
==== Front
Nat Commun
Nat Commun
Nature Communications
2041-1723
Nature Publishing Group UK London

39237573
52206
10.1038/s41467-024-52206-8
Article
Quantum geometry quadrupole-induced third-order nonlinear transport in antiferromagnetic topological insulator MnBi2Te4
Li Hui 1
http://orcid.org/0000-0003-1093-6972
Zhang Chengping 1
Zhou Chengjie 1
Ma Chen 1
Lei Xiao 2
Jin Zijing 1
He Hongtao 2
Li Baikui 1
http://orcid.org/0000-0003-0501-6290
Law Kam Tuen phlaw@ust.hk

1
http://orcid.org/0000-0002-6133-6371
Wang Jiannong phjwang@ust.hk

1
1 grid.24515.37 0000 0004 1937 1450 Department of Physics, the Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China
2 https://ror.org/049tv2d57 grid.263817.9 0000 0004 1773 1790 Department of Physics, South University of Science and Technology of China, Shenzhen, Guangdong China
5 9 2024
5 9 2024
2024
15 777922 11 2023
29 8 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
The study of quantum geometry effects in materials has been one of the most important research directions in recent decades. The quantum geometry of a material is characterized by the quantum geometric tensor of the Bloch states. The imaginary part of the quantum geometry tensor gives rise to the Berry curvature while the real part gives rise to the quantum metric. While Berry curvature has been well studied in the past decades, the experimental investigation on the quantum metric effects is only at its infancy stage. In this work, we measure the nonlinear transport of bulk MnBi2Te4, which is a topological anti-ferromagnet. We found that the second order nonlinear responses are negligible as required by inversion symmetry, the third-order nonlinear responses are finite. The measured third-harmonic longitudinal (Vxx3ω) and transverse (Vxy3ω) voltages with frequency 3ω, driven by an a.c. current with frequency ω, show an intimate connection with magnetic transitions of MnBi2Te4 flakes. Their magnitudes change abruptly as MnBi2Te4 flakes go through magnetic transitions from an antiferromagnetic state to a canted antiferromagnetic state and to a ferromagnetic state. In addition, the measured Vxx3ω is an even function of the applied magnetic field B while Vxy3ω is odd in B. Amazingly, the field dependence of the third-order responses as a function of the magnetic field suggests that Vxx3ω is induced by the quantum metric quadrupole and Vxy3ω is induced by the Berry curvature quadrupole. Therefore, the quadrupoles of both the real and the imaginary part of the quantum geometry tensor of bulk MnBi2Te4 are revealed through the third order nonlinear transport measurements. This work greatly advanced our understanding on the connections between the higher order moments of quantum geometry and nonlinear transport.

MnBi2Te4 is a topological antiferromagnet, and has a second-order nonlinear Hall effect when inversion symmetry is broken. Here, Li, Zhang and coauthors observe a third order longitudinal response and show how the observed non-linear response relates to the quantum geometry quadrupole of the material.

Subject terms

Electronic properties and materials
Topological insulators
https://doi.org/10.13039/501100002920 Research Grants Council, University Grants Committee (RGC, UGC) 16306421 Wang Jiannong issue-copyright-statement© Springer Nature Limited 2024
==== Body
pmcIntroduction

The study of the intricate relationship between quantum geometric properties and high-order nonlinear transport properties in topological materials is at the frontier of condensed matter physics research1–16. The quantum geometric properties of a material are characterized by the quantum geometric tensor Tαβk=∂αuk(1−∣uk⟩⟨uk∣)∣∂βuk⟩. Here, ∣uk⟩ denotes a Bloch state with crystal momentum k and ∂α denotes the derivative with respect to the α-component of k. The imaginary part of T, Ωαβ=2Im(Tαβ), defines the Berry curvature, and the real part gαβ= Re(Tαβ) defines the quantum metric. Importantly, it has been shown that the Berry curvature, the quantum metric, and their dipole distribution can induce novel nonlinear transport phenomena8,13,15–25. For instance, the Berry curvature dipole-induced second-order nonlinear Hall effect has been well demonstrated in bilayer and few-layer WTe2 flakes5,17,26–29. More recently, with MnBi2Te4 thin films, quantum metric dipole-induced second-order nonlinear Hall effect was observed in which an alternating current (a.c.) with frequency ω induces an a.c. Hall voltage with frequency 2ω20,21. However, inversion symmetry must be broken in these MnBi2Te4 films such that the second-order effect can be observed.

