Mn-rich MnSb2Te4: A topological insulator with magnetic gap closing at high Curie temperatures of 45-50 K

Ferromagnetic topological insulators exhibit the quantum anomalous Hall effect that might be used for high precision metrology and edge channel spintronics. In conjunction with superconductors, they could host chiral Majorana zero modes which are among the contenders for the realization of topological qubits. Recently, it was discovered that the stable 2+ state of Mn enables the formation of intrinsic magnetic topological insulators with A1B2C4 stoichiometry. However, the first representative, MnBi2Te4, is antiferromagnetic with 25 K N\'eel temperature and strongly n-doped. Here, we show that p-type MnSb2Te4, previously considered topologically trivial, is a ferromagnetic topological insulator in the case of a few percent of Mn excess. It shows (i) a ferromagnetic hysteresis with record high Curie temperature of 45-50 K, (ii) out-of-plane magnetic anisotropy and (iii) a two-dimensional Dirac cone with the Dirac point close to the Fermi level which features (iv) out-of-plane spin polarization as revealed by photoelectron spectroscopy and (v) a magnetically induced band gap that closes at the Curie temperature as demonstrated by scanning tunneling spectroscopy. Moreover, it displays (vi) a critical exponent of magnetization beta~1, indicating the vicinity of a quantum critical point. Ab initio band structure calculations reveal that the slight excess of Mn that substitutionally replaces Sb atoms provides the ferromagnetic interlayer coupling. Remaining deviations from the ferromagnetic order, likely related to this substitution, open the inverted bulk band gap and render MnSb2Te4 a robust topological insulator and new benchmark for magnetic topological insulators.

The quantum anomalous Hall effect (QAHE) offers quantized conductance and lossless transport without the need for an external magnetic field [1]. The idea to combine ferromagnetism with topological insulators for this purpose [2][3][4] has fuelled the materials science [5]. It led to the experimental discovery of the QAHE in Cr-and V-doped (Bi, Sb) 2 Te 3 [6][7][8][9][10] with precise quantized values of the Hall resistivity down to the sub-part-per-million level [11][12][13][14]. The stable 3+ configuration of V or Cr substitutes the isoelectronic Bi or Sb [3,15,16] enabling ferromagnetism by coupling the magnetic moments of the transition metal atoms. Hence, time-reversal symmetry is broken enabling, through perpendicular magnetization, a gap opening at the Dirac point of the topological surface state [2][3][4][5]. This gap hosts chiral edge states with precisely quantized conductivity. However, the experimental temperatures featuring the QAHE are between 30 mK [6,12] and a few K [17,18] only, significantly lower than the ferromagnetic transition temperatures T C in these systems [19].
One promising approach is the so-called modulation doping in which the magnetic dopants are located only in certain parts of the topological insulator. This implies strong coupling of the topological surface state to the magnetic moments at a reduced disorder level [17,23].
Most elegantly, this has been realized for Mn doped Bi 2 Te 3 and Bi 2 Se 3 . The tendency of Mn to substitute Bi is weak, such that Mn doping leads to the spontaneous formation of septuple layers with MnBi 2 Te 4 stoichiometry. These septuple layers are statistically distributed among quintuple layers of pure Bi 2 Te 3 or Bi 2 Se 3 at low Mn concentration [24,25] and increase in number with increasing Mn concentration [24]. Eventually, only septuple layers remain when the overall stoichiometry of MnBi 2 Te 4 or MnBi 2 Se 4 [25,26] is reached.
A central drawback of the Bi 2 Te 3 and Bi 2 Se 3 host materials is their strong n-type doping.
