Observation of superparamagnetism in coexistence with quantum anomalous Hall C=$\pm$1 and C=0 Chern states

Simultaneous transport and scanning nanoSQUID-on-tip magnetic imaging studies in Cr-(Bi,Sb)$_2$Te$_3$ modulation-doped films reveal the presence of superparamagnetic order within the quantum anomalous Hall regime. In contrast to the expectation that a long-range ferromagnetic order is required for establishing the quantum anomalous Hall state, superparamagnetic dynamics of weakly interacting nanoscale magnetic islands is observed both in the plateau transition regions as well as within the fully quantized C=$\pm$1 Chern plateaus. Modulation doping of the topological insulator films is found to give rise to significantly larger superparamagnetic islands as compared to uniform magnetic doping, evidently leading to enhanced robustness of the quantum anomalous Hall effect. Nonetheless, even in this more robust quantum state, attaining full quantization of transport coefficients requires magnetic alignment of at least 95% of the superparamagnetic islands. The superparamagnetic order is also found within the incipient C=0 zero Hall plateau, which may host an axion state if the top and bottom magnetic layers are magnetized in opposite directions. In this regime, however, a significantly lower level of island alignment is found in our samples, hindering the formation of the axion state. Comprehension and control of superparamagnetic dynamics is thus a key factor in apprehending the fragility of the quantum anomalous Hall state and in enhancing the endurance of the different quantized states to higher temperatures for utilization of robust topological protection in novel devices.


Introduction
The quantum Hall (QH) and the recently demonstrated quantum anomalous Hall (QAH) effects are prime examples of topological states of matter driven by time-reversal symmetry braking (TRSB) 1-3 . In the QH state the TRSB is induced by high magnetic field applied perpendicular to a high-mobility 2D electron gas. The QAH state, in contrast, can be formed in thin films of topological insulators (TI) even in the absence of magnetic field [3][4][5][6][7][8][9][10] . In this case the TRSB is induced by magnetic order attained by doping the TIs with transition metal elements like Cr, V, or Mn [3][4][5][6][7][8]11,12 , which open a mass gap ∆ in the helical Dirac surface states [13][14][15] .
In order to establish a TRSB topological state on a macroscopic scale the magnetic order should be long-ranged and therefore a robust ferromagnetic (FM) state is believed to be a necessary condition for the experimental observation of the QAH effect with dissipationless chiral edge states 2,13,16 . Indeed, a number of studies using global magnetization measurements 14,17,18 and magnetic force microscopy 19,20 have observed FM-like response in magnetically doped TI thin films. Surprisingly, however, recent scanning SQUID magnetic imaging 21 and transport studies 22,23 , as well as analysis of Kerr microscopy imaging 24 , have suggested the presence of a superparamagnetic (SPM) state in Cr-doped (Bi,Sb)2Te3 films in which instead of a long-range FM order the magnetism is comprised of single-domain magnetic islands. These islands are of characteristic size of few tens of nanometers and are only weakly coupled to each other 21 .
The possible presence of a SPM order has important implications for the properties of the QAH state. The first key consideration is that if the QAH state can indeed be induced by a SPM order, such coexistence can be present only at temperatures well below the SPM blocking temperature, ≪ , since at higher temperatures the QAH state will be destroyed by thermally activated magnetization reversals of the SPM islands. At ≪ the overall magnetic response of a SPM is hysteretic similarly to a FM, making it difficult to discern the two types of magnetic order using global magnetization studies. Moreover, the distinction between SPM and FM states may seem to be of minor importance since in both cases a full alignment of the magnetic moments can be attained at low temperatures and at sufficiently high fields above the coercive field . This aligned magnetization may then be preserved upon decreasing the field back to zero. However, in addition to causing nontrivial dynamics 22 13 . Similarly to the QAH = ±1 state, however, the stability of the axion insulator is also crucially dependent on the establishment of a long large magnetic order. Resolving the microscopic magnetic structure in doped TIs is thus paramount for our understanding of the mechanisms leading to the = 0 and = ±1 states and for enhancing the robustness of the QAH effect and of the ZHP for device applications.
The degree of the magnetic order in magnetically doped TI films, whether FM, SPM, or disordered, can critically depend on material parameters, doping level, growth conditions, and temperature 11,18,32,33 . In this context, it is important to note that none of the previous studies of doped TI films have directly probed the microscopic magnetic structure within the QAH state [19][20][21] . The FM state was reported to be present in films that either do not show QAH at all due to non-optimal doping or composition, or at temperatures substantially higher than those required for the formation of the QAH 19,20 . Similarly, the SPM was demonstrated locally in films that do show QAH but at elevated temperatures at which no quantization was present 21,24 .
Here we report scanning SQUID-on-tip magnetic imaging of modulation-doped (Bi,Sb)2Te3 films performed concurrently with transport measurements that show full QAH quantization. We observe unambiguous SPM behavior both in the magnetization reversal process in the plateau transition regions near as well as within the macroscopically quantized state. No conventional FM domain wall motion is detected under any conditions. Moreover, our results indicate that SPM order is present also in the incipient = 0 state providing an essential insight into the microscopic nature of the different Chern number topological states.

