Evidence for a magnetic-field induced ideal type-II Weyl state in antiferromagnetic topological insulator Mn(Bi1-xSbx)2Te4

The discovery of Weyl semimetals (WSMs) has fueled tremendous interest in condensed matter physics. WSMs require breaking of either inversion symmetry (IS) or time-reversal symmetry (TRS); they can be categorized into type-I and type-II WSMs, characterized by un-tilted and strongly tilted Weyl cones respectively. Type-I WSMs with breaking of IS or TRS and type-II WSMs with IS breaking have been realized experimentally, but TRS-breaking type-II WSM still remains elusive. In this article, we report an ideal TRS-breaking type-II WSM with only one pair of Weyl nodes observed in the antiferromagnetic topological insulator Mn(Bi1-xSbx)2Te4 under magnetic fields. This state is manifested by a large intrinsic anomalous Hall effect, a non-trivial $\mathrm{{\pi}}$ Berry phase of the cyclotron orbit and a large positive magnetoresistance in the ferromagnetic phase at an optimal sample composition. Our results establish a promising platform for exploring the physics underlying the long-sought, ideal TRS breaking type-II WSM.

the cyclotron orbit and a large positive magnetoresistance in the ferromagnetic phase at an optimal sample composition. Our results establish a promising platform for exploring the physics underlying the long-sought, ideal TRS breaking type-II WSM.

I. INTRODUCTION
Weyl semimetals (WSMs) have become a forefront research topic in contemporary condensed matter physics [1][2][3]. They provide model platforms for studying concepts in high-energy physics, such as the magnetic chiral anomaly effect. Additionally, they provide a means of realizing potentially useful exotic quantum states of technological relevance, e.g., quantum anomalous Hall insulator [4]. In WSMs, singly-degenerate, linearly-dispersed bands cross at Weyl nodes in momentum space; these nodes always appear in pairs with opposite chirality and can be understood as monopole and anti-monopole of Berry flux. When the Weyl nodes are located at or near the chemical potential, their diverging Berry curvature can give rise to distinct properties such as the large intrinsic anomalous Hall effect (AHE) and anomalous Nernst effect if time-reversal symmetry (TRS) is broken [5][6][7][8]. Moreover, the bulk-edge correspondence principle leads WSMs to possess unique surface Fermi arcs [9][10][11], which can also generate exotic phenomena such as quantum oscillations [12,13] and a bulk quantum Hall effect through the formation of Weyl orbits [14]. Further understanding of Weyl fermion physics requires an ideal WSM in which all Weyl nodes should be symmetry related and at the same energy level (at or near the chemical potential), without interference from any other bands [1][2][3]15]. However, although ideal Weyl states have been long pursued, it has been realized only in bosonic systems (i.e., photonic crystals) [16]. An ideal fermionic WSM with distinct exotic properties is still lacking.
The Weyl nodes of the WSMs discovered to date are either not close to the chemical potential, or not at the same energy level, or interfered with by other bands. These non-ideal WSMs can be divided into two categories: type-I and type-II WSMs. Type-I WSMs, which feature un-tilted Weyl cones, can be further categorized into inversion symmetry-breaking WSMs and TRS-breaking WSMs. The inversion symmetry-breaking type-I WSM state was first discovered in TaAs-class materials [10,11,[17][18][19], and the TRS-breaking type-I WSM was recently demonstrated in several ferromagnetic (FM) materials such as Co3Sn2S2 [5,6,[20][21][22] and Co2MnGa [23,24]. Unlike type-I WSMs, type-II WSMs are characterized by strongly tilted Weyl cones and violate Lorentz invariance [25]. The inversion-symmetry-breaking type-II WSM was found in several non-magnetic materials such as (W/Mo)Te2 [25][26][27][28], LaAlGe [29], and TaIrTe4 [30,31]; however, the TRS-breaking type-II WSM still remains elusive. YbMnBi2 has been claimed to be a member of this category [32], but there has been debate regarding its TRS breaking [33]. In this article, we report experimental evidence for a recently-predicted, magnetic-field-induced ideal TRS breaking type-II Weyl state with only one pair of Weyl nodes in MnBi2Te4 [34,35].
MnBi2Te4 has been demonstrated as the first intrinsic antiferromagnetic (AFM) topological insulator (TI) [34][35][36]. The combination of spontaneous magnetization and non-trivial band topology makes this material accessible to a variety of exotic topological quantum states in its 2D thin layers, including the quantum anomalous Hall insulator [37], the axion insulator [38], and the C = 2 Chern insulator [39]. In addition, MnBi2Te4 is also predicted to host a long-sought, ideal type-II TRS breaking WSM when its AFM order is coerced to FM order under a magnetic field parallel to the c-axis [34,35]. Theoretical studies show that the interlayer hybridization, combined with the protection of C3 rotational symmetry, induces the band crossings in the FM phase, resulting in a single pair of strongly tilted, type-II Weyl cones with the Weyl nodes at ~30 meV above the chemical potential [35]. Here, we provide experimental evidence for such an ideal WSM state using magnetotransport measurements in Mn(Bi1-xSbx)2Te4. We achieve the ideal WSM by tuning the Sb:Bi ratio and observe the following key characteristics: an electronic phase transition across the AFM-to-FM phase boundary, a large intrinsic AHE, a non-trivial Berry phase of cyclotron orbit, and a large positive MR in the FM phase when the carrier density n is close to a minimum. All these results provide strong support for the existence of the ideal Weyl state in this material.

