Recent advancements in the study of intrinsic magnetic topological insulators and magnetic Weyl semimetals

The studies of topological insulators and topological semimetals have been at frontiers of condensed matter physics and material science. Both classes of materials are characterized by robust surface states created by the topology of the bulk band structures and exhibit exotic transport properties. When magnetism is present in topological materials and breaks the time-reversal symmetry, more exotic quantum phenomena can be generated, e.g. quantum anomalous Hall effect, axion insulator, large intrinsic anomalous Hall effect, etc. In this research update, we briefly summarize the recent research progresses in magnetic topological materials, including intrinsic magnetic topological insulators and magnetic Weyl semimetals.


I. Introduction
Magnetic topological materials, including magnetic topological insulators (TI) and magnetic topological semimetals, have attracted broad interests. Magnetic TIs can be achieved in three different ways: magnetic doping in a TI [1,2], proximity of a TI to a ferromagnetic (FM) or an antiferromagnetic (AFM) insulator [3,4,5], or creating intrinsic FM or AFM order in a TI [6].
The spontaneous magnetization induced in magnetic TIs interacts with topological surface states and opens a gap at the surface Dirac point, which can generate a new topological quantum statequantum anomalous Hall insulator (QAHI), when chemical potential is tuned to an appropriate value in thin film samples. Since QAHI features spin-polarized chiral edge state, which can support dissipationless current, it carries great promise for applications in future energy saving electronics. QAHI was first realized in thin TI films of Cr-and/or V-doped (Bi,Sb)2Te3 [1,2].
This pioneering work has generated a great deal of interest and several review articles [7,8,9,10,11] on this topic have been published. In this research update, we will focus on reviewing recent studies on intrinsic magnetic TI MnBi2Te4 and its related materials.
In magnetic topological semimetals, the interplay between magnetism and non-trivial band topology can also generate new exotic quantum states. One remarkable example is time reversal symmetry (TRS) breaking Weyl semimetal (WSM) state in which linearly dispersed, spin-split bands cross at discrete momentum points, thus resulting in Weyl nodes. Low energy excitations near Weyl nodes behave as chiral Weyl fermions. Weyl nodes always come in pairs with opposite chirality and they can be understood as source and drain of Berry curvature in momentum space. When the Weyl nodes are at or close to the Fermi level, net Berry curvature can be present due to broken TRS, which can give rise to new exotic quantum phenomena such as large intrinsic anomalous Hall effect (AHE) [12] and anomalous Nernst effect [13]. Like non-magnetic WSMs, magnetic WSMs are also characterized by topological surface states, i.e.

