Visualization of superparamagnetic dynamics in magnetic topological insulators.

Quantized Hall conductance is a generic feature of two-dimensional electronic systems with broken time reversal symmetry. In the quantum anomalous Hall state recently discovered in magnetic topological insulators, time reversal symmetry is believed to be broken by long-range ferromagnetic order, with quantized resistance observed even at zero external magnetic field. We use scanning nanoSQUID (nano-superconducting quantum interference device) magnetic imaging to provide a direct visualization of the dynamics of the quantum phase transition between the two anomalous Hall plateaus in a Cr-doped (Bi,Sb)2Te3 thin film. Contrary to naive expectations based on macroscopic magnetometry, our measurements reveal a superparamagnetic state formed by weakly interacting magnetic domains with a characteristic size of a few tens of nanometers. The magnetic phase transition occurs through random reversals of these local moments, which drive the electronic Hall plateau transition. Surprisingly, we find that the electronic system can, in turn, drive the dynamics of the magnetic system, revealing a subtle interplay between the two coupled quantum phase transitions.

INTRODUCTION

The integer quantum Hall effect (QHE), first observed in clean two-dimensional electronic systems at high magnetic fields (), is the paradigmatic example of a topological phase: different integer QH states are characterized by identical symmetries but different integer topological quantum numbers η, with the quantized Hall resistance given by R = h/(ηe2) (). However, Hall quantization may also occur in the absence of an external field as long as time reversal symmetry (TRS) is broken (). This quantum anomalous Hall (QAH) state was recently realized experimentally (–), following theoretical proposals based on combining strong spin-orbit coupling with long-range ferromagnetic (FM) order in magnetically doped topological insulators (TIs) (–). The QH plateau transition represents the canonical example of a topological phase transition, described by the divergence of the localization length and a universal critical scaling of transport coefficients (). Recent theoretical calculations show that under certain assumptions, the QAH plateau transition can be mapped onto the same network model used to describe the integer QH plateau transition, leading to the same scaling laws ().

Although 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. The QH plateau transition is a purely electronic effect, in which delocalization occurs against a background of quenched electronic disorder. In contrast, the QAH plateau transition results from two coupled quantum phase transitions: the field-driven magnetic transition of the FM order and the electronic transition that is driven by TRS breaking of the FM transition. The dynamics of the FM reversal can endow the QAH electronic transition with features that do not have an analog in QHE systems. Exactly at the transition, both systems can be understood as a network of domain walls—magnetic domain walls for the QAH and domain walls of a different filling 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 in thermodynamic equilibrium, the FM domain structure in the QAH is metastable, leading to hysteresis and relaxation dynamics that can directly affect the electronic system. Moreover, the energy scale of the electronic delocalization transition and its critical scaling may depend on the 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 magnetic systems, can in turn modify the phase transition of the magnetic system. Most previous studies of QAH systems have used electronic transport measurements to probe the combined effect of magnetic and electronic evolutions, making it difficult to disentangle the individual roles of the two phase transitions. Thus, exploring the reciprocal coupling of the magnetic and electronic quantum phase transitions requires selective measurement tools that can address the two systems independently.

