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

This study reports the experimental observation of higher odd-order (third-, fifth-, and seventh-order) nonlinear Hall effects in magnetic topological insulator Mn(Bi1-xSbx)2Te4 thin flakes, attributing the phenomenon to Berry curvature multipoles and demonstrating its dependence on the Néel temperature and charge neutral point.

The Gist

Imagine electricity flowing through a wire like water flowing through a river. Usually, if you push the water straight ahead, it goes straight ahead. But in certain special materials, if you push hard enough, the water starts to swirl sideways, creating a "Hall Effect."

For decades, scientists have studied how this sideways flow happens. Recently, they discovered that if you push the water with a rhythmic pulse (like a heartbeat), the sideways flow doesn't just happen once; it can happen in complex, repeating patterns. This is called the Nonlinear Hall Effect.

Until now, scientists mostly looked at the first two or three "beats" of this rhythm (the 2nd and 3rd order). But in this new study, researchers at Nanjing University and Nanjing Normal University decided to listen for the fifth, seventh, and even higher beats of the rhythm. They found them!

Here is a simple breakdown of what they did and what it means, using some everyday analogies:

1. The Special Playground: A Magnetic Ice Cube

The researchers used a very special material called Mn(Bi1-xSbx)2Te4. Think of this material as a stack of thin, magnetic pancakes.

• The Magic: These pancakes are "Topological Insulators." Imagine a chocolate bar where the inside is a solid block of chocolate (insulator), but the very thin outer shell is made of liquid gold (conductor). Electrons can only flow on the surface, not through the middle.
• The Magnetism: These pancakes are also magnetic, but they are arranged in a tricky way: the top layer wants to point North, the next one South, the next North, and so on. This creates a unique magnetic dance.

2. The Experiment: Pushing the Rhythm

The team built a tiny device shaped like a disc with 12 electrical contacts around the edge (like a clock face). They sent an alternating current (a back-and-forth electrical pulse) through it.

• The Discovery: When they pushed the current, they didn't just see a simple sideways voltage. They detected a "symphony" of higher-order voltages.
  • They found the 3rd order (a triple-beat rhythm).
  • They found the 5th order (a five-beat rhythm).
  • They even found the 7th order (a seven-beat rhythm)!

The Catch: These higher rhythms are like whispers. The 3rd order is loud, the 5th is quieter, and the 7th is a faint whisper. The researchers had to turn down the temperature to near absolute zero (colder than outer space) to hear them clearly.

3. The "Sweet Spot" and the "Magic Switch"

The researchers noticed two fascinating rules about these whispers:

• The Sweet Spot: The signal is strongest when the material is "neutral"—neither full of extra electrons nor missing them. It's like tuning a radio to the exact frequency where the static disappears and the music is clearest.
• The Temperature Switch: These signals only exist when the material is cold enough for its magnetic order to settle down (below the "Néel temperature"). If you heat it up, the magnetic dance gets chaotic, and the signals vanish.

4. Why Does This Happen? (The "Berry Curvature" Analogy)

This is the most mind-bending part. Why does the electricity swirl in these complex patterns?

Imagine the electrons are cars driving on a highway. In normal materials, the road is flat. But in this material, the road is shaped like a bumpy, twisted landscape (called "Berry Curvature").

• The Dipole (2nd Order): Imagine the road has a gentle slope. Cars roll down one side.
• The Multipoles (Higher Orders): In this new study, the road isn't just sloped; it has complex bumps, dips, and spirals (called "Multipoles").
  • When the electric current pushes the cars, they don't just roll down; they get caught in these spirals and swirl in complex, higher-order patterns.
  • The researchers proved that these complex swirls are caused by the unique "shape" of the quantum landscape on the surface of the material.

5. Why Should We Care?

You might ask, "Who cares about hearing the 7th beat of an electrical rhythm?"

• New Physics: It's like discovering a new instrument in an orchestra. We knew the violin and cello existed (2nd and 3rd order), but finding the 7th order opens up a whole new section of the quantum orchestra. It helps us understand the deep, hidden geometry of the universe at the atomic scale.
• Future Tech: Just as understanding simple electricity led to computers, understanding these complex "nonlinear" rhythms could lead to super-fast, ultra-sensitive electronic devices. Imagine a radio that can detect signals we can't hear today, or computers that process information using these complex quantum swirls.

Summary

In short, the scientists took a special magnetic material, cooled it down, and sent a rhythmic electrical pulse through it. They discovered that the material responds not just with simple twists, but with a complex, high-pitched symphony of electrical swirls (3rd, 5th, 7th order). These swirls are caused by the unique, bumpy shape of the quantum world inside the material. This discovery proves that the quantum world is even more musical and complex than we thought, paving the way for future technologies that can harness these hidden rhythms.

Technical Summary

1. Problem Statement

The nonlinear Hall effect (NLHE) is a significant phenomenon in condensed matter physics, linking transverse voltage generation to the quantum geometry of materials. While the second-order (quadratic) and third-order (cubic) NLHE have been extensively studied and theoretically understood (often attributed to Berry curvature dipoles and quadrupoles), experimental exploration of higher-order odd nonlinear Hall effects (fifth-order, seventh-order, and beyond) remains scarce. Furthermore, the physical mechanisms governing these higher-order responses, particularly in magnetic topological insulators (MTIs) without external magnetic fields, are not well established. This work aims to fill this gap by systematically investigating higher odd-order NLHE in a specific MTI system.

