Current induced magneto-optical Kerr effect as a probe of Dirac carriers in Bi$_{1-x}$Sb$_x$ alloy

This study demonstrates that current-induced magneto-optical Kerr effect (MOKE) in Bi$_{1-x}$Sb$_x$ alloys serves as a powerful probe for identifying Dirac carriers, evidenced by a signal magnitude exceeding that of transition metals and a distinct scaling relationship with resistivity and mobility that aligns with Dirac electron models rather than conventional parabolic band theories.

The Gist

Imagine you have a tiny, invisible river of electricity flowing through a piece of metal. Usually, when this river flows, it's just a stream of charged particles moving in a straight line. But in certain special materials, this river does something magical: it creates a "side current" of invisible magnetic spins. Think of it like a main river of water that, as it flows, secretly generates a side stream of spinning tops.

Scientists want to see these spinning tops, but they are too small to see with regular eyes. To spot them, they use a special trick involving light, called the Kerr Effect. It's like shining a flashlight at the material and watching how the light bounces back. If those invisible spinning tops are there, they twist the polarization of the reflected light, just like a tiny, invisible hand turning a steering wheel.

The Big Discovery
The researchers in this paper decided to test this trick on a special alloy made of Bismuth (Bi) and Antimony (Sb). They treated this alloy like a dial, turning the knob to change the mixture from pure Bismuth to a mix with more Antimony.

Here is what they found:

• Pure Bismuth is a Super-Producer: When the material was pure Bismuth (no Antimony), the "twist" in the light was massive. It was nearly 10,000 times stronger than what they see in common metals like gold or copper.
• Adding Antimony Dampens the Signal: As they added more Antimony to the mix, the signal got weaker and weaker, like turning down the volume on a radio.

The "Why" Behind the Magic
The scientists wanted to know why pure Bismuth was so much better at creating this effect. They looked at how the electricity moved through the material (its resistance and how fast the particles could zip around, called "mobility").

They found a secret code in the numbers:

• In normal metals, the relationship between the signal and the material's properties follows one set of rules (like a standard recipe).
• In this Bismuth alloy, the rules were different. The signal grew much faster as the material became more resistant.

The "Dirac" Analogy
To explain this strange behavior, the researchers used a concept called Dirac electrons.

• Normal Electrons (The Bouncy Ball): In most metals, electrons act like bouncy balls rolling through a field. They bump into things, and their speed is predictable.
• Dirac Electrons (The Light-Speed Skater): In pure Bismuth, the electrons behave differently. They act more like skaters on a frictionless, perfectly smooth ice rink where the rules of physics are slightly different (linear dispersion). They don't just roll; they zip around in a way that makes them incredibly efficient at generating those spinning side-currents.

The paper argues that the massive signal they saw in pure Bismuth is proof that these "Dirac skaters" are the ones doing the work, not the "bouncy ball" electrons found in normal metals.

The Takeaway
This study shows that by simply shining a light on a material and measuring how the light twists, scientists can tell if the material is full of these special "Dirac" electrons. It's a powerful new way to peek inside the electronic world of materials without breaking them open. The paper confirms that this "light-twisting" method works great for detecting these special carriers in semi-metals, distinguishing them clearly from ordinary metals.

Technical Summary

Technical Summary: Current-Induced Magneto-Optical Kerr Effect as a Probe of Dirac Carriers in Bi$_{1-x}$Sb$_x$ Alloy

Problem Statement
While the magneto-optical Kerr effect (MOKE) is established as a method to probe spin magnetic moment accumulation induced by the spin Hall effect (SHE) in semiconductors and metals, the microscopic origins of the signal magnitude remain incompletely understood. Specifically, it is unclear why semiconductors, despite often possessing smaller spin Hall angles than transition metals, exhibit MOKE signals several orders of magnitude larger. Existing theoretical models suggest the MOKE amplitude depends on parameters such as the spin Hall angle, resistivity, spin diffusion length, and the energy derivative of the AC spin Hall conductivity. However, the general validity of these models across different material classes, particularly those with non-parabolic (Dirac-like) band structures, requires experimental verification. Furthermore, distinguishing the contributions of free electrons in parabolic bands from Dirac electrons in semi-metals remains a challenge.