RESULTS

Here, we combine electronic transport with a scanning superconducting quantum interference device (SQUID) of 200-nm diameter that resides on the apex of a quartz tip [SQUID-on-Tip (SOT)] (, ) to simultaneously probe the magnetic and electronic transitions at 3He temperatures in a 7–quintuple layer (QL)–thick Cr0.1(Bi0.5Sb0.5)1.9Te3 film grown by molecular beam epitaxy (MBE) on an SrTiO3 substrate (Fig. 1). The large dielectric constant of the substrate allows effective control of the chemical potential through back gating. The hysteretic R transition occurs at a coercive field of Hc = 130 mT at 250 mK (Fig. 1A). At elevated fields, the longitudinal resistance (R) shows a pronounced dip as a function of the back-gate voltage at Vg ≈ 6 V due to an incipient QAH state (Fig. 1B). At the lowest measurement temperature (250 mK) used here, the sample we discuss does not yet show the fully developed QAH. This is similar to previous experiments on Cr-doped TI films (–) where quantized Hall resistance appears only at dilution refrigerator temperatures, far below the onset temperature of hysteretic magnetic behavior (, ). Figure 1 (E to H) shows images of the local distribution of the magnetic field B(x,y) in the sample at various points along the magnetization loop. We find B(x,y) to be highly inhomogeneous at all fields, with peak contrast at Hc, where the average magnetization vanishes (Fig. 1, F and G). Submicrometer structure is evident even at fields corresponding to saturation of transport coefficients (Fig. 1, E and H), and images corresponding to opposite magnetization are highly anticorrelated on microscopic scales, including at full saturation. This suggests an inhomogeneous distribution of magnetic moments due to a random spatial distribution of the Cr dopants. Although high-resolution transmission electron microscopy measurements on our samples have yet to show any obvious evidence of Cr clustering (see fig. S1), 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 Bi2Se3 thin-film () samples that do not show the QAH state, whereas large nanoscale fluctuations in the local Cr density have been observed in Te-based samples () because of random doping that introduces strong disorder in the material ().

Fig. 1

Electrical transport and scanning magnetic imaging of 7-QL-thick Cr0.1(Bi0.5Sb0.5)1.9Te3 sample at T = 250 mK.

(A and B) Transport measurements showing magnetic field dependence of R (red) and R (black) at Vg = 6 V (A) and the Vg dependence at 1 T (B). The dip in R marked by the arrow shows the incipient QAH state. (C) Optical image of the sample and SOT showing the electrical contacts and the SOT reflection from the sample surface. (D) Electron micrograph of the SOT used for the magnetic imaging. (E to H) Scanning SOT images (5 × 5 μm2) of the out-of-plane magnetic field B(x,y) at ~300 nm above the sample surface at four antisymmetric locations along the magnetization loop marked in (A). Note strong anticorrelation between (E) and (H), and (F) and (G). Pixel size, 50 nm; pixel integration time, 10 ms.

In metallic FM thin films with out-of-plane magnetization, it is well established that magnetization reversal develops via the nucleation and propagation of domain walls separating regions of opposite magnetization. Such domain wall–mediated magnetization reversal has also been imaged in FM semiconductor films (, ) at magnetic dopant concentrations comparable to those in our magnetic TI films. However, scanning SOT microscopy reveals a very different picture of the magnetization reversal process in magnetic TIs. Figure 2A shows a sequence of B(x,y) images acquired for increasing values of μ0H near Hc. The five images appear almost identical; however, numerical subtraction of successive image data [ΔB(x,y), see Fig. 2B] reveals the underlying dynamic process. Instead of the anticipated domain wall motion, magnetization reversal occurs through a series of random events in which isolated nanoscale islands undergo a reversal of their out-of-plane magnetic moment (see movie S1). These islands are a system of small, weakly interacting magnetic moments (potentially FM single domains) that constitutes a superparamagnet. As we discuss below, our findings establish a direct microscopic observation of superparamagnetism in magnetically doped TI films. Our observations caution against drawing conclusions about the FM state solely from macroscopic magnetization probes (SQUID magnetometry, magneto-optical Kerr effect) that show square hysteresis loops with robust zero field remanence ().

Fig. 2

Magnetization reversal dynamics.