Here, epitaxial MnSb 2 Te 4 is studied using spin-and angle-resolved photoemission spectroscopy (ARPES), scanning tunneling microscopy (STM) and spectroscopy (STS), magnetometry, x-ray magnetic circular dichroism (XMCD) and DFT. All experimental methods were performed as a function of temperature to pin down the intricate correlation between magnetism and non-trivial band topology essential for the QAHE. It is revealed that the material unites the favorable properties of a topological insulator with its Dirac point close to the Fermi level E F with that of a ferromagnetic hysteresis with out-of-plane anisotropy and record-high T C , twice as high as the T N previously reported for antiferromagnetic MnBi 2 Te 4 and MnSb 2 Te 4 [30,31]. Moreover, temperature dependent STS finds a magnetic gap of 17 meV at E F for 4.3 K that closes rather exactly at T C as expected for a ferromagnetic topological insulator. By combining DFT, STM, Rutherford backscattering (RBS), and x-ray diffraction (XRD) it is uncovered that a partial substitution of Sb atoms by Mn is decisive to render MnSb 2 Te 4 both ferromagnetic and topologically non-trivial.
Epitaxial MnSb 2 Te 4 films with 200 nm thickness were grown by molecular beam epitaxy (MBE) (Supplementary Section I). Figure 1a shows the cross section of the MnSb 2 Te 4 lattice structure revealed by high resolution scanning transmission electron microscopy (TEM). It consists of septuple layers (SL) with stacking sequence Te-Sb-Te-Mn-Te-Sb-Te, Figure 1c.
The nearly exclusive formation of septuple layers in the entire MnSb 2 Te 4 samples is confirmed by high-resolution XRD (Supplement: Figure 6), revealing only a minute number of residual quintuple layers. In contrast, TEM and XRD analysis of V-doped (Bi, Sb) 2 Te 3 shows quintuple layers only [57]. This highlights that septuple layers are unfavorable for V 3+ . They require the addition of a charge neutral transition metal 2+ /Te 2− bilayer to each quintuple layer as easily possible for Mn 2+ but not for V 3+ . Detailed XRD analysis again points to an exchange of Mn and Sb within the septuple layers in the 10 % range (Supplement: Figure 6). This implies that Mn does not reside exclusively in the center of the septuples, but also to a small extent on Sb sites in the adjacent lattice planes. Indeed, STM images of the atomically flat and Te terminated surface of MnSb 2 Te 4 epilayers ( Figure 1b) exhibit triangular features, pointing to defects in the cation layer beneath the surface [58,59]. These defects occur with an atomic density of 5 − 10 %. Since this is significantly larger than in undoped Sb 2 Te 3 films [58], the triangles are most likely caused by subsurface Mn atoms on Sb sites, in line with the XRD and RBS results. As shown by DFT below, these defects turn out to be decisive for the ferromagnetic interlayer coupling in MnSb 2 Te 4 . Similar conjectures based on scattering methods have been raised previously [41,46,49,50] as well as for MnBi 2 Te 4 based on STM results [60].  Table I), and twice as large as the T N ≤ 25 K obtained for MnBi 2 Te 4 films grown under nearly identical conditions. It has been crosschecked that the displayed remanent magnetization is exactly the same as in the corresponding hysteresis loops.
In particular, the large remanent magnetization observed by the bulk sensitive SQUID measurements excludes that it is caused by uncompensated antiferromagnetic septuple layers only [36]. The M (H) hysteresis curve (Figure 1e), however, shows a rounded shape that persists up to fields much higher than those typical for domain reversals and does not saturate up to ±5 T where the magnetic moment per Mn atom is still less than 2 µ B , similar to the result for antiferromagnetic MnSb 2 Te 4 [30]. Recently, it was found that 60 T are required to fully polarize a MnSb 2 Te 4 bulk-type sample [61]. This suggests additional types of competing magnetic orders. Indeed, a kink in M (T ) is observed at 20−25 K (Supplement: Figure 7), close to the Néel temperature reported earlier for MnSb 2 Te 4 [30]. This implies that the high-temperature ferromagnetism is most likely accompanied by ferrimagnetism as also supported by the relatively large in-plane hysteresis and magnetization, Figure 1e, in line with observations of competing ferro-and antiferromagnetic order in bulk MnSb 2 Te 4 [39,41,46,[48][49][50]61].
