### 2.1 Basic properties of the CT/BSTS heterostructure

Figure 1a shows *θ*-2*θ* x-ray diffraction (XRD) patterns of a single CT film (8 nm) and a CT/BSTS heterostructure (8 nm/100 nm) grown on an Al_{2}O_{3}(0001) substrate [21]. It shows that both of them are highly oriented along the c-axis and that BSTS is grown epitaxially on CT with keeping its own crystal structures despite the lattice mismatch of ~ 12% between BSTS and CT. It is attributed to both the van der Waals bonding nature of the BSTS and the low growth temperature of the top BSTS layer which is ~ 100 K lower than that of the bottom CT layer. The sharp interface of CT / BSTS heterostructure has been confirmed from the transmission electron microscope (TEM) cross-sectional image (Additional file 1: Fig. S1).

Figure 1b shows the sheet resistance (\({R}_{sh}\)) of an 8-nm thick CT film, a 100-nm thick BSTS film, and a CT(8 nm)/BSTS(100 nm) heterostructure as a function of temperature (\(T\)). A schematic illustration of the CT/BSTS heterostructure and the measurement configuration is shown in Additional file 1: Fig. S2. The BSTS film shows an insulating behavior down to ~ 50 K, below which its *R*_{sh} starts to saturate. It indicates that the increase in surface channel mobility dominates the overall transport properties over the decrease in bulk carrier density at low temperatures, which is consistent with the behavior of TI. \({R}_{sh}\left(T\right)\) of the CT film shows a change in the slope around the Curie temperature of \({T}_{c}\) = 170 K corresponding to the ferromagnetic transition, which is consistent with our previous report on CT films with similar thickness [21]. In contrast, \({R}_{sh}\left(T\right)\) of the CT/BSTS heterostructure is observed to be less sensitive to temperature showing a broad peak around 195 K, which is due to the change of the dominating transport channel from the BSTS layer to the CT layer. Looking at the curve more closely (Fig. 1c), a change in the slope is observed around 76 K, which seems to indicate the existence of another ferromagnetic layer. It might be attributed to a proximity-induced ferromagnetic layer inside the BSTS layer or a Cr-doped BSTS layer possibly formed by the intermixing between the BSTS and CT layers [19, 33,34,35]. This will be further discussed in the later part.

Figure 1d shows the field-cooled magnetization of the CT/BSTS film as a function of temperature with the magnetic field (100 Oe) applied parallel (\({H}_{para}\)) and perpendicular (\({H}_{perp}\)) to the film plane, respectively. From the temperature-dependent magnetization curve, clear ferromagnetic transition is observed under the 170 K with the out-of-plane (OOP) direction consistent \({R}_{sh}\left(T\right)\) of the CT shown in Fig. 1b while the in-plane (IP) direction doesn’t show an increment of magnetization. Figure 1e shows the magnetization of the CT/BSTS film as a function of \(H\) at 70 K under both \({H}_{para}\) and \({H}_{perp}\) configurations, showing a large hysteresis in the OOP direction with the coercive field of ~ 0.8 T. On the other hand, the in-plane (IP) magnetization shows the non-saturating behavior up to ± 2T indicating the strong perpendicular magnetic anisotropy energy of the CT [36, 37]. The magnetic properties such as \({T}_{c}\) and strong perpendicular magnetic anisotropy of CT/BSTS are consistent with the previously reported Cr_{2}Te_{3} thin films [14, 36, 38].

### 2.2 Anomalies in the Hall effect of the CT/BSTS heterostructure

Figure 2 shows the Hall resistivity trace (\({\rho }_{xy}\left(H\right)\)) of BSTS, CT, and CT/BSTS in the temperature range of 50 ~ 150 K. For the single BSTS film (Fig. 2a), \({\rho }_{xy}\left(H\right)\) is observed to be linear resulting in the estimations of the carrier density (*n* = 4.7 × 10^{18} /cm^{3}) and the mobility (*µ* = 105 cm^{2}/Vs) at 50 K. In addition, it is observed that the slope of \({\rho }_{xy}\left(H\right)\) increases with lowering the temperature. Together with \({R}_{sh}\left(T\right)\) of the BSTS film, it implies that the bulk conductivity of the BSTS is sufficiently suppressed at relatively low temperatures (*T* < 50 K) to make the topologically-protected surface channel dominate the carrier transport in the BSTS layer. Meanwhile, as shown in Fig. 2b, \({\rho }_{xy}\left(H\right)\) of the single CT film shows a counter-clockwise hysteresis loop whose coercive field increases with lowering the temperature. The amplitude of the AHE decreases with lowering the temperature, which is not common behavior in a magnetic film. Since the anomalous Hall resistance is proportional to the magnetic moment of the film (e.g., \({R}_{AHE} \propto {R}_{s}M\), \({R}_{s}\) is the anomalous Hall coefficient and \(M\) is the magnetic moment), an anomalous Hall resistance should increase until the magnetic moment reaches the saturation magnetic moment as temperature decreases. However, the magnetization of CT/BSTS has almost saturated under 100 K (Fig. 1d), and the anomalous Hall resistance is limited by \({R}_{s}\) which is determined by both the mechanism of AHE and the longitudinal resistivity [12, 38,39,40]. So, the decrease of \({\rho }_{xy}\) of CT could be accounted for both the decrease in \({R}_{sh}\) of the CT film (Fig. 1b) and the extrinsic AHE mechanism [41].