(A) Sequence of SOT magnetic images B(x,y) taken at consecutive magnetic fields in 0.5 mT steps at T = 250 mK. (B) Differential images ΔB(x,y) obtained by subtracting pairs of consecutive B(x,y) images in (A) showing the isolated magnetic reversal events (red) of the superparamagnetic moments (see movie S1). (C) Statistical analysis of 1690 reversal events attained over ranges of magnetic fields centered around four μ0H values: total number of moment reversals N, average magnetic moment , average superparamagnetic island diameter , and rate of the magnetization change dM/d(μ0H) over the given range. (D) Chart of relative contribution of different moment sizes m to the total magnetization change M within two field ranges centered at μ0H = −15 mT (yellow) and μ0H = 154 mT (blue). Inset: Location of the moment reversals within the field range around μ0H = 154 mT. (E) Cumulative magnetization change M due to moment reversals m over four field ranges (left axis, colored symbols) and the simultaneously acquired R (right axis, black). The total magnetization in each range is offset by an arbitrary constant.

To quantify the superparamagnetic dynamics across the Hall plateau transition, we fit each of the local features in the ΔB(x,y) maps with a point-like out-of-plane magnetic moment m (fig. S2). Figure 2C summarizes ~1700 such fits, accumulated over four different ranges of magnetic field. Throughout the measured range, the spatial distribution of reversal events is random (Fig. 2D, inset, and fig. S3), suggesting weak interactions between neighboring islands and supporting the conclusion that superparamagnetism in Cr-doped BiSbTe films is characterized by aggregations of dopant atoms. The deviation of the Hall resistance from its saturation value begins around μ0H = 0 and is accompanied by the reversal of nanoscale moments with average (Fig. 2, C and D). Given an average saturation magnetization of ~ 3μ /Cr atom as obtained from global magnetization measurements (fig. S4), the estimated average diameter of these flipping islands is d = 51 nm for our 7-nm-thick film, considerably below our spatial resolution of ~300 nm. However, as the field is increased toward Hc, a pronounced change in the moment distribution is observed with a shift to higher m values and appearance of a long tail of large moments with m ≥ 105μ (Fig. 2, C and D). The microscopic reversal moments m can be summed to obtain the net change in magnetization M over a continuous field range (Fig. 2E). Comparison with simultaneously acquired R shows a qualitative match, implying that the behavior of the transport coefficients through the transition is mainly determined by the underlying change in magnetization. Superparamagnetic behavior and a similar relationship between measured magnetization and transport coefficients were found in a second sample as well as in an Mn-doped Bi2Te3 film (see movie S2 and figs. S6 to S9).

Hysteretic magnetic transitions are a signature of metastability and typically display temporal relaxation via thermal activation or quantum tunneling. We probe magnetic relaxation in real time by polarizing the system at μ0H = −1 T and then ramping the field to a positive set point μ0Hset. We then acquire repeated images of B(x,y) while simultaneously monitoring electronic transport. No relaxation is observed on laboratory time scales for μ0Hset < 0 in either magnetization or transport coefficients. Spontaneous relaxation begins to be evident at small positive fields (μ0Hset = 63 mT), manifesting as magnetic reversal events ΔB(x,y) and a slow upward drift of R. The frequency and number of these reversals significantly increase near the coercive field at μ0Hset = 126 mT (Fig. 3, A and C).

Fig. 3

Temporal and back gate–induced relaxation of the superparamagnetic state.

(A) Differential image ΔB(x,y) obtained by subtraction of two consecutive images acquired at constant μ0Hset = 126 mT and Vg = 6 V after a field ramp from −1 T. Image acquisition time is 200 s with 50-s wait time between images. (B) Same as (A) with gate excursion progressively increasing from ΔVg = 0.1 to 1.1 V in-between consecutive images. (C) Histogram of the temporal relaxation process showing the moment reversals m attained from four consecutive ΔB(x,y) images at μ0Hset = 63 mT (dark blue) and at μ0Hset = 126 mT (light blue), and of Vg-induced relaxation at μ0Hset = 126 mT acquired after the temporal relaxation of 20 min (green). (D) R as a function of field (black) and during relaxation at a fixed field taken simultaneously with the magnetic imaging. Temporal relaxation over 20 min is more pronounced at 126 mT (light blue) than at 63 mT (dark blue). Vg excursions (green) induce large relaxation at 126 mT. Inset: Full R hysteresis loop showing the region of interest.

