Visualization of superparamagnetic dynamics in magnetic topological insulators
E. Lachman, A. F. Young, A. Richardella, J. Cuppens, Naren HR, Y. Anahory, A. Y. Meltzer, A. Kandala, S. Kempinger, Y. Myasoedov, M. E. Huber, N. Samarth, E. Zeldov
VVisualization of superparamagnetic dynamics inmagnetic topological insulators
E. Lachman , ∗ , A. F. Young , , ∗ , A. Richardella , J. Cuppens , Naren HR ,Y. Anahory , A. Y. Meltzer , A. Kandala , S. Kempinger ,Y. Myasoedov , M. E. Huber , N. Samarth and E. Zeldov Department of Condensed Matter Physics, Weizmann Institute of ScienceRehovot 76100, Israel Physics Department, University of CaliforniaSanta Barbara 93106-9530, USA Department of Physics, The Pennsylvania State UniversityUniversity Park, Pennsylvania 16802, USA Department of Physics, University of Colorado DenverDenver, Colorado 80217, USA ∗ These authors contributed equallye-mail:[email protected]
Quantized Hall conductance is a generic feature of two dimensional electronicsystems with broken time reversal symmetry. In the quantum anomalous Hallstate recently discovered in magnetic topological insulators, time reversal sym-metry is believed to be broken by long-range ferromagnetic order, with quan-tized resistance observed even at zero external magnetic field. Here, we usescanning nanoSQUID magnetic imaging to provide a direct visualization ofthe dynamics of the quantum phase transition between the two anomalousHall plateaus in a Cr-doped (Bi,Sb) Te thin film. Contrary to naive expecta-tions based upon macroscopic magnetometry, our measurements reveal a su-perparamagnetic state formed by weakly interacting magnetic domains with a r X i v : . [ c ond - m a t . m e s - h a ll ] J un characteristic size of few tens of nanometers. The magnetic phase transi-tion occurs through random reversals of these local moments, which drive theelectronic Hall plateau transition. Surprisingly, we find that the electronic sys-tem can in turn drive the dynamics of the magnetic system, revealing a subtleinterplay between the two coupled quantum phase transitions. The integer quantum Hall effect (QHE), first observed in clean two dimensional electronsystems at high magnetic fields ( ), is the paradigmatic example of a topological phase: differ-ent integer quantum Hall states are characterized by identical symmetries but different integertopological quantum numbers η , with the quantized Hall resistance given by R xy = ηh/e ( ).However, Hall quantization may occur also in the absence of an external field as long as time-reversal symmetry (TRS) is broken ( ). This quantum anomalous Hall (QAH) state was recentlyrealized experimentally ( ) following theoretical proposals based on combining strong spin-orbit coupling with long-range ferromagnetic (FM) order in magnetically doped topologicalinsulators (TI) ( ). The QH plateau transition represents the canonical example of a topo-logical phase transition, described by the divergence of the localization length and a universalcritical scaling of transport coefficients ( ). Recent theoretical calculations show that undercertain assumptions, the QAH plateau transition can be mapped onto the same network modelused to describe the integer QH plateau transition, leading to the same scaling laws ( ).While the electronic topological transition in QH and QAH systems, taken in isolation,can be viewed as identical, the two experimental systems differ in several key aspects. TheQH plateau transition is a purely electronic effect, in which delocalization occurs against abackground of quenched electronic disorder. In contrast, the QAH plateau transition resultsfrom two coupled quantum phase transitions: the field-driven magnetic transition of the FMorder and the electronic transition that is driven by TRS breaking of the FM transition. Thedynamics of the FM reversal can endow the QAH electronic transition with features that do not2ave an analog in QHE systems. Exactly at the transition, both systems can be understood as anetwork of domain walls—magnetic domain walls for the QAH, and domain walls of differentfilling factor in the QHE—that host counterpropagating chiral edge states. Crucially, however,in contrast to the QH plateau transition, which describes the phase diagram of the system inthermodynamic equilibrium, the FM domain structure in the QAH is metastable leading tohysteresis and relaxation dynamics that can directly affect the electronic system. Moreover, theenergy scale of the electronic delocalization transition and its critical scaling may depend onthe details of the microscopic FM structure and on the scaling of the magnetic phase transition.Finally, the electronic system, which commonly mediates FM interactions in dilute magneticsystems, can in turn modify the phase transition of the magnetic system. Most previous studiesof QAH systems have used electronic transport measurements to probe the combined effectof magnetic and electronic evolutions, making it difficult to disentangle the individual roles ofthe two phase transitions. Exploring the reciprocal coupling of the magnetic and electronicquantum phase transitions thus requires selective measurement tools that can address the twosystems independently.Here we combine electronic transport with a scanning superconducting quantum interfer-ence device (SQUID) of 200 nm diameter that resides on the apex of a quartz tip (SQUID-on-Tip (SOT)) (
