Magnetically Controllable Topological Quantum Phase Transitions in Antiferromagnetic Topological Insulator MnBi$_2$Te$_4$

TL;DR

This paper predicts magnetically controllable topological quantum phase transitions in MnBi₂Te₄, revealing how magnetic manipulation can induce various topological phases and transitions in this antiferromagnetic topological insulator.

Contribution

It demonstrates, through first-principles calculations, that magnetic orientation control can induce multiple novel topological phases and phase transitions in MnBi₂Te₄.

Findings

Magnetic orientation affects material symmetries and topological phases.

Predicted phase transitions include AFM TI to mirror topological crystalline insulator.

Transitions from type-II to type-I Weyl semimetal and axion to Chern insulator.

Abstract

The recent discovery of antiferromagnetic (AFM) topological insulator (TI) MnBi$_2$Te$_4$ has triggered great research efforts on exploring novel magnetic topological physics. Based on first-principles calculations, we find that the manipulation of magnetic orientation and order not only significantly affects material symmetries and orbital hybridizations, but also results in variant new magnetic topological phases in MnBi$_2$Te$_4$. We thus predict a series of unusual topological quantum phase transitions that are magnetically controllable in the material, including phase transitions from AFM TI to AFM mirror topological crystalline insulator, from type-II to type-I topological Weyl semimetal, and from axion insulator to Chern insulator. The findings open new opportunities for future research and applications of magnetic topological materials.

Figures

Tables

Table 1. Table 1: Material symmetries of MnBi 2 Te 4 bulk with different magnetic configurations.

	$P$	$P𝒯$	$M_x$	$C_{3z}$	$S=T_{1/2}𝒯$
AFM-$z$	$✓$	$✓$	$\times$	$✓$	$✓$
AFM-$x$	$✓$	$✓$	$✓$	$\times$	$✓$
FM-$z$	$✓$	$\times$	$\times$	$✓$	$\times$
FM-$x$	$✓$	$\times$	$✓$	$\times$	$\times$

Full text

The subject of antiferromagnetic (AFM) topological insulators (TIs) has attracted enormous research interests in condensed matter physics and materials science Šmejkal et al. (2018); Tokura et al. (2019), since the first theoretical model proposed in 2010 Mong et al. (2010). AFM TIs are new states of quantum matter characterized by a $\mathbb{Z}_{2}$ topological invariant Mong et al. (2010); Fang et al. (2013); Liu (2013), in analogy to time-reversal invariant (TRI) TIs Hasan and Kane (2010); Qi and Zhang (2011); Bansil et al. (2016). Differently, their topological surface states can be inherently gapped Mong et al. (2010); Fang et al. (2013); Liu (2013). Remarkably, the surface gap is originated from the interplay between massless Dirac-like fermions and intrinsic magnetization, which is able to create the novel half-quantum Hall effect on the surface Qi and Zhang (2011); Tokura et al. (2019) and the long-sought topological axion states Qi et al. (2008); Wilczek (1987); Essin et al. (2009). Therefore, AFM TIs provide an ideal platform to explore exotic topological quantum phenomena, including axion electrodynamics, topological magnetoelectric effects, the quantum anomalous Hall (QAH) effect and topological superconductivity Qi and Zhang (2011); Qi et al. (2008); Wilczek (1987); Essin et al. (2009); Peng and Xu (2018). The same kind of phenomena, in principle, could be realized in TRI TIs, which, however, relies critically on extrinsic magnetic effects (such as magnetic field, doping or heterojunctions) Chang et al. (2013); Nomura and Nagaosa (2011); Morimoto et al. (2015); Wang et al. (2015); Mogi et al. (2017a, b); Xiao et al. (2018), making corresponding material systems difficult to fabricate and control in experiments. In contrast, AFM TIs with intrinsic magnetic effects are much superior for experiments and applications. Very recently, the AFM TI states have been theoretically and experimentally discovered in van der Waals layered material MnBi2Te4 Li et al. (2018); Zhang et al. (2018); Gong et al. (2018); Otrokov et al. (2018), which was proposed to host plenty of intriguing magnetic topological phases from two dimensions (2D) to 3D. The findings enable exploring the emergent quantum physics in real materials, which has stimulated intensive effort in the field Cui et al. (2019); Otrokov et al. (2019); Zeugner et al. (2019); Peng and Xu (2018); Lee et al. (2018); Yan et al. (2019); Vidal et al. (2019); Yu et al. (2019); Chen et al. (2019); Wu et al. (2019); Zhang et al. (2019a).