II. Intrinsic magnetic topological insulator MnBi 2 Te 4 and its related materials
Although quantum anomalous Hall effect (QAHE) has been seen in thin TI films of Crand V-doped (Bi,Sb)2Te3 [1,2], the 'critical temperature' required is below ~2 K, severely constraining the exploration of fundamental physics and technological applications.
Inhomogeneous surface gap induced by randomly distributed magnetic dopants is believed to be the origin of low-temperature requirement for observing QAHE [ 26 ,27 ]. High-temperature QAHE has been predicted to occur in thin films of intrinsic FM or AFM TI materials [15,28].
Nevertheless, despite considerable theoretical and experimental efforts, there has been little progress until the recent discovery of an intrinsic AFM TI MnBi2Te4 [29,30,31]. MnBi2Te4 is a layered ternary tetradymite compound; it crystallizes in a rhombohedral structure (space group R-3m), built of the stacking of Te-Bi-Te-Mn-Te-Bi-Te septuple layers (SLs) (Fig. 1a). SLs are coupled through van der Waals bonding.
The single crystals of MnBi2Te4 can be grown either from the melt with stoichiometric composition [31,32] or using the flux method with excessive Bi2Te3 serving as flux [33]. Since MnBi2Te4 is metastable [34], its single crystals can be obtained only through quenching at a temperature close to 590C. For the melt growth, the stoichiometric mixture first needs to be 4 heated to a high temperature (700-1000C), then slowly cooled down to a temperature close to 590, finally followed by annealing and quenching at this temperature [31,32]. For the flux growth, prolonged slow cooling (~2 weeks) from ~600C to ~590C is necessary, and the excessive flux is separated through centrifuging [33].
MnBi2Te4 enables combination of intrinsic antiferromagnetism with nontrivial band topology, thus giving rise to an intrinsic AFM TI [29,30,31]. Its antiferromagnetism is produced by the Mn-sub-lattice, while its nontrivial band topology is formed by inverted Bi and Te pz bands at the  point due to strong spin-orbital coupling (SOC). Its AFM state shows an A-type AFM order (TN = 25K) [31,33,35], characterized by Mn FM layers stacked antiferromagnetically along the c-axis and the ordered magnetic moments are aligned along the c-axis [33]. A large spin gap as well as magnetic frustration due to large next-nearest neighbor AFM exchange have also been probed in recent inelastic neutron scattering experiments on MnBi2Te4 [36]. On the (001) surface, a large gap (~88 meV [31]) is opened at the surface Dirac node due to the breaking of the S=T1/2 symmetry ( and T1/2 represent the time reversal and primitive translation symmetry respectively) (Fig. 1b). Such a surface gap was probed in ARPES measurements on single crystal samples first by Otrokov et al. [31] (Fig. 1d) and subsequently by several other groups [ 37 , 38 ]. However, there have also been reports on ARPES experiments which [39,40,41,42] show the surface Dirac cone state is gapless either in the paramagnetic or the AFM state (Fig. 1e).
MnBi2Te4 offers an ideal platform to realize new exotic topological quantum states.
Theory predicts it can host not only high-temperature QAHE and axion insulator with topological magnetoelectric effect in thin film samples [29,30,31,43], but also an ideal Weyl semimetal state with one pair of Weyl nodes near the Fermi level in its bulk FM phase driven by 5 external magnetic fields or strain (Fig. 1c) [29,30]. Moreover, chiral Majorana mode is also predicted to be accessible via interaction between MnBi2Te4 and a s-wave superconductor [44].
Recently, remarkable progresses have been made toward realizing these predicted quantum states [45,46].
Deng et al. [45]  This is revealed by a magnetic-field induced electronic phase transition at the AFM-to-FM phase boundary, a large intrinsic anomalous Hall effect (see Fig. 1f), a non-trivial  Berry phase of the cyclotron orbit and a large positive magnetoresistance in the FM phase [50].  [61] or a FM TI [59]. MnBi8Te13 is reported to be an intrinsic FM axion insulator [60].
Another common property of these materials is that they all show large magnetic hysteresis and low spin-flip transition fields. Therefore, they offer a new promising platform to explore novel topological quantum states, including QAHE and axion insulator at high temperatures.  [24] and YbMnBi2 [25]. In this section, we will review the recent research progress in the study of these magnetic WSMs.