In particular, the distribution of Berry curvature in a genetic time-reversal broken system could result in the formation of non-zero Berry curvature quadrupoles even if the Berry curvature monopole and the dipole are zero, as depicted in Fig. 1a. When an electric field is introduced, we reach a steady state in which the electric field induces a Berry curvature dipole along the electric field direction, as demonstrated in Fig. 1b. This electric field-induced dipole will cause a nonlinear response in the third order of the electric field as explained in ref. 13. This Berry curvature quadrupole induced third order effect offers complementary experimental knobs to investigate the high-order nonlinear anomalous Hall effect in magnetic materials. Analogously, a finite normalized quantum metric Gαβ can induce a finite Berry connection and ∂x2Gxx gives rise to the third-order nonlinear response15. Bulk antiferromagnetic topological insulator MnBi2Te4 is an ideal platform for investigating the quantum geometry quadrupole effects as the second harmonics are suppressed by inversion symmetry and the third order harmonics are the leading order nonlinear response.Fig. 1 Berry curvature quadrupole in MiBi2Te4.

a Schematic diagrams of the band structure (the outline) and Berry curvature (color scale) of a generic magnetic material with non-zero Berry curvature quadrupoles. The + and – refer to the sign of Berry curvature. b Electric field induced change of Berry curvature (color scale) relative to the equilibrium state. Due to the Berry curvature distribution of the quadrupole, a field-induced dipole will appear, which contributes to the higher-order nonlinear anomalous Hall effect. The + and – refer to the sign of Berry curvature. c Schematic crystal structure of layered MiBi2Te4. The red arrows in Mn layers indicate the spin direction of Mn atoms.

Antiferromagnetic topological insulator MnBi2Te4 has rich magnetic structures and striking topological band structures7,30–40. The lattice structure of MnBi2Te4 is shown in Fig. 1c. It has a septuple layer structure in each unit cell, which composes a Bi2Te3 quintuple layer and an MnTe layer. Each Mn layer in the unit cell contributes a typical ferromagnetic feature, while the adjacent two Mn layers are antiferromagnetic coupled at low temperatures. The bulk MnBi2Te4 preserves an inversion symmetry (P) and a time-reversal times half-translation symmetry (Tt1/2), giving rise to a combined PT symmetry. The PT symmetry will force the Berry curvature quadrupoles to vanish. However, when a magnetic field (B-field) is applied to induce a finite magnetization, the PT symmetry will be broken and a nonzero Berry curvature quadrupole will emerge, which can induce the third order nonlinear Hall effect13,22. On the other hand, the quantum metric-induced nonlinear effects are allowed by the PT symmetry11, therefore can be observed even in the absence of the external B-fields. In recent experiments20,21 on thin films, inversion symmetry is broken and the second-order nonlinear effects were observed. In contrast, the bulk MnBi2Te4 preserves the inversion symmetry, making the third-order nonlinear effects the leading-order contributions from the quantum metrics.

