Introduction

An axion is a hypothetical particle postulated to resolve the charge conjugation-parity problem in particle physics1,2. It is also an attractive yet unobserved candidate for the missing dark matter in cosmology. Recent theoretical studies find the same mathematical structure of axion electrodynamics, specifically, a coupling term with the form \(\theta {e}^{2}\vec{E}\cdot \vec{B}/2\pi h\), where θ plays the role of “axion”, in a class of topological materials, called axion insulators3,4,5,6. Here \(\vec{E}\) and \(\vec{B}\) are electric and magnetic fields inside a material, respectively. Unlike the hypothetical axions in high-energy physics, the axion insulator has been realized in a three-dimensional (3D) topological insulator (TI), when its top and bottom surfaces have antiparallel magnetization alignment3,4,5. The Hall conductance in an axion insulator on the top and bottom surfaces cancels each other but the value of \(\theta\) is pinned to \(\pi\) by time-reversal symmetry and/or inversion symmetry in the interior. One consequence of the quantized non-zero value of \(\theta\) is the topological magnetoelectric (TME) effect, which refers to the quantized magnetoelectric response of the induced \(\vec{E}\) to applied \(\vec{B}\) and vice versa3,4,5,6,7,8. Besides the TME effect, certain types of axion insulators have been theoretically predicted to host a higher-order TI phase with chiral hinge states9,10.

The axion insulator state has been observed in both molecular beam epitaxy (MBE)-grown magnetically doped TI sandwiches11,12 and exfoliated intrinsic magnetic TI MnBi2Te4 flakes with an even number layer13. Although the axion insulator state by definition must be in a 3D regime11,14, all samples showing the axion insulator signatures to date have a thickness limited to ~10 nm, near the 2D-to-3D boundary3,15,16,17,18. In transport measurements, the axion insulator usually shows a zero Hall conductance plateau under antiparallel magnetization alignment and a well-quantized quantum anomalous Hall (QAH)/Chern insulator state under parallel magnetization alignment3,11,12,13. Therefore, in the literature, the appearance of the zero Hall conductance plateau is frequently associated with the realization of the axion insulator state3,11,12,13. However, the zero Hall conductance plateau can also be formed by the hybridization gap between the top and bottom surfaces when the magnetic TI sandwich structure is thin19,20. Therefore, the appearance of only a zero Hall conductance plateau is not sufficient evidence for the axion insulator phase3,15,16. A recent study16 suggests that the zero Hall conductance plateau observed in vanadium (V)-doped TI/TI/chromium (Cr)-doped TI sandwiches11,12 is a result of the hybridization gap in the 2D regime rather than the formation of the axion insulator state in the 3D regime. Therefore, the observation of the axion insulator state in a thick sample clearly in the 3D regime is an important experimental priority and provides a favorable material platform for the exploration of the TME effect and the high-order TI phase.

In this work, we employ MBE to grow asymmetric magnetic TI sandwich heterostructures by systematically varying the thickness m of the middle undoped TI layer, specifically, 3 quintuple layers (QL) (Bi,Sb)1.89V0.11Te3/m QL (Bi,Sb)2Te3/3 QL (Bi,Sb)1.85Cr0.15Te3. Note that the thickness of 1 QL TI is ~1 nm. Our transport measurements confirm the axion insulator state with both the zero Hall resistance and conductance plateaus persists in the m = 100 sample with a total thickness of ~106 nm. The appearance of the well-quantized QAH effect in the m = 100 sample under high magnetic fields indicates that the side surfaces of the thick axion insulator are horizontally insulating and thus gapped. By varying the thickness m of the middle undoped TI layer, we find that the axion insulator state starts to appear for m ≥ 3. We also find that the two-terminal resistance in the axion insulator regime decreases rapidly with increasing m. We perform theoretical calculations on the side surface gap δ and find that its decay behavior with increasing m is consistent with our experimental observation.