The temporal relaxation measurements further corroborate the superparamagnetic behavior: at temperatures well below the blocking temperature, the magnetization of a superparamagnet is hysteretic, showing minimal relaxation at low fields. On approaching Hc, the magnetic anisotropy barrier U is reduced, leading to relaxation when U ≃ kT. Because U is proportional to the volume of the superparamagnetic particles, smaller islands undergo thermal activation at a lower field. Simultaneous transport measurements (Fig. 3D) indicate that the electronic transition closely tracks the magnetic relaxation, with transport coefficient relaxation evident at μ0Hset = 63 mT and pronounced at 126 mT, in accord with the total temporal change in magnetization extracted from the SOT data (Fig. 3C).

The plateau transition observed in electronic transport appears to be mainly dictated by the underlying magnetic reversal. Surprisingly, however, we find that the dynamics of the magnetic system can, in turn, be influenced by the electronic system. To explore this effect, we perform a sequence of magnetic imaging scans at 126 mT interspersing consecutive scans with small excursions of the back-gate voltage ΔVg (Fig. 3B). All imaging scans are taken at a gate voltage of 6 V. Between scans, the gate is ramped at 500 mV/s to a voltage 6 V + ΔVg and back to 6 V, with a wait time of a few seconds at the extremal value. Remarkably, even small ΔVg ~ 1 V excursions significantly enhance the relaxation of the superparamagnetic islands, as is evident from the statistics of the observed moment reversals (Fig. 3C). This enhancement, in turn, is evident in the transport coefficients, as shown in Fig. 3D.

A full comparison of the effects of applied magnetic field, gate voltage, and time on transport coefficients is presented in Fig. 4C, which shows a unified parametric plot of the transport coefficient vector (R, R). On this plot, the large magnetic hysteresis evident in Fig. 4 (A and B) is absent, demonstrating that the relation between R and R (at a given Vg) is a universal function determined by the magnetization. Within a single constant-Vg arc-shaped plot, zero net magnetization corresponds to the maximum of the arc at R = 0, whereas the varying Hall angle along the arc reflects the varying sample magnetization. Variable Vg traces over ΔVg = ±30 V at μ0H = ±1 T, marked in black, show contours of variable Fermi level at constant saturated magnetization. Note that at full saturation, variable Vg data retrace the same path upon repetition.

Fig. 4

Transport measurements and universal plot of magnetic relaxation.

(A and B) R (A) and R (B) versus applied field at T = 250 mK and different Vg showing magnetic hysteresis with similar Hc. (C) Same data plotted as universal arc-like curves of R versus R at various Vg. 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 μ0Hset = 126 mT and Vg = 6 V (see also fig. S13). Gate sweeps at μ0H = ±1 T (black lines) trace the ends of the arcs and are reversible. Gate sweeps at 126 mT (blue and cyan) are metastable, inducing magnetic relaxation and propagation along the arcs from R < 0 toward positive saturation. (D) R relaxation data (gray, blue, and cyan) and R field sweep at Vg = 6 V (green).

On this plot, the temporal relaxation of transport coefficients (fig. S13) is seen to track constant Vg arcs, implying that temporal magnetic relaxation is the dominant mechanism and consistent with the scanning magnetometry results (Fig. 4, C and D, gray dots along Vg = 6 V arc). Again, consistent with magnetometry data, Vg excursions markedly enhance the magnetization relaxation in the metastable regime (Fig. 4, C and D). The blue trace shows the evolution of the transport coefficients during six successive Vg excursions, with ΔVg increasing from ±1 to ±3 V at a constant field of 126 mT. Successive Vg sweeps do not retrace each other, consistent with an irreversible change in the net magnetization. Repeating the same experiment for larger excursion of ΔVg = ±30 V, the magnetization is observed to rapidly relax, nearly reaching complete saturation (cyan in Fig. 4, C and D). As Fig. 4C makes clear, the gate-induced magnetic relaxation is strongly dependent on maximum extent of the voltage excursion, although we find it to be independent of sweep direction and rate.