The ferromagnetism is confirmed by element specific, zero field XMCD recorded in diffraction geometry. For these measurements, the sample was remanently magnetized at ∼ 0.5 T and 10 K. From spectra recorded with oppositely circularly polarized light at the (0001) Bragg peak, the intensity difference C + − C − (Figure 1h Note that the spin-orbit interaction is crucial for both out-of-plane easy axis and large coercivity [24]. While it is sufficiently strong to turn the magnetization out of plane in Mncontaining Bi 2 Te 3 , the atomic weight of Se in MnBi 2 Se 4 is too weak [24,62]. The present data reveals that Sb 2 Te 3 is sufficiently heavy, i.e., spin-orbit coupling sufficiently large, to maintain the perpendicular anisotropy for high Mn content. To elucidate the origin of the ferromagnetism, DFT calculations are employed (Supplement: Table III). They firstly highlight the differences between MnSb 2 Te 4 and MnBi 2 Te 4 .
In both cases, the in-plane Mn coupling within each septuple layer is ferromagnetic. It is unlikely that the observed difference between antiferromagnetic interlayer coupling in MnBi 2 Te 4 and ferromagnetic interlayer coupling in epitaxial MnSb 2 Te 4 is caused by the lowered spin-orbit interaction, since it remained large enough for out-of-plane anisotropy.
However, the in-plane lattice constant a is by ∼ 2% smaller for MnSb 2 Te 4 . Hence, the DFT based exchange constants of MnSb 2 Te 4 for the in-plane lattice constant a determined by XRD and for a expanded to the value of MnBi 2 Te 4 are compared (Supplement : Table III).
While the in-plane compression increased the in-plane exchange constant J between nearest neighbors by almost a factor of three, T N is barely changed. The reason is that the enlarged in-plane overlap of Mn d states weakens the already small, perpendicular interlayer coupling.
Consequently, the energy gain of antiferromagnetism against ferromagnetism becomes as low as 0.6 meV per Mn atom. This suggests that small structural changes along the interlayer exchange path can induce the transition to ferromagnetic order.   Supplement: Figure 9) to tune the electron wave number perpendicular to the surface, k z , once through the whole bulk Brillouin zone (Supplement : Table II), reveals no dispersion evidencing the 2D character of the Dirac cone. This is contrary to the lower-lying 3D bulk bands that strongly disperse with photon energy (Figure 2f, arrows). Spin-resolved ARPES of the 2D Dirac cone showcases a helical in-plane spin texture when probed away from the Γ zone center, i.e., it exhibits the characteristic reversal of spin orientation with the sign of k ( Figure 2g,i, Supplement: Figure 8). This spin chirality is a key signature of a topological surface state. In addition, a pronounced out-of-plane spin polarization (about 25 %) occurs at the Γ zone center in the vicinity of E F in the remanently magnetized sample and reverses sign when the sample is remanently magnetized in the opposite direction ( Figure 2j). Such out-of-plane spin texture at Γ is evidence for a magnetic gap opening at the Dirac point [63]. Since a Dirac point above E F is not accessible for ARPES, STS is employed to directly assess the ferromagnetic gap formed at T < T C . Figure 3a  by different temperatures of the two measurements (4.3 K versus 300 K) or by larger-scale potential fluctuations [66,67].