21, 22 ) to simultaneously probe the magnetic and electronic transitions at Hetemperatures in a 7 quintuple layer (QL) thick Cr . (Bi . Sb . ) . Te film grown by molecularbeam epitaxy on a SrTiO substrate (Fig. 1). The large dielectric constant of the substrate al-lows effective control of the chemical potential through back gating. The hysteretic transitionbetween two Hall resistance ( R xy ) plateaus occurs at a coercive field of H c = 130 mT at 250mK (Fig. 1A). At elevated fields, the longitudinal resistance ( R xx ) shows a pronounced dip asa function of the back gate voltage at V g ≈ V due to an incipient QAH state (Fig. 1B). Atthe lowest measurement temperature (250 mK) used here, the sample we discuss does not yet3how the fully developed QAH. This is similar to previous experiments on Cr-doped topolog-ical insulator films ( ) where quantized Hall resistance appears only at dilution refrigeratortemperatures, far below the onset temperature of hysteretic magnetic behavior (
4, 23 ). Figures1E-H show images of the local distribution of the magnetic field B z ( x, y ) in the sample at var-ious points along the magnetization loop. We find B z ( x, y ) to be highly inhomogeneous at allfields, with peak contrast at H c , where the average magnetization vanishes (Figs. 1F,G). Sur-prisingly, submicron structure is evident even at fields corresponding to saturation of transportcoefficients (Figs. 1E,H). Images corresponding to opposite magnetization are highly anticor-related on microscopic scales—including at full saturation. This suggests an inhomogeneousdistribution of magnetic moments due to segregation of the Cr dopants. While high resolutiontransmission electron microscopy measurements on our samples have yet to show any obviousevidence of Cr clustering (see Fig. S11), we cannot preclude inhomogeneity at the nanoscale,akin to that found in other magnetically doped semiconductors such as Cr-doped ZnTe ( ).Clear evidence of phase separation has only been seen in Cr doped Bi Se thin film ( ) sam-ples that do not show the QAH state, while large nanoscale fluctuations in the local Cr densityhave been observed in Te based samples ( ) due to random doping that introduces strongdisorder in the material ( ).In metallic FM thin films with out-of-plane magnetization, it is well established that magne-tization reversal develops via the nucleation and propagation of domain walls (DWs) separatingregions of opposite magnetization. Such DW mediated magnetization reversal has also beenimaged in ferromagnetic semiconductor films (
28, 29 ) at magnetic dopant concentrations com-parable to those in our magnetic TI films. However, scanning SOT microscopy reveals a verydifferent picture of the magnetization reversal process in magnetic TIs. Figure 2A shows asequence of B z ( x, y ) images acquired for increasing values of µ H near H c . The five imagesappear almost identical; however, numerical subtraction of successive image data ( ∆ B z ( x, y ) ,4ee Fig. 2B) reveals the underlying dynamic process. Instead of the anticipated DW motion,magnetization reversal occurs through a series of random events in which isolated nanoscale is-lands undergo a reversal of their out-of-plane magnetic moment (see Movie S1). As we discussbelow, this constitutes a direct microscopic observation of superparamagnetism in magneticallydoped TI films. Our observations caution against drawing conclusions about the ferromagneticstate solely from macroscopic magnetization probes (SQUID magnetometry, magneto-opticalKerr effect) which show square hysteresis loops with robust zero field remanence ( ).To quantify the superparamagnetic dynamics across the Hall plateau transition, we fit eachof the local features in the ∆ B z ( x, y ) maps with a point-like out-of-plane magnetic moment m (Fig. S3). Figure 2C summarizes ∼ µ H = 0 and is accompanied by the reversal of nanoscale moments with average ¯ m = 3 × µ B (Figs. 2C,D). Given an average saturation magnetization of ∼ µ B / Cr atom asobtained from global magnetization measurements (Fig. S14), the estimated average diameterof these flipping islands is d = 51 nm for our 7 nm thick film, considerably below our spatialresolution of ∼
