Probing the Chirality of Trigonal Selenium and Tellurium by Spin and Orbital Hall Effects

Using first-principles calculations, this study demonstrates that the spin and orbital Hall conductivities of left- and right-handed trigonal selenium and tellurium exhibit opposite signs due to mirror-symmetry-induced antisymmetry in their Berry curvature, thereby establishing a direct link between measurable transport signals and structural chirality.

The Gist

Imagine you have a pair of gloves: a left-handed one and a right-handed one. They look almost identical, but if you try to put a left glove on your right hand, it just doesn't fit. In the world of crystals, some materials are like these gloves. They come in two "handed" versions (called enantiomers) that are mirror images of each other but cannot be stacked on top of one another perfectly.

This paper is about two specific materials, Selenium (Se) and Tellurium (Te), which naturally form these spiral, "handed" crystal structures. The researchers wanted to see if these two mirror-image versions behave differently when electricity flows through them, specifically looking at how they handle spin (a tiny magnetic property of electrons) and orbit (how electrons move around atoms).

Here is the breakdown of their findings using simple analogies:

1. The Setup: Two Mirror-Image Mazes

Think of the crystal structure of Selenium and Tellurium as a long, twisting helix (like a spiral staircase or a DNA strand).

• One version twists clockwise (Right-handed).
• The other twists counter-clockwise (Left-handed).

Even though the "stairs" look the same, the direction of the twist is different. The researchers used powerful computer simulations (first-principles calculations) to see what happens when they push an electric current through these two different versions.

2. The Discovery: The "Traffic Detour"

When electricity flows through a normal wire, electrons just go straight. But in these chiral crystals, something interesting happens due to the spiral shape and the heavy atoms involved:

• The Spin Hall Effect (SHE): When you push electrons forward, the crystal acts like a traffic cop, forcing some electrons to veer off to the side. Crucially, it forces them to spin in a specific direction as they turn.
• The Orbital Hall Effect (OHE): Similarly, the electrons' "orbit" (their path around the atom) gets pushed to the side.

The paper found that for these specific materials, the direction of the turn depends entirely on which "glove" you are wearing.

• If you use the Left-handed crystal, the electrons get pushed to the side and spin one way (let's say, "Up").
• If you use the Right-handed crystal, the electrons get pushed to the same side, but they spin the opposite way ("Down").

It's like driving a car on a circular track. If the track is built on a left-handed spiral, the car drifts left. If you build an identical track on a right-handed spiral, the car drifts right, even if you drive it the same way.

3. The "Why": The Mirror Rule

Why does this happen? The researchers explained it using the rules of symmetry (math that describes how shapes behave when flipped).

They found that the two crystals are related by a mirror operation. Imagine holding a mirror up to the Left-handed crystal; its reflection looks exactly like the Right-handed crystal.

• The researchers discovered that for a specific type of measurement (called the $\sigma_{yx}$ component), the "spin" and "orbit" properties act like a reversible switch when you look in the mirror.
• The mirror flips the sign of the result. Positive becomes negative. "Up" becomes "Down."
• However, other parts of the measurement don't change; they stay the same in both crystals. Only this specific "side-turn" signal flips.

4. The Takeaway: A Fingerprint for Handedness

The main point of the paper is that Spin Hall Conductivity and Orbital Hall Conductivity can act as a fingerprint for the crystal's handedness.

• In the past, scientists knew these materials had different optical properties (how they bend light).
• This paper shows they also have different transport properties (how they move electricity and spin).

Because the signal flips sign depending on whether the crystal is Left or Right-handed, measuring this electrical signal could theoretically tell you which "glove" you are holding without needing to look at the crystal structure under a microscope.

Summary

The paper demonstrates that in the spiral crystals of Selenium and Tellurium, the direction of a specific electrical "side-current" (spin and orbital) is strictly tied to the crystal's handedness. If you flip the crystal's twist from left to right, the direction of this current flips too. This proves that the "handedness" of the material is a fundamental switch that controls how electrons spin and orbit as they travel through it.

Technical Summary

Technical Summary: Probing the Chirality of Trigonal Selenium and Tellurium by Spin and Orbital Hall Effects

Problem Statement
Chiral crystals exhibit enantiomer-dependent transport phenomena capable of generating pure spin or orbital currents. While the interaction between structural chirality and electronic properties (such as circular dichroism and nonlinear electrical responses) is well-documented, the handedness sensitivity of spin Hall conductivity (SHC) and orbital Hall conductivity (OHC) remains insufficiently understood. Specifically, it is unclear how the structural handedness of non-magnetic chiral semiconductors correlates with specific tensor components of SHC and OHC, and whether these transport signatures can serve as a reliable probe for structural enantiomers.

