Quantum Anomalous Hall Effect by Coupling Heavy Atomic Layers with CrI$_{3}$

TL;DR

This paper demonstrates that coupling heavy atomic layers with monolayer CrI$_{3}$ can induce a quantum anomalous Hall effect with sizable gaps and enhanced ferromagnetic properties, opening new avenues for spintronics applications.

Contribution

It introduces a novel approach of combining heavy-element layers with CrI$_{3}$ to realize quantum anomalous Hall effect and improve magnetic properties.

Findings

CrI$_{3}$/Bi system has a 30 meV bulk gap.

Heavy atomic layers enhance ferromagnetic properties.

Spin orientation tuning affects band structure and Fermi level.

Abstract

We explored the possibility of realizing quantum anomalous Hall effect by placing heavy-element atomic layer on top of monolayer CrI$_{3}$ with a natural cleavage surface and broken time-reversal symmetry. We showed that CrI$_{3}$/X (X = Bi, Sb, or As) systems can open up a sizable bulk gap to harbour quantum anomalous Hall effect, e.g., CrI$_{3}$/Bi is a natural magnetic insulator with a bulk gap of 30~meV, which can be further enlarged via strain engineering or adjusting spin orientations. We also found that the ferromagnetic properties (magnetic anisotropic energy and Curie temperature) of pristine CrI$_{3}$ can be further improved due to the presence of heavy atomic layers, and the spin orientation can be utilized as a useful knob to tune the band structure and Fermi level of CrI$_{3}$/Bi system. The topological nature, together with the enhanced ferromagnetism, can unlock new potential applications for CrI$_{3}$-based materials in spintronics and electronics.

Tables

Table 1. Table 1: Bader charge analysis for CrI 3 /X systems. △ △ \triangle Q represents the charge difference on Cr and iodine after the deposition of heavy atomic layers on CrI 3 monolayer. Magnetic moments of each Cr atom, Heisenberg exchange constants, and Curie temperatures are symbolized by M 𝑀 M , J 𝐽 J , and T C subscript 𝑇 C T_{\rm C} for CrI 3 /X systems, respectively.

System	$△Q$		Magnetic Properties
	$\mathrm{Cr}$	$I$	$M_{\mathrm{Cr}}\left({\mu }_B\right)$	$J\left(\mathrm{meV}\right)$	$T_C\left(K\right)$
${\mathrm{CrI}}_3$	$0$	$0$	$3.0$	$2.59$	$44$
${\mathrm{CrI}}_3/\mathrm{Bi}$	$0.23$	$0.27$	$3.72$	$9.62$	$165$
${\mathrm{CrI}}_3/\mathrm{Sb}$	$0.21$	$0.24$	$3.70$	$7.10$	$122$
${\mathrm{CrI}}_3/\mathrm{As}$	$0.11$	$0.14$	$3.62$	$6.37$	$110$

Full text

Introduction—. The interplay of symmetry, spin-orbit coupling, and magnetic structure together helps realize various topological phases ranging from quantum Hall effect to topological superconductor. The quantum anomalous Hall effect or magnetic topological insulator corresponds to the quantum Hall effect without applying external magnetic field weng2015quantum ; ren2016topological ; he2018topological , and exhibits immense application potential in dissipationless quantum electronics liu2018intrinsic ; ni2018intrinsic ; zanolli2018hybrid ; ren2018quantum ; deng2018quantum . Typically, an element possessing high spin-orbit coupling is usually magnetically inactive, and vice versa. Therefore, the magnetic topological insulator rarely exists in natural materials. Theoretically, many attempts have been implemented to make it realistic in a single system by doping/adsorption tse2011quantum ; qiao2010quantum ; ding2011engineering ; zhang2012electrically ; qiao2012microscopic ; jiang2012quantum ; kou2013interplay ; pan2014valley ; qi2016high ; deng2017realization ; deng2018quantum , chemical functionalization jin2015quantum ; hsu2017quantum , and heterostructure schemes luo2013massive ; yang2013proximity ; qiao2014quantum ; xu2015quantum ; zhang2018strong . However, thus far, experimentally, the quantum anomalous Hall effect has been only observed in magnetic element-doped topological insulators at very low temperatures chang2013experimental ; liu2016quantum ; bestwick2015precise . It is noteworthy that a new temperature record of 3.5 K for observing the quantum anomalous Hall effect was reported in MnBi2Te4 system MnBiTe .