All 3 QL (Bi,Sb)1.89V0.11Te3/m QL (Bi,Sb)2Te3/3 QL (Bi,Sb)1.85Cr0.15Te3 sandwich heterostructures are grown on ~0.5 mm thick heat-treated SrTiO3(111) substrates in a commercial MBE chamber (Omicron Lab10) with a base pressure lower than ~2 × 10-10 mbar (Methods, Supplementary Fig. 1). The Bi/Sb ratio in each layer is optimized to tune the chemical potential of the sample near the charge neutral point21,22,23,24,25. The electrical transport measurements are carried out in a Physical Property Measurement System (Quantum Design DynaCool, 1.7 K, 9 T) and a dilution refrigerator (Oxford Instruments, 70 mK, 8 T) with the magnetic field applied perpendicular to the sample plane. The mechanically scratched six-terminal Hall bars are used for electrical transport measurements. More details about the MBE growth, sample characterizations, and transport measurements can be found in Methods.

Results

We first focus on the magnetic TI sandwich heterostructure with m = 100. Figure 1a shows the cross-sectional scanning transmission electron microscopy (STEM) image of the m = 100 sample and the corresponding energy-dispersive X-ray spectroscopy (EDS) mappings of V and Cr near its top and bottom surface layers, respectively. Our cross-sectional STEM image shows the thicknesses of the top V-doped TI, the middle undoped TI, and the bottom Cr-doped TI layers are ~3 QL, ~100 QL, and ~3 QL, respectively. The total thickness of the m = 100 sample is ~106 nm, easily in the 3D TI regime with no hybridization gap between two surface states as the surface state penetration depth of TI is only around a few nanometers15,16. The EDS mappings of the V and Cr near its top and bottom surface layers show the top and bottom layers of the m = 100 sample are separately doped with V and Cr (Fig. 1a). The larger difference of the coercive fields and the weaker interlayer coupling between the top V-doped and bottom Cr-doped TI layers in thick magnetic TI sandwich will favor the formation of the axion insulator state, but the potential metallic side surfaces are expected to smear the appearance of the axion insulator state3.

Fig. 1: MBE-grown 3 QL V-doped (Bi,Sb)2Te3/100 QL (Bi,Sb)2Te3/3 QL Cr-doped (Bi,Sb)2Te3 sandwich (i.e., the m = 100 sample).
figure 1

a Cross-sectional STEM image. Inset: the EDS map of V (Cr) near the top (bottom) surface layers of the sample. b, c Magnetic field μ0H dependence of the longitudinal resistance ρxx (b) and the Hall resistance ρyx (c) at Vg = Vg0 = 0 V. At T = 70 mK and Vg = Vg0 = 0 V, the observations of the zero ρyx plateau and huge ρxx between coercive fields of the top V- and the bottom Cr-doped TI layers indicate this m = 100 sample is in the axion insulator state.

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Next, we perform electrical magneto-transport measurements on the m = 100 sample at the charge neutral point Vg = Vg0 and temperatures down to T = 70 mK (Fig. 1b, c). At T = 70 mK, the m = 100 sample shows the well-quantized QAH effect when the top and bottom magnetically doped TI layers have the parallel magnetization alignment. Under zero magnetic field, the value of Hall resistance ρyx is ~1.008 h/e2 (Fig. 1c), concomitant with longitudinal resistance ρxx ~ 0.002 h/e2 (~51 Ω) (Fig. 1b). The appearance of the well-quantized QAH state indicates that the side surfaces of the m = 100 sample are insulating along the sample edge direction and thus gapped, which is a prerequisite for the formation of the thick axion insulator state.