In this work, we study the higher-order nonlinear transport properties of bulk MnBi2Te4 which is a topological anti-ferromagnet. As expected, due to the presence of the inversion symmetry in the bulk, both the second order Hall response and the second order longitudinal response are zero, which is different from the observations in thin films20,21. Incredibly, at temperatures below the antiferromagnetic transition temperature of MnBi2Te4 flakes, we observed a third-order longitudinal response in which an a.c. current with frequency ω induces an a.c. voltage Vxx3ω with frequency 3ω along the current direction. Interestingly, in the presence of an out-of-plane magnetic field, an a.c. current induces an a.c. voltage Vxy3ω with frequency 3ω in the transverse direction. These high-order nonlinear responses are tightly related to magnetic transitions of MnBi2Te4 flakes. In addition, the measured Vxx3ω and Vxy3ω show even and odd dependence on the external B-fields respectively. Through the symmetry analysis and the semi-classical Boltzmann equation approach, we show that Vxx3ω is induced by the quantum metric quadrupole ∂x2Gxx (where Gαβ is the normalized quantum metric defined in the Method Section). On the other hand, the third order Hall voltage Vxy3ω is induced by the Berry curvature quadrupole which was predicted recently13. More importantly, the observed magnetic field dependence of Vxx3ω and Vxy3ω match the calculated third order response well. In other words, in bulk MnBi2Te4, the suppression of the second order nonlinear transport by inversion symmetry enabled the third order nonlinear response to be the leading nonlinear response, which is induced by the quadrupole distribution of the Berry curvature and the quantum metric. Our experimental work clearly reveals the higher moments of both the real and the imaginary part of the quantum metric tensor and has deepened our understanding of the relation between quantum geometry, crystal symmetry and the nonlinear transport.

Results

Temperature dependence of nonlinear transport properties of MnBi2Te4 flakes

Figure 2a shows the schematic drawing of the measurement setup for the nonlinear transport properties of MnBi2Te4 flakes. The a.c. with a low frequency of 179 Hz was injected and the longitudinal voltage (Vxx) and transverse voltage (Vxy) signals at different harmonics were collected using lock-in techniques. The thickness of MnBi2Te4 flakes is ~ 60 nm as determined by the atomic force microscope measurement in Supplementary Fig. 1 in Supplementary Note 1. Good ohmic contact between different probes has been achieved, as shown in Supplementary Fig. 2b in Supplementary Note 2. Striking third-harmonic longitudinal and transverse voltage are clearly observed in MnBi2Te4 flakes with their magnitude being increased with increasing first-harmonic current, as shown in Supplementary Fig. 3 in the Supplementary Note 3. On the contrary, no distinct second-harmonic signal has been observed in MnBi2Te4 flakes due to the crystallographic symmetry constraint (Supplementary Fig. 4 in the Supplementary Note 3). We note that the magnitude of the third-harmonic signal of a resistor of ~ 100 Ω is negligible. This rules out the instrumental errors as a possible origin of the observed third-harmonic signal in MnBi2Te4 flakes. In addition, the third-harmonic longitudinal and transverse response is independent of the frequency of the driving current, as detailed in Supplementary Fig. 5 in the Supplementary Note 4.Fig. 2 Temperature dependence of the nonlinear transport properties in MiBi2Te4.

a Schematic drawing of the measurement setup. b Temperature-dependent of the first-harmonic longitudinal resistance at different out-of-plane B-fields with excited a.c. current of 0.6 mA. c Temperature dependence of the Vxx3ω (upper panel) and Vxy3ω (lower panel) at different out-of-plane B-fields with excited a.c. current of 0.6 mA. Inset in lower panel shows the zoom-in view of Vxy3ω at low temperatures.