To prove that the energy gap is of magnetic origin [24], the temperature dependence ∆(T ) is probed by STS. This has not been accomplished yet for any magnetic topological insulator because at higher temperatures k B T ≥ ∆/5 the STS gap ∆ is increasingly smeared by the Fermi-Dirac distribution. This leads to a small, i.e., non-zero tunneling current at voltages within the band gap [68]. A direct deconvolution of the local density of states (LDOS) and the Fermi distribution function would require an assumption on the shape of the LDOS as function of energy. Such an assumption is not justified because a significant spatial variation of the dI/dV curves is observed at 4.3 K (Figure 3a-c), as found consistently in other magnetic topological insulators [64,69]. Therefore, a new method to derive ∆ at elevated T is established. For this purpose, the ratio between dI/dV at V =0 mV and dI/dV at larger Comparing experimental data points with the calculated lines reveals that the gap size continuously decreases as the temperature approaches T C and closes rather precisely at T C = 45-50 K in line with the T C deduced from XMCD and SQUID. As described above, the gap size varies spatially across the surface (Figure 3a-c) due to local disorder. Accordingly, at higher temperatures the STS ratios also exhibit a considerable variation depending on where the STS spectra were recorded. For this reason, larger ensembles of data points have been recorded at four selected temperatures T = 10, 21, 31 and 36 K within an area of 400 nm 2 . From these, the average gap sizes are deduced to ∆(21 K) = 11 ± 5 meV, ∆(31 K) = 6 ± 1 meV, and ∆(36 K) = 4 ± 1 meV, as represented by the large grey and white dots in Figure 3d. Thus, the gap indeed gradually shrinks as the temperature approaches T C and closes above fulfilling the expectations for a ferromagnetic topological insulator [24]. Note that the different grey shades of the small data points mark different cooling runs starting from an initial elevated temperature. Hence, the tip slowly drifts during cooling across the sample surface while measuring at varying temperature and, thus, explores variations of ∆ by T and by spatial position simultaneously. Accordingly, the visible trend of ∆(T ) relies on the sufficient statistics of probed locations that is particularly adequate for the spatially averaged ∆ (large dots).
The conclusion that the gap closes at T C is corroborated by comparing the experimen-  (Figure 2j). Such a magnetic gap of a topological surface state close to E F is highly favorable for probing the resulting topological conductivity and its expected quantization.
To clarify the origin of the discovered ferromagnetic topological insulator, the electronic band structure of MnSb 2 Te 4 has been calculated by various DFT methods, considering different magnetic configurations including chemical and magnetic disorder (Supplementary Section XI). As a general result, the topological insulator is reproduced by introducing magnetic disorder. Calculating the bulk band structure, the perfect ferromagnetic system Disordered local moments 50% spin up   Figures 14e, 15a). This is in agreement with recent calculations [47,53], but obviously disagrees with the STS and ARPES results. On the other hand, the defect-free antiferromagnetic system is found to be a topological insulator with a bulk band gap of 120 meV, Figure 4k. This is evidenced by the band inversion at Γ, indicated by the color code of the spectral function difference between cationic and anionic sites [red  Figure 15). This nontrivial topology induced by magnetic disorder is robust against chemical disorder as Mn-Sb site exchange that proved to be essential for inducing the ferromagnetic order in the system (Supplement : Table III). Indeed, a Mn-Sb site exchange by 5 % does not affect the band topology (Supplement: Figure 14). This implies that, contrary to recent conclusions [54], defect engineering accomplishes simultaneously a nontrival topology and very high Curie temperature for the MnSb 2 Te 4 system. The magnetic gap size turns out to be a rather local property caused by the exchange interaction in near-surface MnSb 2 Te 4 septuple layers. Naturally, the gapped Dirac cone forms also when the surface of an antiferromagnet is terminated by a few ferromagnetic layers (Supplement: Figure 16d). A relatively strong out-of-plane spin polarization at the gap edges (∼ 60%) is found in that case nicely matching the results of the spin-resolved ARPES data measured at 30 K (∼ 25%), if one takes into account the temperature dependence of the magnetization (Figure 1d,f). Moreover, for more random combinations of antiferromagnetic and ferromagnetic layers, the Dirac cone with magnetic gap persists, albeit the bulk band gap vanishes due to the more extended ferromagnetic portions in that structure (Supplement: Figure 16a). Finally, for a system where the magnetic moments of adjacent septuple layers are continuously tilted with respect to each other, a gapped Dirac cone was also observed, in the presence of disorder [71] as experimentally observed [72,73]. Note that such disorder is witnessed in our samples by the spatial gap size fluctuations ( Figure 3)  We modelled the Mn-Sb site exchange by different Mn occupancies C 1 and C 2 within the central cation lattice planes, usually assumed to be occupied by Mn only, and within the outer cation lattice planes, usually assumed to be occupied by Sb only, respectively. For that purpose, the structure factor of MnSb 2 Te 4 reads where f n (Q z ) is the atomic form factor of the n-th atom in the unit cell, z n its coordinate,

VII. RESONANT SCATTERING AND X-RAY CIRCULAR DICHROISM
Resonant scattering and XMCD were measured at the extreme ultraviolet (XUV) diffractometer of the UE46-PGM1 undulator beam line of BESSY II at Helmholtz-Zentrum Berlin.
