Spin and orbital-to-charge conversion in noncentrosymmetric materials: Hall versus Rashba-Edelstein effects

This paper establishes a general macroscopic formalism to distinguish between Hall and Rashba-Edelstein contributions to spin- and orbital-to-charge conversion in noncentrosymmetric materials, demonstrating through a case study of ferroelectric GeTe that the Rashba-Edelstein effect is the dominant mechanism driving charge current generation.

The Gist

Imagine you have a busy highway where cars (electricity) are driving. In the world of modern electronics, we want to do more than just move cars; we want to use those cars to carry "spin" (a quantum property of electrons, like a tiny spinning top) to process information. This is the field of spintronics.

The big challenge is: How do we turn a flow of cars into a flow of spinning tops, and vice versa?

This paper investigates two different "traffic rules" (mechanisms) that allow this conversion in special materials that lack a center of symmetry (noncentrosymmetric materials). The authors focus on a specific material called $\alpha$-GeTe (Germanium Telluride), which is a ferroelectric semiconductor (meaning it has a built-in electric polarity that can be flipped like a switch).

Here is the breakdown of their discovery using simple analogies:

1. The Two Traffic Rules: The "Hall" vs. The "Rashba"

The paper compares two ways to convert charge (cars) into spin (spinning tops):

• The Spin Hall Effect (SHE): The "Roundabout"

  • The Analogy: Imagine a roundabout. When cars enter, the road geometry naturally pushes some cars to the left and some to the right. This happens because of the material's internal structure. It's a "bulk" effect, meaning it happens everywhere inside the material, regardless of which way the material is facing.
  • The Physics: An electric current creates a sideways flow of spin.

• The Rashba-Edelstein Effect (REE): The "One-Way Street"

  • The Analogy: Imagine a street with a strong, one-way wind blowing down the center. If you try to drive against the wind, your car spins wildly. If you drive with it, it spins the other way. Crucially, if you flip the direction of the wind (by flipping the material's electric polarity), the spinning direction of the cars flips too.
  • The Physics: This effect only happens in materials without a center of symmetry. It creates a direct link between the electric current and the spin direction. If you flip the material's "polarity" (like flipping a switch), the spin flips.

2. The Experiment: Flipping the Switch

The authors wanted to know: Which rule is the boss in $\alpha$-GeTe? Is it the "Roundabout" (Hall) or the "One-Way Street" (Rashba)?

In many materials, it's hard to tell them apart. But $\alpha$-GeTe is special because it is ferroelectric. You can flip its internal electric polarity with a voltage.

• If the "Roundabout" (Hall) is dominant, flipping the switch shouldn't change much.
• If the "One-Way Street" (Rashba) is dominant, flipping the switch should completely reverse the direction of the spin current.

3. The Big Discovery: The "Small" Parameter

The authors built a sophisticated mathematical model (a "traffic simulation") combining real-world measurements with computer calculations of the atoms.

They found something surprising:

• Previous studies thought the "Rashba" effect in this material was huge, based on a simple view of the atoms near the center of the energy map.
• The authors found that when you look at the entire map (all the energy bands), the "Rashba" effect is actually much smaller than previously thought. It's like realizing a giant wave was actually just a ripple when you looked at the whole ocean.

However, here is the twist: Even though the Rashba parameter is "small," the Rashba-Edelstein effect still wins.

4. Why the Winner Wins: The "Traffic Jam" Analogy

Why does the "One-Way Street" (Rashba) dominate over the "Roundabout" (Hall) even if the Rashba parameter is small?

• The Hall Effect (Roundabout): It relies on the material's ability to scatter cars sideways. In $\alpha$-GeTe, the "road" is very smooth for driving straight (high conductivity), so the cars don't get pushed sideways very much. The "Roundabout" is weak.
• The Rashba Effect (One-Way Street): Even though the "wind" (the Rashba parameter) is weaker than expected, the injection of spin is incredibly efficient because of the specific way the electrons interact with the material's polarity. It's like having a very efficient funnel that catches every car and spins it, even if the wind isn't a hurricane.