Figure 2b shows the temperature (T) dependence of the first-harmonic longitudinal resistance (Rxxω) of the MnBi2Te4 flakes collected at different out-of-plane B-fields (B ⊥ I) with an a.c. current of 0.6 mA. The driving current dependence of the third-harmonic nonlinear response is detailed in Supplementary Note 5. As temperature decreases, the resistance decreases gradually with an abnormal peak occurring at TN ~ 21.7 K, corresponding to the antiferromagnetic (AFM) transition in MnBi2Te4 flakes30,37. It is noted that the measured AFM transition temperature TN ~ 21.7 K is slightly lower than the measured TN of ~ 23.7 K with the applied current of ~ 1 μA (Supplementary Fig. 7a) and previous studies of ~ 24 K30,37. After safely excluding the heating effect (see details in Supplementary Note 5), such deviation is probably due to the large current caused suppression of the antiparallel alignments of the spins in MnBi2Te4 flakes and the spin-transfer torque effect41,42. In addition, the AFM transition in MnBi2Te4 flakes is strongly related to out-of-plane B-fields. As indicated by the orange curve in Fig. 2b, the AFM transition in MnBi2Te4 flakes shifts to low temperatures of ~ 19 K at 2 T. Further increasing the B-fields to 4 T, a striking resistance upturn occurs at temperature below ~ 18 K and gradually saturates at temperature below ~ 5 K, as indicated by the blue curve in Fig. 2b. Such an intriguing transition is believed to be related to the formation of a canted AFM state in MnBi2Te4 flakes at low temperatures37. At higher B-fields above 7 T, the AFM transition in MnBi2Te4 flakes smears out with resistance continuously decreases with decreasing temperature in the entire temperature ranging from 100 K to 2 K, behaving as a metal (green curve for 7 T and purple curve for 9.5 T)36,37.

The temperature dependence of third-harmonic longitudinal (Vxx3ω) and transverse (Vxy3ω) voltage signals of MnBi2Te4 flakes collected at different out-of-plane B-fields are shown in the upper and lower panel of Fig. 2c, respectively. The Vxx3ω (Vxy3ω) of MnBi2Te4 flakes is obtained by averaging of the sum (difference) of voltages measured at positive and negative B-fields, respectively. Both Vxx3ω and Vxy3ω are strongly dependent on the B-fields. As it can be seen from the upper panel of Fig. 2c, at 0 T, the Vxx3ω decreases slowly with decreasing temperature from 70 K to ~ 37 K. Below ~ 37 K, it starts to increase and reach a peak value at ~ 23.7 K, then decreases sharply (black curve). The rapid change around 23.7 K coincides with the AFM transition of MnBi2Te4 flakes, suggesting the change of Vxx3ω may relate to the AFM transition of MnBi2Te4 flakes. Further decreasing temperature, the Vxx3ω of MnBi2Te4 flakes features a slight upturn at ~ 18.7 K followed by a continuous decrease at temperature below ~ 13.7 K. These striking features remain unchanged as B-field increases to 2 T (orange curve) except the transition temperature shifted to lower temperature by ~ 2 K comparing with that at 0 T. While for higher B-field of 4 T, the Vxx3ω gradually decreases with decreasing temperature from 70 K to 35 K, then increases as temperature below 35 K, and finally saturates with temperature below ~ 5 K (blue curve). In contrast, the Vxx3ω continuously decreases with decreasing temperature from 70 K to 2 K for the large B-field of 7 T (green curve) and 9.5 T (purple curve).

The temperature dependence of Vxy3ω of MnBi2Te4 flakes measured at different B-fields is shown in the lower panel of Fig. 2c. At B-fields of 0 T (black curve) and 2 T (orange curve), the Vxy3ω is almost zero at temperatures far above the AFM transition temperature of MnBi2Te4 flakes with a slight increase with decreasing temperature (see the zoom-in view in the inset of the lower panel). As temperature approaches the AFM transition temperature of MnBi2Te4 flakes, an abrupt change of Vxy3ω occurs within a narrow temperature range from ~ 23 K to ~ 19.5 K at B = 0 T (black curve), where its magnitude changes by ~ 18 μV from ~ 3 μV to ~ − 15 μV. Further decreasing temperature, the Vxy3ω shows a slight upturn followed by a gradual decrease. Similar abrupt changes near the AFM transition of MnBi2Te4 flakes are also observed at B = 2 T (orange curve) and 4 T (blue curve). However, the magnitude of the Vxy3ω is greatly enhanced at high B-fields of 4 T. In contrast, the Vxy3ω exhibits a continuous increase with decreasing temperatures at even higher B-fields of 7 T (green curve) and 9.5 T (purple curve). Notably, the magnitude of Vxy3ω reaches ~ 65 μV at 2 K and 9.5 T, which is almost one order of magnitude larger than that at 0 T.

