Higher odd-order nonlinear Hall effect in magnetic topological insulator Mn(Bi1-xSbx)2Te4

Abstract: The nonlinear Hall effect is a new member of the Hall effect family, which attracts intense research interests, and it is closely related to the quantum geometry of quantum materials. The previous studies primarily concentrate on the second-order and third-order nonlinear Hall effect. However, the experimental study of higher-order nonlinear Hall effect is scarce at present. In this work, we report the observations of the higher odd-order (third-, fifth-, seventh-order) nonlinear Hall effect in magnetic topological insulator Mn(Bi1-xSbx)2Te4 thin flakes. The higher odd-order nonlinear Hall voltage exhibits a twofold angular dependence and exists only below the Néel temperature. It reaches its maximum near the charge neutral point and decays exponentially as the order of the nonlinear Hall effect increases. Furthermore, such higher odd-order nonlinear Hall effect is observed in both odd- and even-layer samples with comparable magnitudes. Theoretical analysis indicates that the higher odd-order nonlinear Hall effect responses may arise from the Berry curvature multipoles. Our work paves the way for the study of the higher-order nonlinear transport phenomena.

• The paper experimentally demonstrates higher odd-order nonlinear Hall effects (3rd, 5th, and 7th) in Mn(BiSb)2Te4, linking these responses to Berry curvature multipoles.
• Key measurements reveal that NLHE signals depend on gate voltage and temperature, manifesting only below the Néel threshold and remaining nearly thickness-independent.
• The findings suggest potential applications in nonlinear transport device engineering and motivate further investigations into quantum geometric effects in magnetic topological insulators.

Higher Odd-Order Nonlinear Hall Effect in Magnetic Topological Insulator Mn(Bi₁₋ₓSbₓ)₂Te₄

Overview and Motivation

This work systematically investigates higher odd-order nonlinear Hall effects (NLHE) in the magnetic topological insulator (MTI) Mn(Bi₁₋ₓSbₓ)₂Te₄, specifically examining thin flakes with Sb doping $x=0.3$. Previous research on NLHE has focused primarily on second- and third-order responses, leaving the experimental demonstration of higher-order effects, particularly above third order, largely unexplored. The study delivers clear experimental evidence of third-, fifth-, and seventh-order NLHE in both odd- and even-layer samples, and presents a coherent theoretical framework attributing these responses to Berry curvature multipoles (BCMs).

Experimental Results

Odd-order NLHEs were observed through measurements of Hall voltages under current-driven excitation at various harmonics (frequencies proportional to odd powers of the driving current). Key findings include:

• Presence of Higher Odd-Order NLHE: Third-, fifth-, and seventh-order NLHE voltages were distinctively detected, satisfying $V_{xy}^{(2n+1)} \propto (I)^{{2n+1}}$ scaling for each respective order.
• Angular Dependence: All NLHE signals display a robust twofold angular periodicity (180º), independent of lattice orientation, interpreted as emergent $C_{2z}$ symmetry due to Sb-doping.
• Temperature Regime: NLHEs manifest only below the Néel temperature ($T_N \approx 24$ K), highlighting the magnetic order’s necessity for the effect. The onset temperature decreases with increasing NLHE order.
• Gate Tunability: Hall voltages peak near the charge neutral point (CNP), and are tunable via gate voltage, illustrating the Fermi level’s control over nonlinear transport responses.
• Layer Dependence: Comparable magnitudes for NLHEs were found in both odd- and even-layer samples, with responses nearly thickness-independent.
• Strong Numerical Results: At $1.8$ K and $5$ μA drive current, the magnitudes of the third-, fifth-, and seventh-order NLHE reached $28.5$ μV, $7.5$ μV, and $2.7$ μV respectively.
• Exponential Decay: The strength of higher-order NLHE drops exponentially with increasing order, quantified as $V_{xy}^{(N)} \propto \exp(-\alpha N)$, with $V_{xy}^{(2n+1)} \propto (I)^{{2n+1}}$0 for the system.
• Absence of Even-Order NLHE: Even-order responses were suppressed, consistent with inversion symmetry present in the doped crystal structure.

Theoretical Analysis

The observed phenomena are theoretically interpreted by considering the quantum geometry of MTI surface states, specifically their Berry curvature multipoles:

• Dirac Surface States Dominance: Near the CNP, electronic properties are governed by gap openings in the Dirac cones at MTI surfaces, induced by magnetism.
• Berry Curvature Multipoles as Origin: The $V_{xy}^{(2n+1)} \propto (I)^{{2n+1}}$1-th order NLHE arises from the $V_{xy}^{(2n+1)} \propto (I)^{{2n+1}}$2-th moment of Berry curvature, with dominant components calculated for quadrupole (third-order), octapole (fifth-order), and dodecapole (seventh-order).
• Consistency with Observed Peak: Theoretical calculations of Berry curvature multipole integrals show peaks near CNP, matching experimental gate dependence.
• Magnetic Order Correlation: Above $V_{xy}^{(2n+1)} \propto (I)^{{2n+1}}$3, the gap closes, Berry curvature vanishes, and NLHE disappears, in direct alignment with experimental data.
• Surface-Bulk Decoupling: Layer-resolved modeling demonstrates that NLHE is nearly insensitive to sample thickness and is predominately a surface state phenomenon, with bulk states contributing negligibly.
• Symmetry Constraints: Sb-doping averages crystal symmetry, maintaining inversion symmetry and thereby suppressing even-order Berry curvature moments and NLHE harmonics.

Implications and Future Directions

The demonstration of higher odd-order NLHE extends the scope of nonlinear transport phenomena in quantum materials, directly linking observable responses to higher moments of Berry curvature. Several key implications and prospects emerge:

• Nonlinear Transport Engineering: NLHE, especially at high orders, presents opportunities for designing rectifiers and RF devices with enhanced sensitivity and selectivity.
• Quantum Geometry Exploration: This work prompts further exploration of quantum geometric effects in other MTIs and noncentrosymmetric systems, including the impact of different types of multipoles and symmetry breaking.
• Temperature and Doping Control: Understanding tunability via gate voltage and temperature opens the door for active control of nonlinear transport in spintronic and topological devices.
• Fundamental Theory Development: The exponential attenuation of higher-order NLHE, diverging from nonlinear optical counterparts, demands theoretical clarification and could guide future mathematical models for quantum materials.
• Multi-Harmonic Detection: The ability to access higher harmonics without external magnetic fields, alongside layer-insensitive responses, expands functional avenues for device integration and material optimization.

Conclusion

This study establishes the first systematic experimental and theoretical analysis of higher odd-order NLHE in Mn(Bi₁₋ₓSbₓ)₂Te₄ thin flakes, with results firmly anchored in Berry curvature multipole physics. The work advances the understanding of nonlinear Hall transport in MTIs, highlights strong numerical behavior in temperature- and gate-dependent regimes, points to practical device possibilities, and lays a foundation for broader investigation of Berry curvature-driven nonlinear phenomena in quantum matter.