Besides the well-quantized QAH effect, the m = 100 sample also shows a clear zero Hall resistance ρyx plateau between the coercive fields of the top V- and bottom Cr-doped TI layers. Here the coercive field μ0Hc1 of the top V-doped TI layer is ~1.058 T, while the coercive field μ0Hc2 of the bottom Cr-doped TI layer is ~0.171 T at T = 70 mK. Sweeping μ0H first reverses the magnetization of the bottom Cr-doped TI layer, resulting in the antiparallel magnetization alignment between the top V- and bottom Cr-doped TI layers. Therefore, the Hall resistance of the top and bottom surfaces of the m = 100 sample cancels each other and a zero Hall resistance ρyx plateau appears between μ0Hc1 and μ0Hc2. The zero ρyx plateau corresponds to the emergence of the axion insulator state, where a large value of ρxx (>40 h/e2) appears (Fig. 1b, c). Note that the axion insulator state can persist at μ0H = 0 T through a minor loop measurement. With increasing T, the value of ρyx gradually deviates from the quantized value, but the maximum value of ρxx is greatly reduced and the zero ρyx plateau is substantially shrunk. The maximum value of ρxx is ~1.787 h/e2 and the value of ρyx at μ0H = 0 T is ~0.785 h/e2 at Τ =1.7 Κ. The zero ρyx plateau changes to a kink feature near the ρyx sign change at T = 5 K (Fig. 1c). We note that the critical temperature of the QAH state in the m = 100 sample is ~4.5 K (Supplementary Fig. 2). Here the critical temperature of the QAH state is defined as that at which the ratio between the Hall and longitudinal resistances at μ0H = 0 T is equal to 1 (Ref. 3). The two quantities track each other closely, indicating the intimate correlation between the zero ρyx plateau under antiparallel magnetization alignment and the QAH effect under parallel magnetization alignment.

To further demonstrate the appearance of the axion insulator state in the m = 100 sample at T = 70 mK, we convert its ρyx and ρxx into Hall conductance σxy and longitudinal conductance σxx. Four sharp peaks of σxx are observed at ±μ0Hc1 and ±μ0Hc2 and two broad zero σxy plateaus appear for −μ0Hc1 ≤ μ0H ≤ −μ0Hc2 and μ0Hc2  ≤  μ0H  ≤  μ0Hc1 (Fig. 2a). The zero σxy plateau and the corresponding nearly vanishing σxx are a result of the cancellation of the contributions from the top and bottom surfaces in the antiparallel magnetization alignment, further confirming the realization of the axion insulator state in the m = 100 sample. To examine the magnetic field-induced quantum phase transition between QAH and axion insulator states, we plot the flow diagram (σxy, σxx) of the m = 100 sample at T = 70 mK (Fig. 2b). Two semicircles of radius e2/2h centered at (σxy, σxx) = ( ± e2/2h, 0) appear, the QAH and axion insulator states correspond to (σxy, σxx) = ( ± e2/h, 0) and (0, 0), respectively. For points (σxy, σxx) = (-e2/2h, 0.58 e2/h) and (e2/2h, 0.55 e2/h), which correspond to the quantum critical points with the appearance of extended states, the value of ρxx is ~0.98 h/e2. This value is close to the universal value h/e2, which has also been observed in the quantum phase transition between quantum Hall and a Hall insulator or QAH and an axion insulator26,27. This flow diagram validates the appearance of the zero σxy plateau as a result of the cancellation of σxy = ±e2/2h on top and bottom surfaces of the m = 100 sample.

Fig. 2: Zero Hall conductance plateau and flow diagram of the m = 100 sample.
figure 2

a μ0H dependence of the longitudinal conductance σxx (red) and the Hall conductance σxy (blue) at Vg = Vg0 = 0 V and T = 70 mK. b Flow diagram of (σxy, σxx) of the m = 100 sample. Two semicircles of radius e2/2h centered at (e2/2h, 0) and (−e2/2h, 0) are shown in dashed lines. Top: schematics of the QAH state at (-h/e2, 0) and (h/e2, 0) and the axion insulator state at (0,0).