The Conclusion: In $\alpha$-GeTe, the conversion of electricity to spin is not driven by the bulk "Roundabout" (Hall effect). It is driven almost entirely by the "One-Way Street" (Rashba-Edelstein effect) caused by the material's ferroelectric polarization.

5. The "Orbital" Bonus: A Second Layer of Spin

The paper also looked at Orbital Angular Momentum.

• Analogy: If "Spin" is the car spinning on its own axis, "Orbital" is the car spinning as it goes around a track.
• The authors found that this "Orbital" effect is also present and quite strong, but again, the Rashba mechanism (the one-way street) is the primary driver for converting charge to these orbital currents.

Summary for the General Public

This paper solves a mystery in a specific type of smart material ($\alpha$-GeTe). Scientists were arguing whether the material converts electricity to spin via a "bulk" mechanism (like a roundabout) or a "surface/interface" mechanism (like a one-way street).

By creating a new, more accurate mathematical model, the authors proved that the "One-Way Street" (Rashba-Edelstein effect) is the dominant force, even though previous estimates of its strength were too high. This is great news for engineers because it means we can control this material's spin properties simply by flipping its electric switch (polarity), making it a prime candidate for future ultra-fast, low-power memory and computing devices.

In short: They found that in this specific material, the "switch" (polarity) controls the traffic flow much more effectively than the "road geometry" (bulk structure), and they corrected the math to show exactly how strong that switch really is.

Technical Summary

1. Problem Statement

The optimization of charge-spin interconversion is a central goal in modern spintronics. In noncentrosymmetric materials (those lacking inversion symmetry), two primary mechanisms govern this conversion:

1. Spin Hall Effect (SHE): Generates a transverse spin current from a charge current (bulk effect).
2. Rashba-Edelstein Effect (REE/SREE): Generates a spin density from a charge current (interface or bulk effect in noncentrosymmetric systems).

The Challenge:

• In many experimental setups (e.g., heavy metal/ferromagnet bilayers), it is difficult to distinguish whether the observed signal arises from the bulk SHE or the interfacial SREE.
• In noncentrosymmetric ferroelectrics like $\alpha$-GeTe, both effects coexist in the bulk.
• Previous theoretical estimates of the Rashba parameter ($\alpha_R$) in such materials often rely on simplified models (e.g., near the $\Gamma$ point) that may not account for the full complexity of the band structure, leading to discrepancies with experimental observations.
• There is a need for a unified theoretical framework that treats SHE and SREE on an equal footing and extends to orbital-to-charge conversion (OHE and OREE), which is an emerging field of interest.

2. Methodology

The authors developed a comprehensive theoretical framework combining macroscopic drift-diffusion modeling, linear response theory (Kubo formula), and first-principles calculations.

• Drift-Diffusion Model:

  • They modeled a bilayer system consisting of a ferromagnetic spin source (pumping spin current) and a ferroelectric conductor ($\alpha$-GeTe).
  • They derived analytical expressions for the charge current ($J_c$) generated by both the Inverse SHE and the Inverse SREE as a function of the material thickness ($d_N$) and spin diffusion length ($\lambda$).
  • They defined a ratio $\Xi$ to compare the magnitude of the SHE-induced current versus the SREE-induced current.

• Macroscopic Observables & Kubo Formalism:

  • Instead of relying solely on phenomenological parameters, they derived conversion coefficients directly from microscopic quantities.
  • SHE: Calculated via the spin Berry curvature (interband transitions).
  • SREE: Calculated via the spin-current correlation function (intraband transitions).
  • Key Innovation: They established a rigorous relationship for the inverse SREE coefficient ($\eta_{s \to c}$) using the Onsager reciprocity theorem, linking it to the spin susceptibility ($\Sigma_B$) and the spin-current correlation, rather than just the density of states ($N$). This allows for a calculation independent of the scattering time ($\tau$).

