Evidence of Spin-Valley Coupling in Dirac Material BaMnBi2 Probed by Quantum Hall Effect and Nonlinear Hall Effect

This study presents experimental evidence of a unique spin-valley locked state in the bulk Dirac material BaMnBi2, characterized by a four-fold degeneracy and supported by observations of stacked quantum Hall and nonlinear Hall effects, thereby establishing a new platform for valleytronic applications distinct from its sister compound BaMnSb2.

The Gist

Imagine you are trying to build a super-fast, super-efficient computer. To do this, you need to find a way to store and move information not just using electric charge (like a standard battery), but also using two other "hidden" properties of electrons: their spin (how they twirl) and their valley (which "valley" in the energy landscape they are sitting in).

This field is called Valleytronics. Think of it like a highway system where cars (electrons) don't just drive forward; they also have a specific "lane" (valley) and a specific "spin direction" (left-hand drive or right-hand drive). If you can lock these two together—so that if a car is in the "North Valley," it must be spinning "Left"—you can control traffic with incredible precision.

Until now, this "locking" mechanism was mostly found in ultra-thin, single-layer materials (like a single sheet of graphene). Finding it in a thick, 3D block of material (a bulk crystal) has been like looking for a needle in a haystack.

Enter BaMnBi₂: The New Highway

In this paper, researchers discovered that a chunky crystal called BaMnBi₂ (Barium-Manganese-Bismuth) has exactly this special "spin-valley locking" property. Here is how they proved it, using two clever experiments:

1. The Quantum Hall Effect: The "Stacked Coin" Test

Imagine you have a stack of coins. If you push a magnetic field through them, the coins usually just slide around randomly. But in a special "Quantum Hall" state, the coins lock into perfect, rigid rows.

• The Experiment: The researchers put the BaMnBi₂ crystal in a massive magnetic field (35 times stronger than a fridge magnet) and cooled it down to near absolute zero.
• The Discovery: They saw the electrical resistance jump up and down in perfect, flat steps (like a staircase). This is the "Quantum Hall Effect."
• The Clue: By counting the size of these steps, they realized there were 4 distinct types of electron lanes moving at once.
  • Analogy: Imagine a highway with 4 lanes. In the older, similar material (BaMnSb₂), there were only 2 lanes. The fact that BaMnBi₂ has 4 lanes means it has a more complex, "locked" system where the spin and valley are tied together in a unique way. It's like upgrading from a 2-lane road to a 4-lane superhighway where every car knows exactly which lane to be in based on its spin.

2. The Nonlinear Hall Effect: The "One-Way Street" Test

Usually, if you push a car forward, it goes forward. If you push it backward, it goes backward. This is "linear." But in this special material, the rules change.

• The Experiment: They sent an alternating current (like a wave pushing back and forth) through the crystal.
• The Discovery: Even though the current was wiggling back and forth, the material generated a steady, one-way voltage on the side. This is called the Nonlinear Hall Effect.
• The Clue: This happens because the "energy landscape" of the electrons is lopsided (like a bowl that is tilted). This tilt creates a "Berry Curvature Dipole"—a fancy physics term for a magnetic-like push that only happens in one direction.
• The Metaphor: Imagine a ball rolling in a bowl. In a normal bowl, if you shake it, the ball rolls back and forth evenly. In this BaMnBi₂ bowl, the shape is so weird that when you shake it, the ball always drifts slightly to the right, no matter which way you shake it. This proves the electrons have a built-in "handedness" (spin-valley locking) that creates this one-way traffic.

Why Does This Matter?

1. It's a Bulk Material: Most of these cool quantum effects only happen in single layers of atoms (2D). BaMnBi₂ is a thick, 3D crystal. This means we can actually hold it, cut it, and wire it up like a real component for a device.
2. It's Different: It behaves differently than its "sister" material, BaMnSb₂. This proves that by tweaking the chemistry (swapping Antimony for Bismuth), we can engineer new types of quantum highways.
3. Future Tech: This opens the door for Valleytronic devices. Imagine computer chips that use these "valleys" to store data. They could be faster, use less energy, and be immune to certain types of errors because the "spin-valley lock" protects the information.

In Summary:
The researchers found a new 3D crystal that acts like a perfectly organized, multi-lane highway where every electron knows its lane and its spin. They proved this by watching how the electrons behave in strong magnetic fields (the staircase steps) and how they generate one-way currents when shaken (the nonlinear effect). This discovery is a major step toward building the next generation of super-efficient, quantum-powered electronics.

Technical Summary

1. Problem Statement

Valleytronics, the field of using the valley degree of freedom to encode information, relies heavily on spin-valley locking (where spin and valley degrees of freedom are coupled). While this phenomenon is well-established in 2D monolayer transition metal dichalcogenides (e.g., MoS2) due to broken inversion symmetry, identifying bulk materials that host spin-valley locked electronic states remains a significant challenge.

• Context: The sister compound BaMnSb2 was previously identified as a bulk material with spin-valley locking, exhibiting a spin-valley degeneracy of 2 near the X-point.
• Gap: Theoretical predictions suggested that BaMnBi2, a related layered compound with stronger spin-orbit coupling (SOC) from Bismuth, possesses a distinct and more complex spin-valley locked state with potentially higher degeneracy (predicted to be 10, but experimentally unverified). Direct experimental evidence of the topological quantum transport signatures and the specific spin-valley degeneracy in BaMnBi2 was lacking.

