Intrinsic magnetic topological insulators in van der Waals layered MnBi 2 Te 4 -family materials
Jiaheng Li, Yang Li, Shiqiao Du, Zun Wang, Bing-Lin Gu, Shou-Cheng Zhang, Ke He, Wenhui Duan, Yong Xu
aa r X i v : . [ c ond - m a t . m t r l - s c i ] A ug Intrinsic magnetic topological insulators in van der Waals layeredMnBi Te -family materials Jiaheng Li , , Yang Li , , Shiqiao Du , , Zun Wang , , Bing-Lin Gu , , ,Shou-Cheng Zhang , Ke He , , ∗ Wenhui Duan , , , † and Yong Xu , , ‡ State Key Laboratory of Low Dimensional Quantum Physics,Department of Physics, Tsinghua University,Beijing 100084, People’s Republic of China Collaborative Innovation Center of Quantum Matter,Beijing 100084, People’s Republic of China Institute for Advanced Study, Tsinghua University,Beijing 100084, People’s Republic of China Department of Physics, McCullough Building,Stanford University, Stanford, California 94305-4045, USA RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan
Abstract
The interplay of magnetism and topology is a key research subject in condensed matter physics andmaterial science, which o ff ers great opportunities to explore emerging new physics, like the quantumanomalous Hall (QAH) e ff ect, axion electrodynamics and Majorana fermions. However, these exoticphysical e ff ects have rarely been realized in experiment, due to the lacking of suitable working mate-rials. Here we predict that van der Waals layered MnBi Te -family materials show two-dimensional(2D) ferromagnetism in the single layer and three-dimensional (3D) A -type antiferromagnetism in thebulk, which could serve as a next-generation material platform for the state-of-art research. Remark-ably, we predict extremely rich topological quantum e ff ects with outstanding features in an experi-mentally available material MnBi Te , including a 3D antiferromagnetic topological insulator withthe long-sought topological axion states, the type-II magnetic Weyl semimetal (WSM) with simplyone pair of Weyl points, and the high-temperature intrinsic QAH e ff ect. These striking predictions,if proved experimentally, could profoundly transform future research and technology of topologicalquantum physics. ff ect showing dissipationless chiral edge states [4–6], the topological axion states displaying quantized magnetoelectric e ff ects [7–9], and Majoranafermions obeying non-abelian statistics [2, 10, 11]. In this context, great research e ff ort has beendevoted to explore the novel topological quantum physics, which is of profound importance tofundamental science and future technologies, like dissipationless topological electronics and topo-logical quantum computation [1, 2].One key subject that is of crucial significance to the whole research community is to developtopological quantum materials (TQMs) showing the coexistence of topology and other quantumphases (e.g. magnetism, ferroelectricity, charge density wave, and superconductivity), coined“composite TQMs” (CTQMs), which include magnetic TQMs (MTQMs) as an important subset.Currently, very limited number of MTQMs are experimentally available, including magneticallydoped TIs and magnetic topological heterostructures [6, 12–14], whose material fabrication, exper-imental measurement and property optimization are quite challenging. Since the major topologicalproperties of these existing MTQMs depend sensitively on delicate magnetic doping or proximitye ff ects, only little preliminary experimental progress has been achieved until now, leaving manyimportant physical e ff ects not ready for practical use or still unproved. For instance, the QAHe ff ect was only observed in magnetically doped (Bi x Sb − x ) Te thin films at very low workingtemperatures by fine tuning chemical compositions [6, 12]. While a previous theory predicted theexistence of topological axion states at the surface of a three-dimensional (3D) antiferromagnetic(AFM) TI [15], no 3D AFM TI has been realized experimentally, as far as we know. For futureresearch and applications, people should go beyond the existing strategy for building