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To investigate the evolution of the zero σxy plateau in magnetic TI sandwiches, we perform electrical transport measurements on 3 QL (Bi,Sb)1.89V0.11Te3/m QL (Bi,Sb)2Te3/3 QL (Bi,Sb)1.85Cr0.15Te3 heterostructures by varying m (Fig. 3, Supplementary Figs. 3 to 14). For m ≥ 12, the sandwich samples show similar behaviors, i.e., the zero σxy plateau and nearly vanishing σxx for μ0Hc2 ≤ |μ0H | ≤ μ0Hc1 (Fig. 3a). The widths of the zero σxy plateau are independent of the thickness of the middle undoped TI layer, which excludes the possibility of unintentional magnetic doping concentration difference. By further decreasing m, the strength of the interlayer magnetic coupling between the top V- and bottom Cr-doped TI layers becomes stronger, and the difference between μ0Hc2 and μ0Hc1 is reduced and thus the zero σxy plateau becomes narrower and disappears for m = 2. The widths of the zero σxy plateau are ~0.70 T, ~0.52 T, and ~0 T for the m = 8, 3, and 2 samples (Fig. 3b–d). We find the zero σxy plateaus in m ≥ 3 samples persist at zero magnetic field by minor loop measurements (Supplementary Fig. 10). The disappearance of the zero σxy plateau for the m ≤ 2 samples indicates that the magnetizations of the top V- and bottom Cr-doped TI layers are strongly coupled, leading to the absence of the axion insulator state. For the m = 1 sample, the zero σxy plateau completely disappears and the sample shows the standard QAH state23,24,28,29,30,31. Only one coercive field observed in the m = 1 sample indicates the collective flipping of the magnetizations of both top V- and bottom Cr-doped TI layers. We note that μ0Hc ~ 0.310 T of the m = 1 sample is much larger than μ0Hc2 ~ 0.171 T of the m = 100 sample. This difference confirms the stronger interlayer exchange coupling with decreasing m (Refs. 32,33,34). Remarkably, a unique feature here is that all samples over the very wide range of m, exhibit the well-quantized QAH state under high magnetic field, confirming that the side surface states are gapped along the sample edge direction.

Fig. 3: Evolution of the zero Hall conductance plateau in magnetic TI sandwiches by varying m.
figure 3

a μ0H dependence of σxx and σxy of the magnetic TI sandwiches with m = 100, m = 75, m = 30, and m = 12, respectively. b–f μ0H dependence of σxx (red) and σxy (blue) of the magnetic TI sandwiches with m = 8 (b), m = 3 (c), m = 2 (d), m = 1.5 (e), and m = 1 (f), respectively. g Flow diagrams of (σxy, σxx) of the m = 100, m = 75, m = 30, and m = 12 samples. h–l Flow diagram of (σxy, σxx) of the m = 8 (h), m = 3 (i), m = 2 (j), m = 1.5 (k), and m = 1 (l) samples. Two semicircles of radius e2/2h centered at (e2/2h, 0) and (-e2/2h, 0) and one semicircle of radius e2/h centered at (0, 0) are shown in dashed lines. All measurements are taken at Vg = Vg0 and T = 70 mK.

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Next, we plot the flow diagram (σxy, σxx) of the magnetic TI sandwich samples with different m at T = 70 mK (Fig. 3g–l). The m ≥ 3 samples show similar flow diagram (σxy, σxx) behavior as the m = 100 sample (Fig. 2b). Two semicircles of radius e2/2h centered at (σxy, σxx) = ( ± e2/2h, 0) appear, in which the QAH and axion insulator states correspond to (σxy, σxx) = ( ± e2/h, 0) and (0, 0), respectively. With further decreasing m, the two semicircle behaviors gradually change to one semicircle of radius e2/h centered at (0,0) for the m = 1 sample. The disappearance of the (0, 0) point and the finite value of σxx when σxy = 0 in the flow diagram for the m ≤ 2 samples confirms the absence of the axion insulator state in these samples. The one semicircle flow diagram in our m = 1 sample with a total thickness of 7 QL also suggests that both the top V- and bottom Cr-doped TI layers are topologically nontrivial21,22. Therefore, the appearance of the zero σxy plateau in our priors studies11,26 should be a result of the antiparallel magnetization alignment rather than the formation of the hybridization gap in the m = 4, m = 5, and m = 6 samples.