300 nm. As the field is increased towards H c , however, a pronounced change inthe moment distribution is observed with a shift to higher m values and appearance of a longtail of large moments with m & µ B (Figs. 2C,D). The microscopic reversal moments m canbe summed to obtain the net change in magnetization M over a continuous field range (Fig.2E). Comparison with simultaneously acquired R xy shows a qualitative match, implying thatthe behavior of the transport coefficients through the plateau transition is mainly determined bythe underlying change in magnetization. Superparamagnetic behavior and a similar relationship5etween measured magnetization and transport coefficients were found in a second sample aswell as in a Mn-doped Bi Te film (see Movie S2 and Figs. S2, S12-13 and S15).Hysteretic magnetic transitions are a signature of metastability, and typically display tem-poral relaxation via thermal activation or quantum tunneling. We probe magnetic relaxation inreal time by polarizing the system at µ H = − T and then ramping the field to a positive setpoint µ H set . We then acquire repeated images of B z ( x, y ) while simultaneously monitoringelectronic transport. No relaxation is observed on laboratory time scales for µ H set < ineither magnetization or transport coefficients. Spontaneous relaxation begins to be evident atsmall positive fields ( µ H set = 63 mT), manifesting as magnetic reversal events ∆ B z ( x, y ) anda slow upward drift of R xy . The frequency and number of these reversals increases significantlynear the coercive field at µ H set = 126 mT (Figs. 3A,C).The temporal relaxation measurements further corroborate the superparamagnetic behavior:at temperatures well below the blocking temperature, the magnetization of a superparamag-net is hysteretic, showing minimal relaxation at low fields. On approaching H c the magneticanisotropy barrier U is reduced, leading to relaxation when U ’ k B T . Since U is proportionalto the volume of the superparamagnetic particles, smaller islands undergo thermal activationat a lower field. Simultaneous transport measurements (Fig. 3D) indicate that the electronictransition closely tracks the magnetic relaxation, with transport coefficient relaxation evidentat µ H set = 63 mT and pronounced at 126 mT, in accord with the total temporal change inmagnetization extracted from the SOT data (Fig. 3C).The plateau transition observed in electronic transport appears to be mainly dictated by theunderlying magnetic reversal. Surprisingly, however, we find that the dynamics of the magneticsystem can in turn be influenced by the electronic system. To explore this effect, we perform asequence of magnetic imaging at 126 mT interspersing consecutive scans with small excursionsof the back gate voltage ∆ V g (Fig. 3B). Remarkably, even small ∆ V g ∼ V excursions enhance6he relaxation of the superparamagnetic islands significantly, as is evident from the statistics ofthe observed moment reversals (Fig. 3C). This enhancement in turn is evident in the transportcoefficients, as shown in Fig. 3D.A full comparison of the effects of applied magnetic field, gate voltage, and time on transportcoefficients is presented in Fig. 4C, which shows a unified parametric plot of the transportcoefficient vector ( R xx , R xy ). On this plot, the large magnetic hysteresis evident in Figs. 4A,Bis absent, demonstrating that the relation between R xx and R xy (at a given V g ) is a universalfunction determined by the magnetization. Within a single constant- V g arc-shaped plot, zeronet magnetization corresponds to the maximum of the arc at zero Hall angle, while the varyingHall angle along the arc reflects the varying sample magnetization. Variable V g traces over ∆ V g = ± V at µ H = ± T, marked in black, show contours of variable carrier density atconstant saturated magnetization. Note that at full saturation, variable V g data retraces the samepath upon repetition.On this plot, the temporal relaxation of transport coefficients (Fig. S8) is seen to trackconstant V g arcs, implying that temporal magnetic relaxation is the dominant mechanism andconsistent with the scanning magnetometry results (Figs. 4C,D, gray dots along V g = 6 V arc).Again consistent with magnetometry data, V g excursions enhance the magnetization relaxationdramatically in the metastable regime (Figs. 4 C,D). The blue trace shows the evolution ofthe transport coefficients during three V g excursions with ∆ V g increasing from ± V to ± V at a constant field of 126 mT. Successive V g sweeps do not retrace each other, consistentwith an irreversible change in the net magnetization. Repeating the same experiment for largerexcursion of ∆ V g = ± V, the magnetization is observed to rapidly relax, nearly reachingcomplete saturation (cyan in Figs. 4C,D). As Fig. 4C makes clear, the gate-induced magneticrelaxation is strongly dependent on maximum extent of the voltage excursion, although we findit to be independent of sweep direction and rate.7trikingly, despite the dramatic effect of gate voltage variations on magnetic relaxation, fieldsweeps at different values of V g show only small deviations in the coercive field (Figs. 4A,B)indicating that carrier density has little effect on the average magnetization. These seeminglycontradictory observations can be qualitatively understood