B-fields dependence of nonlinear transport properties in MnBi2Te4 flakes

Figure 3a shows the out-of-plane B-fields dependent Vxx3ω of MnBi2Te4 flakes measured at different temperatures. The Vxx3ω curves in Fig. 3a are symmetrized by averaging the sum of magnitude collected in the positive and negative B-fields regions to exclude any mixture effect. It is clear that all curves exhibit a distinct even-symmetry with B-fields directions. Such an even symmetry of the Vxx3ω is well evidenced even from the raw data in Supplementary Fig. 8a in Supplementary Note 7. The evolution of Vxx3ω is strongly associated with the magnetic transitions of MnBi2Te4 flakes under applied B-fields. The MnBi2Te4 flakes undergo multiple magnetic transitions from AFM state to canted-AFM state and then to polarized ferromagnetic (FM) state with increasing B-fields, as illustrated by the magnetoresistance curve of MnBi2Te4 flakes at 2 K in Supplementary Fig. 7b in the Supplementary Note 6. At 2 K and small B-fields (< ~ 2.7 T) with MnBi2Te4 flakes in the AFM state, the Vxx3ω is almost unchanged (black curve). With increasing B-fields, a sudden drop changing from approximately − 230 μV to approximately − 460 μV followed by an abrupt flipping from approximately − 460 μV to ~ 175 μV occurs in a narrow B-field range from ~ 2.7 T to ~ 3.2 T. This corresponds to the magnetic transition from AFM state to canted AFM state in MnBi2Te4 flakes, as indicated in Supplementary Fig. 7b in the Supplementary Note 6. Further increasing B-fields, the Vxx3ω decreases continuously through the magnetic transition from canted-AFM state to FM state. Finally, it saturates as MnBi2Te4 flakes enter the polarized FM state at higher B-field above ~ 8 T. Similar behaviors are observed at temperatures below AFM transition temperature of MnBi2Te4 flakes, as indicated by the red curve for 10 K, blue curve for 15 K, and green curve for 20 K, respectively, in Fig. 3a. However, as temperature increases above the AFM transition temperature of MnBi2Te4 flakes, these abrupt changes in the Vxx3ω disappear. As a result, the Vxx3ω only shows a slight and monotonical change with increasing B-fields, for example, the Vxx3ω slightly changes by ~ 30 μV as B-fields increase from 0 to 9 T at 30 K (purple curve) and is almost unchanged at 50 K (brown curve). Further increasing the temperature to 100 K, the magnitude of Vxx3ω only shows a slight increase with increasing B-fields (cyan curve).Fig. 3 B-fields dependence of the longitudinal third-harmonic voltage in MiBi2Te4.

a The out-of-plane B-fields dependence of the Vxx3ω measured at different temperatures. b The calculated σxxxx contributed by quantum metric as a function of the magnetic order parameter Δ. c–e The band splitting and distribution of quantum metric Gxx (color scale) under different magnetic order parameter Δ = 0 (c), Δ = 0.2 (d), and Δ = 0.7 (e), where the black dashed line indicates the chemical potential.

