Non-Hermitian dynamics in quantum anomalous Hall insulators

Abstract

Magnetically doped topological insulators (TIs) exhibit two distinct phases: the quantum anomalous Hall (QAH) phase when the Fermi level resides within the surface gap and a metallic phase outside the gap. The QAH phase hosts unidirectional transport channels known as chiral edge states, while the metallic phase exhibits nonreciprocal transport due to unbalanced bidirectional edge states. Using the chiral edge states in Cr-doped (Bi,Sb)₂Te₃ sandwich structures, we realize non-Hermitian conductance matrices in a one-dimensional Corbino chain with well-defined chirality. By tuning the boundary conditions from periodic to open, we reveal the non-Hermitian skin effect, where eigenstates localize exponentially at one end of the chain. In the metallic phase, we further observe asymmetric, bidirectional coupling between the neighboring sites in the conductance matrix, a direct consequence of the system’s intrinsic nonreciprocity. These results establish magnetic TIs as a powerful platform for investigating emergent non-Hermitian phenomena in topological systems.

Non-Hermitian dynamics and skin effect emerge in a magnetic field–free topological platform with tunable intersite couplings.

INTRODUCTION

Recent advances in non-Hermitian physics have provided a rigorous framework for analyzing dissipative quantum systems, where gains and losses lead to complex energy spectra and violation of energy conservation (1–3). Non-Hermitian systems and their hallmark signatures known as exceptional points, spectral degeneracies where both the eigenvalues and the associated eigenstates of the system coalesce, have been studied both theoretically and experimentally in diverse platforms; these include photonic systems (4–17), atomic gases (18–22), electrical circuits (23–32), magnonic systems (33–35), and quantum systems (36–39). These platforms have enabled applications spanning precision sensing (29, 40–46) and topological insulator (TI) lasers (47, 48). A key manifestation of non-Hermiticity is the non-Hermitian skin effect (NHSE) (49–55), which is characterized by the exponential localization of eigenstates at the system boundaries. Unlike Hermitian Hamiltonians, which satisfy HΨ = EΨ with real eigenvalues, non-Hermitian Hamiltonians allow for complex eigenvalues, HΨ = (E + iα)Ψ, where α is a real number that captures the dissipative effects (3). The connection between non-Hermitian topology and Hermitian topological quantum materials has recently been explored (56–58). In this work, we investigate the interplay between non-Hermitian dynamics (56, 57, 59) and nonreciprocal transport in a quantum anomalous Hall (QAH) insulator (60) as depicted by the schematics in Fig. 1. The QAH insulator used in our study is a magnetically doped Cr-(Bi,Sb)₂Te₃ (61–64) that exhibits two distinct phases, namely, the QAH phase at the charge neutrality point (CNP) and a metallic phase away from the CNP. In the QAH phase, time-reversal symmetry is explicitly broken by magnetic doping, leading to a gap opening in the Dirac surface states and the emergence of a single chiral edge state in the absence of an external magnetic field. The chiral edge state facilitates unidirectional, dissipationless electrical transport along the boundary of the TI (61, 65). In this work, we use this robust chiral edge of magnetic TIs in a Corbino geometry to introduce non-Hermiticity (56). We show that, in the metallic phase, the interplay between the Dirac surface states and the magnetization and the resulting nonreciprocal behavior lead to bidirectional coupling and the NHSE. Furthermore, we implement the Hatano-Nelson (HN) model in the QAH phase of the magnetic TI, where the contact arms in the Corbino geometry (as shown in Fig. 1B) play the role of lattice sites in the HN model. The HN model is a one-dimensional paradigmatic non-Hermitian lattice model that incorporates asymmetric hopping amplitudes, explicitly breaking Hermiticity and leading to NHSE (66). Within the context of tight binding models (or coupled mode theory in optics), the HN Hamiltonian is given by