One key subject essential to the study of topological physics is to effectively manipulate topological material properties and control quantum phase transitions. Various approaches have been developed by tuning electronic structure, for instance, via strain, electric gating, quantum confinement Hasan and Kane (2010); Qi and Zhang (2011); Bansil et al. (2016). For AFM TIs, there exist additional degrees of freedom in magnetic structure, which are well known to be highly tunable and play an important role in determining electronic properties Šmejkal et al. (2018); Tokura et al. (2019). In fact, the variance in magnetic order can significantly change material symmetries. For instance, a transition from AFM to ferromagnetic (FM) state breaks $P\mathcal{T}$ symmetry (i.e. the combination of space inversion $P$ and time reversal $\mathcal{T}$), leading to spin-split bands. Moreover, a change of magnetic orientation can result in distinct symmetry selection rules and invoke different topological classifications Schnyder et al. (2008), which could induce distinct topological phases. The influence of magnetic structure on topological material properties is thus expected to be crucial, which, however, has not been well studied for AFM TIs.

In the work, we systematically study the influence of magnetic structure on electronic and topological properties of AFM TI MnBi2Te4 by first-principles calculations (see Methods in Supplemental Material SM ). We find that the interlayer coupling of MnBi2Te4 is strongly restricted by $P\mathcal{T}$ symmetry and can be significantly enhanced by symmetry breaking, which emphasizes the important role of $P\mathcal{T}$ symmetry in layered materials. Furthermore, we demonstrate that the change of magnetic order and orientation, which is controllable by applying magnetic field, has profound effects on orbital hybridization and band structure of MnBi2Te4. Noticeably, variant exotic topological states, including AFM mirror topological crystalline insulator (TCI) and type-I topological Weyl semimetal (WSM) in 3D as well as Chern insulator in 2D, can be realized in MnBi2Te4 by tuning magnetic structure. Unusual quantum phase transitions between these intriguing topological states are accessible, which could not only greatly enrich our understanding of magnetic topological physics but also inspire the design of new functional devices.

MnBi2Te4 is a van der Waals layered material crystallized in a rhombohedral structure with space group $R\overline{3}m$ Lee et al. (2013), formed by ABC-stacked Te-Bi-Te-Mn-Te-Bi-Te septuple layers (SLs). The ground state of this material has A-type AFM order with an out-of-plane easy axis, displaying FM order within SL and AFM order between neighboring SLs Li et al. (2018); Zhang et al. (2018); Gong et al. (2018); Otrokov et al. (2018), as schematically displayed in Fig. 1. In nonmagnetic state, there exist some key material symmetries, including rotational symmetry $C_{3z}$, mirror symmetry $M_{x}$, space inversion $P$ and time reversal $\mathcal{T}$ Li et al. (2018); Zhang et al. (2018). Here the $x$ and $z$ axes are defined along the in-plane and out-of-plane directions, respectively (Fig. 1). When the AFM order is formed in MnBi2Te4 below the N$\mathrm{\acute{e}}$el temperature ($\sim$25 K) Gong et al. (2018); Otrokov et al. (2018); Cui et al. (2019); Zeugner et al. (2019); Lee et al. (2018); Yan et al. (2019); Chen et al. (2019), $M_{x}$ and $\mathcal{T}$ get broken, whereas $P\mathcal{T}$ is preserved and a new symmetry $S=T_{1/2}\mathcal{T}$ emerges ($T_{1/2}$ represents a half magnetic-unit-cell translation) Mong et al. (2010). In experiments, different magnetic orders and orientations could be realized by applying magnetic field. For instance, magnetic orientation could be tuned from out-of-plane to in-plane under a moderate magnetic field, and even further magnetic order can be driven from AFM to FM under a relatively strong magnetic field ($\sim$7 Tesla) Gong et al. (2018); Cui et al. (2019); Lee et al. (2018). Different magnetic configurations (Fig. 1), labeled as AFM-$z$, AFM-$x$, FM-$z$ and FM-$x$ ($x$ and $z$ represent magnetic orientation), are thus accessible, which manifest different material symmetries as summarized in Table 1. Note that ordered AFM-$x$ states are not easy to realize in MnBi2Te4. However, there are some intrinsic magnetic insulators in the MnBi2Te4 family, like XBi2Te4 (X=V, Ni, Eu), whose native magnetic ground state is AFM-$x$ Li et al. (2018).

