Magnetized topological insulator multilayers Chao Leia, Shu Chena,b , and Allan H. MacDonalda,1 aDepartment of Physics, The University of Texas at Austin, Austin, TX 78712; and bDepartment of Physics, Shanghai University, Shanghai 200444, China Contributed by Allan H. MacDonald, September 18, 2020 (sent for review July 6, 2020; reviewed by Rob McQueeney and Jian Wang) We discuss the magnetic and topological properties of bulk crys- with theoretical predictions (9, 12), neutron scattering mea- tals and quasi–two-dimensional (quasi-2D) thin films formed by surements (17, 26, 30) show that bulk MnBi2X4 (MBX) has stacking intrinsic magnetized topological insulator (for example, A-type antiferromagnetic order with Mn ions ordered ferromag- Mn (SbxBi1−x)2X4 with X = Se,Te) septuple layers and topolog- netically within each septuple layer and antiferromagnetically ical insulator quintuple layers in arbitrary order. Our analysis between adjacent septuple layers. Because the antiferromag- makes use of a simplified model that retains only Dirac cone netic interactions between septuple layers are weak, the Mn degrees of freedom on both surfaces of each septuple or quintu- layer moments in thin films can be aligned by magnetic fields ple layer. We demonstrate the model’s applicability and estimate ∼5 T. Thin films can be obtained either by epitaxial growth its parameters by comparing with ab initio density-functional the- or by mechanical exfoliation from bulk crystals (8, 30, 31, 34– ory (DFT) calculations. We then employ the coupled Dirac cone 39). Magnetic fields establish quantum Hall effects in thin model to provide an explanation for the dependence of thin-film films not by establishing Landau quantization, but by chang- properties, particularly the presence or absence of the quantum ing the magnetic configuration from antiferromagnetic to fer- anomalous Hall effect, on film thickness, magnetic configuration, romagnetic through overcoming the weak interlayer exchange and stacking arrangement, and to comment on the design of Weyl interactions (21, 23, 29). superlattices. In this paper, we develop a simple model that can be used to address the properties of thin films and bulk crys- Sb Bi X MnBi Te j magnetized topological insulator j topological superlattices j tals formed by stacking Mn ( x 1−x )2 4 septuple layers and 2 4 Sb , Bi X quantum anomalous Hall effect j Weyl semimetal ( x 1−x )2 3 quintuple layers in arbitrary order. The model contains only Dirac cone degrees of freedom on each surface of each septuple or quintuple layer. By comparing with ab ini- opological insulator (TI) thin films in which time-reversal tio density-functional theory (DFT) calculations we are able to Tsymmetry is broken by magnetic order (1) have long been establish that this highly simplified description is usually accurate recognized as a promising platform for the interplay between and also to fix values of the model’s material-dependent parame- transport and magnetic properties that powers spintronics. ters. The advantages of the simplified model are that it facilitates Indeed the quantum anomalous Hall (QAH) effect, a high point the descriptions of crystals with complex stacking sequences and of topological spintronics characterized by dissipationless trans- thin films containing many layers and more importantly that it port, was first observed (2, 3) in magnetic topological insulator allows trends in magnetic and topological properties across the (MTI) thin films with ferromagnetic order, and strong magneto- family of materials to be recognized and understood. Important electric response properties are expected in antiferromagnetic materials variations include changes in the Sb fraction x on the TI films (4–6). MTIs were first produced simply by doping pnictogen sites; substitutions of X = S, Se, Te on the differ- (Sbx Bi1−x )2X3 TI thin films with magnetic elements. How- ent chalcogen sites; and growth-controlled defect concentrations. ever, disorder, thought to be due mainly to inhomogeneity of the magnetic dopants (7), leads to complex magnetic order in Significance these systems. As a consequence the QAH effect appears only at extremely low temperatures, for example only at ∼30 mK in Topological insulators have Dirac cone surface states pro- Cr-doped Bi2Te3, even though the Curie temperature is ∼15 K tected by time-reversal symmetry (TRS). Weak magnetic order (3). For this reason the recent identification (8) of the Mn in topological insulator thin films can lead to the quan- (Sbx Bi1−x )2X4 family of layered van der Waals materials, which tum anomalous Hall effect (QAHE), which was first real- can be viewed as MTIs that have magnetic moments on an ized by magnetic doping of topological insulators to estab- ordered lattice, is a promising advance. lish a fragile ferromagnetic state. Recent work has estab- Important progresses have been made in understanding the lished MnBi2Te4 as a structurally ordered magnetic