II. RESULTS AND DISCUSSION
Prior studies of Mn(Bi1-xSbx)2Te4 [40,41] show that the carrier density is minimized near where the carrier type also changes from electron to hole. Sb substitution for Bi also leads to striking changes in its magnetic properties through the Neel temperature shows a relatively small change from MnBi2Te4 (TN = 25 K) to MnSb2Te4 (TN = 19 K) [40,41]. Undoped MnBi2Te4 shows two magnetic transitions upon increasing the magnetic field along the c-axis, i.e., the AFM to canted antiferromagnetic (CAFM) transition at Hc1 and the CAFM-to-FM transition at Hc2 [42].
Both Hc1 and Hc2 are suppressed by increasing Sb content and tend to merge as x approaches 1 [41].
Here, we focus on investigating the magnetotransport properties of Mn(Bi1-xSbx)2Te4 under high magnetic fields (up to 41.5 T) and seek transport evidence for the predicted type-II WSM.
Single crystals of Mn(Bi1-xSbx)2Te4 used in this work were grown by a flux method (see Methods). From magnetization measurements, we find the evolution of magnetism of Mn(Bi1-xSbx)2Te4 with Sb content is consistent with prior studies [40,41]. Figure 1 shows the isothermal magnetization data at T = 2 K for several representative samples (blue curves). The AFM, CAFM, and FM phases determined by these data for each sample are color-coded in Fig. 1.
We conducted high magnetic field Hall resistivity ρxy measurements for these samples and find their magnetic field and temperature dependences of ρxy exhibit drastic changes with increasing Sb Note that the red triangles and purple diamonds in Fig. 3 represent the carrier densities of the samples in the PM states at 75 K and FM states at 2 K, respectively (see more xy data of additional samples in Fig. S2 [43]). Fig. 3 shows the x ~ 0.26 samples have a much lower carrier density; the minimum is ~3 × 10 18 cm -3 , about two orders of magnitude smaller than that of MnBi2Te4 [42,44].
For the x = 0.26(II-III) samples which show the non-linear field dependence of ρxy (Figs. 1(c-d)), their carrier densities are estimated through the two-band model fits (see SM Section S1 [43]). Furthermore, we found that the samples with lower carrier density have higher carrier transport mobility μH. The maximal value of μH is ~1.1  10 3 cm 2 /Vs (see Methods), about two orders of magnitude larger than those of MnBi2Te4 (~58 cm 2 /Vs) and MnSb2Te4 (9.7 cm 2 /Vs) (Fig. 3).
The access to the lower carrier density in Mn(Bi1-xSbx)2Te4 enables the observation of the transport signatures of the predicted ideal FM WSM. As noted above, it has been theoretically predicted that the AFM-to-FM transition driven by a magnetic field along the out-of-plane direction should lead to a topological phase transition from an AFM TI to an FM WSM due to the crossing of spin-split bands induced by the FM exchange interaction [45]. Thus, an electronic phase transition (i.e., band structure reconstruction) is expected to occur concomitantly with the AFM-CAFM-FM transition. This is indeed manifested by the magnetotransport properties of the lightly doped samples with x = 0.26(II-III). Unlike moderately/heavily doped samples (x < 0.26 or > 0.26) whose RH is hardly or weakly temperature-dependent below and above TN If this implicit electronic phase transition corresponds to the transformation from the AFM TI to the FM WSM as the theory predicts [34,35], typical transport signatures of an FM WSM such as a large intrinsic AHE and a  Berry phase of cyclotron orbits should be probed through the transport measurements. In our experiments, we indeed observed these effects in the lightly and moderately doped samples. As indicated above, a large intrinsic AHE of an FM WSM arises from the diverging Berry curvature of the Weyl nodes, which is manifested by large intrinsic anomalous Hall conductivity and anomalous Hall angle ( = / ). Figure 4(a) presents at T = 2 K for those representative samples shown in Fig. 1, which is derived from xy and longitudinal resistivity xx (SM Fig. S3 [43]) via tensor conversion (see Methods). We find strongly depends on carrier type and density and shows values consistent with the theoretically predicted FM WSM state in moderately/lightly hole-doped samples. MnBi2Te4, which is heavily electron-doped, shows negative and its absolute value   rises to a maximum of ~35  -1 cm -1 within the CAFM phase, which can be attributed to the intrinsic AHE due to non-collinear spin structure [42]. However, in the FM phase,   drops to ~18  -1 cm -1 and becomes nearly independent of magnetic field. For the lightly electron-doped sample (x = 0.26(I)), its shows not only a sign reversal from being positive-to-negative in the CAFM phase, but also quantum oscillations in the FM phase, with its non-oscillatory component with the Weyl node separation k being ~0.06 Å -1 [34] (or 0.095 Å -1 [35]) along the kz axis.
of such a Weyl state is proportional to k (i.e. = (e 2 /2h)k) [46] and is estimated to 35  -1 cm -1 (or 55  -1 cm -1 ). The large values seen in the FM phases of the moderately/lightly hole-doped samples are apparently consistent with such a predicted Weyl state.
To further examine the evolution of the AHE with the carrier density, we convert  to anomalous Hall angle (see Methods and SM Fig. S4 [43]) and plot the maximal orof the FM phase at T = 2 K as a function of the carrier density in Fig. 4 GdPtBi (~16 %) [8]. These results imply that the Weyl nodes are near the chemical potential only when the Mn(Bi1-xSbx)2Te4 system is tuned to be lightly/moderately hole-doped. The temperature dependences of xx measured under magnetic fields for the lightly hole-doped samples also agree well for the expected field-induced FM WSM. As shown in Fig. 4(c), when the magnetic field is strong enough to drive the AFM state to the CAFM/FM state, xx(T) exhibits an insulating-like behavior, in contrast with the metallic behavior under zero magnetic field. Given that the magnetic field-induced insulating-like behavior is commonly observed in topological semimetals due to their large magnetoresistance at low temperatures [15], our observation of such behavior is consistent with the predicted transition from the AFM TI to an FM WSM. As noted above, the FM WSM in MnBi2Te4 should belong to type-II [34,35] and has Weyl nodes at the touch points of electron and hole pockets. Therefore, the AFM-FM transition is expected to result in a Fermi surface transformation from one-hole pocket in the AFM state as revealed by ARPES measurements (Fig. 2(b)) to a Fermi surface consisting of both hole and electron pockets in the integer LL indices should be assigned [15] (see Methods). In Fig. 5(b), we present oscillatory conductivity for the x = 0.26(I) and 0.39 samples derived via inverting the resistivity tensor and subtracting the non-oscillatory background. Following the above LL index definition, we plot the LL fan diagram in Fig. 5(c). The linear fits yield intercepts of 0.48 ± 0.03 and 0.64 ± 0.06 for the x = 0.26(I) and 0.39 samples respectively, which correspond to the non-trivial Berry phases of (0.96 ± 0.06)π and (1.28 ± 0.12)π, indicating the carriers participating in the SdH oscillations on both samples are relativistic fermions. Furthermore, we also note the SdH oscillations show a systematic evolution with carrier density (see SM Fig. S5 [43]) and the oscillation frequency decreases with carrier density for both electron and hole doping sides (Fig. 5(d)). However, the lightly hole-doped samples showing a large AHE (see the red data points in Fig. 4