MTQMs andtry to design intrinsic MTQMs in no need of introducing alloy / doping or heterostructures.Noticeably, van der Waals (vdW) layered materials represent a large family of materials withgreatly tunable properties by quantum size e ff ects or vdW heterojunctions [16], in which a varietyof quantum phases in di ff erent spatial dimensions have been discovered by the state-of-the-artresearch [17–22]. For instance, both two-dimensional (2D) and 3D TI states were previously2ound in the tetradymite-type Bi Te -class materials [17, 18], and 2D intrinsic magnetism wasrecently found in ultrathin films of CrI [23] and Cr Ge Te [24] . However, these topologicalmaterials are not magnetic; those magnetic material are not topological. It is highly desirableto incorporate the magnetic and topological states together into the same vdW material, so as toobtain layered intrinsic MTQMs, which are able to inherit advantages of vdW-family materials.In this work, based on first-principles (FP) calculations, we find a series of layered intrin-sic MTQMs from the tetradymite-type MnBi Te -related ternary chalcogenides (MB T : M = transition-metal or rare-earth element, B = Bi or Sb, T = Te, Se or S), in which the intralayerexchange coupling is ferromagnetic (FM), giving 2D ferromagnetism in their septuple layer (SL);while the interlayer exchange coupling is antiferromagnetic, forming 3D A -type antiferromag-netism in their vdW layered bulk. Strikingly, we predict that plenty of novel 2D and 3D intrinsicmagnetic topological states with outstanding features could be naturally realized in an experimen-tally available material MnBi Te , including the 3D AFM TI in the bulk together with the long-sought topological axion states on the (001) surface, large-gap intrinsic QAH insulators ( E g ∼
38 meV) with dissipaltionless and patternable chiral edge states in thin films, the type-II mag-netic WSM with a single pair of Weyl points in the FM bulk, and possibly Majorana fermions ifinteracting with superconductivity.The tetradymite-type MnBi Te crystalizes in a rhombohedral layered structure with the spacegroup R m , and each layer has a triangular lattice with atoms ABC-stacked along the out-of-planedirection, the same as Bi Te . Slightly di ff erently, one layer MnBi Te includes seven atoms in aunit cell, forming a Te-Bi-Te-Mn-Te-Bi-Te SL, which can be viewed as intercalating a Mn-Te bi-layer into the center of a Bi Te quintuple layer (Fig. 1a). Noticeably, the mixing between Mn andBi in this compound would create unstable valence states Mn + and Bi + , which is energeticallyunfavorable. Thus the formation of alloys could be avoided, leading to stoichiometric compounds.This physical mechanism also explains the stability of compounds like SnBi Te and PbBi Te andis applicable to many other MB T -family materials. Importantly, our FP calculations predictedthat a series of MB T compounds are energetically and dynamically stable, including M = Ti, V,Mn, Ni, Eu, etc., as confirmed by FP total-energy and phonon calculations [25], which could befabricated by experiment. These stable compounds are characterized by an insulating band gap intheir single layers, o ff ering a collection of layered intrinsic magnetic (topological) insulators.In fact, MnBi Te is now experimentally available [26], which is selected to introduce proper-ties of the material family. Mn has a valence charge of + s electrons. The re-3aining five 3 d electrons fill up the spin-up Mn- d levels according to the Hund’s rule, introducing5 µ B magnetic moment (mostly from Mn) per unit cell. By comparing di ff erent magnetic struc-tures, we found that the magnetic ground state is a 2D ferromagnetism with an out-of-plane easyaxis in the single layer (depicted Fig. 1a), in agreement with the previous work [27]. Each Mn atomis bonded with six neighboring Te atoms, which form a slightly distorted edge-sharing octahedron.According to the Goodenough-Kanamori rule, the superexchange interactions between Mn-Te-Mnwith a bonding angle of ∼ ◦ are ferromagnetic, similar as in CrI