$${\mathcal{H}}_{\text{HN}}=\sum _i{J}_{\text{left}}{c}_{i-1}^{†}{c}_i+J_{\text{right}}{c}_{i+1}^{†}{c}_i={\mathbf{c}}^{†}{H}_{\text{HN}}\mathbf{c}$$ (1)

where J_left and J_right are the left and right hopping amplitudes between adjacent sites in the one-dimensional HN lattice, respectively, and J_left ≠ J_right induces a preferential accumulation of the eigenstates at one boundary, thereby breaking the conventional bulk-boundary correspondence (67–70). In Eq. 1, $c_i$ and $c_i^{†}$ are the annihilation and creation operators for the lattice site i, respectively. In our studies, we observe that, by tuning the Fermi level, via a back gate, we can continuously move the system from the QAH phase to the metallic phase, allowing in situ control of J_left and J_right. This allows us to implement a generalized HN model, where the coupling between different sites is asymmetric and bidirectional. Furthermore, it lets us explore the interplay between nonreciprocity and non-Hermitian dynamics. Another key advantage of this approach is that it functions without requiring a constant external magnetic field, thereby expanding its applicability to a wide range of devices, particularly important for non-Hermitian sensors (71–73).

Fig. 1. Material and geometric topologies.

(A) A schematic of the quantum anomalous Hall (QAH) effect in a Corbino geometry. The magnetization direction within the TI is pointing up. The dissipationless chiral edge current corresponding to Chern number C = 1 flows counterclockwise along the outer perimeter and clockwise along the inner perimeter. The current injected into one of the terminals will flow counterclockwise to ground. (B) A simple five-site Hatano-Nelson (HN) model with an open boundary condition. The hopping amplitudes J_left and J_right are unequal, resulting in asymmetric, bidirectional coupling between the adjacent sites. In the case of the QAH insulator, the hopping is unidirectional, i.e., J_left = 0 and J_right = 1.

RESULTS

We investigate non-Hermitian dynamics in multiterminal magnetic TI devices made in a Corbino geometry. The magnetic TI is a sandwich of three–quintuple layer (QL) Cr-doped (Bi,Sb)₂Te₃/five-QL (Bi,Sb)₂Te₃/three-QL Cr-doped (Bi,Sb)₂Te₃ sitting on top of a SrTiO₃ (STO) substrate; see Materials and Methods for more details about material growth. Figure 2A shows an optical image of a representative device, along with its cross-sectional schematic. Each contact arm of the device has an inner and outer electrode. We begin by investigating the magneto-transport properties of the device at temperature $T$ = 100 mK. An ac excitation current (I_ac = 1 nA) is injected into the inner arm using a lock-in amplifier as shown in Fig. 2A. The magnetic field is then swept within the ±0.5-T range at a back gate voltage of V_g = V_g⁰ = 4 V, where V_g⁰ corresponds to the CNP. The resulting magnetic field dependences of the longitudinal (R_xx) and Hall (R_xy) resistances, normalized with respect to the resistance quantum, h/e², are shown in Fig. 2B. We observe a hysteresis loop characterized by a quantized value (0.99 h/e²) of R_xy accompanied by vanishing (0.015 h/e²) R_xx. This observation is a characteristic hallmark of the QAH effect (61). The deviation from perfect quantization in R_xy primarily arises from the lack of precise calibration of the lock-in amplifier and the resistors used for high-accuracy measurements, whereas the nonzero R_xx indicates an imperfect QAH phase. For the proceeding measurements, we first magnetize the device at $B$ = +0.5 T and then set the magnetic field to B = 0 T.

Fig. 2. QAH insulator.

(A) Optical image of the device showing the Corbino geometry with five pairs of contact arms. The configuration for current injection and voltage probes used to measure the four-terminal longitudinal and transverse resistances is indicated. A schematic cross section of the device is shown at the top. (B) The longitudinal, R_xx, and Hall, R_xy, resistances measured at the CNP (V_g⁰ = 4 V) at 100 mK. Arrows indicate the directions of the magnetic field sweep.