(b)) do not show
SdH oscillations, implying that its Weyl nodes are indeed near the chemical potential.
Another distinct magnetotransport property of a WSM is chiral anomaly, which should be manifested by negative longitudinal magnetoresistivity (LMR) [48,49]. However, in  Fig. S6(b) and S6(d)). All these observations indicate the scattering mechanism has changed essentially when the carrier density is tuned to be close to a minimum and are consistent with the expected large MR of a WSM.
We note the layered compound EuCd2As2 with an A-type AFM order has also recently been reported to host a WSM state with one pair of Weyl nodes either in its PM state [50] or field-driven FM phase [51,52]. As compared to EuCd2As2, the Weyl state in MnBi2Te4 has distinct natures: i) its Weyl cones are much more strongly tilted, forming a type-II WSM state; ii) the tunability of chemical potential by Sb substitution for Bi enables the observation of remarkable exotic quantum transport properties of Weyl fermions, including large intrinsic AHE and  Berry phase.

A. Sample preparation and measurements
Single crystals of Mn(Bi1-xSbx)2Te4 with x from 0 to 1 were grown by a self-flux method as previously reported [44]. sample whose xx data were reported in our previous work [42]. For those lightly doped samples, since their ρxy of FM states do not show linear field dependence, we extract using  = σyxand yx =xy/(xx 2 +xy 2 ). Here is the normal Hall conductivity, which can approximately be defined as the σyx value at 24 K, slightly below TN (=24.4 K). In Fig. S8, we show the calculated yx of the x = 0.26(II) sample as an example. The difference in  between 2 K and 24 K represents  . The anomalous Hall angle  shown in Fig. 4(b) and SM Fig. S4 [43] is defined as is also obtained via tensor conversion, i.e.  =xx/(xx 2 +xy 2 ).

C. To assign LL indices for SdH oscillations
To obtain the correct Berry phase using the LL fan diagram, integer LL indices should be assigned when Fermi energy lies between LLs, i.e., at the minimal DOS(EF) [53]. It [15,58].    Fig. 1 and SM Fig. S2 [43]), n is obtained from the linear slope of ρxy(H) at T = 2 K for the FM phase (purple diamond) and 75 K for the PM phase (red triangle).