Half-Magnetic Topological Insulator

Topological magnets are a new family of quantum materials providing great potential to realize emergent phenomena, such as quantum anomalous Hall effect and axion-insulator state. Here we present our discovery that stoichiometric ferromagnet MnBi8Te13 with natural heterostructure MnBi2Te4-(Bi2Te3)3 is an unprecedented half-magnetic topological insulator, with the magnetization existing at the MnBi2Te4 surface but not at the opposite surface terminated by triple Bi2Te3 layers. Our angle-resolved photoemission spectroscopy measurements unveil a massive Dirac gap at the MnBi2Te4 surface, and gapless Dirac cone on the other side. Remarkably, the Dirac gap (~28 meV) at MnBi2Te4 surface decreases monotonically with increasing temperature and closes right at the Curie temperature, thereby representing the first smoking-gun spectroscopic evidence of magnetization-induced topological surface gap among all known magnetic topological materials. We further demonstrate theoretically that the half-magnetic topological insulator is desirable to realize the half-quantized surface anomalous Hall effect, which serves as a direct proof of the general concept of axion electrodynamics in condensed matter systems.

Here, we experimentally verified that stoichiometric MnBi8Te13 (MnBi2Te4/(Bi2Te3)n with n = 3), with intrinsic ferromagnetic (FM) ground state, is a "half-magnetic topological insulator", in which the surface magnetization exists at the MnBi2Te4 septuple-layer (SL) surface but at neither of the three Bi2Te3 quintuple-layer (QL) terminated surfaces. Our angle-resolved photoemission spectroscopy (ARPES) measurements have directly revealed at the SL termination (S-termination) a ~ 28 meV surface gap below the Curie temperature of = 10.5 K , which decreases monotonically with increasing temperature and closes right at to form a gapless Dirac cone, proving its magnetic nature. These results represent the first direct spectroscopic evidence of magnetization-induced topological surface gap among all known magnetic topological materials. In sharp contrast, a gapless Dirac cone with negligible FM proximity is observed on the opposite surface terminated by the triple Bi2Te3 QLs, analogous to the situation in nonmagnetic TI Bi2Te3.
Utilizing density functional theory (DFT) calculations, we find half-QAH conductivity well localized at the SL termination, regardless of the cleavage of the other termination. Therefore, the half-magnetic topological insulator provides an ideal platform for observing the half-QAH effect at a single surface and the related axion physics.

II. MnBi8Te13 single crystal with FM ground state
Single crystal MnBi8Te13 has a trigonal structure [38] with a space group of 3 ̅ . The lattice of MnBi8Te13 consists of one MnBi2Te4 SL and three Bi2Te3 QLs stacking alternately along axis ( Figure 1a). These SLs or QLs are coupled through weak van der Waals (vdW) forces. Cleaving the single crystal perpendicular to axis could yield four possible terminations, namely, the Stermination, Q-termination, QQ-termination and QQQ-termination. The crystallinity was examined by X-ray diffraction (XRD). As shown in Figure 1b, all of the diffraction peaks, particularly the low angle ones, can be well indexed by the (00l) reflections with lattice parameter = 132.6 Å , in agreement with previous report [39].
The temperature-dependent anisotropic magnetic susceptibility ( Figure 1c) shows Curie−Weiss behavior above 150 K (inset) with the characteristic temperature = 20 K and 15 K for // and // , respectively, through a fitting with ( ) = 0 + /( − ). Around = 10.5 K , an FM transition was revealed by magnetic susceptibility (Figure 1c) and resistivity measurements ( Figure 1d). The frustration parameter ( / ) for // was calculated to be ~2, indicating a moderate magnetic frustration. For // , the observed larger bifurcation between zero field cooling (ZFC) and field cooling (FC) magnetization ( Figure 1c) and magnetic hysteresis  Figure S1. The negative slope of ( ) in Figure S1f indicates electron-type carriers, and the obvious anomalous Hall effect is observed for // . In a ferromagnet, the Hall resistivity is described by the formula = 0 + = 0 + , where 0 is the ordinary Hall coefficient, is the anomalous Hall resistivity, is the anomalous Hall coefficient and is the magnetization. Above (20 K), ( ) exhibits the same slope at all H (see Figure S1f), indicating a constant 0 which allows us to subtract the ordinary Hall resistivity to obtain the anomalous part as shown in Figure 1f. scales well with thecurve to the anomalous part of the Hall resistivity and is calculated to be = 1.76 × 10 −6 m 3 /C, two orders of magnitude larger than 0 = 1.15 × 10 −8 m 3 /C. Unlike the previous report [39], the from both increasing and decreasing field measurements keeps a near-vanishing value (< 0.1%) and exhibits sharp peaks without any overlap at the appearance of anomalous Hall plateau. This feature is reminiscent of the behavior in Cr-doped (Bi,Sb)2Te3 films when approaching the quantum anomalous Hall region [11,12].

