Ubiquitous Order‐Disorder Transition in the Mn Antisite Sublattice of the (MnBi2Te4)(Bi2Te3) n Magnetic Topological Insulators

Abstract Magnetic topological insulators (TIs) herald a wealth of applications in spin‐based technologies, relying on the novel quantum phenomena provided by their topological properties. Particularly promising is the (MnBi2Te4)(Bi2Te3) n layered family of established intrinsic magnetic TIs that can flexibly realize various magnetic orders and topological states. High tunability of this material platform is enabled by manganese–pnictogen intermixing, whose amounts and distribution patterns are controlled by synthetic conditions. Here, nuclear magnetic resonance and muon spin spectroscopy, sensitive local probe techniques, are employed to scrutinize the impact of the intermixing on the magnetic properties of (MnBi2Te4)(Bi2Te3) n and MnSb2Te4. The measurements not only confirm the opposite alignment between the Mn magnetic moments on native sites and antisites in the ground state of MnSb2Te4, but for the first time directly show the same alignment in (MnBi2Te4)(Bi2Te3) n with n = 0, 1 and 2. Moreover, for all compounds, the static magnetic moment of the Mn antisite sublattice is found to disappear well below the intrinsic magnetic transition temperature, leaving a homogeneous magnetic structure undisturbed by the intermixing. The findings provide a microscopic understanding of the crucial role played by Mn–Bi intermixing in (MnBi2Te4)(Bi2Te3) n and offer pathways to optimizing the magnetic gap in its surface states.


I. INTRODUCTION
The interplay between non-trivial topology and magnetic order has been under the spotlight since the advent of a topological era in condensed matter physics because it may enable versatile and tunable topological phases [1][2][3][4][5].Magnetic topological materials emerged as an ideal platform for harbouring emergent quantum phenomena of technological relevance, including the quantum anomalous Hall effect (QAHE) and axion electro-dynamics [6][7][8][9][10].A layered (van der Waals) compound MnBi 2 Te 4 , which is a magnetic derivative of the prototypical Bi 2 Te 3 topological insulator, has been established as the first intrinsically magnetic TI [11][12][13][14].In this compound, the local moments of the Mn atoms adopt an A-type antiferromagnetic (AFM) order, consisting of a ferromagnetic (FM) alignment within the Mn layer, with AFM stacking in the perpendicular direction [11,14,15].Combination of a layered crystal structure and the Atype AFM order makes MnBi 2 Te 4 to fall under the Z 2 topological classification of AFM insulators [16].The non-trivial value of the invariant, Z 2 = 1, stemming from the spin-orbit coupling driven bulk band gap inversion, categorizes it as a three-dimensional AFM TI [11][12][13][14].
On top of its exciting intrinsic properties, MnBi 2 Te 4 also fosters a highly tunable material platform.Multiple tuning knobs, not only extrinsic such as magnetic field, pressure, and temperature, but also intrinsic such as Mn-Mn interlayer distance, variations of the chemical composition, and defect engineering, result in various magnetic and topological states.For example, various pnictogen or chalcogen substitutions and Mn/Bi/Te stoichiometry alternations give rise to such materials as MnSb 2 Te 4 , MnBi 2 Se 4 , or Mn 2 Bi 2 Te 5 , whose magnetic and topological properties have been studied both theoretically and experimentally [23][24][25][26][27]. Furthermore, the van der Waals nature of MnBi 2 Te 4 tolerates interlacing the adjacent (MnBi 2 Te 4 ) septuple layers with various number n of nonmagnetic (Bi 2 Te 3 ) quintuple layers, resulting in the (MnBi 2 Te 4 )(Bi 2 Te 3 ) n family of stacked structures (n = 1 for MnBi 4 Te 7 , n = 2 for MnBi 6 Te 10 , etc.) [28,29].The increasing distance between the septuple layers progressively weakens the interlayer exchange coupling with an increasing n, which enables an effective tuning of the magnetic structure by moderate magnetic fields [30][31][32][33], or hydrostatic pressure [34], driving these compounds from the AFM to the FM state.The (MnBi 2 Te 4 )(Bi 2 Te 3 ) n materials may host exotic, fieldinduced topological phases [30,35,36] and temperaturedependent metamagnetic states [37].
Native defects in these materials are lately in the center of attention thanks to their strong influence on the magnetic and electronic structure [38][39][40][41][42][43].They are exploited as an effective tuning knob to purposely modify the latter [38,[44][45][46].These defects originate from antisite intermixing between the native manganese and pnictogen crystallographic sites.This phenomenon is favored by similar ionic radii, especially those of the Mn and Sb [47], which enables ca. 3 times stronger degree of intermixing in MnSb 2 Te 4 compared to MnBi 2 Te 4 .Specifically, Mn atoms partially occupy the Sb/Bi 6c Wyckoff site (Mn 6c antisite), while pnictogen atoms swap to the Mn 3a site (Bi 3a antisite).The amounts of swapped cations do not necessarily fulfil the electroneutrality assumptions for Mn(II) and Sb(III)/Bi(III), and the occurrence of cationic vacancies in both sites is debated [48].Strong intermixing in MnSb 2 Te 4 promotes the FM interlayer coupling [49].The magnetic transition temperature jumps from T N =19 K in the AFM-like bulk Mn 1−x Sb 2+x Te 4 single crystals [38] to T C =58 K in the FM-like Mn 1+x Sb 2−x Te 4 ones [46], which is achieved by varying the growth conditions.(Hereinafter, the main magnetic transition is referred to as T m , meaning either T N for AFM or T C for FM samples.)Moreover, the interlayer coupling can become truly FM in MnBi 6 Te 10 via Mn-Bi defects engineering under appropriate growth conditions [44,45].This FM coupling may not only help to realize an FM axion insulator state [35,36], but also the QAHE in contrast to the AFM MnBi 2 Te 4 [22,50].
Low-temperature neutron diffraction measurements, performed on both the AFM-and FM-like MnSb 2 Te 4 bulk single crystals, reveal that the local moments of the Mn 6c atoms are coupled antiparallel to the Mn 3a ones (ferri magnetic structure) [38,51].However, for the (MnBi 2 Te 4 )(Bi 2 Te 3 ) n family the magnetic role of the antisites has not been decisively established yet.Indeed, the available neutron diffraction data [52][53][54] do not shed light on this issue (likely because of the much lower levels of intermixing as compared to MnSb 2 Te 4 ), although the high-field magnetization studies performed on MnBi 2 Te 4 do suggest that there is an AFM coupling between the Mn 6c and Mn 3a sublattices [39].
To close this important gap, this article systematically investigates the magnetic behavior of the antisites in polycrystalline (MnBi 2 Te 4 )(Bi 2 Te 3 ) n with n = 0, 1, 2 and in one MnSb 2 Te 4 sample by means of local magnetic probes, namely nuclear magnetic resonance (NMR) (Sec.II) and muon spin spectroscopy (µSR) (Sec.III).Both techniques probe the dynamic and thermodynamic material properties, as well as the disorder introduced by the antisites.The measurements are performed in applied and in zero (ZF) external magnetic field, and 55 Mn NMR provides direct local evidence of the nearly opposite relative spin alignment of Mn 3a and Mn 6c for all compounds studied here.Importantly, NMR reveals a magnetic order-disorder transition in the Mn 6c sublattice of MnSb 2 Te 4 well below T m = 27 K.This clearly discriminates two regions in the MnSb 2 Te 4 magnetic phase diagram: (i) T < T * < T m , when the Mn 6c and Mn 3a moments are coupled antiferromagnetically; and (ii) T * < T < T m , when the Mn 6c sublattice is paramagnetic-like, whereas the Mn 3a one sustains its intra-and interlayer orders.µSR and bulk magnetometry measurements confirm the same order-disorder transition in the Mn 6c sublattice of all studied (MnBi 2 Te 4 )(Bi 2 Te 3 ) n as well.