• First-Principles Calculations:

  • Material: Ferroelectric $\alpha$-GeTe (Space group $C_{3v}$).
  • Tools: Density Functional Theory (DFT) using Quantum ESPRESSO, Wannier functions (Wannier90), and transport coefficients (WannierBerri).
  • Scope: Calculated longitudinal conductivity, Spin Hall conductivity ($\sigma_{SH}$), Orbital Hall conductivity ($\sigma_{OH}$), Rashba parameters, and susceptibilities across the energy spectrum near the Fermi level.

3. Key Contributions

1. Unified Formalism: Developed a general theory treating SHE and SREE (and their orbital counterparts OHE and OREE) on equal footing within a drift-diffusion model.
2. Rigorous Definition of Rashba Parameter: Derived an expression for the effective Rashba parameter ($\bar{\alpha}_R$) based on the ratio of spin susceptibility to spin-current correlation. This corrects previous approximations that relied on simplified band models.
3. Orbital Extension: Extended the theory to include Orbital Hall Effect (OHE) and Orbital Rashba-Edelstein Effect (OREE), providing the first comparative analysis of spin vs. orbital contributions in $\alpha$-GeTe.
4. Resolution of Discrepancies: Explained why previous theoretical estimates of the Rashba parameter in $\alpha$-GeTe were significantly higher than effective values derived from experiments.

4. Key Results

• Dominance of SREE over SHE in $\alpha$-GeTe:

  • The calculated ratio $\Xi$ (SHE current / SREE current) indicates that the Rashba-Edelstein effect dominates the charge current generation in $\alpha$-GeTe.
  • This dominance is attributed to the strong spin injection driven by the ferroelectric polarization-induced Rashba texture, which outweighs the bulk spin Hall conductivity.

• Effective Rashba Parameter ($\bar{\alpha}_R$):

  • The study found that the effective Rashba parameter in $\alpha$-GeTe is 2 to 3 orders of magnitude smaller than values reported in previous literature.
  • Reason: Previous studies often focused on the Rashba splitting near the $\Gamma$ point. The authors' full-band calculation reveals that cancellations between different bands (away from $\Gamma$) significantly reduce the net effective parameter when all intraband transitions at the Fermi level are considered.

• Orbital vs. Spin Contributions:

  • Orbital Hall Effect (OHE): Exhibits a much larger magnitude than the Spin Hall Effect (SHE), consistent with metals having dominant $d$-orbital structures. Notably, the OHE does not show a sign reversal near the gap, unlike the SHE.
  • Orbital Rashba Effect (OREE): Similar energy dependence to SREE but generally larger in magnitude.
  • Conversion Efficiency: While orbital effects are large, the study highlights that the ferroelectric polarization primarily drives the spin/charge conversion via the Rashba mechanism.

• Thickness Dependence:

  • The theoretical model shows distinct thickness dependencies for SHE and SREE currents. However, in the regime where $d_N < \lambda$ (thin films), distinguishing them experimentally is challenging due to surface roughness and the coexistence of both effects.

5. Significance and Implications

• Experimental Interpretation: The findings provide a crucial correction for interpreting spin-charge conversion experiments in ferroelectric materials. Researchers must account for the full band structure cancellations to avoid overestimating the Rashba parameter.
• Mechanism Identification: The paper confirms that in ferroelectric $\alpha$-GeTe, the charge transport is governed predominantly by the ferroelectric polarization via the Rashba effect, rather than by bulk spin-orbit coupling-induced Hall responses.
• Orbital Spintronics: By quantifying OHE and OREE, the work opens avenues for "orbitalronics," suggesting that orbital angular momentum could be a more efficient carrier for information than spin in certain noncentrosymmetric materials.
• Future Directions: The authors suggest applying this rigorous framework to other noncentrosymmetric systems, such as Type-II Weyl semimetals (e.g., WTe$_2$, TaIrTe$_4$) and other ferroelectrics (SnTe, HfO$_2$), to better understand the interplay between topology, ferroelectricity, and spin-orbit coupling.

In conclusion, this paper establishes a robust theoretical foundation for distinguishing and quantifying spin and orbital conversion mechanisms in noncentrosymmetric materials, revealing that the Rashba-Edelstein effect is the primary driver in $\alpha$-GeTe and correcting long-standing overestimations of the Rashba parameter.