We measure the thickness dependence of two-terminal resistance ρ12,12 of our magnetic TI sandwiches with an alternating current (AC) voltage in the axion insulator regime (Fig. 4b inset)26. Figure 4a shows the μ0H dependence of ρ12,12 of the magnetic TI sandwiches with 8 ≤ m ≤ 100. In the QAH regime, the value of ρ12,12 is ~h/e2 for all samples. However, in the axion insulator regime, there are two systematic trends as m increases, which are absent in prior studies11,12,26. First, for the samples with m ≤ 30, ρ12,12 exhibits a pronounced peak as a function of μ0H in the axion insulator regime. However, for the m = 75 and m = 100 samples, ρ12,12 changes slightly and shows a nearly flat feature as a function of μ0H in the axion insulator regime on a logarithmic scale (Fig. 4a). The different μ0H dependence of ρ12,12 in thin and thick axion insulators indicates that the response of the energy gap to magnetic fields varies in these two regions. As demonstrated with detailed calculations below, for thinner samples, the side surface states exhibit a large confinement gap and the energy gap is sensitive to the change of the magnetization gap of the surface states. On the other hand, for thicker samples, the energy gap is determined by the much smaller confinement gap of the side surfaces which is less sensitive to external magnetic fields. Second, all samples show much larger ρ12,12 values (Fig. 4a) that increase with decreasing m. The maximum values of ρ12,12 (labeled as ρ12,12,max) are summarized in Fig. 4b. We find that the ρ12,12,max value of the m = 8 sample is ~2.7 × 105 h/e2, which is three orders of magnitude larger than that of the m = 100 sample. For the m = 3 sample, the ρ12,12,max value exceeds the reliable range of our measurement setup. The ρ12,12,max value shows a sudden decrease for the m ≤ 2 samples, specifically, ρ12,12,max ~ 29.2 h/e2, ~12.5 h/e2, and ~5.3 h/e2 for the m = 2, 1.5, and 1 samples (Fig. 4b and Supplementary Fig. 14), indicating the absence of the axion insulator state. To examine the QAH and the axion insulator phases in thick magnetic TI sandwiches, we measure the Vg dependence of both the two-terminal resistance and nonlocal resistance in the m = 100 sample when its top and bottom magnetic layers exhibit antiparallel and parallel magnetization alignments (Supplementary Figs. 11 and 12).

Fig. 4: m dependence of the two-terminal resistance of the axion insulator state in magnetic TI sandwiches.
figure 4

a μ0H dependence of the two-terminal resistance ρ12,12 of the axion insulator state in the m ≥ 8 samples. b m dependence of the maximum value of ρ12,12 (labeled as ρ12,12,max). The black circles and red squares indicate the ρ12,12,max values of the m ≥ 8 and m ≤ 2 samples, respectively. The ρ12,12,max value exceeds the reliable range of our measurement setup in the gray shadow area. Inset: the measurements circuit we used to measure the huge ρ12,12 of the axion insulator state. No series resistance is subtracted in these two-terminal measurements.

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Discussion

Our experiments on the m = 100 sample suggest that two key prerequisites for the axion insulator can be fulfilled: (i) the film thickness should be within the 3D regime, allowing the axion parameter to approach a quantized value and (ii) the side surface states should have a gap so that the material can exhibit an insulating behavior. For condition (i), a prior theoretical calculation4 shows that the magnetoelectric parameter \(\alpha\) that characterizes the axion term by \(\alpha \frac{{e}^{2}}{2h}\,{\vec{E}\cdot \vec{B}}\) can exceed 0.9 when the thickness is greater than ~30 nm, and thus is expected to be even closer to the quantized value of \(1\) in our sample with a thickness of ~100 nm. This is an important reason why the observation of the axion insulating state in our thick samples is a crucial step for the investigation of the quantized magnetoelectric effect. For condition (ii), the rapid decrease of ρxx with increasing temperature in the axion insulator regime (Fig. 1b) is typical behavior for an insulator phase, which implies a gap opening for the side surface states for the m = 100 sample. The origin of this side surface gap is likely attributed to the quantum confinement effect. We plot the spectral function of the side surfaces as a function of the momentum kx and energy E for samples with different m (Fig. 5a–c, and Supplementary Fig. 15). It is clear from Fig. 5a–c that there are dispersive side surface states, which are inherited from 3D TI. The confinement gap, partly caused by the finite thickness of the sample, becomes smaller as the thickness increases. Figure 5c, for m = 100, reveals a confinement gap of ~1.24 meV that resides within the magnetization gap of the top and bottom surface states which is about 10 meV. In the following, we study how the side surface gap changes as a function of the magnetization of the surfaces for different samples to explain the behaviors of ρ12,12 in Fig. 4a in the axion insulator regime.

Fig. 5: Quantum confinement-induced surface gaps in thick axion insulators.
figure 5

a–c Surface spectral functions in axion insulators with m = 8 (a), m = 30 (b), m = 100 (c). 2 M = 10 meV is used in a–c. d The side surface energy gap δ as a function of 2 M in axion insulators with different m. e The side surface energy gap δ as a function of m in axion insulators at 2 M = 10 meV. The black circles are the values of ρ12,12,max in Fig. 4b.