II. THE ANTISITES FROM THE NMR PERSPECTIVE
In this and the next section, we describe a unified picture emerging from NMR and µSR, skipping the technical details of its derivation for the sake of clarity.The principles of the two techniques are briefly described in Sec.V and further important details are provided in the Supplemental Material [61].Sec.V also gives details about the sample preparation and characterization and is based on our previous works [30,44,47,62].
In ZF-NMR, 55 Mn nuclear spins (gyromagnetic ratio 55 γ = 10.576MHz/T) precess around a very large hyperfine field B hf , between 40 and 45 T at 1.4 K. B hf is primarily due to the negative coupling, −A, to the onsite moment gµ B S, with much smaller positive couplings B to six nearest neighbors Mn 3a , and even smaller distant dipole contributions, so that in first approximation, assuming parallel S in the layer, the NMR frequency is 55  Therefore two distinct Mn sites experience a different total local field modulus 55 B a,c , i.e. different values of B hf , as it would be expected for the main site Mn 3a and the anti-site Mn 6c .The 5-10% breadth of the frequency peaks is due to disorder in their vicinity, producing small variations of the local electronic environment, reflected in B hf .
The relative area under the two peaks at 3T assigns the lower-frequency, majority peak to Mn 3a and the minority peak to Mn 6c (Fig. 1a-c, the proportionality of the signal amplitude with the number of nuclei may not be guaranteed in ZF but it is recovered in 3T, see Sec.V C).The frequencies confirm the assignment, since the six nonnegligible transferred couplings B of Mn 3a and the three nearly vanishing ones for Mn 6c are both opposite to the on-site coupling A. This simple argument is confirmed by DFT simulations of the hyperfine coupling at the Mn sites (Fig. S.3).

B. Antisite spin alignment
The same 3D plots (Fig. 1 a-c) show that the frequency splitting increases with the applied field.This can be understood from the field vector composition 55 B = µ 0 H + B hf .In simple cases, the relative local orientation of the Mn 3a and Mn 6c magnetic moments is easily inferred from this vector composition, as shown in Fig. 1 (d-f) Namely, in the FM case, or equivalently, for the large magnetization sublattice of a ferrimagnet (↑ FIM) the moments align along the applied field, hence B hf is opposite to H and the frequency decreases with increasing field.On the other hand, they anti-align to H for the small magnetization sublattice of a ferrimagnet (↓ FIM), so there the frequency grows with B hf ∥ H. Finally, in the collinear AFM case, they do not align, and the superposition of all relative vector orientations in our polycrystalline samples leads to a broadening with no first order shift.[61] Figures 1 g,h show the shifts vs. applied field for the (MnBi 2 Te 4 )(Bi 2 Te 3 ) n samples.The majority Mn 3a peak of MnBi 2 Te 4 (red stars) does not shift with H up to 1 T, following the AFM behavior predicted by Eq. 1.Its small shift for 2 T≤ H ≤ 3 T is consistent [61] with a canted antiferromagnetic (CAFM) state [63][64][65].In contrast, MnBi 4 Te 7 and MnBi 6 Te 10 (black and blue symbols, respectively) follow the FM case, like the MnSb 2 Te 4 majority site, shown by the red symbols in Fig. 1-i, all fitted with negative slopes equal to γ = − 55 γ within error bars.
Neutron diffraction measurements on MnSb 2 Te 4 have revealed the antiparallel alignment of the Mn 3a and Mn 6c local moments.[38,51] In our NMR data, this spin alignment is directly demonstrated in Fig. 1-i by the positive slope of 55 ν(H) for the Mn 6c sites.Moreover, we see exactly the same behavior for all (MnBi 2 Te 4 )(Bi 2 Te 3 ) n materials (Fig. 1 g,h), i.e., their septuple layers show the same FIM structure as MnSb 2 Te 4 .While indications of this behavior in MnBi 2 Te 4 were previously seen in highfield magnetometry [39], the presented NMR results provide direct local evidence of the opposite spin alignment of Mn 3a and Mn 6c for all (MnBi 2 Te 4 )(Bi 2 Te 3 ) n with n = 0, 1 and 2. It is likely that the n > 2 members of the (MnBi 2 Te 4 )(Bi 2 Te 3 ) n family [36,66,67] should also display this FIM structure, pretty much as their n = 0 − 2 analogs.Note that the 55 ν(H) slope for (MnBi 2 Te 4 )(Bi 2 Te 3 ) n in Fig. 1-h is slightly reduced [61] by an average canting angle θ between the sublattice magnetization and H, according to γ = 55 γ cos θ.This angle is small for the FM-like materials, but quite sizable for CAFM MnBi 2 Te 4 , reflecting the large powder average angle between the magnetization of its canted Mn 3a sublattices and the field.

