Beyond spin-1/2: Multipolar spin-orbit coupling in noncentrosymmetric crystals with time-reversal symmetry

TL;DR

This work develops a comprehensive multipolar total-angular-momentum framework for strong spin-orbit coupling in time-reversal-symmetric, noncentrosymmetric crystals with $C_{3v}$ symmetry, extending beyond the conventional spin-$1/2$ picture to $j= {1/2, 3/2, 5/2}$. Using a symmetry-based ${\bf k}\cdot{\bf p}$ approach, the authors construct all allowed SOC terms up to fifth order, revealing modified Rashba and higher-rank multipolar couplings that reshape Fermi surfaces and yield TAM textures with winding numbers $|W_n|=1,2,5$ and band-dependent anisotropy. They demonstrate that multipolar SOC can enhance and nonmonotonically modulate current-induced TAM polarization (Edelstein effect) as the chemical potential is tuned, with distinct signatures across TAM multiplets. The framework is presented as a practical tool for predicting TAM textures and spintronic responses in heavy-element noncentrosymmetric materials, with PtBi$_2$ and BiTeI highlighted as promising platforms for experimental exploration and orbitronic applications.

Abstract

We develop a general multipolar theory of strong spin-orbit coupling for large total angular momentum $j$ in time-reversal-symmetric, noncentrosymmetric crystals. Using a $j\in\{1/2,3/2,5/2\}$ multiplet basis appropriate for heavy-element \textit{p}- and \textit{d}-bands, we systematically construct all symmetry-allowed spin-orbit coupling terms up to fifth order in momentum and generalize the usual spin texture to a total-angular-momentum texture. For $j>1/2$, multipolar spin-orbit coupling qualitatively reshapes Fermi surfaces and makes the topology of Bloch states band dependent. This leads to anisotropic high-$j$ textures that go beyond a single Rashba helix. We classify these textures by their total-angular-momentum vorticity $W_{n}$ for every energy band and identify distinct $|W_{n}|=1,2,5$ phases. We show that their crossovers generate enhanced and nonmonotonic current-induced spin-polarization responses, namely the Edelstein effect, upon tuning the chemical potential. Our results provide a symmetry-based framework for analyzing and predicting multipolar spin-orbit coupling, total-angular-momentum textures, and spintronic responses in heavy-element materials without an inversion center.

Figures (7)

• Figure 1: Bulk band dispersion for $j=3/2$ electrons near $\Gamma$ point preserving time-reversal and $C_{3v}$ symmetry, shown for two high-rank spin-orbit coupling configurations from decomposition (a) $A_{1,+}\otimes A_{1,+}$ and (b)--(d) $A_{1,+}\otimes A_{1,+}+E_{-}\otimes E_{-}$. Model parameters are (a) $(\mu_{2},\alpha_{2,1},\alpha_{2,2})/E_{F}=(-1,1,1)$, (b) $\gamma_{1,1}=0.3/E_{F}$, (c) $\gamma_{1,2}=0.6/E_{F}$, and (d) $\gamma_{1,3}=0.3/E_{F}$. Other parameters are $(\alpha_{1,1},\alpha_{1,2})/E_{F}=(-2,-1)$ and $E_{F}=\mu_{1}$. The inset in (a) illustrates Brillouin zone with a ditrigonal pyramid in a hexagonal setting Texture_2010_1.
• Figure 2: Magnitude of $E_{-}\!\otimes\!E_{-}$ intraband and interband spin splittings for $j=1/2$ and $j=3/2$ electrons near the bulk $\Gamma$ point, evaluated along a ditrigonal-pyramidal path in the hexagonal Brillouin zone. Model parameters are fixed to $\gamma_{1,i}=\gamma_{2,i}=\gamma_{3,i}=\gamma_{4,i}=0.5\,E_{F}$. (a) Pure modified Rashba $\hat{\mathcal{H}}_{\text{MR}}$ with $i=1$, (b) pure $\hat{\mathcal{H}}_{\mathrm{HS}}^{(1)}$ with $i=2$, (c) pure high-spin term $\hat{\mathcal{H}}_{\text{HS}}^{(2)}$ with $i=3$, and (d) full SOC $\sum_{i=1}^{3}\hat{\mathcal{H}}_{i}$ with $\hat{\mathcal{H}}_{i}$ denoting pure terms in panels (a)--(c).
• Figure 3: Phase diagrams of total-angular-momentum vorticity as function of (a) $\mathbf{q}=(q_{1},q_{2},q_{3})$ and (b) $\mathbf{q}=(q_{1},q_{2},0.5)$. Polar plots of the $j$ texture for (c) single winding, $W=-1$, at $\mathbf{q}=(1.5,0,0.5)$; (d) twofold winding, $W=2$, at $\mathbf{q}=(0.5,1.5,0.5)$; and (e) fivefold winding, $W=5$, at $\mathbf{q}=(0.2,0.2,0.5)$. The color bar indicates the winding number. Black planes and lines in panels (a) and (b) denote boundaries of regions with different winding number.
• Figure 4: Two-dimensional constant-energy contours with TAM texture in $j=3/2$ basis for light-mass band, i.e., $|3/2,-1/2\rangle$. (a) Linear-quintic mixture $E_{-}\!\otimes\!E_{-}$ with $\gamma_{1,1}/E_{F}=0.3$ and $\gamma_{4,1}/E_{F}=1.5$, (b) linear-cubic mixture $A_{2-}\!\otimes\!A_{2-}+E_{-}\!\otimes\!E_{-}$ for slice $k_{z}=0.5$ with $\gamma_{1,1}/E_{F}=1$, $m_{1,1}/E_{F}=1.5$, and $m_{2,1}/E_{F}=1.2$, (c) linear-cubic mixture $E_{-}\!\otimes\!E_{-}$ with $\gamma_{1,1}/E_{F}=0.3$ and $\gamma_{2,1}/E_{F}=1.5$, and (d) full term with $\gamma_{\nu,1}/E_{F}=m_{2,1}/E_{F}=0.5$ for $\nu\in\{1,2,3\}$, $m_{1,1}/E_{F}=1$ and $k_{z}=1$. The yellow solid circles denote the phase transition boundaries where the winding changes. The out-of-plane component of TAM texture is marked in dark and light colors. The heavy-mass bands carry similar information. for other helical energy branch $W\rightarrow-W$. Other parameters are set to zero.
• Figure 5: (a) Three-dimensional bulk spectrum with constant-energy contours and (b) corresponding vorticity phase diagram. Both panels are evaluated for the light mass band with positive helicity, i.e., $|3/2,-1/2\rangle$. Model parameters: $(\gamma_{1,1},\gamma_{2,1},\gamma_{3,1},\gamma_{4,1})/E_{F}=(-0.5,-1,0.4,0.2)$ and $|\mathbf{k}_{\perp}|=(k_{x}^{2}+k_{y}^{2})^{1/2}$. In panel (a), the constant-energy contours are taken in the range $E/E_{F}\in[-1,1]$. Heavy-mass bands $\lvert3/2,\pm3/2\rangle$ carry similar information to the light-mass bands $\lvert3/2,\pm1/2\rangle$. Moreover, $W\rightarrow-W$ for opposite helicity.