Let us first consider 2-SL MnBi2Te4 with magnetic configurations of AFM-$z$ and FM-$z$ and discuss the influence of magnetic order on interlayer coupling. The influence is visualized by the differences in their band structures (Figs. 2a and 2b), since both systems would approach the same monolayer limit in the absence of interlayer coupling. It is well known that FM interlayer coupling gives spin-split bands, while AFM interlayer coupling introduces spin-degenerate bands Tang et al. (2016). Generally the coupling between two separated FM SLs, each of which is spin-split by itself, would create four bonding and antibonding states. However, only two copies of states are generated by AFM interlayer coupling, which seems unusual. Detailed analysis finds that with $P\mathcal{T}$ symmetry, the top-layer spin-up state $|t,\uparrow\rangle$ and the bottom-layer spin-down state $|b,\downarrow\rangle$ are degenerate in energy, which would typically couple strongly with each other. While based on $|b,\downarrow\rangle=P\mathcal{T}|t,\uparrow\rangle$ and $(P\mathcal{T})^{2}=-1$, $\langle t,\uparrow|H|b,\downarrow\rangle=\langle t,\uparrow|HP\mathcal{T}|t,\uparrow\rangle=-\langle t,\uparrow|H|b,\downarrow\rangle$, implying that the two states are forbidden to couple by $P\mathcal{T}$ (Fig. 2d). This unique feature, as far as we know, has not been discussed before. The analysis suggests that the interlayer coupling is strongly restricted by $P\mathcal{T}$ symmetry. This mechanism generally works for layered materials, which well demonstrates symmetry-determined interactions in quantum materials.

Conversely, interlayer coupling would be significantly enhanced by breaking $P\mathcal{T}$ symmetry, which is well demonstrated by changing magnetic order from AFM to FM. A topological band inversion between Bi $p_{z}^{+}$ and Te $p_{z}^{-}$ appears in the FM state due to the enhanced interlayer coupling (Fig. 2e), leading to a Chern insulator phase. Edge-state calculations find gapless chiral edge modes within the bulk gap (Fig. 2h), in agreement with previous results Otrokov et al. (2019). In contrast, there is no such kind of band inversion for the AFM state, explained by restricted interlayer coupling. Therefore, when applying an out-of-plane magnetic field, a magnetic transition from AFM to FM state would be accompanied by a quantum phase transition from trivial insulator to Chern insulator.