C. Antisite moment temperature dependence
The 55 Mn NMR signal quickly disappears when increasing the temperature above T = 1.4 K, as the NMR T 2 relaxation time gets shorter than the instrumental dead-time.For MnSb 2 Te 4 the Mn 3a ZF NMR frequency (red symbols in Fig. 1 j) is detected up to t = T /T m ∼ 0.6, whereas the Mn 6c ZF NMR frequency (black symbols) is detected only up to t = 0.4.They both decrease towards the second order transition at T m , following the order parameter.However, the ratio of their peak areas (blue symbol) decreases much more quickly over the same range, vanishing around t = 0.4.This implies that the Mn 6c peak disappears in a first-order-like fashion at a temperature T * well below T m .
Summarizing, NMR at 1.4 K directly detects the opposite alignment of the Mn 3a and Mn 6c sublattices in all of the systems studied here, and it demonstrates that FIG.1: (a-c), 55 Mn NMR spectra of MnBi 2 Te 4 (β sample), MnBi 4 Te 7 and MnSb 2 Te 4 respectively, at T = 1.4 K in increasing applied fields, starting from ZF; (d-f) polycrystal vector composition 55 B = µ 0 H + B hf and their resulting spectral shifts for three simple cases: soft ferromagnet (FM, d), soft ferrimagnet, minority spin (↓FIM, e), antiferromagnet (AFM, f); (g,h) field dependence of the Mn 3a and Mn 6c mean frequency peaks for the (MnBi 2 Te 4 )(Bi 2 Te 3 ) n family; (i,j) Mn 3a and Mn 6c peaks for MnSb 2 Te 4 , field dependence (i) and temperature dependence (j) of their frequencies (black, red dots), temperature dependence of the ratio of their areas (blue dots, j). in the Sb-based compound the antisite Mn 6c disorders above T * , well below T m .The same information is not accessible for (MnBi 2 Te 4 )(Bi 2 Te 3 ) n , due to a combination of lower ordering temperatures and shorter T 2 .

III. µSR RESULTS
Spin-polarized muons implanted in polycrystalline samples stop in few lowest energy interstitials.In ZF and for T < T m the muon spin precesses around the local magnetic field B µ , due to ordered moments, producing coherent oscillations at a frequency γ µ B µ in the asymmetry of the muon decay at early times (γ µ = 135.554MHz/T, see Sec.V).
Selected ZF-µSR early-time asymmetries A(t) are shown in Fig. 2 a-c for the (MnBi 2 Te 4 )(Bi 2 Te 3 ) n materials at different temperatures, with their best fit to a minimal model, Eq. 2 in Sec.V.The model describes the internal field distribution p(B µ ) probed by muons as a few Gaussian components of very broad width ∆B i .Starting from MnBi 2 Te 4 (Fig. 2 a), the asymmetry shows a fast, overdamped initial relaxation and a second damped oscillation below T m .Both components correspond to appreciable internal fields, the fast initial Gaussian decay to a mean value smaller than its width 0 ≲ B 1 < ∆B 1 , and the visible oscillation to an observable mean value B 2 > ∆B 2 .At low temperature a third oscillating component (B 3 > ∆B 3 ) appears.By comparison, the n = 1, 2 members differ from n = 0 in i) the absence of the B 1 < ∆B 1 fast initial decay and ii) a lower field B 2 value, whereas, they also display iii) a high field B 3 component, that sets in only at lower temperatures.The presence of two distinct oscillations, both heavily damped, is more evident in the low temperature best fits of Fig.

A. Magnetic transitions
The temperature dependence of the internal fields, ∆B 1 (T ) (the width of that distribution), B 2 (T ) and B 3 (T ), are shown in Fig. 2 d-f.It reveals two common features among all three family members: ∆B 1 and B 2 correspond to the order parameter and vanish at the second-order magnetic transitions, T m ; in contrast, B 3 vanishes abruptly at T * , inside the ordered phase, without any corresponding anomaly in ∆B 1 , B 2 .
Weak transverse field (WTF) µSR provides the amplitude of the spin precession in a small applied field (µ 0 H ≪ ∆B 1 , B 2 , B 3 ).Below T m this amplitude drops abruptly.The (MnBi 2 Te 4 )(Bi 2 Te 3 ) n magnetic volume fraction m wtf shown in Fig. 2 g-i is obtained from WTF measurements (Eq. 4 in Sec.V) and demonstrates that MnBi 2 Te 4 and MnBi 6 Te 10 undergo very sharp transitions, at T m , (the width of the transition for a 90%-10% volume reduction is ∆T < 1 K), despite their relatively large atomic disorder implied by the presence of antisites.
The n = 1 sample displays a sharp transition as well, but a 20% contribution, which we attribute to intergrowths of the MnBi 2 Te 4 phase, is also visible (Fig. 2 h).
Similar results were obtained for MnSb 2 Te 4 .[68] Notably, we measured two distinct MnBi 2 Te 4 samples, labeled α and β (open and filled symbols, respectively, in Fig. 2 d,g,j), which readily show distinguishable transitions (T m = 26.0(3),24(1) K), due to slightly different preparation conditions.The residual WTF amplitude well below T m is due to a small fraction of muons implanted outside the sample, that do not experience its spontaneous internal magnetic field distribution.Separate magnetic volume fractions m i are also derived from the ZF normalized amplitudes at each internal field B i (Eq. 5, Sec.V D).Independently, the total volume fraction m L is obtained from the longitudinal amplitude (Eq.6). Figure 2   (a) Half the primitive cell with Mn 3a only, (b) Te-µ-Mn@Bi and c) Te-µ-Bi@Mn.
The muon sites were identified and their respective T = 0 K field values, B µ , were computed using a standard protocol, known as DFT+µ, [69][70][71][72].These are explained briefly in Sec.V E and discussed in more details in the Supplemental Material.[61] In the ideal MnBi 2 Te 4 crystal, this protocol identifies three stable sites, shown in Fig. 3 as Te-µ-Mn (red sphere), Te-µ-Bi (blue sphere), and Te-µ-Te (yellow sphere) and reported in the top box of Tab.I.The mean field values at these sites agree nicely with the experimental component B 1 (Te-µ-Bi and Teµ-Te) and B 2 (Te-µ-Mn), respectively.Notice that the correspondence between sites and fields is not bijective (more sites may contribute to the same field distribution) and that the uncertainty in the DFT+µ derived values is below 25%.TABLE I: T = 0 local field at the representative muon sites in MnBi 2 Te 4 , identified by color as in Fig. 3 (see Supplemental for details [61]).Boxes refer to muons sites: top, far from antisites; bottom, nn to an antisite.
Colored dashes refer to colors of symbols in Fig. 2 d. • Te-µ-Mn@Bi Mn6c 527 550(30) − B3 • Te-µ-Bi@Mn Bi3a 95 0(80) − ∆B1 The presence of intermixing modifies these findings in two ways: (i) inherent disorder broadens considerably all field distributions, producing large widths ∆B i , and (ii) the extra moment modifies significantly the mean field values at muon sites nearest neighbor (nn) to Mn 6c and Bi 3a .We label these two modified sites as Te-µ-Mn@Bi and Te-µ-Bi@Mn, respectively, and report their properties in the lower box of Tab.I.Both these consequences are observed experimentally, in particular the calculation for the muon site Te-µ-Mn@Bi agrees with local field B 3 , while Te-µ-Bi@Mn contributes to B 1 .
Let us now turn to the (MnBi 2 Te 4 )(Bi 2 Te 3 ) n compounds with n = 1, 2, where the sites closest to the Mn 3a layer are predicted to be very similar to those of n = 0.The farther high symmetry Te-µ-Te site, instead, is replaced by more than one site in the intervening quintuple layers.Since all of them are far removed from Mn 3a , we expect a large majority of muon sites characterized by very small or vanishing local field values.However, n = 1, 2 samples show non-vanishing net magnetic moment , hence B µ has an additional Lorentz field term B L = 4πM/3 ≈ 70, 50 mT, respectively.This contributes negligibly to the high field of the Te-µ-Mn@Bi site, B 3 ≫ B L , but significantly to the low field Te-µ-Bi@Mn, Te-µ-Bi and Te-µ-Te sites.This justifies both the second experimental field value B 2 ≈ B L (Fig. 2 e,f), its large fraction f 2 and the disappearance of the f 1 , B 1 = 0 signal.
The agreement of all these predictions, shown as hatched color bands in Fig. 2 d-f, with T → 0 K experimental data is altogether remarkable.It is the more so, in as much the same three DFT muon sites support a coherent, simple interpretation of the data for up to three experimental fit components, in four different compounds, over two magnetic phases.