The out-of-plane B-fields dependent Vxy3ω of MnBi2Te4 flakes at different temperatures are shown in Fig. 4a. The Vxy3ω curves are symmetrized by averaging the magnitude difference between the positive and negative B-fields regions to exclude the mixing of Vxx3ω. The Vxy3ω is of odd-symmetry, which is in striking contrast to that of Vxx3ω, as demonstrated in Fig. 4a and the raw data in Supplementary Fig. 8b in Supplementary Note 7. In addition, the measured third-harmonic nonlinear Hall voltage for two pairs of Hall bar shows a good consistency and reproducibility, as shown in Supplementary Fig. 2c in Supplementary Note 2. The evolution of Vxy3ω is also strongly related to the magnetic transitions of the MnBi2Te4 flakes, like that in the longitudinal direction. At a low temperature of 2 K (black curve), the magnitude of Vxy3ω closes to 0 and is almost unchanged in the AFM state of MnBi2Te4 flakes in small B-fields <~2.7 T. With increasing B-fields, the magnitude of Vxy3ω decreases sharply by ~ 50 μV in a narrow B-fields ranging from ~ 2.7 T to ~ 3.2 T. This coincides with the abrupt change occurring in the Vxx3ω (Fig. 3a), which is related to the magnetic transition from AFM state to canted AFM state of MnBi2Te4 flakes in the intermediate B-fields36,37. Further increasing B-fields from ~ 3.2 T to ~ 7.5 T, the Vxy3ω increases continuously through the magnetic transition from canted AFM state to FM state in MnBi2Te4 flakes. Then, the Vxy3ω saturates at even higher B-fields above 7.5 T corresponding to the formation of the polarized FM state in MnBi2Te4 flakes. The overall features of Vxy3ω are the same at temperatures below AFM transition temperatures except that the fields corresponding to the sharp decrease shift to lower B-fields with increased temperatures, as demonstrated in Fig. 4a for 10 K (red curve), 15 K (blue curve) and 20 K (green curve), respectively. In contrast, the Vxy3ω features a monotonic increase as temperature further increases above the AFM transition temperature of MnBi2Te4 flakes. The magnitude of Vxy3ω increases by ~ 22.5 μV with increasing B-fields from 0 to 9 T at 30 K (purple curve) and becomes temperature-independent with further increasing temperatures, as shown by the brown and cyan curves at 50 K and 100 K, respectively.Fig. 4 B-fields dependence of the transverse third-harmonic voltage in MiBi2Te4.

a The out-of-plane B-fields dependence of the Vxy3ω measured at different temperatures. b The calculated Berry curvature quadrupole Qxxz as a function of the magnetic order parameter Δ. c–e The band splitting and distribution of Berry curvature quadrupole Ωz (color scale) at different magnetic order parameter Δ = 0 (c), Δ = 0.2 (d), and Δ = 0.7 (e), where the black dashed line indicates the chemical potential.

Discussion

The distinct symmetry properties of the Berry curvature and quantum metric contributions allow us to distinguish the two effects as both are observed in our experiments. The third-order nonlinear response contributed by the Berry curvature quadrupoles is odd under the time-reversal symmetry. Furthermore, the Berry curvature quadrupoles only contribute to the third-harmonic Hall voltage Vxy3ω without longitudinal contributions, so it can account for the observed Vxy3ω in MnBi2Te4 flakes with odd-symmetry with respect to the B-fields directions. On the other hand, the third-order nonlinear response contributed by the quantum metric is even under the time-reversal symmetry and only contributes to the longitudinal third-harmonic voltage Vxx3ω, therefore, it is responsible for the observed Vxx3ω in MnBi2Te4 flakes with even-symmetry with respect to the B-fields directions.

