Figure S2c). Thus, the special twomeron pesudospin texture shown in Supplementary Figures S2a and b is the origin of the QAH effect in the nonDirac electron system of the Cu_{2}S/MnSe heterostructure, with the normal band order. The result shown in Supplementary Figure S2c may be found in other nonDirac electron material systems containing interesting degenerate bands composed of p_{x, y} or d_{xz, yz} instead of d_{x}^{2}_{−y}^{2}_{, xy}.
The exchange field in the monolayer Cu_{2}S is needed for the QAH effect and is sensitive to the interface distance (d) between the monolayer and the substrate (Figure 2a). Here, we reveal the effect observed when the exchange field is enhanced by decreasing the interface distance (d) from −1% to −8% along the caxis, which may be realized experimentally by applying an external vertical stress to the heterostructure.
Figures 5a–c illustrate the band structures without SOC under various compressive strains. Under a compressive strain of −2%, the Cu spinup d_{z}^{2} band close to the Г point is at about −0.05 eV, denoted by the red hexagons (Figure 5a). By increasing the compressive strain to −4%, the d_{z}^{2} band moves up drastically in energy and touches the degenerate point of d_{x}^{2}_{−y}^{2}_{, xy} (the blue squares) at the Г point (Figure 5b). When increasing the compressive strain to −6%, the Cu spinup d_{z}^{2} band inverts with one of the branches of the degenerate bands (the blue squares),and a gap appears between the degenerate bands and the inverted d_{z}^{2} band (Figure 5c). The Chern number as a function of the compressive strains is calculated and plotted in Figure 5d, in which there is a jump of Chern number from −1 to 0 at the compressive strain of −4%. This indicates a topological phase transition under the compressive strain, that is, the QAH effect in the heterostructure will be destroyed by compressive strains larger than −4%. The origin of the Cu spinup d_{z}^{2} band energy upshift can be ascribed to the stronger spin polarization of the Cu d_{z}^{2} band with the increase of the compressive strain in the caxis direction. Thus, the exchange field becomes large, with the Cu spinup d_{z}^{2} moving up and spindown d_{z}^{2} shifting down in energy. By analyzing the band components, the mechanism of the topological phase transition is displayed in Figure 5e, in which the green dashed lines represent the E_{F}. Because the degenerate bands of Cu d_{x}^{2}_{−y}^{2}_{, xy↑} invert with the Cu d_{z}^{2}_{↑} band, both degenerate Cu d_{x}^{2}_{−y}^{2}_{, xy↑} bands around the Г point become occupied. Additionally, a trivial gap is generated around the E_{F}. Our results suggest that the novel Chern insulator predicted in the βCu_{2}S/MnSe heterostructure can survive very well with small compressive strains.
Based on the achieved halfmetallic nontrivial edge states, the dissipationless, 100% spinpolarized, and highspeed electronic transport can be realized in a single device, as shown in Figure 6a. Based on the phase transition given in Figure 5d, we predict that the on–off of this device can be tuned effectively by applying a small vertical stress (Figure 6b) because such stress can make the nontrivial edge states of the sample disappear.
In summary, we proposed a novel Chern insulator with unique halfmetallic chiral edge states in the βCu_{2}S/MnSe heterostructures. This type of Chern insulator is achieved in nonDirac electrons with a normal band order, implying that many ordinary semiconductors may become Chern insulators via this mechanism. The interface charge transfer is found to be important for giving rise to the topological behavior. The robust nontrivial edge state is found to be 100% spinpolarized and to have a very high Fermi velocity (0.87 × 10^{6} m s^{−1}), close to that of graphene. The k·p model shows that the topology in the system originates from the exotic twomeron structure of the pesudospin texture. These results indicate that the βCu_{2}S/MnSe heterostructure has great potential to be fabricated into an advanced dissipationless, 100% spin polarized, and highspeed spintronic device. Our work also creates a rational path to design novel Chern insulators with very special halfmetallic chiral edge states.
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Acknowledgements
The authors are grateful to Dr Quansheng Wu for very helpful discussion. This work was supported by the National Natural Science Foundation of China under Grant No. 11574051 and the Natural Science Foundation of Jiangsu Province of China (Grant No. BK20160007). All calculations were performed at the HighPerformance Computational Center (HPCC) of Department of Physics at Fudan University.
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Xue, Y., Zhao, B., Zhu, Y. et al. Novel Chern insulators with halfmetallic edge states. NPG Asia Mater 10, e467 (2018). https://doi.org/10.1038/am.2017.240
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DOI: https://doi.org/10.1038/am.2017.240
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