and Cr Ge Te [23, 24]. In-tralayer ferromagnetic exchange couplings was also found in other stable MB T -family members,but the easy axis can be varied to in-plane (e.g. for M = V), displaying rich 2D magnetic features.Intriguingly, magnetic and topological states are well incorporated together into MnBi Te ,where Mn introduces magnetism and the Bi-Te layers could generate topological states similar asBi Te [17], as schematically depicted in Fig. 1b. The exchange splitting between spin-up andspin-down Mn d bands are extremely large ( > d -bands are far away from the band gap, and only Bi / Te p -bands are close to theFermi level. The single layer has a 0.73 eV direct (1.36 eV indirect) band gap when including(excluding) spin-orbit coupling (SOC) (Figs. 1c and 1d). While the single layer is topologicallytrivial ferromagnetic insulator, extremely interesting topological quantum physics emerges in thebulk and thin films, as we will demonstrate.For the layered bulk, we found that the interlayer magnetic coupling is AFM, giving a A -type AFM ground state (depicted in Fig. 2a) [25], which is similar as in Cr Ge Te [24] andexplained by the interlayer super-superexchange coupling. The spatial inversion symmetry P (i.e. P centered at O ) is preserved, but the TRS Θ gets broken. There exist two new symmetries: P Θ and S = Θ T / , where P is an inversion operation centered at O and T / is a lattice translation(depicted in Fig. 2a). The band structures with and without SOC are presented in Figs. 2b and 2c.Here every band is (at least) doubly degenerate, which is ensured by the P Θ symmetry [28]. In thepresence of S , a Z classification becomes feasible [15]. However, in contrast to the time-reversalinvariant case, there is a Z invariant for the k z = k z = π plane. Z = Z due to the P symmetry [29]. Interest-ingly, the parities of the valence band maximum (VBM) and conduction band minimum (CBM)at Γ are opposite, and both change signs by the SOC e ff ects (Figs. 2b and 2c), implying a bandinversion. By varying the SOC strength, the band gap first closes and then reopens at Γ (Fig.4d), showing simply one band inversion and thus implying a topological phase transition. TheSOC-induced band reversal is happened between Bi p + z and Te p − z , essentially the same as forBi Te [17]. Our Wannier charge center (WCC) calculations for the k z = Z = Z =
0) for with (without) SOC, confirming that MnBi Te is a 3D AFM TI. The global bandgap is ∼ Z , and the direct gap at Γ is ∼ S , which is confirmed by the surface-state calculations (Figs. 2f and 2g). The surface states are indeed gapless on the (100) termination.However, they become gapped on the (001) termination due to the S symmetry breaking.The intrinsically gapped (001) surfaces are promising for probing the long-sought topologi-cal axion states, which give the topological quantized magnetoelectric e ff ect related to an axionfield with θ = π [7–9, 15]. Previous work proposed to probe the novel states by adding oppositeout-of-plane ferromagnetism onto the bottom and top surfaces of 3D TIs [7, 14]. Such kind ofTRS-broken surface states are naturally provided by even-layer MnBi Te films (when with neg-ligible hybridizations between top and bottom surface states), benefitting from their A -type AFMstructure. An additional requirement is to open a band gap at side surfaces, which is realized inrelatively thin films or by breaking the S symmetry on the side surfaces (e.g. controlling surfacemorphology). The simplified proposal together with the suitable material candidate MnBi Te could greatly facilitate the research of axion electrodynamics.The AFM ground state of MnBi Te could be tuned, for instance, by applying an externalmagnetic field, to other magnetic structures. Then the material symmetry changes, leading todistinct topological phases. This concept is demonstrated by studying the simple FM orderingalong the out-of-plane direction (Fig. 3a). The FM structure has P but neither Θ nor P Θ , leadingto spin-split bands, nonzero Berry curvatures and possibly nonzero