Results and discussion
In this work we studied (Bi,Sb)2Te3 heterostructures with Cr modulation doping which gives rise to a significantly more robust QAH state 34 sustained up to 2 K and in addition renders a clearly developed = 0 state 27 . Figure 1a shows a schematic structure of the film containing two 2 nm thick 12%-Cr doped layers separated by a 3 nm undoped layer. The bottom doped layer is separated from the InP substrate by a 1 nm undoped layer of (Bi,Sb)2Te3.
The heterostructure was coated with 23 nm of AlOx and 10 nm of Ti/Au top-gate electrode and patterned into a Hall bar structure (Fig. 1b). Imaging of the local out-of-plane magnetic field ( , ) employing a scanning SQUIDon-tip (SOT) [35][36][37] was carried out simultaneously with transport measurements at = 300 mK. The Indium SOT with effective diameter of 115 nm (Fig. 1c) was scanned at a constant height of about 300 nm above the sample surface at a typical scan rate of 9 μm/sec.  Since the SOT response is periodic in magnetic field (see Fig. S1) the value of the measured local magnetic field can be determined only up to a constant offset. Here we are interested in studying the variations in local field ( , ) stemming from the spatial fluctuations in the local magnetic structure in the film. We therefore define = 0 at ~3 µm from the edge of the sample (at the left end of the imaged area) where the stray field from the magnetized film, which decays as ~/ , where is the distance from the edge and = 7 nm is the magnetic thickness of the film, is negligible. The attained ( , ) thus describes the magnetic field above the surface of the sample which arises from the variations in local magnetic structure, while the total magnetic field is given by 0 + ( , ). At 0 = 166.4 mT > 0 the net average magnetization of the film is positive resulting in the negative stray field immediately to the left of the sample edge (blue in Fig. 1e). The interior of the sample, however, shows a highly inhomogeneous structure with both positive and negative values of ( , ) reflecting microscopic regions with correspondingly positive and negative out-of-plane magnetization .
The magnetism in Cr doped (Bi,Sb)2Te3 films is strongly anisotropic with an out-of-plane easy magnetization axis 17 .
The inhomogeneous magnetic structure in By integrating the reversing moments for each field increment we can derive the magnetization curve ( ) = ∑ (Fig. 2h) along with the simultaneously measured . Comparison of ( ) and ( ) curves unveils two significant observations. The first is that in the = −1 plateau region for 0 < 80 mT the magnetization grows appreciably with while remains quantized. A similar behavior is observed in the = +1 plateau at ≅ 250 mT. Figure 2f shows that at these plateau fields substantial island flipping occurs (blue and red distributions) providing a direct observation of SPM dynamics within the QAH = ±1 plateau regions. This implies that the global QAH state is preserved in the presence of oppositely magnetized islands as long as their density is low. In Fig. 2h the onset of deviation from quantized value occurs at 0 ≅ 80 mT (blue arrow) where the relative magnetization of the oppositely magnetized islands reaches about 5%. Above this fraction the edge states surrounding the islands of opposite magnetization apparently start percolating across the sample 13 giving rise to finite and a drop in | |. Our results thus show that the establishment of quantized conductance requires a very large fraction of over 95% of the SPM islands' magnetization to be aligned, significantly higher than required for conventional percolation threshold. The second observation in Fig. 2h is that near the coercive field ( ) displays a sharper plateau transition than the ( ) curve. This behavior could be ascribed to the enhancement of in the = 0 state, similarly to the behavior in the regular anomalous Hall effect where ∝ 2 is commonly found 25 .
We now address the = 0 plateau and its magnetic state. Moreover, while in the = ±1 states in Figs. 3b,d the magnetization is not entirely uniform due to variations in local doping and remaining low density of oppositely oriented islands (see the statistical analysis in Fig. 2f), the degree of the inhomogeneity is relatively small (Fig. 3e). In the = 0 state, in contrast, the variations in the local field are substantially larger and ( , ) crosses zero many times, revealing a random distribution of SPM islands with positive and negative values of .
In order to realize a robust axion state in TI heterostructures with two doping layers, the magnetic properties of the layers have to meet two requirements: i) The coercive field of one of the layers, , has to be lower than ℎ of the second layer, and ii) in the field range < < ℎ , where the two layers have opposite magnetization, the magnetic structure within each of the layers has to be sufficiently homogeneous similarly to the structure attained in the = ±1 states with over 95% island alignment. Since in the SPM case the width over which the global magnetization reversal occurs is finite, the second requirement necessitates the separation between the two coercive fields, ∆ = ℎ − , to be substantially larger than the width of the magnetization reversal transition of each of the layers, ∆ ≫ . In such a case, upon sweeping the magnetic field from = −1 state with both layers negatively magnetized (Fig. 3h), the first layer undergoes a transition at accompanied by reversal events of the SPM islands. The axion state is established when most of the islands in the first layer have reversed their magnetization, accompanied by a drop in the rate of the magnetization reversals. As the field is further increased upon approaching ℎ , the SPM reversal events in the second layer will set in, leading to the destruction of the axion state. Since our SOT imaging cannot distinguish between the SPM islands in the two layers, we expect to observe two peaks in the rate of reversals at and ℎ and the integrated magnetization curve ( ) = ∑ should display two step-like features and a plateau at intermediate fields < < ℎ . The transport data showing two peaks in in Fig. 3a is indicative of the positions of the two coercive fields and ℎ . In our magnetic imaging data, however, we observe only one broadened range of fields over which the rate of events is large (Fig. S3a) and the corresponding ( ) curve shows a single smoothed step-like behavior with no observable intermediate plateau (Fig. S3b). These results thus indicate that in our modulation doped Cr-