by invoking the strongly disorderednature of the superparamagnetic state. In our strongly disordered system, the global gap in thedensity of states is bridged by a proliferation of subgap states induced by the random polar-ization of the superparamagnetic islands. These localized states can in turn mediate the FM,in addition to the proposed global van Vleck mechanism that arises from band states ( ). V g excursions may modify the magnetic anisotropy energy of individual islands through the strongdependence of local density of states on position and energy, randomly changing the magneticpotential landscape without significantly changing the average magnetization and coercive field.Unlike in QHE systems, however, this disorder is not quenched: the weakly interacting super-paramagnetic islands are metastable. At any positive field, some local configurations are sep-arated from the ground state by small energy barriers U , and any local perturbation can causeirreversible flipping of some of the islands leading to rapid stimulated relaxation upon repeatedexcursions of V g . This mechanism is of course enhanced near H c . Our statistical analysis of theflipping moments at various V g in Fig. S5 qualitatively supports this mechanism.The observed superparamagnetic state and its rich dynamics emphasize the intricate cou-pling between the electronic and magnetic sub-systems in magnetic TIs. Our results suggestthat the quantum phase transition between QAH states can be strongly affected by the nature ofthe underlying magnetic transition leading to deviations from the expected universal scaling ofthe plateau transition as indicated by recent transport studies ( ). Even though the topologicalstate is robust to the presence of magnetic disorder at very low temperatures, nanoscale super-paramagnetism may be responsible for the fragility of the QAH state at elevated temperatures.8 eferences
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Electrical transport and scanning magnetic imaging of 7QL thickCr . (Bi . Sb . ) . Te sample at T=250 mK. (A-B) Transport measurements showing mag-netic field dependence of R xx (red) and R xy (black) at V g = 6 V (A) and the V g dependence at 1T (B). The dip in R xx marked by the arrow shows the incipient QAH state. (C) An optical imageof the sample and SOT showing the electrical contacts and the SOT reflection from the samplesurface. (D) Electron micrograph of the SOT used for the magnetic imaging. (E-H) ScanningSOT images × µ m of the out-of-plane magnetic field B z ( x, y ) at ∼ nm above the sam-ple surface at four anti-symmetric locations along the magnetization loop marked in (A). Notestrong anti-correlation between (E) and (H), and (F) and (G). Pixel size 50 nm, pixel integrationtime 10 ms. 14igure 2: Magnetization reversal dynamics. (A) Sequence of SOT magnetic images B z ( x, y ) taken at consecutive magnetic fields in 0.5 mT steps at T = 250 mK. (B) Differential images ∆ B z ( x, y ) obtained by subtracting pairs of consecutive B z ( x, y ) images in (A) showing the iso-lated magnetic reversal events (red) of the superparamagnetic moments (see Movie S1). (C) Sta-tistical analysis of 1690 reversal events attained over ranges of magnetic fields centered aroundfour µ H values: total number of moment reversals N m , average magnetic moment m , averagesuperparamagnetic island diameter d , and rate of the magnetization change d M/ d ( µ H ) overthe given range. (D) Chart of relative contribution of different moment sizes m to the totalmagnetization change M within two field ranges centered at µ H = − mT (yellow) and µ H = 154 mT (blue). Inset: location of the moment reversals within the field range around µ H = 154 mT. (E) Cumulative magnetization change M due to moment reversals m over fourfield ranges (left axis, colored symbols) and the simultaneously acquired R xy (right axis, black).The total magnetization in each range is offset by an arbitrary constant.15igure 3: Temporal and back gate induced relaxation of the superparamagnetic state. (A)Differential image ∆ B z ( x, y ) obtained by subtraction of two consecutive images acquired atconstant µ H set = 126 mT and V g = 6 V after a field ramp from -1 T. Image acquisition timeis 200 sec with 50 sec wait time between images. (B) Same as (A) with gate excursion progres-sively increasing from ∆ V g = 0 . to . V in-between consecutive images. (C) Histogram ofthe temporal relaxation process showing the moment reversals m attained from four consecu-tive ∆ B z ( x, y ) images at µ H set = 63 mT (dark blue) and at µ H set = 126 mT (light blue),and of V g -induced relaxation at µ H set = 126 mT acquired following the temporal relaxationof 20 minutes (green). (D) R xy as a function of field (black) and during relaxation at fixed fieldtaken simultaneously with the magnetic imaging. Temporal relaxation over 20 minutes is morepronounced at 126 mT (light blue) than at 63 mT (dark blue). V g excursions (green) inducelarge relaxation