To explore the NHSE, we use spectral reconstruction whereby the conductance matrices, G, are retrieved from a series of electrical measurements. This is achieved by first measuring the resistance matrices, R, and obtaining G from G = R⁻¹. The experimental setup for measuring a particular R matrix element, R₃₂, is shown in Fig. 3A. Contact arms labeled 1 to 5 are used for measurements. The inner electrode of each arm is used for current injection and voltage measurement, while the outer electrode is grounded. The current through each arm is monitored via the outer electrode using a current-to-voltage preamplifier, which provides a virtual ground. The contact arm labeled BC is used for tuning the boundary condition of the HN lattice. To measure R₃₂, we inject I_ac = 1 nA into the inner electrode of the contact arm 2 using a lock-in amplifier and measure the voltage drop, V₃, across the inner electrode of the contact arm 3 and ground. Due to the presence of multiple grounds, current will be distributed across the device, and the distributed current will flow from the device to the measurement setup through the fridge lines, each having a resistance of R_F = 2.95 kΩ. To obtain an accurate estimate for R₃₂, the voltage drop across the fridge line needs to be subtracted from V₃ (see Materials and Methods). A variable resistor, R_BC, is also connected between the inner electrode of the contact arm BC and the ground (see Fig. 3A). By tuning the value of R_BC, we can continuously tune the boundary condition of the finite lattice from OBC (R_BC = 0) to periodic boundary condition (PBC; R_BC = $\infty$). However, in our cryostat, R_F = 2.95 kΩ limits the lowest value of $R_{\text{tot}}=R_{\text{BC}}+R_{\mathrm{F}}$. To distinguish this scenario from the true OBC with $R_{\text{tot}}=0$ (see section S4), in the following, we refer to $R_{\text{tot}}=R_{\mathrm{F}}\ne 0$ as the pseudo-OBC.

Fig. 3. Non-Hermitian dynamics in the QAH regime.

(A) Optical image of the device and measurement setup for the resistance matrix element $R_{32}$. Contact arms used for current/voltage probes are labeled 1 to 5 in the clockwise direction. A variable resistor $R_{\text{BC}}$ is connected from the inner electrode of contact arm BC to ground, enabling continuous tuning of the circuit boundary condition. Due to the fridge line resistance, the effective resistance between the inner electrode and ground is $R_{\text{tot}}=R_{\text{BC}}+R_{\mathrm{F}}$, where R_F = 2.95 kΩ. (B and C) Color maps of the G matrices corresponding to pseudo-OBC and PBC, respectively. Site index corresponds to the contact arm labels. The matrix elements are normalized to $e^2/h$. (D) Conductance matrix element $G_{15}$ versus $R_{\text{tot}}$. “Float” denotes $R_{\text{BC}}=\infty$. (E) Eigenvalues of the conductance matrices for different values of $R_{\text{BC}}$, plotted in the complex plane. The real parts of the eigenvalues are shifted by 2$e^2/h$, centering the distribution at the origin. (F) Sum of the probability density (SPD) plotted against site index. During the transition from OBC to PBC, the SPD evolves from exponential localization at a single site to uniform distribution across the sites, a hallmark of the NHSE. (G) Normalized first (j = 1) eigenvector versus site index showing exponential localization similar to the one obtained for SPD in (F). In (E) to (G), R_F ≠ 0 and R_BC = 0 correspond to pseudo-OBC, whereas R_F = 0 and R_BC = 0 correspond to near-ideal OBC. (H) Normalized evolution vector, $\mid \mathrm{\Psi }\left(N\mathrm{\Delta }t\right)⟩$, versus the site index for different number of iterations, N’s. As N increases, $\mid \mathrm{\Psi }\left(N\mathrm{\Delta }t\right)⟩$ becomes more exponentially localized.

