Quantification of electronic asymmetry: chirality and axiality in solids

TL;DR

This work introduces a relativistic framework to quantify electronic asymmetry in solids by defining the electron chirality density $\tau^Z(\mathbf r) = \Psi^\dagger(\mathbf r) \gamma^5 \Psi(\mathbf r)$ and the axiality $\mathbf X = \int d\mathbf r\, \mathbf r \tau^Z(\mathbf r)$, along with the spin-derived electric polarization $\boldsymbol{\mathcal P}_S(\mathbf r)$. In the non-relativistic limit, $\tau^Z$ reduces to the helicity projection $\tau^Z(\mathbf r) = \frac{1}{2mc} \Psi^\dagger \leftrightarrow \mathbf p \cdot \boldsymbol{\sigma} \Psi$, and $\boldsymbol{\mathcal P}_S(\mathbf r) = -\frac{1}{4mc} \Psi^\dagger \leftrightarrow \mathbf p \times \boldsymbol{\sigma} \Psi$, both expressible via the spin current tensor $j_{Sij}$. First-principles calculations for Te (chiral) and K$_2$Zr(PO$_4$)$_2$ (axial) reveal distinct spatial patterns of $\tau^Z$ and quantify $C = \int d\mathbf r \langle \tau^Z(\mathbf r)\rangle$ and $\mathbf X = \int d\mathbf r \langle \mathbf r \tau^Z(\mathbf r) \rangle$, with $C$ changing sign between enantiomorphs and potentially reversing with energy, implying doping-tunable handedness. The authors show that circular dichroism in photoemission, via $I_{CD}^{\mathbf q}(E,\mathbf k)$, directly probes $\tau^Z$ through the spin-current decomposition, enabling experimental access to electron chirality; isotropic CD isolates chirality, while SOC-induced band splitting in CoSi demonstrates the link between $\tau^Z$ and band structure. Overall, the framework provides intrinsic metrics for chirality, axiality, and polarity in solids and suggests new pathways for discovering and tuning chiral/axial materials, with potential extensions to time-reversal-broken systems.

Abstract

Chiral and axial materials offer platforms for intriguing phenomena, such as cross-correlated responses and chirality-induced spin selectivity. However, quantifying the properties of such materials has generally been considered challenging. Here, we demonstrate that the spatial distribution of the electron chirality, represented by $Ψ^\dagger γ^5 Ψ$ with the four-component Dirac field $Ψ$, characterizes the chirality and axiality of materials. Furthermore, we reveal that spin-derived electric polarization can serve as an effective indicator of material polarity. We present quantitative evaluations of electron chirality distribution and spin-derived electric polarization based on first-principles calculations. Additionally, we propose that electron chirality can be directly observed via circular dichroism in photoemission spectroscopy, which measures the difference between right- and left-handed circularly polarized light. Electron chirality and spin-derived electric polarization provide a new framework for quantifying chirality, axiality, and polarity in asymmetric materials, paving the way for the exploration of novel functional materials.

Figures (7)

• Figure 1: Quantification of asymmetric crystals. (Left) The charged (ionic) state and polar crystal are characterized by the charge density $\rho(\bm r)$ distribution. (Right) By contrast, the chiral and (ferro-)axial crystals are quantified based on the electron chirality $\gamma^5$, i.e., $\tau^Z(\bm r)$ defined in Eq. \ref{['eq:chirality_dirac']}, which measures the difference between right- and left-handed particles. The symbol "Inv.$=+(-)$" indicates that each physical quantity is even (odd) under SI. For completeness, we show Al crystal as an example of a highly symmetric simple substance in (a).
• Figure 2: Spatial distributions of electron chirality $\tau^Z(\bm r)$ for (a) the chiral crystal Te and (b) the axial crystal K$_2$Zr(PO$_4$)$_2$. The black shadows indicate cross-sections of the unit cell.
• Figure 3: Energy dependences of (a) the electronic charge and (b) electron chirality for $\mathrm{Te}$. The derivatives with respect to energies are also shown for both quantities.
• Figure 4: (a) Schematic illustration of photoemission spectroscopy with circularly polarized light. (b) Circular dichroism spectra for chiral crystal CoSi (c) Band dispersion with a characterization based on the electron chirality in $\bm k$-space for CoSi. We set the parameters as $\hat{\bm q} = \hat{\bm z}, \bm e_{\mathrm{L}} = \hat{\bm x} + \mathrm{i}\hat{\bm y}, \hbar\nu = 100 \, \mathrm{eV}$ in (b). The magnitude of electron chirality in (c) is normalized by its maximum value in this energy range.
• Figure S1: (a) Spin-derived electric polarization density (b) Energy dependences of total polarization for $\mathrm{BaTiO_3}$.