1a. Kagome-lattice WSM Co 3 Sn 2 S 2:
The kagome lattice is known to host exotic quantum states such as spin liquid [78]. Recent studies show a layered FM compound Co3Sn2S2 with Kagomelattice (space group, R-3m) hosts a TRS breaking WSM state [16,17,18,19]. The magnetic properties of this material originate from the kagome-lattice of cobalt, whose magnetic moments order ferromagnetically and are oriented along the out-of-plane direction in the ground state ( Fig.   3a) [16]. Recent SR experiments showed such an out-of-plane FM order sustains up to 90K, 9 and then evolves into a mixed phase of the out-of-plane FM and the in-plane AFM order in the 90-172K range, and finally to a mixed phase of paramagnetic and FM in the 172-175K range [79]. The C3v-rotation and inversion symmetries of this material generates a total of six nodal rings without considering SOC. When SOC is considered, the linear crossing points of nodal rings split into three pairs of Weyl modes as shown in Fig. 3d. These Weyl nodes are only about 60 meV above the Fermi level according to theoretical calculations [18].
The experimental evidence for such a TRS breaking WSM state of Co3Sn2S2 was first revealed in magnetotransport measurements [16,17]. This material exhibits not only negative longitudinal magnetoresistance (LMR) [16], but also large intrinsic anomalous Hall effect (AHE) [16,17] and large anomalous Nernst effect (ANE) [80,81,82]. Negative LMR is the manifestation of the chiral anomaly arising from the charge pumping between paired Weyl nodes with opposite chirality under parallel electric and magnetic fields. The intrinsic origin of AHE in Co3Sn2S2 is evidenced by the observations that its anomalous Hall conductivity  is nearly independent of longitudinal conductivity  below 90K [16] and linearly increases with magnetization [17].
Besides large  (~1130  -1 .cm -1 at ~90K), the anomalous Hall angle (θH =  / ) of Co3Sn2S2 was also found to be large, ~ 20% at 90K, about one order of magnitude larger than those of typical magnetic systems [83]. As shown in Fig. 3b, such large values of  and θH can be attributed to large net Berry curvature of occupied states [16]. The steep decrease of  above 90 K is due to the fact that the out-of-plane FM phase coexists with the in-plane AFM phase and the volume fraction of the FM phase decreases with increasing temperature [79].
Furthermore, systematic studies on the ANE of Co3Sn2S2 by Ding et al. [80] show that the anomalous Nernst response (the ratio of transverse electric field to the longitudinal temperature gradient) is inversely proportional to the carrier mobility , contrasted with the ordinary Nernst response 0 , which is  . This indicates that anomalous transverse thermoelectricity  in Co3Sn2S2 is determined by the Berry curvature, rather than the mean free path [80].
The hallmark of the electronic structure of a WSM phase is the surface Fermi arcs (SFAs), which connect the projected Weyl nodes with opposite chirality on the surface Brillouin-zone.
Such an expected feature for Co3Sn2S2 has recently been demonstrated by the angle-resolved photoemission spectroscopy (ARPES) [18] and scanning tunneling spectroscopy (STS) experiments [19]. As shown in Fig. 3c and 3d( [18,19]. Moreover, the STS experiments also show the surface Fermi-arc contour and Weyl node connectivity is termination dependent [19]. By means of in situ electron doping, the ARPES experiments also detected bulk Weyl nodes (Fig. 3e) [18].

1b. Heusler alloy FM WSM Co 2 MnGa and Co 2 MnAl: Recent theoretical work predicted that
Co-based Heusler compound Co2XZ (X=V, Zr, Nb, Ti, Mn, Hf; Z =Si, Ge, Sn, Ga and Al) can host unique FM WSM phases [84,85,86]. First, its Weyl states can have the least number of Weyl nodes (two), which can make the interpretation of spectroscopic and transport properties much easier. Second, the Weyl node separation in momentum space is large, giving rise to large anomalous Hall effect, which is of great use for applications. Third, the Weyl node location in momentum space can be manipulated by controlling the magnetization direction [84,85]. These characteristics make Co2XZ a promising platform for exploring novel magnetic Weyl physics and potential applications.