By systematically permuting the injected current and voltage probes along the labeled arms, we obtain a complete set of measurements that allow us to reconstruct each column of the matrix R. Figure 3B shows the color map of the resulting G = R⁻¹ matrix, with the matrix elements normalized with respect to the conductance quantum, e²/h (see fig. S1 for values of the matrix elements in the Supplementary Materials). At the CNP, the upper- and lower-diagonal elements of G are populated with values of 0 and −1, respectively, which is consistent with a unidirectional (chiral) edge state. In the case of pseudo-OBC, the matrix element G₁₅, which corresponds to the coupling strength between the first and the last sites of the chain, is −0.11, which is different from G₁₅ = 0 required for ideal OBC with R_tot = 0, as can be obtained from a typical Landauer-Büttiker scattering matrix for a five-terminal mesoscopic setup (74). However, the slightly negative G₁₅ value in pseudo-OBC indicates a weak cross-link due to nonzero $R_{\text{BC}}=2.95$ kΩ. By floating the inner electrode of the contact arm BC, we can achieve PBC with G₁₅ ≈ −1, as shown in the color map of matrix G in Fig. 3C. The deviation from the ideal value of G₁₅ = −1 can be attributed to the calibration issues as was discussed previously. Figure 3D shows a smooth and continuous transition of G₁₅ from the pseudo-OBC to PBC as the R_BC varies from R_BC = 0 to R_BC = ∞, respectively. Furthermore, the eigenvalues of the G matrices corresponding to the PBC spread around a circle in the complex plane. As we gradually move toward pseudo-OBC, the radius of the eigenvalue circle becomes progressively smaller, as shown in Fig. 3E, which is consistent with the existing theoretical calculations (58). Without loss of generality, we can write the conductance matrix as

$$G=\left(\begin{array}{ccccc}2+t& 0& 0& 0& G_{15}\\ -1+s& 2+t& 0& 0& 0\\ 0& -1+s& 2+t& 0& 0\\ 0& 0& -1+s& 2+t& 0\\ 0& 0& 0& -1+s& 2+t\end{array}\right)$$ (2)

where t and s are real numbers denoting the deviations from the quantized values 2 and −1. The eigenvalues of G are ${\mathrm{\lambda }}_{k=\{0,\dots 4\}}=2+t+{\left[G_{15}{(s-1)}^4\right]}^{1/5}{\mathrm{\omega }}_k$, where ${\mathrm{\omega }}_k=e^{i2\mathrm{\pi }k/5}$. This means that ${\mathrm{\lambda }}_k$’s lie on a circle with radius $r={\left[G_{15}{(s-1)}^4\right]}^{1/5}$. As G₁₅ approaches zero, r shrinks with the scaling of $G_{15}^{1/5}$, and the circle contracts toward the center located at 2 + t. In the perfect OBC case with $G_{15}=0$, the radius becomes r = 0 and all ${\mathrm{\lambda }}_k$’s collapse to a single point ${\mathrm{\lambda }}_k=2+t$ on the real axis, resulting in an eigenvalue degeneracy of fifth order. To minimize the total resistance and approach a near-ideal OBC in our experiments, we connect the inner electrode of contact arm BC directly to the ground cavity of the sample chip carrier, eliminating the fridge line resistance entirely. However, $G_{15}$ still remains nonzero in our experiment due to imperfect QAH state in the magnetic TI (i.e., R_xx ≠ 0); see fig. S4. Consequently, ${\mathrm{\lambda }}_k$’s do not fully collapse onto a single point on the real axis as seen in Fig. 3E, but our experiments show that, as the system approaches OBC, the eigenvalues shift progressively toward the real axis.