The XMCD signal was obtained by measuring the difference of the (0001) Bragg peak intensities for incident photons with opposite circular polarization and the photon energy tuned to the Mn-L 3 resonance. In this setup, the sample was field cooled down to 10 K in an external field of about 0.5 T provided by a removable permanent magnet. Subsequent po-   For the measurement in Fig. 2j, main text, the sample was magnetized in situ at 30 K by applying a pulsed magnetic field from a removable coil of ∼ ±0.5 T in the direction perpendicular to the surface. As usual, the spin-resolved ARPES experiment is conducted in remanence. The movement between the ARPES chamber and the preparation chamber, where the coil is situated, is a vertical movement of the cryostat so that the sample is always at 30 K. the Dirac point can be regarded as a rather good measure of the chemical potential. Figure 9 shows that the Dirac cone surface state has no dispersion with the wave vector perpendicular to the surface plane, in contrast to the bulk valence band. Figure 9 is the input for Fig. 2b, main text, with linear interpolation between the 9 photon energies. Table II shows the momentum values in kinetic energy.

IX. SCANNING TUNNELING MICROSCOPY AND SPECTROSCOPY
For measurements above the base temperature of 4.3 K, the STM body was exposed to thermal radiation via opening of a radiation shield until a maximum T 60 K was achieved. Then, the shield was closed, and dI/dV (V ) curves were recorded while the sample temperature slowly decreased back to 4.3 K. Measurements at constant T > 4.3 K were performed with partly open shield. A more frequent stabilization between subsequent dI/dV curves was necessary due to the remaining thermal drift of the tip-sample distance during the cooling process. This implies shorter recording times that have been compensated by a more intense averaging of subsequently recorded curves.

B. Band Gap Determination at 4.3 K
The band gaps at T = 4.3 K were determined as follows. First, the noise level of the dI/dV curves was reduced by averaging 3 × 3 curves covering an area of (1.2 nm) 2 . Subsequently, an averaging of dI/dV (V ) across ±2 mV in bias direction was employed. Then, we estimated the remaining dI/dV noise level as the maximum of |dI/dV | that appears with similar strength positively and negatively. For that purpose, we employed about 50 randomly selected dI/dV (V ) curves. Afterwards, the threshold was chosen slightly above the determined noise level. This threshold is marked in Fig. 3a #1 as dashed line and in Fig. 3c as yellow line in the color code bar. The voltage width, where the dI/dV (V ) spectra stayed below this threshold, defines the measured gap ∆ as used in Fig. 3a-c, main text.
We crosschecked that dI/dV values below the noise level appeared exclusively close to the determined band gap areas. The resulting gap size ∆ turned out to barely depend on details of the chosen noise threshold. Figure 11a and b display two additional ∆(x, y) maps like the ones in Fig. 3b, main text, but recorded on different areas of the sample surface. They exhibit a similar range of spatial fluctuations of ∆ as shown in Fig. 3b, main text. Sometimes, ∆ could not be determined from the dI/dV (V ) curves as, e.g., in the bright area marked by a black circle in Fig. 11a.