at 126 mT. Inset: full R xy hysteresis loop showing the region of interest.16igure 4: Transport measurements and universal plot of magnetic relaxation. (A) R xy and(B) R xx vs. applied field at T = 250 mK and different V g showing magnetic hysteresis withsimilar H c . (C) The same data plotted as universal arc-like curves of R xx vs. R xy at various V g . Extrema of the arcs correspond to saturation magnetization at -1 T (+1 T) on the lower left(right) end of each arc. Gray dots indicate 60 min of temporal relaxation at µ H set = 126 mTand V g = 6 V (see also Fig. S9). Gate sweeps at µ H = ± T (black lines) trace the ends ofthe arcs, and are reversible. Gate sweep at 126 mT (blue and cyan) are metastable, inducingmagnetic relaxation and propagation along the arcs from R xy < towards positive saturation.(D) The R xy relaxation data (gray, blue, cyan) and the R xy field sweep at V g = 6 V (green).17 upplementary materialVisualization of superparamagnetic dynamics inmagnetic topological insulators
E. Lachman , ∗ , A. F. Young , , ∗ , A. Richardella , J. Cuppens , Naren HR ,Y. Anahory , A. Y. Meltzer , A. Kandala , S. Kempinger ,Y. Myasoedov , M. E. Huber , N. Samarth and E. Zeldov Department of Condensed Matter Physics, Weizmann Institute of ScienceRehovot 76100, Israel Physics Department, University of CaliforniaSanta Barbara 93106-9530, USA Department of Physics, The Pennsylvania State UniversityUniversity Park, Pennsylvania 16802, USA Department of Physics, University of Colorado DenverDenver, Colorado 80217, USA ∗ These authors contributed equallye-mail:[email protected]
Contents V g dependence of the dynamics of magnetic relaxation 85 Transport measurements 96 Growth and structural characterization 15 a r X i v : . [ c ond - m a t . m e s - h a ll ] J un Transport and superparamagnetic dynamics in 10QL Cr-doped ( Bi, Sb ) T e sam-ple 178 Global magnetization studies 199 Transport and superparamagnetism in Mn-doped BiTe 2210 Atomic Force Microscopy of Cr-doped samples 23 Movies of the magnetic moment reversal dynamics
The fabrication of the SOT devices and the scanning SOT magnetic imaging were performedas described in Refs. . The scanning magnetic images of B z ( x, y ) and the differences be-tween consecutive images ∆ B z ( x, y ) were acquired at constant increments of applied magneticfield and compiled into movies (Movies S1 and S2). Figures S1 and S2 show representativeframes from the movies of two Cr-doped (Bi,Sb) Te samples.3igure S1: Movie snapshot of magnetization reversal process in 7QL Cr . (Bi . Sb . ) . Te film at T=250 mK. (A) Scanning SOT image × µ m of the out-of-plane magnetic field B z ( x, y ) at ∼ nm above the sample surface. The running title indicates the value of theapplied field which is increased in steps of 0.5 mT between consecutive images (see Movie S1).Image acquisition time was 200 seconds with pixel size of 50 nm. The median backgroundvalue of the signal is subtracted in each image. (B) Differential image ∆ B z ( x, y ) obtained bysubtracting the preceding B z ( x, y ) image from the current one in (A). Isolated magnetic reversalevents of the superparamagnetic moments are visible as red-green peaks. The characteristic sizeof these peaks is determined by the size of our SOT (200 nm) and by the height of the SOT abovethe sample, while the physical size of the flipping FM islands is only few tens of nm (see Fig.2) and is below our resolution. The moments flip in the positive direction of the ascendingapplied field giving rise to the positive value of the local ∆ B z ( x, y ) field peaks. Note the muchsmaller span of the color code in (B) as compared to (A). (C) R xy of the sample measuredsimultaneously with the magnetic imaging. The red circle shows the current value while theblack circles are the data at previous field values. (D) Same as (C) for R xx . Figures 2A and 2Bin the main text show few images from the movie while the statistical analysis of the flippingmoments in the movie are presented in Figs. 2C-E for field range µ H = 154 mT.4igure S2: Movie snapshot of magnetization reversal process in 10QL Cr-doped(Bi,Sb) Te film at T=250 mK. (A) Scanning SOT image × µ m of the out-of-plane mag-netic field B z ( x, y ) similar to Fig. S1 but on a different sample and on descending magneticfield in steps of 1 mT (see Movie S2). Image acquisition time was 180 sec with pixel size of 50nm. (B) Differential image ∆ B z ( x, y ) obtained by subtracting a previous B z ( x, y ) image fromthe one in (A). The superparamagnetic moments flip in the direction of the descending magneticfield giving rise to the negative values of the local peaks in ∆ B z ( x, y ) . (C) R xy of the samplemeasured simultaneously with the magnetic imaging. (D) Same as (C) for R xx .5 Magnetic moment fitting procedure