In order to capture the effect induced by the out-of-pane B-field, we introduce a model Hamiltonian with a tunable magnetic order parameter Δ (see Methods for details). Figure 3b plots the calculated quantum metric induced third-harmonic conductivity σxxxx as a function of the magnetic order parameter Δ, see calculation details in Methods. As it can be seen, the σxxxx is strongly dependent on Δ, and its variation with Δ reproduces the key features of our experimental measured B-fields dependence of Vxx3ω. Figure 3c–e illustrates the physical origin of the evolution of the σxxxx with increasing Δ. At zero B-field or Δ = 0, all bands of MnBi2Te4 are doubly degenerate due to the combined PT symmetry (Fig. 3c). Each band contributes a finite quantum metric with its sign opposites for conduction band (positive) and valance band (negative), as shown in Fig. 3c. Such finite quantum metric will result in the observation of finite third-harmonic voltage in the longitudinal direction Vxx3ω, as observed in the experimental measurements in Fig. 3a. When applying a B-field or Δ≠0, the broken PT symmetry in MnBi2Te4 will split the degenerated bands and modify the distribution of the quantum metric, as illustrated in Fig. 3d. The changes of the quantum metric distribution will lead to the observation of abrupt changes in Vxx3ω, as plotted in Fig. 3b in the intermediate magnetic order parameters and evidenced from the experimental measurements in the transition from the AFM state to canted-AFM state in MnBi2Te4 in Fig. 3a. Further increasing the magnetic order parameters, the splitting between bands becomes larger, as shown in Fig. 3e. This will modulate the quantum geometric properties of the electrons close to Fermi surfaces and lead to the occurrences of sharp change, as demonstrated by theoretical calculation in Fig. 3b and experimental observations in Fig. 3a. Furthermore, the scaling law analysis suggests the third-harmonic longitudinal voltage could be attributed to Drude-like tensor contribution and quantum metric quadrupole with comparable magnitude at low temperatures, as detailed in Supplementary Note 9. However, the Drude-like tensor contribution is insensitive to the magnetic phase transition induced by the external B-fields, which indicates the experimental observed B-fields dependence of third-harmonic nonlinear longitudinal voltage in bulk MnBi2Te4 is mainly contributed by quantum metric quadrupole with a constant background contributed by Drude-like tensor.

On the other hand, the experimentally observed odd-symmetry of Vxy3ω can be explained by the nonzero Berry curvature quadrupoles when the PT symmetry is broken by the non-zero magnetic order parameter Δ or the external B-fields. Figure 4b shows the calculated magnetic order parameter Δ dependence of Berry curvature quadrupoles in MnBi2Te4. Remarkably, it exhibits a good agreement with our experimental observations in Fig. 4a. Figure 4c–e shows the calculated band structure evolution and the strength distribution of the Berry curvature quadrupole. The combined PT symmetry of MnBi2Te4 flakes forces each band to be doubly degenerated in the absence of external B-fields, as indicated by the black curve in Fig. 4c. This band degeneracy makes the Berry curvature quadrupoles to zero, resulting in the zero Vxy3ω in the antiferromagnetic state of MnBi2Te4 flakes. When the external B-field is applied, the PT symmetry is broken, lifting the band degeneracy. Thus, a non-zero Berry curvature quadrupole emerges, as illustrated in Fig. 4d. This coincides with our experimental observations in Fig. 4a, where an abrupt change occurs in the transition from AFM state to canted-AFM state at increased B-fields. With further increasing the magnetic order parameter Δ, the band splitting enhances and eventually pushes the upper band away from the Fermi level (Fig. 4e). Consequently, the sharp change of the Vxy3ω occurs again, as shown in the theoretical calculation in Fig. 4b and the experimental observations in Fig. 4a at larger B-fields. The detailed theoretical derivations are presented in Methods. It is noted that the skew scattering may also contribute to the third-harmonic transverse voltages. As detailed in the scaling law analysis in Supplementary Note 9, the magnitude of skew scattering contribution is calculated to be comparable to that of the Berry curvature quadrupoles. However, the skew scattering contribution should be insensitive to the external B-fields, the dependence on the order parameter Δ strongly reinforces our claim of observing the third-harmonic transverse nonlinear conductivity stemming from the Berry curvature quadrupoles contributions.

In conclusion, our systematic study of nonlinear transport properties of the antiferromagnetic topological insulator MnBi2Te4 flakes has evidenced the existence of third-harmonic voltages in both longitudinal and transverse directions at temperatures below the antiferromagnetic transition temperature of MnBi2Te4 flakes. The observed third-order nonlinear transport properties are related to the magnetism of MnBi2Te4 flakes, with the distinct changes of the third-harmonic voltages closely following the magnetic transitions. The longitudinal third-harmonic response Vxx3ω is attributed to the quantum metric origin. In contrast, the transverse third-harmonic response originated from the emergency of non-zero Berry curvature quadrupoles induced by the external B-fields. Our findings provide the connection between the quantum geometry quadrupole and the high-order nonlinear transport properties, while lower-order nonlinear responses are suppressed by inversion symmetry. We expect that even higher order (such as the seventh order) responses, can be the leading order nonlinear responses given the appropriate symmetry13. This work opens a new venue to explore quantum geometry multipoles induced nonlinear Hall effect.