Strikingly, despite the marked effect of gate voltage variations on magnetic relaxation, field sweeps at different values of Vg show only small deviations in the coercive field (Fig. 4, A and B), indicating that carrier density has little effect on the average magnetization. These seemingly contradictory observations can be qualitatively understood by invoking the strongly disordered nature of the superparamagnetic state. In our strongly disordered system, the global gap in the density of states is bridged by a proliferation of subgap states induced by the random polarization of the superparamagnetic islands, possibly leading to electron-hole puddles () rather than a hard gap. These localized states can, in turn, mediate local FM interactions. Vg excursions modify the magnetic anisotropy energy of individual islands through the strong dependence of density of states on position and energy, randomly changing the magnetic potential landscape without significantly changing the average properties of the magnetic disorder potential (and thus the coercive field measured at fixed Vg). However, unlike in QH systems, the magnetic disorder is not quenched: the weakly interacting superparamagnetic islands are metastable and subject to irreversible relaxation. When the applied field is oriented opposite to the net magnetization, some local configurations are separated from the ground state by a small energy barrier U. Any local perturbation, including a modulation of local density of states, can cause an irreversible flip, leading to rapid stimulated relaxation upon repeated excursions of Vg. Our statistical analysis of temporal relaxation (fig. S15) shows that gate voltage does modulate the average size of the flipping moments for the same applied magnetic field. This again suggests that carriers mediate magnetic interactions.

DISCUSSION

In conclusion, we have imaged the magnetic structure of Cr-doped thin films of (Bi,Sb)2Te3 and found it to be strongly disordered. The magnetic reversal transition and its temporal relaxation are mediated by isolated nanoscale island reversals rather than domain wall motion, pointing to a superparamagnetic state composed of nearly noninteracting magnetic domains. Quantitative analysis shows that the observed dynamics account for most of the total magnetic moment, excluding the possibility of an underlying uniform FM state. Unexpectedly, gate voltage modulation strongly enhances the magnetic relaxation, suggesting that the charge carriers are involved in the mediation of magnetic interactions. Our results suggest that the quantum phase transition between QAH states can be strongly affected by the nature of the underlying magnetic transition, leading to deviations from the expected universal scaling of the plateau transition as indicated by recent transport studies (). In particular, the magnetic disorder potential lowers the local QAH gap, lowering energy barriers for hopping conductivity across the sample bulk. Moreover, the large distribution in the flipping energy barrier U guarantees that, for zero magnetic field and finite temperature, some magnetic reversals will occur because of thermal fluctuations, further decreasing the average bulk gap. Although topological states are robust to disorder in the low temperature limit, as the temperature increases toward the blocking temperature TB of the superparamagnetic islands, the enhanced dissipation due to thermally excited magnetization reversals is bound to destroy the protected state. Thus, nanoscale superparamagnetism may be one of the sources responsible for the fragility of the QAH state at elevated temperatures.

MATERIALS AND METHODS

Samples were grown by MBE on commercial SrTiO3 substrates with dimensions of 5 × 5 × 0.5 mm3. Substrates were typically annealed in oxygen at 925°C for 2.5 hours and were screened using atomic force microscopy (AFM) to make sure that the surface was atomically ordered with a root mean square roughness of less than 0.3 nm (fig. S16). Substrates were indium-mounted for growth in the MBE chamber and outgassed at a substrate temperature of 550°C for about 1 hour to remove any residual contamination before growth. The TI films were grown using elemental materials of at least 5N purity evaporated from Knudsen-type thermal cells at a growth rate of about 0.4 QL per minute at a temperature of 250°C, as measured by a pyrometer. The Cr, Bi, and Sb compositions were estimated from secondary ion mass spectrometry (SIMS) measurements on a similarly grown sample. The SIMS atomic percentages were calibrated using another sample of known composition that was measured at the same time.