III. Gapped and gapless TSS Dirac cone in MnBi8Te13
We employ a µ-Laser-ARPES system [43] with a focused laser spot size of ~5 µm to measure the termination-sensitive band structure of MnBi8Te13. Figures S2 and S3 present the spectra at a highsymmetry direction as well as a set of constant energy contours for all four terminations. Here in Figure 2 we highlight the band structure of the S-termination and its opposite cleaving plane, the QQQ-termination. Shown are spectra taken at 7 K and 20 K, which correspond to FM and PM phases, respectively. In the FM phase, the S-termination shows an unambiguous energy gap of about 28 meV at the Dirac point ( Figure 2c). This is in sharp contrast to other Mn-Bi-Te family members such as MnBi2Te4 [23][24][25]33], MnBi4Te7 [27,33,35] and MnBi6Te10 [31,33], whose S-terminations consistently show no apparent gap-opening at the Dirac point below the magnetic transition [44]. Above in the PM phase, a gapless Dirac cone is observed (Figure 2d). Comparison between the FM and PM phases suggests that the origin of the surface gap for the S-termination is magnetism.
The gap opening is captured by an effective massive Dirac Hamiltonian , where the first two terms describe a Dirac cone, and the last the effective Zeeman field induced by the ferromagnetically-ordered Mn atoms. The gap size, ~0.28 , is in qualitative agreement with our DFT prediction ( Figure 2b). The detailed comparison between the DFT and the ARPES results is provided in Figure S2.
At the QQQ-termination, the gapless Dirac surface states appear to persist below . At first sight, this seems to contradict with the broken of time-reversal symmetry. However, given the considerable spatial separation between the top Bi2Te3 QL and the magnetic MnBi2Te4 SL, it is reasonable to assume a negligibly small effective Zeeman field for the surface states. Such a conjecture is indeed supported by our DFT calculation, which also reproduces the gapless Dirac cone despite a magnetic ground state (Figure 2f). We note here that the DFT surface-only spectra agree with the ARPES spectra better than the DFT bulk spectra, likely due to the limited photoemission probing depth [45].
Similar gapless Dirac cones have been observed by ARPES at the FM phase for both Q-and QQterminations, which is owing to the hybridization between the TSS and the bulk bands that buries the Dirac point [31], shown in Figure S4. In all, MnBi8Te13 is a half-magnetic topological insulator.
The time-reversal symmetry is broken at the S-termination where shows a temperature-dependent gap, while approximately preserved at the other surface where shows a gapless state.

IV. The nature of the TSS Dirac gap at S-termination
Having established a TSS Dirac point gap opening in the S-termination of FM MnBi8Te13, we now demonstrate that this gap is indeed opened due to the long-range FM order of the magnetic moments.
Zoom-in ARPES − map of the S-termination in the PM state (15 K) is shown in Figure 3a while that in the FM state (7 K) is shown in Figure 3d. In Figure 3g, systematic Lorentzian fitting to the Γ ̅ EDCs at various temperatures below and above the bulk PM-FM transition (10.5 K) are presented. The constant energy mapping and dispersions corresponding to each temperature are shown in Figure S5 and S6, allowing us to unambiguously extract the dispersion at the Γ point. At the lowest temperature (6 K), similarly three Lorentzian peaks are needed to fit the EDC, of which two dark blue peaks ( 1 and 2 ) correspond to the split Dirac cone. The Dirac cone gap size ∆ = 1 − 2 and its temperature evolution is plotted in Figure  3h. With increasing temperature, 1 and 2 move closer to each other ( ∆ decreases) and eventually merge into one Lorentzian peak at 11 K (gap closes), strongly suggesting a clear correlation between the size of this Dirac point gap and the FM exchange interaction. It is worth noting that, while we can also assume similarly two dark blue peaks ( 1 and 2 ) for the EDCs measured at ≥ 11 K, the fitting iterations always result in vanishing or even negative area of peak 2 , and ∆ = 1 − 2 ≤ 3 meV, which is negligible compared to the width of the Lorentzian peaks. In short, the gapless-ness of the Dirac cone at temperatures above 11 K is well established.
Assuming a linear relation between this exchange splitting and the magnetic moment, the gap should be well described by a power law curve [46] ∆ ~ 0 • (1 − / 0 ) 2 , where 0 represents the saturated exchange splitting energy at = 0 K . Fitting the ∆( ) curve with this power law function yields 0 = 11 ± 1 K and = 0.23 ± 0.02 . The fitted onset temperature 0 matches the susceptibility-derived Curie temperature well within the fitting error. The saturated exchange splitting energy is fitted as 0 = 33 ± 1 meV. We thus established an FM-induced Dirac point gap in the S-termination of MnBi8Te13. It is noteworthy that, although Dirac point gaps have been reported for other members of the Mn-Bi-Te family [28,[35][36][37], these observations are still controversial [23][24][25]27,31,33]. In particular, all the reported gaps remain open above the magnetic transition temperature, contradicting the scenario of the restoration of time reversal symmetry. Consequently, our results that a TSS Dirac cone gap decreases monotonically with increasing temperature and closes right at TC forming a gapless Dirac cone represent the first smoking-gun evidence of TSSs gapped by the magnetic order among all known magnetic topological materials.