It has been proposed garrity2013chern that magnetic topological insulators may be realized by depositing heavy-element layers on top of the surface of non-van der Waals (non-vdW) magnetic insulators. However, such a strategy has several experimental complications, such as surface reconstruction due to existence of dangling bonds on the surface of substrates. Owing to the presence of strong chemical bonding between constituent layers, it is experimentally not easy to downsize the vertical dimension of non-vdW three-dimensional (3D) ferromagnetic/anti-ferromagnetic insulators while preserving the lateral surface geometry (or only changing it slightly). To push away all these practical barriers and make the ideal environment more realistic, here, we suggest a versatile experimental welcoming platform based on 2D vdW ferromagnetic insulators as a substrate for the deposition of heavy spin-orbit coupling components. Unlike a non-vdW substrate, 2D vdW ferromagnetic insulators, in principle, possess a natural cleavage plane with a perfect surface geometry free from dangling bonds, have no surface reform complication, are free of spin alignments, and are more accessible to being synthesized experimentally with excellent stability. This may attract the experimental community to invest their efforts in probing magnetic topological insulators. In addition, we address the following key queries to advance new, and possibly superior, routes for magnetic topological insulators. (i) Can the addition of spin-orbit coupling drive trivial 2D vdW ferromagnetic systems to become large-bandgap magnetic topological insulators in a fundamental and controlled way? (ii) Can they improve the ferromagnetic properties of 2D vdW ferromagnetic insulators, such as the Curie temperature in these systems?

In this Letter, we theoretically reported that several candidate systems CrI3/X (X = Bi, Sb, or As) present non-trivial magnetic topological behaviors with a sizeable bandgap, by employing the state-of-art first-principles calculation methods. Besides a robust magnetic topology with non-zero Chern numbers, notable enhancements of the ferromagnetic properties of pristine CrI3 include the local magnetic moments, Heisenberg exchange interactions, and magnetic anisotropic energy and Curie temperature after coupling with heavy-atomic layers. Furthermore, we showed that either the ferromagnetic orientation or the strain effect can be utilized to tune the topologically nontrivial band gaps.

Ferromagnetic Properties of CrI3/X—. We launched our search for stable configurations comprised of a heavy-atoms monolayer, such as Bi, Sb, and As on the single-layer ferromagnetic insulator, CrI3, to realize the quantum anomalous Hall effect in these systems. Upon a careful examination, we found that deposited heavy elements on CrI3 finally formed a monolayer with a triangular lattice structure, as displayed in Fig. 1(a). The evaluated interlayer binding energies for CrI3/Bi, CrI3/Sb, and CrI3/As were $-0.65$, $-0.62$, and $-0.58$ eV per unitcell, respectively. This moderate interface interaction ensures that these atomic layers do not significantly affect the atomic structure of the pristine CrI3. To our surprise, we found a significant improvement in the magnetic moment of Cr in CrI3/X systems, as listed in Table 1. This is understandable, because the apparent interface interaction can change the charge distribution in CrI3 when heavy atomic layers are deposited on CrI3. As a result, the magnetic moments can be modified. This is indeed the case. Table 1 displays that the charges of Cr in CrI3 and CrI3/Bi are 2.17 and 1.94 e, respectively, which indicate an additional 0.23 e transferring from Bi to Cr after the deposition of Bi monolayer. Same situation occurs for CrI3/Sb and CrI3/As systems. However, the magnitude of charge transfer decreases from Bi to As atomic layers. This is reasonable because the attraction ability of the nucleus on the valence electrons becomes weaker, and then, the probability of charge transfer is increased when we move from As to Bi in the periodic table. Thus, the magnetic moment of Cr atoms in a CrI3 single layer can be obviously enlarged by coupling with heavy atomic layers.

By depositing heavy atomic layers, the local magnetic moments of Cr in CrI3 are considerably increased. However, what about the magnetic coupling between these enhanced magnetic moments? We found that the resulting ferromagnetic coupling in CrI3 can also be enhanced. From our calculations, the ferromagnetic stabilities were $\triangle E_{\rm{FM-AFM}}$=-130, -96 and -86 meV for CrI3/Bi, CrI3/Sb and CrI3/As systems, respectively, where $\triangle E_{\rm FM-AFM}$ represents the energy difference between ferromagnetic and anti-ferromagnetic arrangements. This indicates that the ferromagnetic nature of monolayer CrI3 can be further stabilized by interacting with X(=Bi,Sb,As) atomic layers. In principle, the improvement of ferromagnetic stability boosts the Heisenberg exchange constant $J$. Our estimated Heisenberg exchange constants are respectively 9.62, 7.10, and 6.37 meV for Cr$I_{3}$/Bi, CrI3/Sb, and CrI3/As. These are much higher than that of monolayer CrI3 ($\sim 2.59$ meV) webster2018strain . Based on the above data and the mean field theory, the estimated Curie temperatures are 165, 122, and 110 K for CrI3/Bi, CrI3/Sb, and CrI3/As, respectively SM . These are greatly enhanced from $\sim 44~{}\rm K$ of monolayer CrI3. In particular, a high Curie temperature is tall-demandable for quantum anomalous Hall system to operate at moderate temperatures in future device applications.