Methods

Devices fabrication and magnetoresistance measurements

To study the magnetotransport properties, a Hall-bar geometry was patterned through standard electron-beam lithography and lift-off techniques. Au/Cr electrodes with a thickness of 100 nm/10 nm were deposited using thermal evaporation methods. The magnetotransport properties of the devices were then measured in a Quantum Design Physical Property Measurement System using a standard lock-in technique.

Theoretical calculations of third-order nonlinear conductivity

The generic third-order conductivity can be defined as jα=σαβγδEβEγEδ. In particular, the third-order longitudinal conductivity induced by the quantum metric is151 σxxxx=e4τℏ2−∫k∂x2Gxxf0+12∫kvx2Gxxf0″

Where v=∇kεk is the band velocity, ∂x≡∂/∂kx, and Gab=2Re∑m≠n(Aa)nm(Ab)mnεn-εm is the band-energy normalized quantum metric tensor2,11,20,21. In order to get a gauge-independent result, we have used the relationship (Aa)nm(Ab)mn=⟨n∣∂aH∣m⟩⟨m∣∂bH∣n⟩(εn−εm)2 in our calculations.

On the other hand, the third-order Hall conductivity induced by the Berry curvature quadrupole is ref. 132 σyxxx=e4τ2ℏ3Qxxz

where Qxxz=∫k∂x2Ωzf0 is the Berry curvature quadrupole.

Next, we introduce the effective Hamiltonian of MnBi2Te4 flakes in our theoretical calculations for third-order nonlinear effects induced by the quantum metric and Berry curvature quadrupole. ref. 43 provides a model Hamiltonian of ferromagnetic MnBi2Te4, where the ferromagnetic state is stabilized by the external B-field. Here, in order to study the effect of the increase of the B-field, we introduce a tunable magnetic order parameter Δ, and the effective Hamiltonian reads,3 Hk=H0k+ΔHMk

where H0k respects the PT symmetry, and HMk is the part which is induced by the finite magnetization and breaks the time-reversal symmetry. The detailed parameters of the Hamiltonian can be found in Supplementary Note 9.

Odd- and even-symmetry of B-fields dependent third-harmonic nonlinear voltages

The B-fields dependence of third-harmonic nonlinear voltages are symmetrized for longitudinal direction and antisymmetrized for transverse direction, as indicated below,4 Vxx3ωB=Vxx3ωB+Vxx3ω−B2

5 Vxy3ωB=Vxy3ωB−Vxy3ω−B2

Supplementary information

Supplementary Information

Peer Review File

Source data

Source Data

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-024-52206-8.

Acknowledgements

This work was supported in part by the Research Grants Council of the Hong Kong SAR under Grant Nos. 16306421, AoE/P-701/20-3, and C6025-19G.

Author contributions

J.N.W. conceived the research; C.M. and X.L. fabricated devices with support from H.H.; H.L. performed transport experiments with support from C. Zhou, C.M., Z.J., and B.L.; C. Zhang and K.T.L. provided theoretical support; H.L., C. Zhang, H.H., K.T.L., and J.N.W. analyzed experimental data and wrote the manuscript with input from all co-authors.

Peer review

Peer review information

Nature Communications thanks Weibo Gao, and the other anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Data availability

The data represented in Figs. 2, 3, and 4 are available as Source Data files. All other data that support the plots within this paper and other findings of this study are available from the corresponding author on request. Source data are provided in this paper.

Competing interests

The authors declare no competing interests.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

These authors contributed equally: Hui Li, Chengping Zhang.
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