V. Surface anomalous Hall conductance as a signature of axion insulator
Till now we have demonstrated a magnetic gap at the S-termination and gapless feature at the QQQtermination of MnBi8Te13, rendering the material a "half-magnetic topological insulator". To further identify its topological nature, we next theoretically analyze the surface anomalous Hall conductance (AHC) of this gapped surface and corresponding experimental signatures. Due to the inversion symmetry, the band structure of MnBi8Te13 may be characterized by a higher-order topological invariant, i.e. the Z4 number (the symmetry indicator of inversion [47,48]). Our explicit computation shows that MnBi8Te13 has Z4 = 2, in agreement with a previous study [39] ( Figure S8 and Table S1). For an FM compound, while Z4 = 1 or 3 implies a Weyl semimetal, Z4 = 2 corresponds to an axion insulator or a 3D Chern insulator, with distinct surface AHC behaviors [13]. Therefore, we compute the surface AHC by integrating the local Chern numbers through surface-related layers for two-dimensional slabs of MnBi8Te13, expressed as with a period of the unit cell thickness (four vdW layers) starting from the fourth layer from the surface. Therefore, the half-quantized AHC of MnBi8Te13 is a local quantity at the Stermination, indicating an axion insulator phase.
Though direct experimental measurement of the half-quantized surface AHC is challenging due to various reasons, including sample quality, possible electrode scattering, actual size of the surface gap and the electron chemical potential in the sample, numerical validation in MnBi2Te4/(Bi2Te3)n [13] and an experimental proposal based on non-local transport measurement have recently been provided [14], with a hexagonal six-contact-probing set-up shown in Figure 4c.
In the case of MnBi8Te13, as discussed above, magnetism opens a gap in the S-termination surface.
In the transport measurement the half-quantized AHC acts as a chiral state carried by the hinge of this surface. We computed the spectral functions at the hinges and the center of the top surface terminated by SL (Figure 4d), validating the existence of the chiral hinge state at the S-termination.
It clearly shows that although the center of the S-termination (point 2 in Figure 4c)

Sample growth
MnBi8Te13 single crystals were grown by the conventional high-temperature solution method using

Transport and magnetic measurements
The structure of the crystals was characterized by x-ray diffraction with Cu ɑ radiation at room temperature using a Rigaku Miniex diffractometer. The diffraction pattern can be well indexed by the (00l) reflections with ∆ 2 ~ 2° for adjacent peaks especially at lower angles ( < 20° ).

Resistivity measurements were performed by a Quantum Design (QD) Physical Properties
Measurement System (PPMS) with a standard six-probe method, using a drive current of 8 mA.
The current flows in the plane and the magnetic field is perpendicular to the current direction.

ARPES measurement
µ-Laser-ARPES [43] measurements were performed at the Hiroshima Synchrotron Radiation Center (HSRC), Hiroshima, Japan with a VG Scienta R4000 electron analyzer and a photon energy of 6.36 eV. The energy and angular resolution were better than 3 meV and less than 0.05°, respectively. Samples were cleaved in situ along the (001) crystal plane under ultra-high vacuum conditions with pressure better than 5 × 10 −11 mbar and temperatures below 20 K.

First-principles calculations
DFT calculations [49,50] were conducted by using the projector-augmented wave (PAW) pseudopotentials [51] and exchange-correlation was described by the Perdew-Burke-Ernzerhof (PBE) version of the GGA functional [52,53] as implemented in the Vienna ab-initio Simulation Package (VASP) [54]. Considering the transition-metal element Mn in MnBi8Te13, PBE + U functional with = 5 eV was used for Mn 3d orbitals for all the results in this work [55]. The kmesh, energy cutoff and total energy tolerance for the self-consistent calculations were 5 × 5 × 5, 500 eV, and 10 −5 eV, respectively. The experimental lattice constants ( 0 = 4.37 Å, and 0 = 132.32 Å) were taken, while the atomic positions were fully relaxed until the force on each atom is less than 10 −2 eV/Å. Spin−orbit coupling was included in the calculations self-consistently. We constructed Wannier representations by projecting the Bloch states from the first-principles calculations of bulk materials onto Mn-d, Bi-p, and Te-s orbitals. The topological surface states as well as the surface anomalous Hall conductivity were calculated in tight-binding models constructed by these Wannier representations, as implemented in the WannierTools package [56][57][58][59].   Figure S6 integrated over a ±0.001 Å −1 momentum ∥ window. Furthermore, the spectra for each temperature are extracted from the corresponding map shown in Figure S5 integrated over a ±0.002 Å −1 momentum ( ) window.
(h), TSS Dirac cone gap size (blue triangles) evolution with temperature and its fitting (red solid line) using a power law curve. The error bar of the gap size is defined as = √ 1 2 + 2 2 , where 1 and 2 represent the half-width-at-half-maximum for peak 1 and 2 , respectively. We note that the EDC fitting in (g) yields standard deviation of the peak positions much smaller (< 1 ) than the error bars shown in (h).