Methodology
The authors employed first-principles calculations based on density functional theory (DFT) using the Vienna ab initio simulation package (VASP).

• Electronic Structure: Calculations utilized the generalized gradient approximation (PBE) and the hybrid functional HSE06, with spin-orbit coupling (SOC) included in all cases. Lattice constants were refined using experimental values to ensure physical fidelity.
• Transport Properties: Intrinsic SHC and OHC were calculated using the Kubo formula implemented in the WANNIERTOOLS package. The Bloch states from HSE06+SOC calculations were projected onto maximally localized Wannier functions (via WANNIER90) to facilitate dense Brillouin zone integration ($100 \times 100 \times 100$ k-point mesh).
• Orbital Calculation: The OHC was computed by modifying the WannierTools code to replace the spin operator ($S$) with the orbital angular momentum operator ($L$) in the Hall response routine.
• Symmetry Analysis: The study focused on the prototypical chiral semiconductors trigonal Selenium (Se) and Tellurium (Te). The two enantiomers were modeled using space groups $P3_221$ (left-handed) and $P3_121$ (right-handed). The relationship between these structures was analyzed via the $M_{xy}$ mirror operation, which relates the two enantiomers but is not a symmetry operation of the individual chiral crystals.

Key Contributions and Results

1. Handedness-Dependent Sign Reversal: The study demonstrates that for trigonal Se and Te, specific SHC and OHC tensor elements exhibit opposite signs between left- and right-handed enantiomers. Specifically, the tensor components $\sigma^{S_y}_{yx}$ (spin) and $\sigma^{L_y}_{yx}$ (orbital) reverse sign upon switching structural handedness.

   • For Se, $\sigma^{S_y}_{yx}$ shows a value of $+98.6$ $(\hbar/e)(S/cm)$ for the left-handed structure and $-98.6$ $(\hbar/e)(S/cm)$ for the right-handed structure at the Fermi energy.
   • For Te, the magnitude is larger ($\sim 157.7$ $(\hbar/e)(S/cm)$) due to stronger SOC, but the sign reversal pattern remains identical.
   • Similar sign reversals are observed for the orbital component $\sigma^{L_y}_{yx}$, with values of $\pm 224.1$ $(\hbar/e)(S/cm)$ for Se and $\pm 382.9$ $(\hbar/e)(S/cm)$ for Te.

2. Symmetry Origin: The sign reversal is attributed to the antisymmetric transformation of the spin and orbital Berry curvatures under the $M_{xy}$ mirror operation.

   • While the band structures of the two enantiomers are identical, the spin/orbital Berry curvature ($\Omega^{S_y}_{yx}$ and $\Omega^{L_y}_{yx}$) changes sign under the mirror mapping.
   • Symmetry analysis reveals that under $M_{xy}$, the spin/orbital component $X_y$ (where $X=S$ or $L$) changes sign, while the velocity components $\nu_x$ and $\nu_y$ remain invariant. Consequently, the current operator $\hat{j}^{X_y}_y$ reverses sign, leading to the observed reversal in the conductivity tensor $\sigma^{X_y}_{yx}$.
   • Other non-zero tensor components (e.g., $\sigma^{S_z}_{xy}$, $\sigma^{S_y}_{zx}$) do not exhibit this sign reversal and remain identical for both enantiomers.

3. Magnitude and Anisotropy: The calculations reveal strong anisotropy in the SHC and OHC of Se and Te. Te exhibits significantly larger responses than Se due to its stronger atomic SOC ($5p$ vs. $4p$). The orbital Hall conductivity in Se reaches magnitudes comparable to those in conventional semiconductors like Si and Ge.

Significance and Claims
The paper claims that SHC and OHC provide a transport signature directly correlated with structural handedness in trigonal Se and Te. The primary significance lies in clarifying the symmetry origin of handedness-dependent spin and orbital transport. The authors assert that:

• The sign of the measured $\sigma^{S_y}_{yx}$ or $\sigma^{L_y}_{yx}$ component can be directly correlated with the structural handedness (left- vs. right-handed) provided the experimental axes align with the theoretical tensor definition.
• This behavior is a symmetry-governed consequence for mirror-related enantiomorphic structures, rather than a universal criterion for all chiral materials.
• The results suggest a potential route for correlating measurable SHC/OHC signals with structural handedness in specific chiral materials, offering a microscopic link between chirality and spin-orbit coupling effects.

The authors maintain a modest scope, noting that while their analysis focuses on the $P3_121/P3_221$ pair, the underlying symmetry principles could guide the identification of handedness-sensitive tensor components in other enantiomorphic space groups through group-theoretic analysis and material-specific calculations. They do not propose specific experimental setups or commercial applications, but rather highlight the potential for chiral materials as platforms for SHE and OHE studies.