Relativistic and nonrelativistic spin splitting above and below the Fermi level in a $g$-wave altermagnet

Abstract

Nonrelativistic spin splitting (NRSS) challenges conventional wisdom about antiferromagnets by allowing spin-split electronic bands even in collinear orders with zero net magnetization. This sub-class of antiferromagnets, recently dubbed "altermagnets," enforces distinctive spin textures via spin-group symmetries in the crystal. However, direct experimental evidence for such symmetry-driven magnetism remains scarce, and distinguishing it from relativistic spin splitting presents additional challenges. Here, we combine first-principles calculations, symmetry analysis, and two spin-resolved spectroscopies--angle-resolved photoemission (spin-ARPES) and our newly developed spin- and angle-resolved electron reflection spectroscopy (spin-ARRES)--to achieve the first complete momentum-resolved mapping of relativistic (RSS) and nonrelastivistic (NRSS) spin splitting in CoNb$_4$Se$_8$. By probing both the occupied (spin-ARPES) and unoccupied (spin-ARRES) electronic states in a single experiment, we uncover a series of momentum-dependent spin splitting phenomena each of which switch sign under sixfold rotations and persists far above and below the Fermi level. Crucially, distinct nodal planes in momentum space distinguish NRSS from RSS features. Additionally, the observed collapse of NRSS and the persistence of RSS above the Néel temperature, distinguishes a genuine magnetic phase transition from inversion symmetry breaking. Our work demonstrates, for the first time, the combined power of spin-ARPES and spin-ARRES in capturing the full spin texture across an extended energy range, positioning CoNb$_4$Se$_8$ as a prototype for exploring spin-group-based phenomena. These findings open new routes for engineering spin-based functionalities ranging from neuromorphic computing to unconventional superconductivity in layered antiferromagnets.

Figures (19)

• Figure 1: Real-space magnetic structure of CoNb$_4$Se$_8$. (A) Magnetic structure of $\mathrm{CoNb_4Se_8}$ along the $a$-axis, and (B) along the $c$-axis at $c$ = 0 (sublattice I) and $c$ = 1/2 (sublattice II). The DFT-calculated pink and blue magnetization clouds correspond to spin up and down, respectively, indicating the $\{6^{\pm}_{001}\,\vert\, 0\, 0 \, \tfrac{1}{2}\}$ screw relating the two sublattices. We show only the bottom Se atoms for clarity. Solid black (dashed grey) lines correspond to glide-reflection (mirror) planes connecting opposite (identical) spin sublattices. (C,D) Momentum-space spin splitting behavior for NRSS (B-) and RSS (C-) predicted by symmetry analysis, for $k_z \ne 0$. Nodal planes, across which the sign of NRSS (RSS) flips, and corresponding mirror planes in B, are indicated by the solid black (dashed grey) lines. (E,F) Electronic Structure from DFT (E-) and measured in ARPES (F-) along the direction of dotted white line in (C).
• Figure 2: Tight-binding band structure of NRSS(A) Results of the model without including CF and exchange terms, indicated with bare Co sites. Top (bottom) row of the bands shows the sublattice polarization (spin splitting). In both cases, the bands are unpolarized. (B) Results where CF is included, indicated with two octahedrons related by the screw operation $\{6^\pm_{001} | 0\,0\, \frac{1}{2}\}$. Sublattice polarization arises, while spin splitting remains zero. (C) Results where both CF and exchange interactions are included, indicated by Co sites colored with pink and blue denoting up and down magnetization, respectively. Here, sublattice polarization remains finite and NRSS also develops.
• Figure 3: ARPES measurement of nonrelativistic spin-splitting in occupied electronic structure in CoNb$_4$Se$_8$.(A,B) Fermi surface calculated from DFT (A-) and measured in ARPES (B-) for $k_z$ near $0.25c^*$ as indicated by the inset. (C) 13 K ARPES spectra along $\mathrm{M^{\prime\prime} - \Gamma{^\prime}}$ , as indicated by the sketch in the inset. Spin up (down) DFT bands from A are overlaid in pink (blue). (D,E) Spin-resolved EDCs spectra (D-) and corresponding spin polarization along the $z$ axis (E-) at momenta indicated by black vertical lines in B. Measurement uncertainty out to to $1\sigma$ outlined in grey. Red (navy) arrows indicate DFT energies of the $\delta_1$($\delta_2$) band. (F-H) Same as (C-E) but for electronic structure along $\mathrm{ \Gamma{^\prime} -M^{\prime}} \equiv (6^{\pm}_{001})^3(\mathrm{M^{\prime\prime} - \Gamma{^\prime}})$ direction in momentum space. (I) Spin-resolved nonrelativistic DFT calculations of electronic structure along $\mathrm{M^{\prime\prime} - \Gamma{^\prime} - M^{\prime}}$ projected on the Co ions. The color indicates cobalt contribution to each band multiplied by $\pm1$ for spin up and down. (J)$S_\mathrm{z}$ polarization as a function of momentum along $\mathrm{M^{\prime\prime} - \Gamma{^\prime} - M^{\prime}}$, extracted from binding energy depicted by dotted lines in (E,H).
• Figure 4: ARPES measurement of relativistic spin-splitting in occupied electronic structure in CoNb$_4$Se$_8$.(A) 13 K ARPES spectra along electron pockets surrounding $\mathrm{K^{\prime\prime}} \equiv (6^{\pm}_{001})^3 \mathrm{K^{\prime}}$ (left) and $\mathrm{K^{\prime}}$ (right) as indicated by the inset momentum space cartoon. (B) First-principles calculation of spin-resolved band structure projected on one $\mathrm{NbSe_2}$ layer (C) Sketch of photoelectron probe depth governed by electron mean-free path at $h\nu = 55\, \rm{eV}$. Grey dashed rectangle indicates one NbSe$_2$ layer. (D, E) Spin-resolved EDCs spectra along momenta indicated by black (white) vertical lines in A. Red (blue) ticks indicate peak locations for bands polarized spin up (down), extracted as a parabolic fit within the vicinity of spectral maximum. (F) Representative momentum-dependent spin splitting, extracted from the difference in spin up and spin down band locations in D and E .
• Figure 5: ARRES measurement of spin-split unoccupied electronic structure in CoNb$_4$Se$_8$(A, B) Schematic of sp-ARRES experiment: Spin-polarized electrons incident on the sample are preferentially absorbed if there is an unoccupied electronic state with corresponding spin $\vec{s}$, energy $E$ and momentum $\vec{k}$. (C) Schematic of BZ unfolding in CoNb$_4$Se$_8$ onto the parent BZ of NbSe$_2$. (D) DFT calculations of spin-resolved unoccupied electronic structure near high symmetry points 11 eV above $E_\mathrm{F}$. (E, F) Spin-integrated (E) and spin-polarized (F) ARRES spectra at constant energy surface 11 eV above $E_\mathrm{F}$ at T = 30 K. Band locations, indicated by maximal band curvature, overlaid in black. (G) Spin contrast in the ARRES experiment along the region indicated by the grey ring in F centered at $|k_\parallel|= 0.8|\rm{\bar{\Gamma M}}|_{\mathrm{NbSe_2}}$.