Methodology
The authors investigated the current-induced MOKE in semi-metallic Bi$_{1-x}$Sb$_x$ alloy thin films grown via molecular beam epitaxy (MBE) on quartz and Si substrates. The study utilized films with a thickness of approximately 60 nm, significantly larger than the optical penetration depth, to focus on bulk semi-metallic properties rather than topological surface states.

• Sample Characterization: Transport properties (resistivity $\bar{\rho}_{xx}$, carrier density $n_c$, and mobility $\mu_c$) were extracted using four-point probe and Hall resistance measurements across varying Sb concentrations ($x$). Optical constants (refractive index $n$ and extinction coefficient $k$) were determined via ellipsometry.
• MOKE Measurement: A lock-in detection scheme was employed using a 633 nm HeNe laser. An AC current was applied to Hall bar devices, and the resulting change in the polarization state (rotation angle $\theta_K$ and ellipticity $\eta_K$) of the reflected light was measured. The signal was normalized by current density ($j$) to isolate the MOKE response.
• Theoretical Modeling: The experimental data were compared against a phenomenological model derived from Maxwell's equations and the Kubo formula. The model calculates the MOKE signal based on the DC and AC spin Hall conductivities, spin diffusion length ($l_s$), and the energy derivative of the AC spin Hall conductivity at the Fermi energy. Crucially, the authors modeled the scaling relations for two distinct carrier types: Dirac electrons (characteristic of Bi-rich Bi$_{1-x}$Sb$_x$) and free electrons in parabolic bands.

Key Results

1. Magnitude and Composition Dependence: The current-induced MOKE signal was found to be largest in pure Bi ($x=0$), exceeding that of typical transition metals by nearly four orders of magnitude. The signal magnitude decreased monotonically with increasing Sb concentration ($x$).
2. Scaling Relations: The MOKE signal per unit current density ($|\theta_K + i\eta_K|/j$) exhibited specific scaling behaviors with transport parameters:
   • Resistivity ($\rho_{xx}$): Scales as $\rho_{xx}^{1.7 \pm 0.6}$.
   • Mobility ($\mu_c$): Scales as $\mu_c^{2.0 \pm 0.2}$.
   • Carrier Density ($n_c$): Scales as $n_c^{-0.47}$.
3. Model Validation:
   • Dirac vs. Free Electron Models: The experimental scaling exponents align with a model assuming Dirac electrons, where the mobility scales with carrier density as $\mu_c \propto n_c^{\alpha-1/3}$ with $\alpha \approx -1/3$. In contrast, a free electron model (parabolic band) predicts a scaling of $\rho_{xx}^2$ and $\mu_c^{-2}$, which contradicts the experimental observations for Bi$_{1-x}$Sb$_x$.
   • Spin vs. Orbital Contributions: Calculations including both spin and orbital Hall effects showed that while the orbital contribution is significant, the observed scaling trends are primarily consistent with the Dirac electron model. The model successfully reproduced the order of magnitude of the signal, though it slightly underestimated the signal for intermediate $x$ values, potentially due to unmodeled topological surface state contributions.
4. General Applicability: The scaling of the MOKE amplitude with resistivity was found to partly explain the order-of-magnitude differences in signal strength observed across metals, semi-metals, and semiconductors.

Significance and Claims
The paper demonstrates that current-induced MOKE serves as an effective probe for characterizing the nature of spin current generation in materials with diverse electronic structures. The primary claim is that the distinct scaling laws observed in Bi$_{1-x}$Sb$_x$ (specifically the positive correlation between MOKE signal and mobility/resistivity) provide direct evidence that Dirac electrons are responsible for the generation and accumulation of the spin magnetic moment in these semi-metals. This contrasts sharply with the behavior predicted for free electrons in parabolic bands. The authors conclude that the established theoretical framework for current-induced MOKE, originally developed for metals, can be extended to semi-metals, provided the specific band structure (Dirac-like dispersion) and resulting transport scaling relations are accounted for. The study does not claim to definitively resolve the intrinsic versus extrinsic origin of the spin Hall effect or the specific spin relaxation mechanism (Dyakonov-Perel vs. Elliott-Yafet) within the Bi$_{1-x}$Sb$_x$ system, noting that the current data makes these distinctions difficult.