There, the spectra stayed below the threshold on the positive V > 0 mV side up to 100 mV while a gap edge is observed only on the negative side, for an unknown reason. These spectra (∼ 5 % of all spectra) are discarded from further analysis including the histogram of Fig. 3b, main text. The correlation length ξ of ∆(x, y) is calculated as FWHM of the correlation function resulting in ξ 2 nm as given in the main text.
The central energy within the gap, E 0 , is deduced as the arithmetic mean of all voltages where the dI/dV signal remains below the threshold. Figure 11c-f displays maps E 0 (x, y) for the three studied areas as well as a resulting E 0 histogram. Favorably, the average of anticorrelated with the gap size ∆. This is demonstrated in Fig. 13b for T = 40 K. All dI/dV (V ) curves recorded at 4.3 K (area 1-3, 8000 curves) are convoluted with the derivative of the Fermi-Dirac distribution of 40 K (as in Fig. 13a, yellow curve) before R is determined.
Subsequently, each R is related to its corresponding ∆ (same dI/dV (V ) curve) as deduced via the method described in subsection IX B. The anticorrelation of R and ∆ appears in  Fig. 13c-d displaying the same data set as Fig. 3d, main text, but using two different V ref .

X. ELECTRIC TRANSPORT MEASUREMENTS
Transport measurements were performed in the van der Pauw geometry with applied magnetic fields ranging from -3 T to +3 T and oriented parallel to the rhombohedral axis of the epilayers. A mini cryogen-free system was employed for the magnetotransport investigations at temperatures between 2 K and 300 K. For temperatures below about 50 K an anomalous Hall effect appears in the transport data that becomes hysteretic for magnetic fields of less than about 1 T such as the sheet resistance of the samples (compare  Fig. 14, Fig. 15a,b) used a Green function method within the multiple scattering theory [76,77]. To describe both localization and interaction of the Mn 3d orbitals appropriately, Coulomb U values of 3-5 eV were employed within a GGA+U approach [78]. Heisenberg model [79].
Different types of disorder were treated within a coherent potential approximation (CPA) [80,81]. Chemical disorder is modelled by mixing various atomic species on the same atomic site (substitutional alloys vacancies, that were simulated as empty spheres, with Te (Sb) atoms on the same site.
Magnetic moment disorder was modeled by CPA via mixing two Mn atoms with opposite magnetic moment on the same atomic site. This approach has been proven to be very successful in mimicking spin moment fluctuations, e.g., to account for elevated temperatures.
The calculations in Fig. 15c-f were performed with the full-potential linearized augmented plane-wave method as implemented in the FLEUR code. Also here, GGA [82] with a Hubbard U correction using U = 6 eV and J = 0.54 eV was used and spin-orbit coupling was included self-consistently in the non-collinear calculations [83]. Thin film calculations were also performed with the FLEUR code (Fig. 4c, main text, and Fig. 16b-e). These calculations are restricted to relatively small and simple unit cells in order to retain high enough accuracy to derive the Dirac cone dispersion and the size of its magnetic gap by DFT. Instead, the surface band structure for the antiferromagnetic ground state in Fig. 4d-e, main text, for the ferromagnetic ground state in Fig. 4b, main text, and for the mixed state in Fig. 16a was calculated by the projector augmented-wave method [84] using the VASP code [85,86]. The exchange-correlation energy was treated using the GGA [82]. The Hamiltonian contained scalar relativistic corrections and the spin-orbit coupling was taken into account by the second variation method [87]. In order to describe the van der Waals interactions, we made use of the DFT-D3 [88,89] approach. The Mn 3d states were treated employing the GGA+U approximation [78] within the Dudarev scheme [90]. The U eff = U − J value for the Mn 3d states was chosen to 5.34 eV as in previous works [27,28,32,56].