We now explore the existence of the NHSE by determining the sum of probability densities (SPD) calculated from the eigenstates of the G matrices. SPD at the site i of the HN chain is defined as

$$\text{SPD}\left(i\right)={\sum }_j{\left|V_i^j\right|}^2$$ (3)

where $V_i^j$ is the ith component of the jth normalized eigenvector, ${\mathbf{V}}^j$, of a G matrix (56). Figure 3F shows the SPD as a function of the chain site index. We observe that, under near-ideal OBC, the SPD decays exponentially with the site index, whereas, under PBC, it remains approximately constant across all sites. This behavior indicates that the eigenstates are exponentially localized at the right-most site (contact arm 5) for pseudo-OBC, which is a hallmark of the NSHE. Measurements on a six-arm device corroborate this finding (see section S2). To further confirm NHSE in our device, we have calculated individual normalized eigenvectors, $\mid {V_i}^j\mid =\frac{\mid {V_i}^{j2}}{{\sum }_i\mid {V_i}^{j2}}$. Figure 3G plots the normalized eigenvector, $\mid {V_i}^1\mid$ for j = 1, showcasing exponential localization trend as similar to that seen in SPD as we move toward near-ideal OBC (normalized eigenvectors for j = 2 to 5 are shown in section S5). Alternatively, the emergence of NHSE can be checked using the discrete time evolution of the eigenstates of the G matrices. The time evolution operator, U(Δt), associated with the Hamiltonian, H_HN = G − $E\mathbb{l}$, where E is the diagonal term of the G matrix, is given by

$$U(∆t)=\text{exp}(-\frac{i{H}_{\text{HN}}∆t}{\hslash })=1-\frac{i∆t}{\hslash }{H}_{\text{HN}}+\dots$$ (4)

and the corresponding evolution vectors evolve as $\mid \mathrm{\Psi }\left(\mathrm{\Delta }t\right)⟩=U\left(\mathrm{\Delta }t\right)\mid \mathrm{\Psi }\left(0\right)⟩$, where $\mid \mathrm{\Psi }\left(0\right)⟩={\sum }_n{c}_n\mid {n⟩}_{\mathrm{R}}$ and $c_n={}_{\mathrm{L}}⟨n\mid \mathrm{\Psi }\left(0\right)⟩$. $\mid {n⟩}_{\mathrm{R}}$ and $\mid {n⟩}_{\mathrm{L}}$ are the left and right eigenstates of H_HN that satisfy ${}_{\mathrm{L}}{⟨m\mid n⟩}_{\mathrm{R}}={\mathrm{\delta }}_{\mathit{mn}}$. For a finite-sized HN chain in real space, H_HN takes the form

$$H_{\text{HN}}=\left(\begin{array}{ccccc}0& J_{\text{left}}& 0& 0& 0\\ J_{\text{right}}& 0& J_{\text{left}}& \ddots & 0\\ 0& J_{\text{right}}& 0& \ddots & 0\\ 0& \ddots & \ddots & \ddots & J_{\text{left}}\\ 0& 0& 0& J_{\text{right}}& 0\end{array}\right)$$ (5)

After N time steps, the evolution vectors become

$$\mid \mathrm{\Psi }\left(N\mathrm{\Delta }t\right)⟩\approx {(1-\frac{i∆t}{\hslash }{H}_{\text{HN}})}^N\mid \mathrm{\Psi }\left(0\right)⟩$$ (6)

and the probability density, ${\mid \mathrm{\Psi }\left(N\mathrm{\Delta }t\right)\mid }^2$, converges to one end for a large enough N for any given initial state $\mid \mathrm{\Psi }\left(0\right)⟩$ [see numerical simulation in section S6, and the $\mid \mathrm{\Psi }\left(N\mathrm{\Delta }t\right)⟩$ versus site index for different numbers of iterations in Fig. 3H]. This behavior can be intuitively understood by considering a random walk with biased transition probabilities, where the likelihood of jumping left exceeds that of jumping right, or vice versa (75). Regardless of the initial condition, the bias leads to a net drift toward the favored direction over time. In continuum models, this effect becomes particularly evident in the limit of extreme asymmetry, where the transition amplitude in one direction vanishes entirely. In that case, the dynamics are governed by a first-order differential equation, whose solutions describe unidirectional wave-like propagation (76).