The change of magnetic orientation also has important effects on electronic and topological properties. This is exemplified by studying 2-SL MnBi2Te4 with different magnetic orientations. We define a polar angle $\theta$ to quantify magnetic orientation and set $\theta=0^{\circ}$ ($\theta=90^{\circ}$) for FM-$z$ (FM-$x$). Their band structures are noticeably different, especially for valence bands (cf. Figs. 2b and 2c), implying strong influence of magnetic orientation on orbital hybridizations. Importantly, $M_{x}$ is broken for FM-$z$, while it is preserved for FM-$x$. The existence of $M_{x}$ requires a vanishing Hall conductance $\sigma_{xy}$ Liu et al. (2013), which thus suggests that FM-$x$ system has Chern number $\mathcal{C}=0$. The symmetry argument is supported by orbital analysis that finds no topological band inversion (Fig. 2f) as well as by edge-state calculations that show no chiral edge state (Fig. 2i). Remarkably, by gradually tuning magnetic orientation from out-of-plane to in-plane, a topological quantum phase transition from $\mathcal{C}=1$ to $\mathcal{C}=0$ and further to $\mathcal{C}=-1$ would occur, which is accompanied by closing and reopening of band gap (Fig. 2g). The magnetically tunable band gap and topological quantum phase transition could be useful for designing new functional devices, such as quantum topological transistors Molle et al. (2017).

Next we turn to discuss properties of MnBi2Te4 bulk. Band structures of AFM-$z$ and AFM-$x$ states differ noticeably (Fig. 3a). This is possibly caused by their different symmetries (Table 1), which impose different symmetry selection rules and thus result in distinct orbital hybridizations. Nevertheless, topological band inversion between Bi $p_{z}^{+}$ and Te $p_{z}^{-}$ remains unaffected. Thus, AFM-$x$ is an AFM TI characterized by $\mathbb{Z}_{2}=1$ and protected by $S$, the same as AFM-$z$. They both have gapless surface states on the side surfaces that have preserved $S$ symmetry (Fig. 3b). However, significant differences occur on the (111) surface. The surface states are gapped for AFM-$z$ order, which give a half-integer quantum Hall conductance $\sigma_{xy}=e^{2}/2h$Qi and Zhang (2011); Tokura et al. (2019). In contrast, the surface gap of AFM-$x$, if existing, would also induces nonzero $\sigma_{xy}$, which contradicts with $M_{x}$ symmetry. The surface gap thus should be zero, as shown by our surface-state calculations (Fig. 3c). These surface states are nearly isotropic in 2D momentum space, in contrast to anisotropic Dirac-like states on the side (110) surface (Fig. 3d).

The AFM-$x$ state of MnBi2Te4 bulk hosts a new topological phase (see detailed discussions in Supplemental Material SM ), called AFM mirror TCI, which has gapless (111) surface protected by $M_{x}$ symmetry and gapless side surfaces protected by $S$ symmetry. The gapless (111) surface of AFM mirror TCI is of particular interest from the point view of symmetry and topology. The surface magnetism typically introduces a mass term into the Dirac-like surface states by breaking $\mathcal{T}$. However, this mass term is enforced to zero by mirror symmetry. Based on this unique feature, one can obtain $\mathcal{T}$-breaking mass term by simply tuning mirror symmetry breaking, such as via strain, or gradual change of magnetic orientation (supplementary Fig. S1). Moreover, one may introduce another type of mass term, for instance, by superconducting proximity effect, which could induce topological superconductivity Peng and Xu (2018). Therefore, rich topological quantum physics would emerge by controlling symmetry breaking. Note that some intrinsic magnetic insulators, like XBi2Te4 (X=V, Ni, Eu), have magnetic ground state AFM-$x$ Li et al. (2018). The novel AFM mirror TCI phase might be realized intrinsically in these candidate materials.

Topological properties of bulk MnBi2Te4 greatly change when tuning magnetic order from AFM to FM. The FM-$z$ state corresponds to type-II WSM Li et al. (2018); Zhang et al. (2018), in which Lorentz invariance is violated and some parts of electron pockets are located below hole pockets in the Weyl cone Soluyanov et al. (2015). Importantly, only one single pair of Weyl points (WPs) exist in the material, which are located along the $\Gamma$-$Z$ direction as ensured by $C_{3z}$ rotational symmetry and related to each other by $P$ Li et al. (2018); Zhang et al. (2018). When rotating magnetic orientation away from $z$ axis by applying magnetic field, $C_{3z}$ gets broken. Hence, WPs would shift away from $\Gamma$-$Z$ to general $\bf{k}$ points. Figure 4a displays the motion of WPs in momentum space when varying magnetic orientation (see detailed data in Supplemental Material SM ). Quantum phase transition from type-II to type-I WSM phase happens at $10^{\circ}<\theta<20^{\circ}$. When $\theta=90^{\circ}$ (FM-$x$), two WPs eventually merge, leading to annihilation of Weyl fermions and a topological transition to trivial phase. Such kind of phase transition behaviors are rationalized by the fact that Zeeman field induced band splitting is much larger for FM-$z$ than for FM-$x$ in this layered structure, and thus varying $\theta$ considerably changes band dispersions near WPs (Fig. 4b).