For all three methods, the crystal structure of ideal MnSb 2 Te 4 was fully optimized to obtain the equilibrium lattice parameters, namely cell volume, c/a ratio as well as atomic positions. Using this structure yields the antiferromagnetic topological insulator state for MnSb 2 Te 4 within both the projector augmented-wave and full-potential linearized augmented plane-wave methods (VASP and FLEUR, respectively). For the Green function method both the experimental crystal structure as determined by XRD (Fig. 6) and the theoretically optimized one result in the antiferromagnetic topological insulator phase.
In order to visualize the topological character within bulk-type band structure calculations, we analyze the spectral function difference between anion and cation contributions of each state: The resulting A i k (E) are displayed as color code in Fig. 4f-k, main text, Fig. 14, and Fig. 15a,b with red color for A i k (E) > 0 and blue color for A i k (E) < 0. Band inversion can, hence, be deduced from a mutually changing color within adjacent bands. Table III summarizes      to be trivial [51][52][53][54]. We assume that the topological insulator state of antiferromagnetic MnSb 2 Te 4 was missed because structural optimization was either not performed in favor of experimental lattice constants [52,54] or performed without van der Waals forces [51,53] which both gave by ∼ 3% larger c parameters than in the structurally optimized equilibrium lattice. Ferromagnetic MnSb 2 Te 4 instead, is a Weyl semimetal without band gap that remains a Weyl semimetal for moderate Mn-Sb site exchange (Fig. 14b,d), but becomes topologically trivial at large site exchange (Fig. 14f). Hence, site exchange alone does not reveal the experimentally observed ferromagnetic topological insulator in contrast to the magnetic disorder as presented in Fig. 4g-j, main text. Figure 15 corroborates the gap opening by magnetic disorder. Figure 15b displays the complete band structure at maximum spin mixture (50 % spin-up, 50 % spin down) for each Mn lattice site as partially presented in Fig. 4j, main text. An inverted band gap of about 100 meV is found at Γ. Note that the purely ferromagnetic phase in Fig. 15a is calculated for a unit cell of a single septuple layer only and, hence, differs from Fig. 14e employing a unit cell with two septuple layers, due to backfolding.

B. Magnetic Ground State for Different Disorder Configurations
The Weyl point observed for the ferromagnetic MnSb 2 Te 4 can also be opened by rotating the spins of adjacent Mn layers, while keeping a collinear spin order within each layer ( Fig. 15c-f). Thus, magnetic disorder renders a dominantly ferromagnetic MnSb 2 Te 4 a topological insulator as already discussed for Fig. 4g-j, main text. Figure 16 shows the band structures of slab calculations for different magnetic disorder configurations, such that surface states are captured. They are performed, e.g., for a combination of ferromagnetic interior layers surrounded by a few antiferromagnetic layers on top and bottom (Fig. 16a). This configuration reveals a Dirac-type surface state with a gap around the Dirac point of 16 meV, very close to the average gap size observed by STS. The opposite configuration with antiferromagnetic interior surrounded by ferromagnetic surfaces also exhibits a gapped Dirac cone, here with 40 meV gap size, that might be enhanced by the thickness of this slab of only 7 septuple layers (Fig. 16d). Indeed, the spin-polarized states at the gap edge penetrate about 3 septuple layers into the bulk of the thin film (Fig. 16b).
The out-of-plane spin polarization near the Dirac point amounts to ∼ 60%, nicely matching the experimentally found out-of-plane spin polarization in spin-resolved ARPES (Fig. 2j, main text). The latter amounts to ∼ 25 % at 30 K in line with the reduced magnetization at this elevated T (Fig. 1d,f, main text). Note that the band structure in Fig. 16d also features an exchange splitting of bulk bands as visible by the different colors around −0.2 eV.
The pure antiferromagnetic configuration (Fig. 16e) shows a small band gap of the topological surface state as well, while the pure ferromagnetic order leads to a gapped Weyl cone ( Fig. 16c), likely being an artifact of the finite slab size of 7 septuple layers only.