Unconventional relativistic spin polarization of electronic bands in an altermagnet

TL;DR

This work investigates unconventional relativistic spin polarization in altermagnets, focusing on MnTe with $d$, $g$ or $i$-wave altermagnetic order and symmetries that combine spin-space rotations with real-space operations. It combines spin- and angle-resolved photoemission spectroscopy (SARPES/ARPES) on single-domain MnTe, supported by non-relativistic spin-symmetry and relativistic magnetic-symmetry analyses and ab initio ground-state and photoemission theory, highlighting the spin-orbit coupling origin through the term $ \sim \frac{1}{c^2}\,\mathbf{s}\cdot(\mathbf{k}\times \mathbf{E})$. On the $k_z=0$ nodal plane, the relativistic spin polarization is collinear along the $z$-axis, even-parity, and time-reversal-odd, with spins orthogonal to the magnetic-ordering vector and reversing sign under time reversal. These results broaden the phenomenology of spin-polarized spectra, suggesting potential for robust spin-current phenomena and implications for topological transport and dissipationless nanoelectronics in altermagnets, where the unconventional polarization can influence spin-to-charge conversion and spin-Hall effects.

Abstract

Altermagnetism is a recently identified phase with a d, g or i-wave spin symmetry of magnetic ordering. Its discovery opens new research fronts at intersections of magnetism and spintronics with fields ranging from superconductivity to topological and relativistic quantum physics. Here we demonstrate an unconventional relativistic spin polarization in an altermagnet by spin and angle resolved photoemission spectroscopy of electronic bands in single-domain MnTe. The relativistic spin-orbit coupling origin is revealed by observing that the alternating momentum-dependent spin polarization is orthogonal to the magnetic-ordering vector. The collinearity, even-parity and time-reversal-odd nature of the demonstrated relativistic spin polarization in the altermagnet is unparalleled in conventional forms of the relativistic spin polarization. Our experimental results and methodology are supported by non-relativistic spin-symmetry and relativistic magnetic-symmetry analyses, and microscopic ab initio ground-state and photoemission theory.

Figures (4)

• Figure 1: Theory of the unconventional relativistic spin polarization in altermagnetic MnTe. (a) Top panel: schematic side-view of the crystal and magnetic structure of MnTe in the $y–z$ plane. $\left[C_{2}||M_{z}\right]$ marks the altermagnetic spin-group symmetry connecting the two opposite Mn sublattices colored in purple, denoted by opposite black arrows, and surrounded by Te octahedra shaded in grey. The relativistic magnetic symmetry group contains mirror plane symmetry $\mathcal{M}_{z}$, marked in magenta. Bottom panel: schematic top-view of the crystal and magnetic structure of MnTe in the $x–y$ plane. (b) Relativistic ab initio band structure of MnTe calculated at the $k_{z}$=0 plane along the $-\bf{K}_{1} - \boldsymbol{\Gamma} - \bf{K}_{1}$ path (for Brillouin zone notation see panel (c)). The Néel vector is oriented along the crystal $y$-axis (see bottom panel (a)). Red - white - blue color bar marks the computed expectation value of the $z$ - component of spin. (c) Ab initio valence band structure and spin-polarization expectation value calculated at the $k_{z}$=0 plane. In the hexagonal $k_{z}$=0 plane, three wavevectors $\bf{K}_{1}$, $\bf{K}_{2}$, and $\bf{M}$ are highlighted.
• Figure 2: Transmission electron microscopy (TEM) imagining and preparation of the single magnetic domain state in the thin films of MnTe. (a) TEM image of the thin films of $\alpha$-MnTe(0001) grown by molecular-beam epitaxy on an InP(111) substrate. (b) Demonstration of preparation of a single-domain state of MnTe in our pre-patterned sample by X-ray magnetic linear and circular dichroism photoemission electron microscopy. Left panel, before field cooling, and right panel, after field cooling. (c) Angle-resolved photoemission spectroscopy of the pre-patterned sample on the $k_z=0$ momentum-space iso-surface measured at energy 0.28 eV below the top of the MnTe valence band, using a photon energy of 78 eV.
• Figure 3: Spin-resolved and angle-resolved photoemission spectroscopy (SARPES) observation and one-step photoemission theory of unconventional out-of-plane spin polarization $S_{z}$. The $S_{z}$ sign reversal under 180$^\circ$, and 60$^\circ$ rotation of the Néel vector, respectively, demonstrates the unconventional out-of-plane, time-reversal broken and alternating nature of the spin polarization. The first column (a, e, i) shows three experimental geometries with the orientation of the patterned sample (patterning direction marked by grey lines), the Néel vector, and the ab initio calculated constant-energy isosurface at 0.28 eV below the top of the valence band. The second column (b, f, j) presents the SARPES spin-current experimental data, and the third column (c, g, k) shows the corresponding experimental out-of-plane $S_z$ spin polarization. The last column (d, h, l) displays one-step photoemission calculations of the out-of-plane spin-polarization asymmetry, which are consistent with the experimental results. The first two rows demonstrate the reversal of the spin-polarization sign along the $-\bf{K}_{1} - \boldsymbol{\Gamma} - \bf{K}_{1}$ path upon reversing the Néel vector direction, while the last row shows the reversal when measuring along the $-\bf{K}_{2} - \boldsymbol{\Gamma} - \bf{K}_{2}$ direction.
• Figure 4: The ARPES measurement and one-step photoemission theory of band maps along $-\bf{K}_{1} - \boldsymbol{\Gamma} - \bf{K}_{1}$ wavevector path. Measured ARPES band maps with a p-polarized (a) and s- (b) polarized photons, overlayed with ab initio spin-polarized band structure. (c,d) One-step photoemission calculation of the intensity considering photoemission final states in the form of (c) free-electron like and (d) time-reversed low energy electron diffraction (TR-LEED). (e,f) One-step photoemission calculations with TR-LEED final states for (e) the average of the two time-reversed domains and (f) one domain only, as indicated in the insets.