The differential images ∆ B z ( x, y ) were fitted by a distribution of point-like magnetic momentswith out-of-plane polarization as described in Fig. S3.Figure S3: Fitting procedure of magnetic moments . (A) Differential image ∆ B z ( x, y ) in 7QLCr . (Bi . Sb . ) . Te film obtained by subtracting B z ( x, y ) image at µ H = 154 . mT fromimage at µ H = 154 . mT (same as the bottom image in Fig. 2B). Numerical procedure is thenused to find local peaks marked by × that are above a threshold of 4 µ T set by the measurementnoise level. (B) Best fit procedure is then applied in which each peak (five in this case) is fittedto the magnetic field generated by a point dipole m at height h below the SOT convoluted withthe active area of the SOT (the factor 2 arises from the fact that the superparamagnetic islandmagnetization flips from − m to + m ). (C) The quality of the fit is demonstrated by subtractingthe fit (B) from the image (A). (D) The resulting magnitudes m of the fitted dipoles at a fixedoptimal h = 345 nm. The diameter d of the superparamagnetic islands is then calculated fromtheir moment m assuming an average saturation magnetization of µ B / Cr atom (obtained fromglobal magnetization measurements presented in Fig. S14), Cr doping of 5%, unit cell size of0.43 nm, and thickness of 7 nm. The resulting size of the islands is smaller than SOT diameteror the height h justifying the point dipole aproximation. A full simulation taking into accountthe finite size of the islands gave similar results.6 Spatial distribution of the magnetization reversals
Figure S4:
Spatial distribution of the magnetization reversal process in 7QLCr . (Bi . Sb . ) . Te sample at T=250 mK. (A-D) Location of all the moments that flippedupon increasing the applied field from µ H = − mT to 1 mT (A), µ H = 60 to 100 mT(B), µ H = 126 to 140 mT (C), and µ H = 140 to 167 mT (D) corresponding to the four fieldranges in Figs. 2C-E. The random distribution of the moment locations further demonstratesthat the magnetization transition occurs through reversal of weakly interacting superparamag-netic islands rather than through FM domain wall motion.7 V g dependence of the dynamics of magnetic relaxation Figures 4A,B show that V g has only a small effect on the coercive field H c while ∆ V g excursionsnear H c relax the magnetization dramatically as shown in Figs. 4C,D. We have also investigatedthe temporal relaxation process at different constant values of V g as shown in Fig. S5.Figure S5: Statistical analysis of temporal moment relaxation for different V g values in7QL Cr . (Bi . Sb . ) . Te film at T=250 mK. The histogram shows the relative contributionof moments of different magnitudes m to the total magnetization relaxation M for three valuesof V g . For each histogram the magnetic state was initialized at a fixed V g by sweeping theapplied field from -1 T to 125 mT and consecutive B z ( x, y ) images were acquired for 25 minat fixed µ H set = 125 mT similar to Fig. 3A. While for V g = − and 6 V the distributionsare similar, at V g = 37 V a clear shift of the distribution to larger moments is visible showingthat at the same applied field the dynamics of the relaxation process has some dependence on V g . These results indicate that the magnetic anisotropy energy of individual islands is affectedby V g apparently through variations in the disorder-induced random local potential or densityof states. Since the local variations are random their average effect on the global coercive fieldcould be small as observed in Figs. 4A,B. 8 Transport measurements
The samples were mechanically patterned into a Hall-bar geometry and the transport measure-ments were performed using three synchronized SR830 lock-in amplifiers for simultaneousmeasurement of the current, longitudinal voltage, and transverse voltage (Fig. S6). The appliedcurrent was 10 to 20 nA at a frequency of 17.71 Hz. In order to minimize the mixing betweenlongitudinal and transverse resistances switching between the current and voltage leads in VdPconfiguration was performed using Keithly7001 switch box for most of the presented data. Thegate voltage was applied using a DC source (Keithley2400). The sweeping rate of V g was lim-ited to 0.5 V / sec to ensure absence of heating, and after reaching the target V g there was a 3seconds waiting time before the measurement.The simultaneous magnetotransport and SOT imaging measurements were performed inOxford Heliox He cryostat equipped with 12 T magnet. In order to prevent heating the mag-netic field ramping was kept at a low constant rate of 0.4 T/min in all experiments. This ratewas used for magnetotransport measurements as well as for preparation of the initial magneti-zation state for temporal and gate-induced magnetic relaxation measurements. Figures S7 to S9show the dependence of transport coefficients on temperature, back gate voltage, and the tem-poral relaxation. Since large ∆ V g excursions cause pronounced relaxation of the magnetizationwe have also compared the hysteretic behavior of the transport coefficients in presence of gateexcursions as shown in Fig. S10. 9igure S6: Schematics of transport measurements.