topolog- bulk and epitaxial thin film properties of this family of materi- ical insulator. We report on a theoretical study of superlat- als, both theoretically (8–13) and experimentally (14–31). The tices formed from MnBi2Te4 and Bi2Te3 building blocks that quantum anomalous Hall effect has now been observed in the employs a simplified model validated by ab initio density- presence of a relatively weak magnetic field ∼5 T for thick- functional theory. We use it to shed light on the depen- nesses ranging from 3 to 10 septuple layers (21, 23, 29), and large dence of the quantum anomalous Hall effect on film thick- (almost quantized) anomalous Hall effects have been observed ness, magnetic configuration, the stacking sequences of the in the absence of an external magnetic field in high-quality 5- van der Waals coupled building blocks, and bulk electronic septuple-layer MnBi2Te4 films (23), all at temperatures exceed- structure. ing 1 K. The ratio of the quantum anomalous Hall temperature to the magnetic-ordering temperature, ∼20 K, is much higher Author contributions: C.L. and A.H.M. designed research; C.L., S.C., and A.H.M. per- (29) than in doped MTI films. Some (10, 16, 18, 32) [but not all formed research; C.L., S.C., and A.H.M. analyzed data; and C.L. and A.H.M. wrote the (26–28, 33)] photoemission experiments have identified the large paper.y surface state gaps ∼100 meV that are generally expected (10, 15) Reviewers: R.M., Iowa State University; and J.W., Peking University.y theoretically in MTIs. The authors declare no competing interest.y Mn (Sbx Bi1−x )2X4 is a layered material composed of seven- Published under the PNAS license.y layer X-(B, Sb)-X-Mn-X-(Bi, Sb)-X units that are coupled by 1 To whom correspondence may be addressed. Email: [email protected] weak van der Waals interactions. These septuple layers may This article contains supporting information online at https://www.pnas.org/lookup/suppl/ be viewed as (Bi, Sb)2X3 quintuple layers in which the mid- doi:10.1073/pnas.2014004117/-/DCSupplemental.y dle X layer is replaced by an X-Mn-X trilayer. In agreement First published October 19, 2020. 27224–27230 j PNAS j November 3, 2020 j vol. 117 j no. 44 www.pnas.org/cgi/doi/10.1073/pnas.2014004117 Downloaded by guest on September 27, 2021 Our model can also be used as a starting point for theories that cones result from exchange interactions with Mn local moments account for gating electric fields, external magnetic fields, disor- and break time-reversal symmetry der, and other perturbations that are difficult to describe using X ab initio approaches. mi = JiαMα, [2] In this paper we focus on Bi2X3 and MnBi2X4 with X = α Se, Te, which have received greatest attention to date. We find that bulk ferromagnetic MBX is a nearly ideal Weyl semimetal, where α is a layer label. We limit our attention here to the that thin-film ferromagnets are two-dimensional (2D) Chern case of magnetization perpendicular to the van der Waals layers insulators with Chern numbers that grow and gap sizes that so that Mα = ±1 specifies the sense of magnetization on layer decline with film thickness, and that antiferromagnetic thin films α if it is magnetic, and Mα = 0 on nonmagnetic layers. Each with sufficiently large odd layer numbers are Chern insula- Dirac cone can have two nearby Mn layers, one from the same tors with Chern number jC j = 1. We further find that ferro- layer with exchange splitting JS and one from the adjacent layer magnetic [MnBi2X4]M [Bi2X3]N superlattices formed by insert- with exchange splitting JD , shown as in Fig. 1, if these layers ing nonmagnetic quintuple layers in the stack are ordinary are magnetic. In our surveys of MTI superlattice properties we insulators for M =N smaller than about 3, but become Weyl retain only ∆S , ∆D , JS , and JD , which are normally dominant, semimetals for larger M =N . For superlattices with (M,N) = as model parameters; the model allows different values of ∆S (1,1), there is a large chance to achieve a Weyl semimetal in TI and MTI layers and allows ∆D at magnetic/nonmagnetic phase with MnBi2Te4 as magnetic topological insulator lay- heterojunctions to differ its value at magnetic/magnetic or ers and Bi2Se3, Bi2Sb3, or Sb2Te3 as topological insulator nonmagnetic/nonmagnetic heterojunctions. layers. The model simplifies when only one type of layer is present. When that layer is magnetic and the magnetic configuration is Coupled Dirac Cone Model ferromagnetic, our model reduced to the topological insulator We construct a model for a magnetized topological insulator multilayer model proposed by Burkov and Balents (42). Instead multilayer by including Dirac cone degrees of freedom not only of inserting a normal insulating layer between magnetic topolog- on the surface layers (40, 41), but also on the top and bottom of ical insulator layers to form a flexible three-dimensional system, each magnetic septuple layer and nonmagnetic quintuple layer as imagined in ref.