After determining that CrI3/X systems favor the ferromagnetic ground state, we then proceeded to investigate the magnetic anisotropy. Our results showed that $z$ axis is an easy magnetization direction for the CrI3/Bi system, which is approximately 5 meV (per unit-cell with two Cr atoms) lower than that of $x$ axis. This value is much higher than that of monolayer CrI3 ($\sim$ 1.5 meV). This obvious improvement in magnetic anisotropic energy arises from the strong spin-orbit coupling in CrI3/Bi, which has been reported in various systems ujsaghy1996spin ; yi2016atomic .

Quantum Anomalous Hall Effect—. The enhancement in ferromagnetic properties of pristine CrI3, including the local magnetic moments, Heisenberg exchange interactions, magnetic anisotropic energy and Curie temperature, will be beneficial to realizing the quantum anomalous Hall effect in CrI3 after coupling with heavy-atomic layers. Figure 2(a) displays the spin-polarized band structure of CrI3/Bi. The spin-up and -down bands cross around $\Gamma$ to form D1/2 points, which are closely located near the Fermi level. Considering the presence of Bi states at the Fermi level SM , together with the conceptual framework of magnetic topological insulators, one may expect the formation of topological non-trivial insulator around these avoided band-crossing points. When spin-orbit coupling is further considered in CrI3/Bi, the degenerate states at the avoided band-crossing points repel each other to open up an energy gap (see Fig. 2(b)). Our achieved direct and indirect (global) bandgaps in the CrI3/Bi are respectively 30 and 75 meV. We also confirmed the non-trivial behavior inside the band gap by calculating the Chern number using the approach implemented in Wannier90 and WannierTools packages mostofi2008wannier90 ; wu2018wanniertools . Our calculations demonstrate that CrI3/Bi indeed harbours a quantum anomalous Hall effect associated with a nonzero Chern number of $\mathcal{C}=-1$ thouless1982quantized .

Figure 2(c) displays the spin-polarized band structure for CrI3/As. We found that the spin-up band crosses the spin-down bands twice within the energy window from 3.4 to 3.7 eV around $\Gamma$ point. These band-crossing points are labeled as $\delta_{1/2}$. Interestingly, the band structures around these crossing points resemble the ideal cone (Dirac point) in graphene. Due to the relatively strong spin-orbit coupling of As, one may expect a non-trivial bandgap to open around these Dirac points when spin-orbit coupling is invoked. Figure 2(d) presents the band structure in the presence of spin-orbit coupling, where bandgaps open up around the crossing points of $\delta_{1/2}$, which clearly indicate the existence of non-trivial band topologies in the conduction bands of CrI3/As. By using the above mentioned approach, we obtained the Chern number of $\mathcal{C}=2$ for the corresponding opened band gaps. Additionally, the two valence bands touch each other around K point SM . When the spin-orbit coupling is switched on, bandgaps are opened at these touching points, which signifies the topological nature of the CrI3/As in valence bands. In principle, the non-trivial local bandgap in the conduction bands of CrI3/As can be converted into an insulator by charge doping technique, which is equivalent to the electrostatic gating in experiment. To tune the Fermi level into the conduction band, we applied a homogenous background charge to electrostatically control the electron doping using the self-consistent simulation approach. We noticed that the Fermi level was correctly located into the spin-orbit coupling induced bandgap without affecting the band structure pattern when the homogeneous charge of $\triangle Q$=1e was functionalized SM . Notably, the non-trivial bandgap is also widened from 18 to 37 meV upon the functionalization of electrostatic doping SM . Thus, CrI3/As can also become a quantum anomalous Hall insulator upon moderate electrostatic gating.