IV. DISCUSSION AND CONCLUSIONS
An important result from NMR is that the size of the moment on Mn obtained from the hyperfine field, in first approximation, is proportional to the on-site coupling A, which is roughly 10 T/µ B for all 3d ions and for 55 Mn in particular.[73][74][75] An estimate of this coupling comes from the ZF NMR frequency of Mn 3a and Mn 6c .Taking the value of the latter for its much smaller transferred terms B, and, conservatively, half their difference as the uncertainty, we get roughly the same moment for all three (MnBi 2 Te 4 )(Bi 2 Te 3 ) n samples, µ Mn = 4.3(2) µ B , in agreement with neutron diffraction [76].
Importantly, fast Mn 6c spin fluctuations above T * justify the disappearance of the f 3 component, assigned consistently to the Te-µ-Mn@Bi site and mostly due to the nn Mn 6c moment.This observation is common to all three compositions.We recall that the same conclusion is drawn from the NMR findings on MnSb 2 Te 4 (Sec.II), confirmed by µSR as well [61].In addition, µSR results confirm the first order character of T * , since the internal field B 3 , proportional to the Mn 6c moment, is still large when the signal amplitude vanishes (Fig. 2 d-f, mimicking the NMR results of Fig. 1 j).
The anomaly at T * is confirmed by magnetization data, [61] which, however, are obtained in an applied field.Recall that in MnBi 6 Te 10 the local field B 2 is assigned to the low field sites Te-µ-Bi, Te-µ-Te and Te-µ-Bi@Mn, [61], in the presence of an additional comparable Lorentz field, B L , due to the net domain magnetization.The smooth behavior of B 2 across T * indicates that B L survives also in zero applied field above the transition, suggesting that a dominant FM stacking of Mn 3a layers persists above T * in zero field, therefore it is not induced by the external field.
The orientation of the antisite static moments successfully shows the sign of their dominant local exchange, displaying a universal behavior in this family, but this does not tell us the full magnetic structure of each material, that varies in the family.Actually, for MnBi 2 Te 4 we can assume that T m is a Néel transition and the sample is AFM in zero field.Here, T * (detected in ZF) clearly merges with the low temperature transition seen in M (T ) (Fig.S1 [61]).For MnBi 4 Te 7 the presence of a cusp in M (T ) suggests AFM bulk behavior at T m , but µSR shows the presence of an n = 0 contribution and we avoid further considerations.Finally, MnBi 6 Te 10 was already shown to be FM under certain growth conditions.[44,45] FIG.4: MnBi 2 Te 4 phase diagram summary: temperature dependence of the Mn 3a magnetic moment, rescaled from the µSR fields ∆B 1 (T ), B 2 (T ), and T → 0 value obtained from ZF Mn 3a NMR (shaded bands translate uncertainty in the latter).Insets: magnetic structures in one septuple layer of the primitive cell.
Figure 4 summarizes our findings in a schematic phase diagram for the A-type antiferromagnet MnBi 2 Te 4 , where the T → 0 magnitude of the static Mn moment is taken from NMR and its temperature behavior from the interpolation by Eq. 7 on the ZF µSR fields ∆B 1 (T ), B 2 (T ), of Fig. 2 d.The blue region is characterized by the static ordering of the diluted Mn 6c moments, aligned antiparallel to the main Mn 3a moments, as NMR demonstrates.The static moment at Mn 6c vanishes in the red region.
A similar plot for MnBi 6 Te 10 is reported in the Supplemental Material [61] (MnBi 4 Te 7 data are less reliable due to the presence of a sizable fraction of MnBi 2 Te 4 layer intergrowths, Fig. 2 h).Remarkably, in all three compositions, the mean Mn 3a order does not change appreciably across T * .
The red-shaded high-temperature phase is characterized by a very sharp second-order transition, as it is witnessed in all samples by the abrupt vanishing of the magnetic volume fraction, both by WTF and ZF µSR (Fig. 2  g-i).We highlight the unique ability of µSR, as opposed to both magnetometry and neutron scattering, to distinguish the reduction of the magnetic moment, encoded in the local field, from the volume fraction, encoded in the amplitude of the signal.Local disorder, like that expected from magnetically coupled, random antisites, could yield a distribution of transition temperatures, and hence a more progressive reduction of the volume fraction than that displayed in all our samples.This suggests that, from a magnetic point of view, the hightemperature ordered phase approaches closely the ideal, intermixing-free material.This provides both the antiferromagnetic and the ferromagnetic versions of a closeto-ideal magnetic topological insulator.
Early sublattice decoupling is well documented, for instance in intermetallic compounds, [77][78][79][80][81][82][83][84] and it often involves exchange coupling frustration, which may also play a role in the present case [85].Since intermixing is a common feature for all materials, including cation species with similar radii, we speculate that early antisite disordering may take place more often than one thinks.Therefore, our findings may have a more general impact than just on the present materials.The MnBi 2 Te 4 α-sample was prepared from a mixture of pre-synthesized MnTe and Bi 2 Te 3 taken in the ratio 0.87:1.05.The powders were handled in an argon-filled dry glovebox (MBraun), homogenized in a dry ball-mill (Retsch, MM400) at 20 Hz for 20 min and then pressed into a 6-mm pellet (2 tons, 30 sec).The pressling was placed into a quartz ampoule, sealed off under dynamic vacuum (3 × 10 −3 mbar) and annealed in a temperaturecontrolled tube furnace (Reetz GmbH) following the procedure developed in [62].
The MnBi 4 Te 7 and MnBi 6 Te 10 samples were obtained from a corresponding, stoichiometric mixture of the powdered binaries which were handled similar to the ones above.The former was annealed at 585 • C for 10 days (heating rate 1 • / h) and the latter was annealed at 575 • C for 4 days (heating rate 90 • / h).Both samples were water-quenched.
The polycrystalline MnBi 2 Te 4 β-sample was synthesized by co-melting of pre-synthesized MnTe and Bi 2 Te 3 in an evacuated quartz ampoule at a temperature of about 980 • C for 12 h, followed by slow cooling down to 580 • C at the rate of 5 • /h.This temperature was kept for 12 h followed by air-quenching.Then, the polycrystalline sample was ground as a fine powder and converted into a pellet and was then sealed in a quartz container under the pressure of 10 −4 Pa.The ampoule was further heated up to 585 • C for 8 h, then kept at this temperature for about 240 h followed by air-quenching.The grinding and annealing process was repeated twice to achieve a homogenized phase-pure compound.
The MnSb 2 Te 4 sample was synthesized from a stoichiometric mixture of the elements (Sigma-Aldrich, 9N5, Mn reduced prior to synthesis) that was ball-milled at 20 Hz for 20 min, pelletized and annealed at 550 • C for 8 days, and finally quenched.