Next, we explore the non-Hermitian dynamics in the metallic phase of the magnetic TI, away from the CNP, where the dissipationless chiral edge state is no longer present. Figure 4A shows the longitudinal and transverse resistance versus the back gate voltage V_g. We observe a deviation from resistance quantization away from the CNP ($V_{\mathrm{g}}$ = −80 V). Owing to the paraelectric nature of the STO substrate used for back-gating, the CNP shifts slightly away from V_g = 4 V during the back-gate sweep. Following the same procedure described earlier, we reconstruct the G matrices for pseudo-OBC and PBC at V_g = −80 V. Figure 4 (B and C) depicts the resulting G color maps. As the chemical potential is tuned out of the bulk gap, both off-diagonal elements depart from their limiting values of −1 (lower off-diagonal element) and 0 (upper off-diagonal element), with the lower element increasing toward 0 and the upper element becoming more negative. These deviations grow systematically as the gate is tuned further away from the CNP (see the comparison between the G matrices in section S1). We further observe finite coupling terms populating previously zero elements of the G matrices, consistent with the absence of unidirectional transport. Instead, the dissipative conduction channels dominate transport at this gate voltage. In this regime, we observe that the lower diagonal elements are reduced by approximately half, while the upper diagonal elements range from −0.2$e^2/h$ to −0.3$e^2/h$, indicating asymmetric, bidirectional coupling between adjacent lattice sites. To further illustrate this point, Fig. 4D presents the evolution of G₁₅ with back gate voltage. For pseudo-OBC, G₁₅ remains ~−0.12 $e^2/h$, whereas, under PBC, it changes from −1.1 $e^2/h$ at the CNP to −0.56 $e^2/h$ at V_g = −80 V. This behavior establishes a direct link between non-Hermitian dynamics and nonreciprocal charge transport, providing a robust framework for understanding asymmetric transport phenomena in TIs. Furthermore, the eigenvalues evolve with the back gate voltage in pseudo-OBC, as shown in Fig. 4E, which exhibits a transition from circular shape in the complex plane, to approximately a line on the real axis as V_g becomes more negative. The change in the shape formed by the eigenvalues is more pronounced at lower V_g’s.

Fig. 4. Non-Hermitian dynamics outside of the QAH regime.

(A) Gate dependence of the longitudinal ($R_{\mathit{xx}}$) and Hall ($R_{\mathit{xy}}$) resistances. A dissipationless chiral edge state exists at the CNP, as depicted in the right inset for electrodes 1 and 2. As the gate is tuned away from the CNP, bidirectional edge states with asymmetric coupling emerge, illustrated in the left inset for the same electrodes. Due to the paraelectric nature of the STO substrate, the CNP is slightly shifted from $V_{\mathrm{g}}$ = 4 V. (B and C) Color maps of the G matrices corresponding to pseudo-OBC (blue dots) and PBC (red dots) at $V_{\mathrm{g}}$ = −80 V. (D) Gate dependence of the conductance matrix element $G_{15}$ for pseudo-OBC and PBC. (E) Gate dependence of the eigenvalues of the conductance matrices in pseudo-OBC. Away from the CNP, the circular distribution of eigenvalues in the complex plane progressively collapses toward the real axis.

In the HN model, the non-Hermicity is realized when the hopping amplitudes between the neighboring sites are direction dependent. Accordingly, the asymmetric, bidirectional coupling between the adjacent chain sites remains consistent with the HN model. The presence of nonreciprocal transport in our devices is further corroborated by second harmonic measurements of the longitudinal resistance, R_xx, in a Hall bar device fabricated on a separately grown magnetically doped TI (see fig. S3). Skew scattering between the chiral edge state and the Dirac surface states has been proposed as a possible mechanism for the nonreciprocity (60).

Last, we have conducted temperature-dependent measurements that revealed that temperature-dependent evolution of the eigenvalues differs from that observed under gate tuning (see section S4): As the temperature increases, bulk conductance gradually turns on and the magnetization alters, causing cross-talk between nonadjacent arms, along with the emergence of bidirectional coupling, which drives the diagonal and off-diagonal matrix elements away from their ideal values. We note that topological edge states may still exist even after QAH breakdown induced by increasing temperature (77). This contrasts with the mechanism by which QAH breakdown occurs when the chemical potential is shifted through gate tuning.