On the other hand, in Fig. 2c, the CT/BSTS heterostructure shows intriguing features such as the sign-reversal of the AHE at 50 K (from the counter-clockwise in CT to the clockwise in CT/BSTS) and a hump structure around \(H\) = 1 T which is the similar value of the coercive field clearly observed at 50 and 70 K, the aforementioned THE-like feature.

The measured \({\rho }_{xy}\left(H\right)\) in Fig. 2c can be decomposed into \({\rho }_{xy}\left(H\right)= {\rho }_{OHE}\left(H\right)+ {\rho }_{AHE}\left(H\right)+ {\rho }_{hump}\left(H\right)= {\rho }_{OHE}\left(H\right)+ {\rho {\prime }}_{xy}\left(H\right),\) where the three terms in the middle represent the ordinary Hall resistivity, the anomalous Hall resistivity, and the hump-related resistivity in order. To focus on the THE-like feature, we have subtracted \({\rho }_{OHE}\left(H\right)\) from \({\rho }_{xy}\left(H\right)\) by assuming that \({\rho }_{OHE}\left(H\right)\) is not related to the spontaneous magnetization and the hump-related mechanism, that is to say, \({\rho }_{OHE}\left(H\right)\) should cross the origin (for extracting the nonlinear \({\rho }_{OHE}\left(H\right)\) from the measured \({\rho }_{xy}\left(H\right)\), we used the two-band model fitting method. For the details, see Additional file 1: Fig. S4 ). \({\rho {\prime }}_{xy}\left(H\right)\) of the CT/BSTS heterostructure is plotted at temperatures from 50 to 80 K as shown in Fig. 3a. Note that the hump structure is the most conspicuous at 60 K with \({\rho }_{hump}\left(H\right)\) overwhelming \({\rho }_{AHE}\left(H\right)\). Furthermore, at the same temperature, it is found that the sign of the AHE is reversed from the clockwise direction to the counter-clockwise direction. Above 60 K, the hump structure decays slowly, completely disappearing above 80 K. Interestingly, this temperature is near 76 K where the longitudinal resistance of the CT/BSTS structure shows a change in slope as shown in Fig. 1c. The amplitude and the coercivity of \({\rho {\prime }}_{xy}\left(H\right)\) in the temperature range of 50 ~ 150 K are summarized in Fig. 3b and c, respectively. It is observed that \({H}_{c}\left(\text{T}\right)\) can be described by Kneller’s law [42] as \({H}_{c}= {H}_{0} {(1-T/{T}_{B})}^{\alpha }\) with *α* = 1/2, implying the single magnetic domain structure [43] (Fig. 3d).

Figure 3e shows the minor loops of \({\rho {\prime }}_{xy}\left(H\right)\) of the CT/BSTS heterostructure at 60 K with varying negative bound (\({H}_{max}^{n}\)) of \(H\) and keeping the positive bound (\({H}_{max}^{p}\)) at 3 T. In previous studies [17, 32], it was reported that the minor-loop has an appearance depending on the mechanism of the hump structure, the skyrmion-based or the 2AHE-based model. When \({H}_{max}^{n}\) = −1 T, slightly higher than the field at the peak of the hump (= \(-{H}_{hump}\)), it is clearly observed that the minor loop of \({\rho {\prime }}_{xy}\left(H\right)\) forms a square-shaped hysteresis curve. Furthermore, in that case, \({\rho {\prime }}_{xy}\left(H\right)\) does not show a dip structure at \(H={H}_{hump}\), which is a conjugate to the hump feature at \(H={-H}_{hump}\) and appears in the full-loop measurement. Finally, when \({H}_{max}^{n}\)= -1.5 T slightly higher than the magnetic field at the saturated magnetization(\({-H}_{sat}\) (≈ -2 T)), \({\rho {\prime }}_{xy}\left(H\right)\) curve for the \(H\)-sweep in the positive direction is offset from the full-loop measurement, resulting in the size of the dip at \(H={-H}_{hump}\) smaller than that in the full-loop measurement. All these observations, Figs. 1c and 3e, are consistent with the sum of hysteresis loops of ferromagnetic domains or layers, supporting the 2AHE-based model for the hump structure in our CT/BSTS heterostructure.