2. Methodology

• Material System: The researchers utilized thin flakes of Mn(Bi$_{1-x}$Sb$_x$)$_2$Te$_4$ (specifically with Sb doping $x \approx 0.3$). This material is a magnetic topological insulator with A-type antiferromagnetic ordering. The Sb doping was crucial for tuning the Fermi level into the band gap (charge neutral point) while preserving topological surface states.
• Device Fabrication: Disc-shaped devices with 12 electrodes were fabricated using electron beam lithography. A graphite/hexagonal boron nitride (h-BN) stack was dry-transferred to serve as a top gate ($V_{tg}$) for carrier density modulation.
• Transport Measurements:
  • An alternating current (AC) driving current ($I_\omega$, frequency $\omega = 23$ Hz) was applied at various angles ($\theta$).
  • Transverse Hall voltages ($V_{xy}$) were measured simultaneously with longitudinal voltages.
  • Lock-in detection techniques were employed to isolate higher harmonic components ($V_{xy}^{(2n+1)\omega}$) corresponding to the 3rd, 5th, 7th, 9th, and 11th orders.
  • Measurements were conducted across a range of temperatures (down to 1.8 K), magnetic fields, and gate voltages.
• Theoretical Analysis: The authors performed theoretical modeling based on Dirac cone surface states and Berry curvature multipoles (BCMs). They calculated the contributions of Berry curvature moments (quadrupole, octapole, dodecapole) to the nonlinear transport response, considering the symmetry breaking induced by surface magnetic moments.

3. Key Contributions

• First Observation of High-Order Odd NLHE: The paper reports the experimental observation of third-, fifth-, seventh-, ninth-, and eleventh-order nonlinear Hall effects in a magnetic topological insulator.
• Identification of Physical Origin: The study attributes these higher-order responses to Berry curvature multipoles (BCMs) (specifically the quadrupole, octapole, and dodecapole moments) arising from the topological surface states, rather than extrinsic scattering or simple dipole effects.
• Thickness Independence: A significant finding is that the higher odd-order NLHE exhibits comparable magnitudes in both odd-layer and even-layer samples, challenging the conventional expectation that surface states in even-layer MTIs might cancel out due to antiferromagnetic coupling. This is explained by a mismatch in out-of-plane magnetic moments between top and bottom layers.
• Exponential Decay Law: The authors establish an empirical exponential decay law for the magnitude of the nonlinear Hall voltage as the order of the nonlinearity increases ($V \propto e^{-\alpha N}$), a behavior distinct from nonlinear optics.

4. Key Results

• Current Dependence: The higher odd-order Hall voltages ($V_{xy}^{(2n+1)\omega}$) scale strictly with the driving current as $(I_\omega)^{2n+1}$, confirming their nonlinear nature.
• Angular Symmetry: All observed higher-order signals exhibit a twofold angular dependence (180° periodicity), which is attributed to the emergent $C_{2z}$ rotational symmetry induced by Sb doping and the specific magnetic configuration.
• Temperature Dependence:
  • The effects exist only below the Néel temperature ($T_N \approx 24$ K).
  • The onset temperature decreases with increasing order: $T_{onset} \approx 20$ K for 3rd order, $\approx 10$ K for 5th order, and $\approx 7$ K for 7th order.
  • The signals vanish above $T_N$, confirming the necessity of magnetic ordering (and the resulting Berry curvature) for the effect.
• Gate Tunability: The signals are strongly modulated by the top gate voltage, reaching a maximum near the Charge Neutral Point (CNP). This aligns with the theoretical prediction that the integrated Berry curvature multipoles peak when the Fermi level is within the surface state gap.
• Magnitude: Under optimal conditions (5 K, 5 $\mu$A current), the maximum amplitudes were:
  • 3rd Order: ~25–28.5 $\mu$V
  • 5th Order: ~3.5–7.5 $\mu$V
  • 7th Order: ~0.7–2.7 $\mu$V
  • Higher orders (9th, 11th) were also detected at lower temperatures.

5. Significance

• Fundamental Physics: This work expands the understanding of nonlinear transport phenomena beyond the second and third orders, providing experimental validation for the role of higher-order Berry curvature multipoles in quantum materials.
• Material Platform: It establishes Mn(Bi$_{1-x}$Sb$_x$)$_2$Te$_4$ as a robust platform for studying high-order nonlinearities without the need for external magnetic fields, distinguishing it from previous studies on WTe$_2$ or other systems.
• Device Applications: The observation of strong nonlinear responses at relatively low orders and the ability to tune them via gating and temperature opens new avenues for frequency multiplication, radio-frequency rectification, and terahertz detection in magnetic topological devices.
• Theoretical-Experimental Bridge: The study successfully bridges the gap between theoretical predictions of BCMs and experimental observation, offering a clear framework for interpreting complex nonlinear transport data in topological magnets.