B. X-ray diffraction and energy-dispersive X-ray spectroscopy
Phase purity of all samples but β was analyzed by powder X-ray diffraction (PXRD) on a Malvern Panalytical Empyrean 3 diffractometer, employing CuKα 1 radiation (λ = 1.54059Å) and set in Bragg-Brentano geometry.Lattice parameters refinement was conducted by Le Bail method to ensure the correct assignment of the (MnBi 2 Te 4 )(Bi 2 Te 3 ) n phases.A PXRD pattern of the βsample was taken on a Bruker D2 PHASER diffractometer using CuKα 1 radiation within the scanning range of 2θ = 5 − 75.All (MnBi 2 Te 4 )(Bi 2 Te 3 ) n samples were found to be single-phase by comparing with the published results [28-30, 44, 62, 86], without admixtures of other binary or ternary phases.
The MnSb 2 Te 4 sample contained a very small imprurity of MnTe 2 assessed as 2 wt.% by Rietveld refinement.The lattice constants of the main MnSb 2 Te 4 phase were refined as a = 4.2416(1), c = 40.883(2)within the R 3m space group.This material can flexibly accommodate strong Mn/Sb intermixing [46,47,87] and, therefore, adopt various total compositions.To achieve the highest accuracy in the determination of the chemical composition of our powdered sample, we conducted calibrated EDX measurements.For that, parts of the sample were cast into synthetic resin (versosit) pucks, sputtered with a gold layer and painted with conductive silver paint to avoid charge accumulation.EDX spectra were recorded with a high-resolution SEM EVOMA 15 (Zeiss) equipped with a Peltier-cooled Si(Li) detector (Oxford Instruments) employing 30 kV acceleration voltage.Element quantification was obtained from least-square fitting of edge models (Mn-K, Te-L, Sb-L) invoking k-factor calibration from the stoichiometric samples of similar composition (Sb 2 Te 3 and MnTe).To assess systematic errors stemming from the different edge and reference choices, we included Sb 2 Te 3 and MnTe references for Te in our quantification statistics.The determined composition of the sample was Mn 0.87(1) Sb 2.03(1) Te 4.00 (1) which is in line with its magnetic properties reported in [68].

C. NMR
The NMR spectra were measured in a He-flow cryostat by means of the HyReSpect home-built phase coherent broadband spectrometer [88].Spin-echoes were excited at discrete frequency points by refocusing P-τ -P Hahn radio-frequency (rf) pulse sequences, with optimized pulse duration and intensity to maximize the resonance signal, and shortest τ delay (limited by the apparatus dead time of few µs).
Spectra are reconstructed from the maximum of the spin-echo Fourier transform amplitude at each frequency, corrected for the frequency-dependent sensitivity and nuclear Boltzmann factors.The normalized values correspond to the spectral distribution of hyperfine fields at the 55 Mn nuclei.When the best fit of the two spectra (ζ = Mn 3a , Mn 6c ) require more than one Gaussian component each, their mean frequency is calculated from the corresponding weights A α,i as the first moment ZF nuclear echoes were collected with a non-resonant probe circuit, by virtue of the large rf enhancement η = H * 1 /H 1 , where H 1 is the applied field at the radio frequency ω, and H * 1 is the ω oscillating component of the huge hyperfine field B hf , following the electronic moment response to H 1 .[89,90] The factor η is very sensitive to nanoscopic and mesoscopic changes in the nucleus environment and this does not guarantee a uniform proportionality between spectral area and number of resonating nuclei.Relative proportionality is recovered in the more uniform resonant conditions obtained at high static applied fields.