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Figure 5d shows the side surface energy gap δ as a function of the magnetic exchange gap 2 M for m ≥ 8. We identify two typical regimes in these samples with varying m: (i) a linear region characterized by a significant slope of δ versus 2 M at small 2 M values; (ii) a saturated region where δ remains constant at large 2 M values. These two regimes provide an understanding of the nearly flat feature observed in thick axion insulators and the peak behavior in thin axion insulators (Fig. 4a and Supplementary Fig. 13). At 2 M = 10 meV, the m \(\le\) 30 samples fall within the linear region, indicating a dramatic change of δ when 2 M varies, resulting in a peak feature of ρ12,12 in the axion insulator regime. In contrast, the m \(\ge\) 75 samples are situated in the saturated region, where δ remains more robust against variations. Figure 5e shows the values of the side surface energy gap δ as a function of m for a fixed 2 M. The calculated δ value reveals a decaying behavior with increasing m, similar to the thickness dependence of ρ12,12,max observed in our experiments (Fig. 4b) under a logarithmic scale. This similarity suggests that the side surface energy gap δ determines the excitation gap in this system. The qualitative and semi-quantitative agreement between our theoretical calculations and experimental results suggests that the confinement-induced gap is likely the primary source of the side surface gap. Note that the disorder effect may also play a role, but it will not alter the overall qualitative picture.

To summarize, we realize the axion insulator state showing the coexistence of the zero Hall resistance and conductance plateaus in a magnetic TI sandwich sample with a total thickness of ~106 nm. By varying the thickness of the middle undoped TI layer, we find that the 3 QL undoped TI layer is thick enough to produce a weak enough interlayer exchange coupling for the formation of the axion insulator state in magnetic TI sandwiches. The axion electrodynamics from the bulk \(\theta\)-term, which is unique in 3D, gives rise to many topological responses such as the topological magnetoelectric effect4,5,6, image magnetic monopole35, and quantized optical response6. Our hundred-nanometer-thick magnetic TI sandwiches with the axion insulator state in the 3D regime ( ~ 106 nm thick) provide a better material platform for the exploration of these topological responses4,5,6,35, as well as the higher-order TI phase9,10. Moreover, our thick magnetic TI sandwiches can also be employed to explore the existence of the half-quantized counter-propagating Hall current in axion insulators36,37,38,39.

Methods

MBE growth

All magnetic TI sandwich heterostructures [i.e., 3 QL (Bi,Sb)1.89V0.11Te3/m QL (Bi,Sb)2Te3/3 QL (Bi,Sb)1.85Cr0.15Te3] used in this work are grown in a commercial MBE system (Omicron Lab10) with a base vacuum better than ~2 × 10-10 mbar. To achieve satisfactory magnetization, both the top and bottom 3 QL (Bi,Sb)2Te3 layers doped with Cr or V are employed. The heat-treated insulating SrTiO3(111) substrates with a thickness of ~0.5 mm are first outgassed at ~600 °C for 1 hour before the growth of the magnetic TI sandwich heterostructures. High-purity Bi (99.9999%), Sb (99.9999%), Cr (99.999%), V (99.999%), and Te (99.9999%) are evaporated from Knudsen effusion cells to grow the samples. During the growth of doped and undoped TI, the substrate is maintained at ~230 °C and the Bi/Sb ratio is fixed at ~0.5 for all three layers. The flux ratio of Te per (Bi + Sb + Cr/V) is set to be greater than ~10 to prevent Te deficiency in the films. The Bi/Sb ratio in each layer is optimized to tune the chemical potential of the entire magnetic TI sandwich heterostructure near the charge neutral point. The growth rate of both magnetically doped TI and undoped TI films is ~0.2 QL per minute.