2. Methodology

The authors employed a combination of high-quality crystal growth and advanced quantum transport measurements under extreme conditions:

• Sample Preparation: Single crystals of BaMnBi2 were grown using the Bi-flux method. Due to the material's reduced air stability compared to BaMnSb2, samples were handled in an argon-filled glovebox and encapsulated with GE varnish for high-field measurements.
• High-Field Transport: Measurements were conducted at the National High Magnetic Field Laboratory (NHMFL) up to 35 Tesla.
  • Quantum Hall Effect (QHE): Longitudinal ($\rho_{xx}$), transverse ($\rho_{xy}$), and out-of-plane ($\rho_{zz}$) resistivities were measured at temperatures ranging from 1.4 K to 75 K. Angular dependence was tested by tilting the magnetic field relative to the crystal's $ab$-plane.
  • Nonlinear Hall Effect (NLHE): Second-harmonic Hall voltage ($V_{2\omega}^{\perp}$) measurements were performed using a lock-in technique with AC currents to detect intrinsic Berry curvature dipole (BCD) effects.
• Analysis: Data was analyzed using Shubnikov–de Haas (SdH) oscillation theory (Lifshitz-Kosevich formula) to extract effective mass and carrier density, and QHE quantization rules to determine spin-valley degeneracy.

3. Key Contributions

• Discovery of Stacked QHE: The first experimental demonstration of a stacked Quantum Hall Effect in bulk BaMnBi2, confirming its quasi-2D Dirac nature.
• Determination of Spin-Valley Degeneracy: Quantitative evidence establishing a spin-valley degeneracy of 4 in BaMnBi2, contrasting with the degeneracy of 2 found in BaMnSb2.
• Observation of Intrinsic NLHE: Detection of a bulk nonlinear Hall effect, providing direct evidence for a valley-contrasted Berry curvature dipole, a hallmark of spin-valley locked states.
• Structural-Electronic Correlation: Linking the unique electronic properties to the specific orthorhombic distortion and Bi zig-zag chain structure, distinguishing it from the Sb-based analog.

4. Key Results

A. Quantum Hall Effect (QHE) Analysis

• Quantization: The Hall resistivity ($\rho_{xy}$) exhibited distinct plateaus at specific magnetic fields (e.g., ~10 T and ~23 T) accompanied by minima in $\rho_{xx}$ and maxima in $\rho_{zz}$.
• Angular Independence: The quantized plateaus remained constant for magnetic field angles ($\theta$) up to 40°, depending only on the perpendicular field component ($B_{\perp}$). This confirms the 2D nature of the transport within the Bi layers.
• Degeneracy Calculation: By analyzing the step size of the inverse Hall resistivity ($1/\rho_{xy}$) and comparing the SdH oscillation frequency ($B_F$) with the Hall carrier density, the authors calculated the spin-valley degeneracy ($s$).
  • Result: $s \approx 3.7$, approximated to 4.
  • Interpretation: This indicates the presence of two pairs of spin-valley locked Dirac cones near the Fermi level (likely near the X and Y points), differing from the single pair (degeneracy 2) in BaMnSb2.
• Effective Mass: The effective mass of Dirac fermions was extracted as $m^* = 0.027 m_0$, indicating highly mobile, nearly massless carriers.
• Valley Splitting: An additional feature in $\rho_{xx}$ near 33 T was attributed to valley splitting, likely caused by a second-order $q$-quadratic term rather than Zeeman splitting, given the lack of angular dependence in $\sigma_{zz}$.

B. Nonlinear Hall Effect (NLHE) Analysis

• Second-Harmonic Response: A clear second-harmonic Hall voltage ($V_{2\omega}^{\perp}$) was observed, scaling quadratically with the input AC current ($I_{\omega}$) at low currents.
• Intrinsic Origin: The signal showed:
  • Frequency independence (up to 1800 Hz).
  • A nonlinear angle ($\theta_{NLHE}$) of ~74° (close to the ideal 90° for pure transverse response).
  • Persistence up to room temperature (300 K), though weaker.
• Mechanism: These features confirm an intrinsic origin driven by a Berry Curvature Dipole (BCD), arising from the valley-contrasted Berry curvature inherent to the spin-valley locked state.
• Domain Suppression: The magnitude of the NLHE signal was noted to be suppressed by the presence of 90° and 180° domains in the bulk crystal, suggesting that single-domain micro-devices could yield significantly stronger signals.

5. Significance and Implications

• New Platform for Spin-Valley Physics: BaMnBi2 is established as a distinct bulk platform for studying coupled spin-valley physics, offering a higher degeneracy (4) than its Sb counterpart (2). This expands the material library beyond TMDCs and BaMnSb2.
• Topological Protection: The observation of stacked QHE and NLHE confirms the topological nature of the electronic states, which are protected against backscattering, a crucial feature for robust electronic devices.
• Valleytronic Applications: The presence of a strong Berry curvature dipole and spin-valley locking suggests BaMnBi2 is a promising candidate for valleytronic logic gates and memory devices.
• Future Potential: The detection of NLHE at room temperature, even in multi-domain bulk crystals, highlights the potential for developing terahertz detectors and energy harvesting devices based on this material, particularly if single-domain thin films or micro-devices are engineered to maximize the signal.

In conclusion, this work provides the first direct experimental proof of a unique, high-degeneracy spin-valley locked state in bulk BaMnBi2, bridging the gap between theoretical predictions and experimental realization in polar Dirac materials.