Remarkably, the type-I WSM phase in MnBi2Te4 is the simplest WSM, which has only a single pair of WPs located exactly at the charge neutral point. For comparison, we show band structures for $\theta=10^{\circ}$ and $\theta=50^{\circ}$ as representative examples of type-II and type-I WSMs (Figs. 4c and 4d), and present their calculated surface states on the ($1\overline{1}0$) termination (Figs. 4e and 4f). While Fermi arcs and residual electron pockets around $\Gamma$ simultaneously appear at the isoenergy level of WPs in the type-II case, only Fermi arcs with no interfering signal from bulk states can be obtained in the type-I case. In contrast to other WSM materials with multiple pairs of WPs and complex Fermi surfaces Wan et al. (2011); Weng et al. (2015); Soluyanov et al. (2015); Armitage et al. (2018), this material hosts the simplest WSM phase, which is more advantageous for transport measurements and other studies of Weyl physics.

Finally we discuss topological properties of MnBi2Te4(111) films in the FM-$z$ state (see results for other magnetic states in Supplemental Material SM ). Their ultrathin films show similar band structures (supplementary Fig. S3), which are in the Chern insulator phase as shown by our calculations of anomalous Hall conductance (Fig. 5) and bandgap as a function of spin-orbit coupling strength (supplementary Fig. S4). Chern numbers and bandgaps (about 50-80 meV) for varying thickness in the FM-$z$ state are summarized in supplementary Table S3. In principle, thick films can have high Chern numbers, which is an inherent feature of topological WSM. In the long wavelength limit, Chern number of MnBi2Te4 films is estimated to be $\mathcal{C}\approx|k^{z}_{\textrm{W}}|d/\pi$, where $k^{z}_{\textrm{W}}$ is the $z$-component wave vector of WP in MnBi2Te4 bulk and $d$ is the film thickness. Therefore, when changing magnetic order from AFM to FM, MnBi2Te4 films can display topological quantum phase transition from axion insulators to Chern insulators of varying Chern numbers.

In conclusion, through an example study of MnBi2Te4, we demonstrate that the symmetry, orbital hybridization and band dispersion of AFM TI are highly controllable by magnetic field, thus leading to quantum phase transitions to variant exotic topological phases, such as AFM mirror TCI and type-I topological WSM. This creates new opportunities for future research. For instance, the tunable Weyl fermions in FM MnBi2Te4 can host some novel quantum phenomena, like the black-hole-horizon analogy emerging at the Lifshitz transition between type-I and type-II WSM phases Volovik (2017); Huang et al. (2018) and 3D quantum Hall effects Zhang et al. (2019b); Tang et al. (2019) without Landau levels.

Note added: Recently, magnetic field induced Chern insulators  Deng et al. (2019) and quantum phase transition from axion insulator to Chern insulator  Liu et al. (2019) were realized experimentally in few-layer MnBi2Te4.

We thank Yayu Wang and Jinsong Zhang for helpful discussions. This work is supported by the Basic Science Center Project of NSFC (Grant No. 51788104), the Ministry of Science and Technology of China (Grants No. 2016YFA0301001, No. 2018YFA0307100, and No. 2018YFA0305603), the National Natural Science Foundation of China (Grants No. 11674188 and No. 11874035) and the Beijing Advanced Innovation Center for Future Chip (ICFC).