DISCUSSION

In conclusion, we have implemented the HN model in a magnetic TI and investigated the dynamics of the resulting non-Hermitian system. We find that, due to the dissipationless nature of the chiral edge state at the CNP, we can obtain non-Hermitian G matrices, exhibiting NHSE where the eigenstates are exponentially localized at one extremity of the HN chain in the pseudo-OBC case, while they are uniformly distributed along the chain in the PBC case. Away from the CNP, we observe asymmetric, bidirectional coupling between adjacent sites of the chain, where the Dirac surface states come into play, giving rise to nonreciprocity that is captured by the nondiminishing NHSE. Recent studies suggest that transport phenomena in the QAH phase are more complex than previously understood, revealing gate-tunable incompressible bulk strips that contribute to charge transport alongside the edge (78). Notably, edge currents of nontopological origin can persist even after the QAH state breaks down due to changes in magnetization created by gradients in the quasichemical potential and the electrochemical potential (79). Our findings add to this ongoing exploration of bulk-boundary correspondence, non-Hermitian topological phases, and their implications for quantum transport in magnetic TIs.

MATERIALS AND METHODS

Material synthesis

STO (111) substrate was prepared by a 1.5-hour soak in deionized water at 80°C followed by a 3-hour annealing at 982°C in a tube furnace with flowing oxygen gas. The substrate was then transferred into an ultrahigh vacuum (~2 × 10⁻¹⁰ torr) VEECO 620 molecular beam epitaxy chamber. The substrate was heated to 630°C for 1 hour to outgas. Afterward, the substrate was cooled to 255°C and exposed to a Te overflux for 15 s to create a Te-rich environment for growth. A three-QL Cr-(Bi,Sb)₂Te₃ was grown at a rate of ~0.25 QL/min. The Cr shutter was then closed to grow a five-QL (Bi,Sb)₂Te₃ at the same growth rate. Last, the Cr shutter was reopened to deposit three-QL Cr-(Bi,Sb)₂Te₃. The cell temperatures were controlled to achieve the desired beam equivalent pressure ratios (BEPRs) between high-purity Cr (5 N), Bi (5 N), Sb (6 N), and Te (6 N) Knudsen effusion cells. The Te/(Bi+Sb) BEPR was ~17 to mitigate the formation of Te vacancies. The BEPR of Sb/Bi was ~1.2 to achieve an intrinsic CNP. The nominal Cr BEP was 4.7 × 10⁻¹⁰ torr.

Device fabrication

A multiterminal Corbino ring was first defined on the magnetic TI film with Heidelberg MLA150 maskless aligner. The magnetic TI film was subsequently etched away in a mixture of H₂O₂ (30%)/H₃PO₄ (85%)/H₂O with a ratio by volume of 1:1:20 for 20 s at room temperature. Another photolithographic step was then performed to define the contacts. Last, Cr/Au (10/50 nm) were deposited onto the sample using electron beam evaporator. Indium was applied to the backside of the substrate after device fabrication to create the back-gate electrode.

Electrical transport

All the electrical transport measurements were performed inside a Bluefors LD250 dilution fridge, equipped with QDevil filtered dc lines, at the base temperature of ~12 mK, unless specified otherwise. The magnetic TI film was magnetized by applying a perpendicular magnetic field up to 0.5 T at 12 mK. To accurately measure the voltage, V, at each contact arm, the current, I, flowing into the inner electrode of the contact arm was measured. The voltage across the fridge line was subsequently subtracted from the measured voltage. The resistance was then calculated from R = (V − R_FI)/I_ac. The current was probed using a low-noise current-to-voltage preamplifier and a lock-in amplifier.

Supplementary Materials

This PDF file includes:

Sections S1 to S6

Figs. S1 to S5