Tunable interplay of orbital and spin magnetization in trigonal tellurium

This study provides systematic experimental evidence and a theoretical framework demonstrating the tunable coexistence of current-induced spin polarization and orbital magnetization in trigonal tellurium, thereby establishing a pathway for advancing orbitronics and spintronics in chiral crystals.

The Gist

Imagine you are trying to understand how a car moves. For a long time, scientists only looked at the engine (the spin of electrons) to explain how the car works. They thought the engine was the only thing that mattered. But recently, they realized there's a hidden part of the car—the chassis and suspension (the orbital motion of electrons)—that also plays a huge role in how the car handles, turns, and reacts to the road.

This paper is about discovering that hidden part in a very special material called Tellurium (a shiny, silvery element).

Here is the story of what they found, explained simply:

1. The Special Material: A Twisted Spiral

Think of the atoms in Tellurium not as a neat grid of bricks, but as spiral staircases or helixes (like a DNA strand or a corkscrew). Because the whole crystal is twisted, it has a property called "chirality" (handedness). It's either a right-handed screw or a left-handed screw.

In these twisted structures, when electricity flows, it doesn't just push electrons forward; it makes them spin and swirl in specific ways.

2. The Two "Dancers": Spin vs. Orbit

The researchers were looking at two different ways electrons can move, which they call "Spin" and "Orbit."

• Spin: Imagine a figure skater spinning in place. This is the electron's intrinsic "spin."
• Orbit: Imagine the skater running in a circle around the rink. This is the electron's "orbital" motion.

For decades, scientists focused almost entirely on the "spin" (the skater spinning). They thought the "orbit" (the skater running) was too weak to matter. But this paper says: "Wait a minute! The orbit is actually doing a lot of the heavy lifting here!"

3. The Experiment: The "L" Shaped Track

To test this, the scientists built tiny devices shaped like the letter "L".

• They sent an electric current down the arms of the "L".
• They spun a giant magnet around the device, changing the angle of the magnetic field like a spotlight moving around a stage.
• They measured how the electricity resisted the flow (resistance) at every angle.

4. The Surprise: The "Wobble"

If only the "Spin" (the skater) were doing the work, the electricity would react in a very predictable, symmetrical way. It would be like a perfectly balanced wheel.

But, they saw a wobble.

• When they used a weak magnetic field, the "sweet spot" where the electricity flowed best was tilted away from where it should be.
• It was as if the skater wasn't just spinning; they were also leaning to the side, pulling the whole system off-center.

This "tilt" or "wobble" is the fingerprint of Orbital Magnetization. It proved that the electrons' orbital motion (running in circles) was creating its own magnetic force that was fighting against the spin.

5. The Remote Control: Tuning the Balance

The coolest part of the discovery is that they found a "remote control" to switch between these two forces.

• They used a gate voltage (like turning a dial on a radio) to change the number of electrons in the material.
• Turn the dial one way: The "Spin" force gets stronger, and the "Orbit" force fades away. The wobble disappears, and the material behaves like the old textbooks predicted.
• Turn the dial the other way: The "Spin" gets weaker, and the "Orbit" force takes over. The wobble becomes huge!

This means they can tune the material to be either "Spin-heavy" or "Orbit-heavy" just by applying electricity.

Why Does This Matter?

Think of the future of computers and electronics.

• Spintronics (using electron spin) is already a big field, used in hard drives and new types of memory.
• This paper opens the door to Orbitronics (using electron orbit).

By proving that we can control the "orbit" just as easily as the "spin," the scientists are saying: "We have a new tool in our toolbox." This could lead to:

• Faster, more efficient computers.
• New types of sensors that are incredibly sensitive.
• Devices that use less energy because they can switch between these two states easily.

The Bottom Line

For a long time, we thought the "engine" (spin) was the only thing driving the car. This paper shows us that the "chassis" (orbit) is just as important, and in Tellurium, we can actually steer the car by switching between the two. It's a fundamental shift in how we understand the quantum world, turning a hidden, overlooked force into a powerful new technology.

Technical Summary

1. Problem Statement

While the spin degree of freedom has been the cornerstone of spintronics, the orbital degree of freedom has historically been underexplored despite its fundamental significance and potential for novel applications. A major challenge in studying orbital effects is the inherent entanglement between orbital and spin degrees of freedom in materials with strong spin-orbit coupling (SOC). In many chiral materials, observations previously attributed solely to spin polarization (such as the Rashba-Edelstein effect) may actually contain substantial orbital contributions.

Specifically, in chiral crystals like trigonal Tellurium (Te), which possesses a unique helical structure, the intrinsic three-fold rotational symmetry ($C_3$) theoretically forbids net transverse magnetization. However, experimental anomalies (such as angular shifts in magnetoresistance) suggest the presence of symmetry-breaking mechanisms and orbital magnetization components perpendicular to the helical axis. There is a lack of systematic experimental evidence to disentangle current-induced spin polarization from orbital magnetization, particularly regarding their tunability and interplay.