D. µSR
The µSR experiments were carried out at the Paul Scherrer Institute, Villigen, Switzerland, on the GPS spectrometer.
A minimal choice for the best-fit function of the time domain ZF asymmetry, arising from parity violation in the weak muon decay, is the following distinguishing the fast relaxing, precessing fractions f T = 3 i=1 f Ti (T for transverse with respect to the initial muon spin direction) from the slow relaxing fraction f L (L for longitudinal), which corresponds to local field components parallel to the initial muon spin direction.The precession relaxation rates are due to the width of each field distribution, σ i = 2πγ µ ∆B i , their fractions obey f T + f L = 1 and the maximum experimental asymmetry, A 0 , is calibrated at high temperature.Polycrystalline averaging leads to 2f L = f T , but a very fast transverse decay and a small fraction of muons stopping outside the sample may relax this ideal condition.
The WFT time domain signals are best fitted by an oscillatory and two relaxing functions where ϕ is an initial phase and f p is the muon fraction that does not experience strong local hyperfine fields, which includes muons stopping outside the sample and inside paramagnetic microdomains.The former corresponds to the low temperature residual value f p0 , whereas for T ≪ T m , one has f p = 1, f T = f L = 0.The two relaxing function represent the transverse and longitudinal fractions of muons experiencing internal fields.Three independent determinations of the magnetic volume fraction are given by where f T0 = lim T →0 f T (T ) and Eq. 5 neglects f p0 , since in ZF f L is indistinguishable from f p .They are used in Fig. 2 g.h (Eq.4) and j (Eq. 5 colored and gray symbols, for Eq. 5 and 6, respectively).Lastly, the internal fields ∆B 1 = σ 1 /2πγ µ , and B 2 proportional to the order parameter are fitted to a standard phenomenological function [91] b used both in the shaded colors of Fig. 2 d-f and in Figs. 4, S.8.

E. DFT
The DFT calculation protocols for determining the muon implantation sites [69] and the contact contri-bution to the hyperfine interactions in magnetic compounds [70,71] are well established.State of the art DFT calculations are performed on a magnetic supercell, including an extra impurity hydrogen atom.Sampling of the starting impurity supercell coordinates and lattice relaxation by force minimization yield minimum total energy sites and their spin couplings, that allow the calculation of the local field [72].Full calculation details are reported in the Supplemental Material [61].

VII. ACKNOWLEDGMENTS
This work was supported by the Deutsche Forschungsgemeinschaft (DFG) within the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter -ct.qmat (EXC 2147, project-id 390858490) and the SFB 1143 (project-id 247310070).IJO and RDR acknowledge financial support from PNRR MUR project ECS-00000033-ECOSISTER and also acknowledge computing resources provided by the STFC scientific computing department's SCARF cluster and CINECA under ISCRA Project ID IsCa4.The cost of the HyRe-Spect equipment used for this investigation was partly supported by the University of Parma through the Scientific Instrumentation Upgrade Program 2020.This work is partially based on experiments performed at the Swiss Muon Source (SµS), Paul Scherrer Institute, Villigen, Switzerland.MMO acknowledges the support by MCIN/ AEI /10.13039/ 501100011033/ (Grant PID2022-138210NB-I00) and "ERDF A way of making Europe" as well as MCIN with funding from European Union NextGenerationEU (PRTR-C17.I1) promoted by the Government of Aragon.We acknowledge MSc.Fabian Lukas and MSc.Laura Mengs (Technische Universität Dresden) for their contributions to inorganic synthesis as student assistants.We are grateful to the group of Prof. A. Lubk (TU Dresden, IFW Dresden) and to Mrs. G. Kreutzer for the calibrated EDX measurements and insightful discussions.In the MnBi 2 Te 4 sample, for µ 0 H= 10 mT, a cusp in M (T ) is observed at T N ≃ 24.6K (see Fig. S.1a, in coincidence with a peak in dM/dT ), indicating the onset of long-range antiferromagnetic order.Upon further cooling, both ZFC and FC curves exhibit an anomaly, appearing first as an upturn below 15 K, while below 11.5 K the ZFC and FC curves separate, denoting the onset of a ferromagnetic-like character at T * = 11.5 K, as indicated by a dip in dM/dT .This second transition for MnBi 2 Te 4 is suppressed in higher applied fields around 0.5 T, which shows the field-induced character of the transition.
A similar double transition is evidenced by the double dip in dM/dT observed in MnBi 4 Te Blue dashed lines show the field dependence of the mean frequency for the Mn 3a and Mn 6c peaks.

II. NMR
A. Field dependence of 55 Mn NMR in MnBi2Te4 The dashed lines on the horizontal plane in Fig. 1 a, main paper, show that the mean frequencies of the NMR Mn 3a and Mn 6c peaks shift slightly with applied field.These frequencies are plotted as red stars in Fig. 1 g, h, for Mn 3a and Mn 6c respectively.Their slope can be appreciated by eye, an effective γ = dν/dH /µ 0 that vanishes for µ 0 H ≤ 1 T as predicted by Eq. 1 for the AFM case.Indeed, the hyperfine field in an AFM polycrystalline sample, depicted as red arrows in the cartoon (Fig. 1 f) adds in all possible crystal grain orientations to the external field (blue arrow), leading to different total local fields B hf (green arrows), evenly distributed in modulus about the external field value.Recalling that 55 ν = 55 γ| 55 B| this leads to a vanishing shift in first order.The absolute value of γ increases around 2 T, as expected in a canted antiferromagnet (CAFM) polycrystal.Indeed, a crystal grain whose hexagonal c axis forms an angle θ with the field undergoes a spin flop transition at 2 ≲ µ 0 H(θ) ≤ 3.57 T. [1][2][3] The powder average results in a distribution of 55 B values (like in the pure AFM case), albeit with a mean shift, due to the coupling to the field (γ < 0 for ↑Mn 3a , γ > 0 for ↓Mn 6c ).
For comparison, in the FIM case the spin alignment to the field is complete at any orientation and the corresponding Mn 3a (Fig. 1 g) and Mn 6c (Fig. 1 h) effective γ values for MnBi 4 Te 7 and MnBi 6 Te 10 (black and blue symbols, respectively) coincide with ± 55 γ within errors, as expected for the ↑ and ↓ sublattice respectively.
B. 55 Mn NMR spectra of MnBi6Te10 A 3D view of the spectra for our polycrystalline sample of MnBi 6 Te 10 in the frequency range 350 to 500 MHz, is shown in Fig. S.2, with ZF at the back and increasing fields µ 0 H towards the front.