Electrical transport measurements

All magnetic TI sandwich heterostructures grown on 2 mm × 10 mm insulating SrTiO3(111) substrates are scratched into a Hall bar geometry using a computer-controlled probe station. The effective area of the Hall bar is ~1 mm × 0.5 mm. The electrical ohmic contacts are made by pressing indium dots onto the films. The bottom gate is prepared by flattening the indium dots on the back side of the SrTiO3(111) substrates. Transport measurements are conducted using both a Physical Property Measurement System (Quantum Design DynaCool, 1.7 K, 9 T) for T ≥ 1.7 K and a dilution refrigerator (Oxford Instruments, 70 mK, 8 T) for T < 1.7 K. The bottom gate voltage Vg is applied using a Keithley 2450 meter. The excitation currents are 1 μA and 1 nA for the PPMS and the dilution four-terminal measurements, respectively. The two-terminal measurements are performed by a standard lock-in technique with a fixed voltage of ~0.1 mV. All magneto-transport results shown in this paper are symmetrized or anti-symmetrized as a function of the magnetic field to eliminate the influence of the electrode misalignment. The contact resistance including wires used in our dilution fridge at room temperature is ~34 Ω. No electronic filter is involved in our dilution refrigerator. More transport results can be found in Supplementary Figs. 2 to 14.

Theoretical modeling

We perform theoretical calculations of the side surface gap based on a sandwich model4:

$${{{{{\mathcal{H}}}}}}({{{{{\bf{k}}}}}})=\left[\begin{array}{cccc}M({{{{{\bf{k}}}}}}) & -i{A}_{1}{\partial }_{z} & 0 & {A}_{2}{k}_{-}\\ -i{A}_{1}{\partial }_{z} & -M({{{{{\bf{k}}}}}}) & {A}_{2}{k}_{-} & 0\\ 0 & {A}_{2}{k}_{+} & M({{{{{\bf{k}}}}}}) & i{A}_{1}{\partial }_{z}\\ {A}_{2}{k}_{+} & 0 & i{A}_{1}{\partial }_{z} & -M({{{{{\bf{k}}}}}})\end{array}\right]+{H}_{X}$$
(1)

where \({k}_{\pm }={k}_{x}\pm i{k}_{y},\,M\left({{{{{\bf{k}}}}}}\right)={M}_{0}+{B}_{1}{\partial }_{z}^{2}-{B}_{2}({k}_{x}^{2}+{k}_{y}^{2}),\) and \({H}_{X}=\Delta (z){{{{{{\rm{\sigma }}}}}}}_{z}\otimes {\tau }_{0}\). To simulate the axion insulator state, we discretize it into a tight-binding model along the z-axis between neighboring QL from \({{{{{\mathcal{H}}}}}}({{{{{\bf{k}}}}}})\). We also assume the spatial-dependent exchange field \(\Delta \left(z\right)\) takes the values M in the top three layers and -M in the bottom three layers, and zero in the middle m layers, respectively. The parameters in our model are as follows: \({M}_{0}=0.28{{{{{\rm{eV}}}}}},\,{A}_{1}=2.2{{{{{\rm{eV}}}}}}\cdot {{{{{\text{\AA }}}}}},\,{A}_{2}=4.1{{{{{\rm{eV}}}}}}\cdot {{{{{\text{\AA }}}}}},\,{B}_{1}=10{{{{{\rm{eV}}}}}}\cdot {{{{{{\text{\AA }}}}}}}^{2},\,{B}_{2}=56.6{{{{{\rm{eV}}}}}}\cdot {{{{{{\text{\AA }}}}}}}^{2}\). The lattice constants are \({{{{{\rm{a}}}}}}=4.14{{{{{\text{\AA }}}}}},\,{{{{{\rm{c}}}}}}=9.57{{{{{\text{\AA }}}}}}\). We choose the magnetic exchange gap 2 M to be ~10 meV for our magnetic TI sandwiches. In our calculations, the side surface spectral function A(kx, ω) is calculated from the recursive Green’s function approach40, which characterizes the density of states on the side surface (Fig. 5). We impose periodic boundary conditions along the x-direction with a good quantum number kx, and calculate the semi-infinite lead surface Green’s function \({G}_{{{{{\mathrm{1,1}}}}}}\) along the y-direction to expose the side surface. By analyzing the side surface spectral function \(A\left({k}_{x},\omega \right)=-{{{{{\rm{ImTr}}}}}}({G}_{{{{{\mathrm{1,1}}}}}})\), we can determine the side surface energy gap δ.