topological Chern numbers thatcorrespond to novel topological phases, like 3D QAH insulators [30] and WSMs [3, 22, 31–33].The band structure of FM MnBi Te (Fig. 3b) displays a pair of band crossings at W / W ′ along the Z - Γ -Z line. The band crossings are induced by interlayer orbital hybridizations and protected bythe C rotational symmetry. Our WCC calculations found that W is a momentum-space monopolewith a topological charge of + π ) (Fig. 3e), and its time-reversal partnerW ′ has an opposite topological charge of -1, indicating that the system is a topological WSM.While in most of the momentum space the electron pocket is located above the hole pocket (e.g.along the F-W-L line), the Weyl cones get tilted along the Z - Γ -Z line (Fig. 3d), leaving somepart of the electron pocket below the hole pocket, which is the characteristic feature of the type-II5SM [22]. In contrast to the time-reversal-invariant WSMs that must have even pairs of Weylpoints, this ferromagnetic WSM represents the simplest one, hosting only a pair of Weyl points.Moreover, our surface-state calculations clearly demonstrated the existence of Fermi arcs on the(100) and (¯100) terminations (Figs. 3f and 3g), which is the fingerprint of the WSM. Importantly,the Weyl points are well separated in the momentum space and very close to the Fermi level,advantageous for experimental observations.The vdW layered materials are featured by tunable quantum size e ff ects. For AFM MnBi Te films, even layers do not possess P and Θ symmetries, but have P Θ , ensuring double degeneracyin every band. Di ff erently, odd layers have P but neither Θ nor P Θ , leading to spin-split bands.The di ff erent symmetries lead to distinct topological properties in even and odd layers. Specifi-cally, the topological Chern number C = P Θ in even layers, C , ff ects, we found that the 5-layer film is an intrinsic QAH insulator with C =
1, as confirmed by the appearance of an quantized Hall conductance and chiral edge stateswithin the bulk gap (Figs. 4a-c). We also calculated a 7-layer film, which is also a QAH insulatorwith C = ff ects in the Bi-Te bands cooperatively induce the QAH e ff ect, similar as in the magneticallydoped Bi Te -class TIs [5]. However, the present material has an intrinsic magnetism and doesnot need uncommon mechanisms (e.g. the Van-Vleck mechanism [5]) to form ferromagnetism ininsulating states. Moreover, since the magnetic doping is not needed, disorder-induced magneticdomains and potential fluctuations, that deteriorate the QAH e ff ect, are avoided. Furthermore, theQAH gap of the 5-layer film is 38 meV, greater than the room-temperature thermal energy of 26meV, enabling a high working temperature.The thickness dependence behaviors can be understood as follows. There are intrinsicallygapped surface states on both sides of thick films, as obtained for the (001) semi-infinite surface.These surface gaps are opened by the TRS-breaking field near the surface, which have half quan-tized Hall conductances σ xy = e / h or − e / h when the magnetism in the surface layer is up- ordown-oriented, respectively. Thus, the Hall conductances of bottom and top surfaces is cancelledin even layers, giving axion insulators with C = C =
1. This physical picture,consistent with our calculation results, suggests an oscillation of C in even and odd layers. Thus6hiral edge states always appear near step edges (Fig. 4d), which can be used for dissipationlessconduction. Based on this unique feature, the chiral edge states could be selectively patterned bycontrolling the film thickness or step edges, advantageous for building dissipationless circuits.Looking back to the history of the TI research, the first- and second-generation TIs are theHgTe / CdTe quantum wells [34] and Bi-Sb alloys, respectively, which are very complex and dif-ficult to study theoretically and experimentally. Research interests have been increased exponen-tially since the discovery of the third-generation TIs in the intrinsic Bi Te -class materials [17, 18].A very similar situation is faced