III.
DFT SIMULATION OF 55 Mn-NMR SPECTRA We have calculated the hyperfine field at the Mn nucleus for both MnBi 2 Te 4 and MnSb 2 Te 4 , first, without site intermixing, by an all electron DFT+U protocol as implemented in the ELK code [4].We have used a dense 15×15×3 Monkhorst-Pack Mesh for MnBi 2 Te 4 and 15×15×7 for MnSb 2 Te 4 .The muffin-tin radius of 2.4 a.u.for Mn, 2.8 a.u.for Bi, 2.6 a.u.for Sb and 2.6 a.u.for Te were used.The maximum length or cutoff for the G + K vectors is 8.0 a.u., divided by the average of the muffin-tin radii.The hyperfine field at the Mn 3a nucleus in the ideal MnBi We have used the GIPAW code [5] to model the intersite mixing in MnBi 2 Te 4 and MnSb 2 Te 4 , and generate the distribution of 55 Mn hyperfine fields at the Mn 3a and Mn 6c nuclei.The calculation details for muon calculations in Supplementary Sec.V were adopted.Hundreds of different structural configurations were generated in a 2×2×2 supercell of the magnetic primitive cell, according to typical values of the Mn/Bi occupancies for the Wyckoff positions 6c (94% Bi and 6%Mn) and 3a (74% Mn, 21% Bi and 5% voids) [6], by making use of a supercell code [7].Of all the generated configurations, one hundred were randomly selected for the DFT calculations of the hyperfine field, involving a total of 800 distinct Mn sites.The resulting distributions of hyperfine fields are shown in

A. Methods
We determine the muon implantation sites in MnBi 2 Te 4 and MnSb 2 Te 4 by performing collinear spinpolarized density functional theory (DFT) calculations by means of the Quantum Espresso code [8].We use the projector augmented pseudopotentials [9] and the semilocal generalized-gradient-approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) [10] functional and the DFT-D3 van der Waals dispersion energy-correction method [11] to capture more accurately the effects of the long-range interactions in the structural optimization.
The role of electron-electron interaction on the Mn−d orbitals are considered within the DFT+U scheme [12][13][14][15], where we have used a value of U eff = (U − J) = 4.0 eV [16].The DFT+U relaxations were performed in an interlayer antiferromagnetic cell (primitive cell doubled along c) for MnBi 2 Te 4 [17].To accommodate the effect of the spurious artificial interactions of the positive muon impurity modeled with a hydrogen pseudopotential, we have used a 2×2×1 supercell, positively charged, with a uniform charge background to restore charge neutrality.The plane-wave kinetic energy and charge density cut-offs used are 70 Ry and 630 Ry, respectively.For structural optimizations, a 3×3×1 k-point mesh was used for the 57-atom supercell, while it was doubled along all the axes for the calculation of the contact hyperfine field via a self-consistent calculation.The atomic positions were optimized till force and energy thresholds of 1×10 −3 a.u(Ry/Bohr) and 1×10 −4 Ry while the lattice parameters remain fixed at the DFT+U obtained values (i.e., without hydrogen).(yellow sphere), Te-µ-Bi (blue sphere) and Te-µ-Mn (red sphere).The Mn (at 6c) and Bi (at 3a) site mixing has been considered for (b) the next nearest 6c site of Te-µ-Te, (c) the 6c site of Te-µ-Bi (labelled Te-µ-Mn@Bi), (d) the 3a site of Te-µ-Mn (labelled Te-µ-Bi@Mn), and (e) the next nearest 6c site of Te-µ-Mn (labelled Te-µ-Mn(+n.nMn@Bi)).The value of the µ-Mn/Bi distance is also shown.All sites have energies within 0.35 eV, suggesting that all of them are likely to be occupied, although a low en-TABLE S.III: MnBi 2 Te 4 : muon bond labels and muon sites 1-5 (3, 5 represents the average of 3, 5), DFT total energy difference (∆E) to the lowest energy and local fields in mT; B c contact (along the bulk magnetization), B dip dipolar, and B µ total fields in mT.All fields are rounded to the mT.Magnetic moments on Mn are aligned along the Cartesian z axis ([0001] direction) [18] in the primitive cell.Top section: MnBi 2 Te 4 muon sites without Mn-Bi intermixing; bottom section: same muon sites affected by the intermixing.ergy barrier between close enough sites, or a flat potential landscape may lead to partial delocalization, i.e. the muon wave function could be broad, spreading over the flat potential region.To investigate the stability of these candidate positions, we have performed nudged elastic band (NEB) [19,20] calculation, mapping the energy landscape along straight muon paths or polylines across close sites (typically, less than 2 Å apart).
The results are shown in Fig. S.6 c for straight path 3-5.Considering that the zero point energy of muons is typically E ZP ≈ 0.5 eV, the much smaller barriers in the energy landscape shows that muons must partially delocalize over both sites 3 and 5, experiencing an average interaction.Similar occurrence happens for the minimum energy path (MEP) of 4 ′ -1-4-2, S.6 b, where from the energy landscape, two distinct muon sites are observed, one among the 4 ′ -1-4 region and the other at site 2. The small energy barrier between 4 and 1 and between the symmetric equivalent 1 and 4 ′ , indicates partial delocalization of the muon over 4 ′ -1-4.
In the following analysis we consider only three main muon sites, labelled Te-µ-Te (delocalized over sites 1, 4 and 4 ′ ), Te-µ-Bi (site index 2) and Te-µ-Mn (delocalized over sites 3 and 5) according to their chemical bonds.However, the MnBi 2 Te 4 sample is characterized by the presence of Bi 3a and Mn 6c antisites.This generates a large number of local configurations.Guided by experiments we approximate it distinguishing three components with widths smaller than the separation of their means, determined in first approximation by the presence or absence of a nn antisite.
Therefore, we additionally consider the effect of nearest neighbor (nn) antisites on Te-µ-Bi and Te-µ-Mn muon sites.They are listed in the second section of Tab.S.III as Te-µ-Mn@Bi (where we consider the Te-µ-Bi site and Bi 6c is replaced by Mn 6c ), Te-µ-Bi@Mn (where we consider the Te-µ-Mn site and Mn 3a is replaced by Bi 3a ) and Te-µ-Mn(+n.nMn@Bi) (where we consider the Teµ-Mn site and the closest Bi 6c is replaced by Mn 6c ).The next 3a or 6c site in Te-µ-Te has a distance above 3.5 Angstrom and the small effect on its local field is neglected, in first approximation.These structures and resulting bond distances are show in Figs.S.7 b-e.