Conclusion
In conclusion, by performing simultaneous transport and magnetic imaging we reveal that in modulation doped Cr-(Bi,Sb)2Te3 films the SPM state coexists with the QAH state in the = ±1 plateaus, in the plateau transition regions, and in the incipient = 0 state. The size of the SPM islands in the modulation doped heterostructures is substantially larger than in comparable homogeneously doped TI films. This points to a more uniform magnetization which apparently lessens the suppression of ∆ in the inter-island matrix leading to the observed large enhancement in the temperature robustness of the QAH in modulation doped structures 34 . In order to attain a full conductance quantization over 95% of the SPM islands' magnetization in each layer have to be magnetized along the same direction. In the = ±1 state such an alignment can be attained by applying magnetic field that is substantially higher than . A similarly high degree of island alignment is required in order to attain a robust axion state. To achieve this, however, the difference in the coercive fields of the two doped layers should be significantly larger than the width of the SPM transition in each of the layers, ∆ ≫ . The difficulty in attaining this requirement in modulation doped TI films in which the two layers are both Cr doped renders the = 0 state to be significantly more fragile than the counterpart = ±1 states. Very recently, two independent studies [38][39] have attained much larger ∆ by doping one layer by Cr and the other layer by V leading to a significantly more robust ZHP providing strong support to this conclusion. Control of the materials properties in terms of doping homogeneity, enlarging the size and the uniformity of the SPM islands, and increasing the difference in the coercive fields of the top and bottom magnetically doped layers thus emerges as the key challenge for attaining robust QAH and axion states and extending their temperature range for novel electronic applications.

Fabrication of modulation doped Cr-(Bi,Sb)2Te3 heterostructures
The Cr-(Bi,Sb)2Te3 films were grown by molecular beam epitaxy (MBE) on semi-insulating InP(111) substrates using the same procedures as described in Ref. 27. The 3-nm-thick AlOx capping layer was deposited by atomic layer deposition (ALD) system immediately after the removal from the MBE chamber. Using standard photolithography and Ar ion milling processes, the films were patterned into the Hall-bar geometry: 600 m long and 300 m wide (Fig. 1b). The top 20-nm-thick AlOx dielectric layer and 10-nm-thick Ti/Au electrode were deposited by ALD and electron beam evaporation, respectively.

SOT imaging
The magnetic imaging was performed using Indium SOT device [35][36][37] with an effective diameter of 115 nm (see Supplementary Note 1). The SOT was mounted in a home-built scanning probe microscope employing a series SQUID array amplifier 40 for signal readout. Scanning was performed using Attocube integrated xyz scanner with xy range of 30 μm and z range of 15 μm. Magnetic imaging measurements were performed at 300 mK in Oxford Heliox He 3 refrigerator.
The datasets generated during the current study are available from the corresponding authors on reasonable request.  Cumulative magnetization change = ∑ due to island reversals upon sweeping the applied field in seven ranges (color coded, left axis) and the simultaneously acquired (black, right axis). The blue arrow at 0 ≅ 80 mT indicates the field at which starts deviating from quantized value, corresponding to a cumulative change in of ~5% of the full saturation. states. In the incipient = 0 axion state (g) majority of the islands are positively (negatively) magnetized in the top (bottom) layers but the degree of alignment is insufficient for attaining quantization of transport coefficients.