by the research of magnetic topological physics. Currently, ex-perimental works are majorally based on magnetically doped TIs and magnetic topological het-erostructures, which are quite challenging and have led to little preliminary progress. Looking for-wards, the research progress is expected to be greatly prompted by discovering intrinsic MTQMsthat are simple and easy to control. The vdW layered MnBi Te -family materials satisfy all thesematerial traits. More importantly, this material family could host extremely rich topological quan-tum states in di ff erent spatial dimensions (like 3D AFM TIs, time-reversal invariant TIs and topo-logical semimetals, 2D QAH and QSH insulators, etc.) and are promising for investigating otherexotic emerging physics (like Majorana fermions), which are thus perfect next-generation MTQMsfor future research. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected][1] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. , 3045 (2010).[2] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. , 1057 (2011).[3] N. Armitage, E. Mele, and A. Vishwanath, Rev. Mod. Phys. , 015001 (2018).[4] F. D. M. Haldane, Phys. Rev. Lett. , 2015 (1988).[5] R. Yu, W. Zhang, H.-J. Zhang, S.-C. Zhang, X. Dai, and Z. Fang, Science , 61 (2010).[6] C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, et al.,Science , 167 (2013).[7] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B , 195424 (2008).[8] F. Wilczek, Phys. Rev. Lett. , 1799 (1987). IG. 1: Monolayer MnBi Te (MBT). (a) The side view of monolayer MBT. (b) A schematic diagram ofband structure in MBT including thin films and 3D bulk. (c,d) Band structures of monolayer MBT calculatedby HSE06 without / with spin-orbital coupling (SOC). The blue (green) curve in the (1c) represents the spin-up (spin-down) states. (e) A schematic diagram of XBT materials family. The red arrows represent themagnetic moment, whose length and direction represent the magnitude and easy magnetic axis of magneticatoms. IG. 2: AFM MBT. (a) The side view of bulk AFM MBT crystal structure.The black dot O representsinversion center located at the magnetic Mn atoms. The black dot O represents symmetry operation whichcombines inversion operator and the lattice translation T / along c axis. The schematic diagram showsMBT is stacked in the way combining the AFM order and ABC-stacking sequence. Brillouin zone used inthe calculation are showed below. (b), (c) Band structures of 3D bulk AFM MBT without / with SOC. Theparity of wavefunction at high-symmetry points Γ is showed by the minus and plus signs. Minus (plus) signmeans odd (even) parity. (d) Band gap at Γ calculated by artificially changing the strength of SOC. Bandgap closing happens at artificial SOC strength around 81%. (e) Wannier charge centers along k on the k = , 146805 (2009).[10] L. Fu and C. L. Kane, Phys. Rev. Lett. , 096407 (2008).[11] Q. L. He, L. Pan, A. L. Stern, E. C. Burks, X. Che, G. Yin, J. Wang, B. Lian, Q. Zhou, E. S. Choi,et al., Science , 294 (2017).[12] C.-Z. Chang, W. Zhao, D. Y. Kim, H. Zhang, B. A. Assaf, D. Heiman, S.-C. Zhang, C. Liu, M. H.Chan, and J. S. Moodera, Nat. Mater. , 473 (2015).[13] F. Katmis, V. Lauter, F. S. Nogueira, B. A. Assaf, M. E. Jamer, P. Wei, B. Satpati, J. W. Freeland,I. Eremin, D. Heiman, et al., Nature , 513 (2016). IG. 3: FM MBT. (a) Atomic structure of FM MBT. Dashed rhombus represents the unit cell. (b) Bandstructure of FM MBT. (c) Brillouin zone of FM MBT. The red and blue points represent a pair of type-IIWeyl points W and W ′ . (d) Zoom-in band structure around Weyl point W from di ff erent directions. (e)Motion of the sum of WCCs on the sphere around Type-II Weyl point W. (f, g) Fermi arcs on the (100) and(¯100) terminations. Fermi energy is fixed at the energy level of Weyl point W.FIG. 4: MBT thin films. (a) Band structure, (b) Hall conductance σ xy as a function of energy and (c) edgestates of the 5-layer MBT. (d) A schematic diagram showing dissipationless edge channels on step edges ofMBT.
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