Figure 2(e) displays the spin-polarized band structure of CrI3/Sb. The spin-up and -down bands cross each other multiple times in the energy range between -1.30 and -1.15 eV. A non-trivial bandgap opens at these crossing points upon the application of spin-orbit coupling [see Fig. 2(f)]. The global bandgap is approximately 65 meV, and exhibits a non-trivial topological nature with a Chern number of $\mathcal{C}=1$. Similar electrostatic gating approach can be used to tune the Fermi level into the bandgap.

Spin Orientation Effect—. The orientation of spin moments along different crystallographic directions provides one way to control the band structure and its associated topologically non-trivial properties. In a broader context, a strong spin-orbit coupling, together with a less symmetric crystallographic structure and a suitable magnetic behavior, is the main parameter to explore the spin orientation effect. Fortunately, 2D CrI3/X systems fulfill all these criteria, and hence, are promising candidate platforms for demonstrating the spin orientation effects. In Fig. 3, our calculations show that tuning the spin moment from $c$ to $a$ axis in CrI3/Bi system remarkably modifies the band structures, including substantially changing the Fermi surface and bandgaps. In particular, one can observe that when all spin moments were adjusted in a ferromagnetic pattern from 0∘ to ${}^{6}8\circ$, the bandgap first gradually increases and then quickly decreases to be closed. Beyond $\theta=68\circ$, the CrI3/Bi system undergos a topological phase transition from quantum anomalous Hall effect to a magnetic metallic state.

Strain Effect—. Strain is one of the significant approaches for engineering electronic and topological properties. Fortunately, the softness level of monolayer CrI3 is higher than that of graphene and MoS2, indicating that monolayer CrI3 can be easily stretched or compressed experimentally zheng2018tunable . From this perspective, we examined the CrI3/Bi system under the application of several biaxial strains ranging from -5 to 5% (see Fig. 4). We observed that both direct and indirect bandgaps are much sensitive to the strain. In the upper panels, we considered the out-of-plane magnetization along the vertical direction with $\theta=0^{\circ}$. It is observed that the global bandgap considerably increases when the lattice is allowed to compress within the horizontal plane. Interestingly, the nature of the global bandgap changes from indirect to direct at an approximately 3.15% compressive strain (see Fig. 4). While for the in-plane magnetization with $\theta=90^{\circ}$ [see lower panels], one can find the the global bandgap remains closed in the typical range of tensile strains. Similarly, the global bandgap remains zero against the compressive strain. However, surprisingly, the zero nature of global bandgap converts into a non-zero one when the compressive strain is increased beyond 4%, as shown in Fig. 4(h). This bandgap progressively enlarges upon a further increase in compressive strain. Therefore, in CrI3/Bi, the quantum anomalous Hall effect with a Chern number of $\mathcal{C}=1$ can also be realized even in the presence of in-plane magnetization at a compressive strain slightly larger than 4%.

Substrate Prototype—. Finally, we considered CrI3/Bi as a representative example to search for an experimentally suitable non-magnetic substrate that guarantees the survival of Dirac points around the Fermi level. We found that MoTe2 (001) can serve as a good non-magnetic substrate SM . Our projected principles for discovering quantum anomalous Hall materials can be straightforwardly extended to other vdW 2D ferromagnetic systems. Because of the epitaxial techniques, it becomes much more convenient to artificially enlarge the family of 2D vdW ferromagnetic materials. Compared to traditional spintronic devices that are comprised of magnetic materials with some finite thickness, vdW ferromagnetic materials can simplify the device structure and shrink the device dimensions.

Summary—. Based on density functional theory, we confirmed that CrI3/Bi system is a quantum anomalous Hall insulator with a direct bandgap of approximately 30 meV at the Fermi level. We observed that CrI3/As and CrI3/Sb can open up topologically nontrivial band gaps in their conduction and valence bands, which can be properly tuned close the Fermi level via moderate electrostatic gating. Apart from the stable topology, we also found a major development in the ferromagnetic properties of CrI3, including local magnetic moments, Heisenberg exchange interactions, magnetic anisotropic energy, and Curie temperatures by coupling with atomic layers of Bi/As/Sb. Furthermore, we showed that the band gaps of our proposed systems can be efficiently manipulated via tuning spin orientation or applying strain. Our study of CrI3/X system can open new routes to explore quantum anomalous Hall effect based on 2D ferromagnetic insulators together with the potential application of CrI3-based materials in spintronics.

Acknowledgements—. This work was financially supported by the National Key R & D Program (2017YFB0405703), the NNSFC (11474265), and Anhui Initiative in Quantum Information Technologies. We also thank the supercomputing service of AMHPC and the Supercomputing Center of USTC for providing the high-performance computing resources.