Three panels showing the scheme of thescratched Hall bar appearing in Fig. 1C. The width of the central channel is 500 µm , the innerdistance between two voltage probes on the same side is 1000 µm and the width of each voltagechannel is 200 µm . The orange part marks the two current leads and the channel between themi.e. where bulk current flows. The red lines indicate measurement configuration of V xx or V xy . (A) Hall bar measurement configuration for simultaneous measurements of I , V xx and V xy using three lock-in amplifiers resulting in R xx = V xx I and R xy = V xy I . (B) VdP configuration formeasurement of R xx with switching between current and voltage leads using a switch box. (C)VdP configuration with switching for measuring R xy .10igure S7: Temperature dependence of the transport coefficients in 7QLCr . (Bi . Sb . ) . Te film. Hysteretic loops of R xx and R xy vs. applied field at V g = 8 V attwo temperatures T = 240 mK and 1.5 K. The coercive field at 1.5 K is reduced to about athird of its value at the base temperature. 11igure S8: Gate voltage dependence of R xx in 7QL Cr . (Bi . Sb . ) . Te sample. Thelongitudinal resistance R xx at µ H = 1 T vs. V g at T = 250 mK and 1.6 K. The incipient QAHstate is clearly visible at T = 250 mK as a pronounced dip around V g = 8 V marked by thearrow. A smaller dip is resolved also at T = 1 . K.12igure S9:
Temporal relaxation of transport coefficients near the coercive field in 7QLCr . (Bi . Sb . ) . Te sample. Time dependence of R xx and R xy at µ H set = 126 mT at V g = 6 V after ramping the field from -1 T. The relation between R xx and R xy follows theuniversal arc-like scaling as shown in Fig. 4C by the gray dots.13igure S10: Transport coefficients in 7QL Cr . (Bi . Sb . ) . Te sample with continuousgate excursions. Hysteretic loops of R xx and R xy vs. applied field at V g = 6 V without (blue)and with (red) gate excursions. In the former case the transport coefficients were measuredcontinuously while sweeping the applied field. In the latter case at each value of the appliedfield, gate excursion of ∆ V g = ± V was applied for ten times prior to the measurement of R xx and R xy . Increasing the number of ∆ V g repetitions did not induce additional change in thetransport coefficients. The effective coercive field is seen to be reduced substantially by the gateexcursions in contrast to Fig. 4A,B which shows no appreciable change in H c for field sweepsat various constant gate voltages over the same range of V g values. Note that the resultingreduction in H c is much larger than what can be achieved by temporal relaxation (see Figs. S9and 4D). Despite the large reduction in the hysteresis by the gate excursions a fully reversibleequilibrium state could not be reached. 14 Growth and structural characterization
Samples were grown by molecular beam epitaxy (MBE) on commercial SrTiO substrates withdimensions of 5 x 5 x 0.5 mm . Substrates were annealed in oxygen at 925 ◦ C for 2 hour30 min typically and were screened using atomic force microscopy (AFM) to make sure thesurface was atomically ordered with an RMS roughness of less than 0.3 nm. Substrates wereindium mounted for growth in the MBE chamber and outgassed at a substrate temperature of550 ◦ C for about 1 hour to remove any residual contamination before growth. The TI films weregrown using elemental materials of at least 5N purity evaporated from Knudsen type thermalcells at a growth rate of about 0.4 QL per minute at a temperature of 250 ◦ C, as measured by apyrometer.Transmission electron microscopy (TEM) studies were carried out on a sample grown undersimilar conditions. The TEM sample was prepared by focused ion beam (FIB) and measuredin an FEI Titan double aberation corrected TEM at 200 keV. High angle annular dark fieldscanning TEM (HAADF) images showed a well ordered crystal with the expected quintuplelayer structure. Two slightly bright lines in the middle of each QL correspond to the locationof the heavier Bi atoms. A disordered amorphous region about 1 nm thick was observed at theinterface with the SrTiO . We have observed a similar layer in undoped films as well. Energydispersive spectroscopy (EDS) was used to map the spatial distribution of elements. Withineach of six regions studied the distributions of elements appeared uniform with the exceptionthat Bi appeared deficient in the interfacial region.15igure S11: STEM imaging and EDS elemental mapping of a Cr doped (Bi,Sb) Te filmon SrTiO . (A) HAADF image showing the crystal structure and EDS elemental maps of thevarious atomic species. Less Bi is observed in the disordered layer at the interface with theSrTiO . (B) Linecut made by averaging over the area shown. The Te-Bi(Sb)-Te-Bi(Sb)-TeQL structure is clearly seen. Te, Bi, Sb and Cr are shown in green, cyan, magenta, and bluerespectively. (C) Large scale HAADF image of the film.16 Transport and superparamagnetic dynamics in 10QL Cr-doped ( Bi, Sb ) T e sample Similar studies were performed on an additional 10QL Cr-doped sample. Figures S2, S12, andS13 show that the transport coefficients in this sample are far from QAH quantization, howeverthe superparamagnetic behavior is qualitatively the same.Figure S12:
Magnetization reversal dynamics and transport coefficients in 10QL thickCr-doped (Bi,Sb) Te thin film at T=250 mK. (A) Sequence of × µ m scanning SOTmagnetic images B z ( x, y ) taken at consecutive descending magnetic fields in 1 mT steps. (B)Differential images ∆ B z ( x, y ) obtained by subtracting pairs of consecutive B z ( x, y ) images in(A) showing the isolated negative magnetic reversal events of the superparamagnetic moments(blue). A larger sequence of images is available in Movie S2 as described in Fig. S2. (C)Transport coefficients vs. applied field at three values of V g .17igure S13: Scaling of the cumulative magnetization change and the transverse resistancein 10QL Cr-doped (Bi,Sb) Te film at T=250 mK. Evolution of magnetization M (left axis,blue circles) on descending magnetic field (after sweep down from 1 T) obtained by cumulativeaddition of the superparamagnetic moment reversals M = P m , derived from a sequence ofimages like in Fig. S12 and the simultaneously measured R xy (right axis, red). The data pointswere acquired only at ranges of fields where the SOT had high sensitivity resulting in the gapsbetween the ranges. The magnetization in each range is offset by an arbitrary constant. Throughmost of the transition region a close relation between R xy and M is observed.18 Global magnetization studies
Magnetization measurements were taken of a 40 nm thick sample using a commercial SQUIDmagnetometer with the applied field perpendicular to the film (Fig. S14). For the temperaturesweeps the sample was cooled from 305K to 3.5K in either a 9000 Oe field for the field cooled(FC) curve or 1.3 Oe field for the zero field cooled (ZFC) curve (1.3 Oe is the remenance inthe magnet after demagnetization, so this corresponds to zero field). The applied field is thenset to 50 Oe and the magnetization is measured as the sample is warmed. The same procedurewas followed using an annealed STO substrate with no film. The measurements from the STOsubstrate were subtracted from the measurements of the film to remove the contribution fromthe substrate. After this a small constant offset was subtracted from the FC and ZFC curves sothat the magnetization is zero at 50K. A clear peak is seen in the ZFC data below T C at around 7K, suggestive of a blocking temperature associated with superparamagnetism. This peak is alsoseen in thin ( ∼
10 QL thick) samples as well as in samples grown on InP(111)A substrates. Suchpeaks are typically associated with superparamagnetic or spin glass materials but have also beenoccasionally observed in ferromagnets. Further studies are needed to identify its origin.For the field sweep, the sample is cooled in field to 5K and the applied field is swept from9000 Oe to -9000 Oe and back. Again, the substrate contribution is subtracted. All measuredmagnetizations are normalized by the sample area. It should be noted that the features in thefield sweep at +/-2500 Oe are artifacts due to the small magnetization signal in the raw datawhen the strong diamagnetic contribution of the substrate causes the total magnetization to crossthrough zero at these values. Using a composition of Cr . (Bi . Sb . ) . Te and a high fieldsaturation magnetization of 7.25e-7 emu/mm gives a moment of approximately 3.15 µ B /Cr,close the expected value of 3.Given these results, a 5x5 µm area of a 7QL sample of Cr . (Bi . Sb . ) . Te should showa magnetization change of ∼ × µ B when integrating from zero magnetization to saturationvalue. This number is comparable to M obtained by cumulative addition of m in Fig. 2E. For10QL sample the total change in M from negative to positive saturation should be ∼ × µ B SQUID magnetometry measurements of a ∼ (A) M vs. T fieldcool and zero field cool magnetization curves showing a peak in the ZFC below T C . (B) M vs.H hystersis curve at 5 K after subtracting the STO substrate contribution.201 Transport and superparamagnetism in Mn-doped BiTe
Figure S15:
Transport and scanning magnetic imaging of 70 nm thick Mn-doped BiTesample at T=250 mK. (A) R xy vs. applied field showing hysteretic loop with coercive fieldof H c ’ . T in 70 nm thick Mn-doped BiTe sample grown by MBE on InP substrate. (B-F,I) Scanning SOT images . × µ m of the magnetic field B z ( x, y ) at six points along themagnetization loop marked in (A). Images (B) and (C) were acquired near zero applied field (45mT and 23.5 mT respectively) on the opposite branches of the loop. The magnetic structure ishighly anticorrelated similar to the behavior in Fig. 1. Consecutive images at fields of 180 (D),195 (E), and 200 mT (F) show superparamagnetic behavior with no domain wall motion. (G)Differential image ∆ B z ( x, y ) obtained by subtracting (D) from (E) showing isolated momentreversals. (H) Same as (G) for images (E) and (F). (I) B z ( x, y ) at 0.6 T - nearly at the closing ofthe loop - demonstrating that though with lower contrast, the magnetic nano-structure prevailsup to high fields. 22 Figure S16:
AFM topography images of the two studied Cr-doped (Bi,Sb) Te samples.samples.