### 2.3 Analyses based on the two-channel AHE model

Based on the 2AHE model for the hump structure observed in the CT/BSTS structure, we have further analyzed the properties of the second ferromagnetic layer other than the CT layer. As mentioned in the discussion of the result in Fig. 1c, the layer might be attributed to a proximity-induced ferromagnetic layer inside the BSTS layer or a Cr-doped BSTS layer. To differentiate each component in the two-channel AHE, we have fitted the measured curve to an approximate form given by the sum of two AHE components, each of which is empirically described by the hyperbolic tangent function.

$$\rho _{{AHE}}^{{tot}} \left( H \right) = \rho _{{AHE}}^{{neg}} \left( H \right) + \rho _{{AHE}}^{{pos}} \left( H \right) = ~ - R^{{neg}} \tanh \left[ {\omega ^{{neg}} \left( {H - H_{c}^{{neg}} } \right)} \right] + R^{{pos}} \tanh \left[ {\omega ^{{pos}} \left( {H - H_{c}^{{pos}} } \right)} \right]$$

(1)

Here, the expressions of “*neg*” and “*pos*” as the superscript are used for representing the negative and the positive AHE, respectively, with the positive meaning the counter-clockwise AHE as observed for the CT single film as shown in Fig. 2b. \({R}^{neg}\), \({ R}^{pos}\), \({\omega }^{neg}\), \({\omega }^{pos}\), \({H}_{c}^{neg}\), and \({H}_{c}^{pos}\) are all positive constants as fitting parameters.

Figure 4a shows the result of the curve fitting at 50 K as a representative example, releasing \({\rho }_{AHE}^{pos}\left(H\right)\) and \({\rho }_{AHE}^{neg}\left(H\right)\). Repeating the same at various temperatures, we have obtained the temperature dependence of each AHE component as shown in Fig. 4b and c (for the results of the curve fitting at various temperatures, see Additional file 1: Fig. S5). The fitting parameters \({H}_{c}^{i}\) and \({{R}^{i}}_{ }\)(*i* = pos, neg) are plotted in Fig. 4d and e as a function of temperature, respectively. Note that \({R}^{neg}\) and \({R}^{pos}\)show the temperature dependence opposite to each other while both \({H}_{c}^{neg}\) and \({H}_{c}^{pos}\) decrease with increasing temperature. In addition, note that the temperature dependence of \({\rho }_{AHE}^{pos}\left(H\right)\) resembles that of the CT single film as shown in Fig. 2b. Therefore, we believe that \({\rho }_{AHE}^{pos}\left(H\right)\) is attributed to the CT layer. On the other hand, considering that \({\rho }_{AHE}^{neg}\left(H\right)\) decreases with increasing temperature, we believe that \({\rho }_{AHE}^{neg}\left(H\right)\) is associated with the BSTS layer whose longitudinal resistivity decreases with increasing temperature. Therefore, in Fig. 4f and g, we have plotted \({R}^{pos}\) and \({R}^{neg}\) as a function of the longitudinal resistivities (\({\rho }_{CT}\) and \({\rho }_{BSTS}\)) of the CT film and the BSTS film, respectively. Indeed, it is observed that \({R}^{pos}\) is linearly proportional to \({\rho }_{CT}\) while \({R}^{neg}\) shows a superlinear dependence on \({\rho }_{BSTS}\) (\({R}^{neg}\tilde{\left({\rho }_{BSTS}\right)}^{1.8}\)).

It is theoretically known that \({\rho }_{AHE}\) is proportional to the longitudinal resistivity, \({\rho }_{0}\), with being described by a power law [41], \({\rho }_{AHE}\tilde{{(\rho }_{0})}^{\beta }\). Here, the exponent *β* depends on the mechanism of the AHE. *β* = 1 corresponds to the extrinsic skew scattering mechanism. *β* = 2 represents the AHE resulting from the intrinsic mechanism related to Berry’s phase which is only affected by the electronic band structure of the material. Therefore, the obtained result of *β* = 1.8 seems to support the intrinsic picture that the negative AHE component might be the proximity-induced ferromagnetic layer inside BSTS which has a nonzero Berry’s phase [9, 44,45,46,47].