2. Methodology

The authors employed a comprehensive approach combining advanced experimental techniques with symmetry-guided theoretical modeling:

• Material System: Single-crystal trigonal Tellurium (Te) flakes, synthesized via hydrothermal growth, possessing a right-handed helical structure (space group $P3_121$).
• Device Fabrication: Pre-patterned 'L'-bar devices were fabricated on Si/SiO$_2$ wafers. This geometry features two arms aligned along the crystallographic $c$-axis (helical axis) and $a$-axis, allowing for the examination of all relative orientations of current and magnetic fields within a single device.
• Measurement Techniques:
  • Angle-Resolved Magnetotransport: Measurements were performed at 2 K with the sample rotated in situ over 360° in two orthogonal directions.
  • Harmonic Detection: Both fundamental ($\omega$) and second-harmonic ($2\omega$) voltages were measured using AC bias currents to extract linear magnetoresistance ($R^{(1)}$) and nonlinear magnetoresistance ($\delta^{(2)}$).
  • Electrostatic Gating: A back-gate configuration (degenerately doped Si) was used to tune the Fermi level across the valence band, modulating the effective SOC strength and carrier density.
  • Weak Localization/Anti-localization (WL/WAL): Perpendicular field measurements were used to characterize the SOC strength and confirm the transition between weak and strong SOC regimes.
• Theoretical Framework: A symmetry-guided Boltzmann transport analysis was developed. The model incorporates Zeeman coupling to both spin texture ($\mathbf{s}$) and orbital magnetization ($\mathbf{m}$), explicitly testing scenarios where rotational symmetry is reduced to allow transverse components.

3. Key Contributions

• Disentanglement of Spin and Orbital Effects: The study provides a general framework to distinguish between current-induced spin polarization and orbital magnetization in chiral crystals by analyzing angular shifts in magnetotransport.
• Identification of Transverse Orbital Magnetization: The authors demonstrate that orbital magnetization in Te is not limited to the helical axis but possesses a significant transverse component ($M_{orb, x}$) perpendicular to the chain, enabled by symmetry reduction (extrinsic strain or intrinsic interactions).
• Gate-Tunable Interplay: The work establishes that the competition between spin and orbital contributions can be precisely controlled via electrostatic gating, offering a pathway to tune material properties for orbitronic applications.
• Resolution of Angular Shift Anomalies: The paper provides a symmetry-allowed explanation for the frequently observed angular shifts in nonlinear conductivity, previously often dismissed as experimental misalignment.

4. Key Results

• Angular Shifts in Nonlinear MR:
  • In Device D1 (higher native doping, stronger effective SOC), the nonlinear magnetoresistance ($\delta^{(2)}$) along the helical axis ($c$-axis) showed an angular shift at low magnetic fields that diminished as the field increased. This indicates a mixing of spin and orbital effects, where strong SOC eventually dominates, locking the magnetization to the current direction.
  • In Device D2 (lower doping, weaker SOC), the angular shift was more pronounced and increased linearly with the magnetic field. This behavior cannot be explained by collinear spin polarization alone and strongly suggests a dominant orbital magnetization component perpendicular to the helical axis.
• Symmetry Breaking: The observed transverse magnetization implies a reduction of the intrinsic $C_3$ rotational symmetry. The authors argue this arises from extrinsic factors (interface strain, gate fields) or intrinsic mechanisms (electron-electron interactions), allowing the orbital magnetization to acquire a transverse component ($M_{orb, x}$) while spin remains locked to momentum.
• Gate Tunability:
  • As the gate voltage became more negative (moving the Fermi level deeper into the valence band), the system transitioned from an orbital-dominant regime (large angular shifts) to a spin-dominant regime (vanishing angular shifts, collinear behavior).
  • This transition correlates with the WL-to-WAL crossover, confirming that deeper Fermi levels induce stronger SOC, which suppresses the pure orbital contribution and enhances spin polarization.
• Theoretical Validation: Boltzmann transport calculations confirmed that a model including an orbital magnetization component odd in $k_z$ (perpendicular to the helix) qualitatively reproduces the experimental angular shifts, whereas a pure spin-texture model (collinear) fails to do so.

5. Significance

• Fundamental Physics: This work challenges the conventional view that chiral materials only exhibit collinear spin polarization. It establishes that orbital magnetization is a robust, tunable, and often dominant effect in chiral semiconductors, particularly when SOC is weak.
• Orbitronics and Spintronics: By demonstrating the ability to tune the interplay between spin and orbital degrees of freedom via electrostatic gating, the study opens new avenues for orbitronics. It suggests that chiral Te can serve as a versatile platform for developing devices that utilize orbital currents for information processing and storage.
• Methodological Advance: The combination of 'L'-bar geometry, harmonic detection, and symmetry analysis provides a robust protocol for identifying and characterizing orbital effects in other quantum materials where spin and orbital degrees of freedom are entangled.

In conclusion, the paper successfully identifies and characterizes a tunable interplay between spin and orbital magnetization in trigonal Tellurium, revealing that orbital effects can generate significant transverse magnetization and compete with spin polarization, a finding that reshapes the understanding of transport phenomena in chiral crystals.