11
Although Co2XZ allows for many different element combinations, experimental studies on their possible exotic properties induced by the expected WSM states are sparse, which is possibly due to the difficulty of the single crystal growth of this family of materials. Co2MnGa is the first confirmed member that have distinct properties associated with the FM WSM state [20,76]. This material is a room temperature ferromagnet with the Curie temperature of 690 K and possesses a cubic structure with the space group of Fm-3m (Fig.4a) [20]. It shows a giant AHE, with its  being as large as 2000 Ω -1 cm -1 at low temperature (Fig.4b) [20]. Furthermore, Co2MnGa was also found to show a giant ANE [20,76]. The Nernst signal Syx increases with elevating temperature, reaching a record high value of Syx ≈ 6 μV K -1 at room temperature and approaching 8 μV K -1 at 400 K [20,76], which is more than one order of magnitude larger than the typical values known for the ANE in other magnetic conductors. These results, together with the unsaturated positive longitudinal magnetoconductance (i.e. chiral anomaly) [20], provide transport evidence for Weyl fermions in Co2MnGa.
Recent ARPES studies on Co2MnGa [21] unveil its characteristics of Weyl state. The combination of mirror symmetries and FM ordering of this material leads to 3D Weyl nodal lines with 2-fold degeneracy, which form Hopflike links and nodal chains [21,87]. These nodal lines are protected by mirror symmetries and give rise to drumhead surface states. Both Weyl nodal lines and drumhead surface states have been visualized in the ARPES experiments [21]. The top panel in Fig. 4c shows an ARPES constant energy surface [21], from which the projection of Weyl nodal lines on the ka-kb plane can be seen clearly. The distribution of calculated Berry curvature on the ka-kb plane (bottom panel of Fig. 4c) matches well the shape of Weyl nodal line projection [21], indicating that the Berry curvature of Co2MnGa predominantly stems from Weyl 12 nodal lines. The  calculated from the Berry curvature is indeed consistent with the experimental value [21].
Like Co2MnGa, the L21 structural phase of Co2MnAl is also predicted to be a FM WSM candidate [86] and early Berry curvature calculations suggest it has the largest AHE among the Co2XZ Heusler alloys [88]. The recent success of single crystals growth of this material has enabled further experimental studies on this material. Li et al. [22] indeed observed a tunable giant AHE in Co2MnAl single crystals. Its  is as large as 1300 Ω −1 cm −1 at room temperature; more noticeably, its room temperature anomalous Hall angle reaches a record value among magnetic conductors, with tanθH = 0.21, which brings the promise for practical device applications. Theoretical studies have further clarified the intrinsic mechanism of such a giant AHE [22]. As shown in Fig gapped. Therefore, the rotation of magnetization leads to a cos-like angular dependence in  , which is indeed observed in experiments (Fig. 4f) [22]. Since Co2MnAl is a soft ferromagnet, the rotation of magnetization can be driven by a weak magnetic field. As such, this material offers an ideal platform to explore band topology tuning by magnetization.  [92]. The first example of Weyl nodes generated by external magnetic fields in zero gap semiconductors is

2)Antiferromagnetic
GdPtBi [24,93,94], which is a half Heusler compound and possesses a cubic structure consisting of interpenetrating face centered cubic lattices and exhibits antiferromagnetic ordering with TN = 9.2 K [24]. The hallmarks of Weyl state in transport, including negative LMR caused by chiral anomaly [24], large intrinsic AHE [93] and planar Hall effect (PHE) [ 95 ], have been demonstrated in this material. Such a magnetic field-driven Weyl state is believed to originate either from the Zeeman splitting by the external magnetic field [24] or from the exchange splitting of the conduction bands [94]. The finding of field-induced Weyl state in GdPtBi has inspired studies on other isostructural half Heusler compounds like NdPtBi [94] and TbPtBi [96,97]. TbPtBi was found to show an exceptionally large AHE with the anomalous Hall angle of 0.68-0.76 [96] ( about a few times larger than that in GdPtBi [93]), though its other transport signatures of Weyl state (e.g. PHE) are not significant. The first-principle electronic structure and the associated anomalous Hall conductivity calculations show that the exceptionally large AHE in TbPtBi does not originate from the Weyl points but that it is driven by the large net Berry curvature produced by the anticrossing of spin-split bands near the Fermi level [96].

Outlook
From the above overview, it can be seen clearly that the interplay between band topology and magnetic states can create unique topological quantum states which are potentially useful for technology applications. QAHI is the most promising example. However, device applications require such a state to be realized at room temperature. Although theoretical studies show this is possible, efforts are needed to discover more promising candidate materials which combine band topology with room temperature magnetism. Among the current magnetic TIs, the magnetic transitions all occur at low temperatures (below 50 K). TIs with room temperature magnetism is highly desired. Materials design by theory and computations could play a key role in this regard.
Since FM WSMs can also evolve into QAHI in the 2D limit, discovering room temperature ideal FM WSMs may be another route to realize high temperature QAHI. An ideal WSM generally refers to a Weyl state with all Weyl nodes being symmetry related and at or close to the chemical potential, without interfered with by any other bands; current magnetic WSMs are not ideal. Data Availability Statements ： The data that support the findings of this study are available from the corresponding author upon reasonable request.