C. Local fields in MnBi2Te4
In a ZF µSR measurement, the total local magnetic field at the muon site consists of the following contributions: B µ = B C + B dip + B L , [21] where B c is the isotropic contact contribution originating from the Fermi interaction, which requires a quantum treatment of the electronic wavefunction and has been obtained here with DFT calculations [22].The last two terms are obtained by the summation of the long-range dipoles in real space using the Lorentz sphere approach, where B dip is the dipolar field obtained from the contributions of the Mn magnetic moments within the sphere, while those outside the sphere contribute to the Lorentz term B L [23], assuming that the outer sample surface is always unmagnetized in ZF.For the antiferromagnetic order of MnBi 2 Te 4 , the B L term vanishes.Table S.III reports our results, where the Mn magnetic moment has been fixed at 4.3 µ B , the value obtained from our NMR measurements.The nonnegligible effect of the muon induced displacements on the host atomic positions is included in calculating B dip .Due to the approximations inherent in DFT+µ, absolute agreement with experimental values is not expected to be better than within 25%.
Since the energy minima of sites 3 and 5 are rather close, compared to E ZP , a rough estimate of their local field is just the value, B µ , obtained from the average of their local components, reported in Tab.S.III as 3, 5 .This value is actually in close agreement with the experimental field B 2 at zero temperature.
For more than one site, namely, Te-µ-Te, Te-µ-Bi and Te-µ-Bi@Mn, the predicted local field value is close to zero (within the second moment of the field for component 1, ∆B 1 ).
Most notable is Te-µ-Mn@Bi, where, due to the short µ-Mn 6c distance ≈ 2. The MnBi 2 Te 4 • n(Bi 2 Te 3 ) structure of the two compounds, is formed by quintuple Bi 2 Te 3 layers intercalated between the septuple layers of MnBi 2 Te 4 , therefore we consider the same sites of Tab.S.III, plus further muon sites, similar to Te-µ-Bi and Te-µ-Mn, in the additional quintuple layers.In the FM structure the hyperfine field is never vanishing by symmetry at any site, as it is the case for Te-µ-Te in the AFM structure.Still, most of these site are far away from the Mn 3a layer and have small local fields.Therefore they contribute to a broad field distribution centered at the Lorentz field value, B L ≈ 40 mT The field value corresponding to site Te-µ-Mn, the highest energy site in MnBi 2 Te 4 , is not observed experimentally.This is not surprising, in view both of its higher The calculation yields similar results and the muon site classification is similar to that of MnBi 2 Te 4 , but the actual sample interlayer order is ferromagnetic, giving rise to a Lorentz contribution B L ≈ 70 mT.Here, the difference between the Sb and the Mn radii is even smaller than in the Bi case, therefore the occupancy of the Mn 6c antisite is larger, and all local field distributions become much broader.The B 2 component agrees with the prediction for all the sites that have low local field values in MnBi 2 Te 4 , namely, Te-µ-Te , Te-µ-Bi and Teµ-Bi@Mn.

A
. NMR peak assignment A 3D view of the spectra for polycrystalline samples of MnBi 2 Te 4 , MnBi 4 Te 7 and MnSb 2 Te 4 at T = 1.4 K, in the frequency range 350 to 500 MHz, is shown in Fig. 1ac (MnBi 6 Te 10 in the Supplemental Material, Fig. S.2) [61]) with ZF at the back and increasing fields µ 0 H towards the front.They all show two broad peaks patterns, each centered at a distinct frequency: ν a,c = 55 γ | 55 B a,c |.
j shows the cumulative sum of the three transverse fractions for MnBi 2 Te 4 -α.Both m L (grey symbols) and m 1 + m 2 (red symbols) drop sharply at the transition T m , in agreement with m wtf .In contrast, m 3 disappears abruptly at T * =12(1) K, suggesting that this component originates from muon sites sensitive to the subtle change that takes place across that point.This is reminiscent of our NMR MnSb 2 Te 4 results.For the (MnBi 2 Te 4 )(Bi 2 Te 3 ) n samples, the temperature coincides with a clear anomaly appearing in magnetization (Fig. S.1 and Tab.S.I), suggesting that Mn 6c moment are undergoing fast (paramagnetic-like) reorientations above T * .Further insight crucially requires the correct identification of the muon-stopping sites.
VSM) was utilized to conduct bulk DC magnetic measurements on the same samples employed for the NMR and µSR experiments.The temperature dependence of the magnetization is shown in Fig.S.1, for both zerofield cooled (ZFC) and field-cooled (FC) protocols, as measured over the temperature range 1.8 K to 60 K.

7 and MnBi 6
FIG. S.2:55 Mn NMR spectrum of MnBi 6 Te 10 at T = 1.4 K in increasing applied fields, starting from ZF.Blue dashed lines show the field dependence of the mean frequency for the Mn 3a and Mn 6c peaks.

FIG. S. 3 :
FIG. S.3: 55 Mn-hyperfine spectra calculated by DFT in MnBi 2 Te 4 (a) and MnSb 2 Te 4 (b), distinguishing the contributions from Mn 3a (green bars) and Mn 6c (yellow bars).In the calculation, the Mn 6c moment is assumed antiparallel to the moment of the nearest neighbor Mn 3a in both cases.The solid black line is the probability density distribution estimated by the Kernel density function (KDE).
3 Å, (Fig. S.7c) the internal muon field increases considerably to 527 mT (see Table S.III).This value agrees very closely with the experimental value of B 3 at zero temperature.The site assignment for MnBi 2 Te 4 is summarized in Tab.I. D. Local fields in MnBi2Te4 • n(Bi2Te3), n = 1, 2 (30 mT)  for MnBi 4 Te 7 (MnBi 6 Te 10 ).This corresponds closely to the experimental B 2 value at zero temperature.Its temperature dependence is shown in Fig. S.8 for n = 2, rescaled to the magnetic moment per site with the ZF NMR calibration at 1.4 K, and T * is the temperature where the B 3 term disappears (see Fig. 2 f).

FIG. S. 8 :
FIG. S.8: MnBi 6 Te 10 temperature dependence of the magnetic moment on Mn 3a , low temperature value from NMR, temperature dependence from µSR, ∆B 2 (T ).The inset shows the smallest unit cell portion sufficient to highlight the change.

2
Te 4 and MnSb 2 Te ment of the van der Waals interactions produce five low total energy candidate muon positions, indexed 1, 2, 3, 4 and 5 in TableS.II and Fig. S.6.Muon site 1 has the lowest total DFT energy, taken as the origin of the energy scale.The reported total DFT energy of the other sites are actually the difference ∆E with this reference energy.They are listed in the upper section of TableS.III, together with the local field contributions at each site.