Evidence of magnetism-induced topological protection in the axion insulator candidate EuSn₂P₂

Single crystals of EuSn₂P₂ were grown using the Sn-flux method (22). EuSn₂P₂ crystallizes in a rhombohedral structure in which Eu atoms and the Sn-P hexagonal network form hexagonal layers and stack along the c direction, as shown in Fig. 1A, with the corresponding surface and bulk Brillouin zones presented in Fig. 1B. The space group of the lattice is $R\overline{3}m$ (No. 166) and the lattice is centrosymmetric. The crystal structure was determined by X-ray diffraction (see SI Appendix, Tables S1–S3), confirming the phase identity; topographic images obtained by scanning tunneling microscopy (see SI Appendix, Fig. S1) show that the cleaved surface is composed of large, homogeneous terraces, hundreds of nanometers wide, interrupted by steps. The step height corresponds to the distance between homonuclear layers, thus indicating that all terraces have the same atomic termination.

The angle-resolved photoemission spectroscopy (ARPES) characterization of the electronic states of EuSn₂P₂ compared to first-principle calculations is presented in Fig. 1. The symmetry-resolved theoretical bulk band structure along the $\overline{\mathrm{\Gamma }}-\overline{K}$ and $\overline{\mathrm{\Gamma }}-\overline{M}$ directions (left halves of Fig. 1 C and D) shows a very good agreement with the experimental E vs. k dispersion maps (Fig. 1 E and F) (22). Several dispersive features are experimentally observed (labeled as $\alpha$, $\beta$, $\gamma$, and $\delta$) and attributed through the calculations to the P and Sn bulk states, while the almost nondispersive $\zeta$ states, located at binding energy ≃ −1.5 eV, originate from the Eu 4f orbitals, in excellent agreement with the calculations. Knowing that electronic states with weak overlap/hybridization favor spin unpairing, hence a localized magnetic character, this observation further corroborates the role of Eu as the magnetic driver in EuSn₂P₂. The calculations shown in Fig. 1 C and D indicate that the P 3p bands ( $\alpha$) display hole-like band dispersion close to the Fermi level. Remarkably, the linearly dispersing bands labeled $\sigma$ and $P$, observed in the experimental ARPES maps (highlighted in the enlarged views in Fig. 1 G and H), find no match with the bulk calculations. The $P$ band can be reproduced in calculations only when considering the phosphorous-terminated surface (right halves of Fig. 1 C and D). It appears clearly outside of the projected bulk band edge in the surface calculation. The $\sigma$ band, close to the bulk band edge, has no evident match, yet it can be identified as a surface resonance (24). The comparison between experiment and theory suggests that the $\sigma$ and $P$ bands are surface states with main contribution from P 3p orbitals.

The identification of extra bands as a specific feature of the P-terminated surface is made even clearer in Fig. 1I, where experimental isoenergetic Fermi surfaces for different terminations are shown compared to the P-terminated surface calculation. Horizontal cuts (k_y = 0 Å⁻¹) of the experimental data clearly highlight the presence of the “extra” $\sigma$ and $P$ bands, the latter corresponding to a bright intense warped hexagon surrounding the bulk structure in the (k_x − k_y) plot. This intense P-derived band has a Fermi velocity of 3.24 eVxÅ (4.92 × 10⁵ m/s) and k_F = 0.23 Å⁻¹. Experimental Fermi surface measurements for the P-terminated surface at different photon energies are shown in SI Appendix, Fig. S5.

EuSn₂P₂, similarly to EuSn₂As₂ and EuIn₂As₂, is classified as an A-type antiferromagnet, as determined by theoretical predictions and bulk-sensitive magnetometric measurements (8, 20, 22). The 4f electrons of Eu are tightly bound and preserve a localized character when passing from the atom to the solid, as indicated by the nondispersive$\zeta$ band in Fig. 1. The proximity of the 4f levels to the Fermi energy may, however, lead to instabilities of the charge configuration of the Eu, i.e., valence fluctuations, which would have a strong impact on the ferromagnetic character of the material. In order to avoid ambiguity in the interpretation of the magnetic properties that might result from uncertainty of the Eu valence, we performed X-ray absorption spectroscopy (XAS) and X-ray magnetic circular dichroism (XMCD) in total yield mode [average probing depth 6 nm (25)] at the Eu M_4,5 edges. The lineshape of the XAS spectrum measured at 15 K (below T_Néel) is compared in Fig. 2A with theoretical calculations for the Eu²⁺ and Eu³⁺ valences (26), revealing that this material has a pure Eu²⁺ character.

For a layered AFM system, one might expect an XMCD signal that averages to zero, due to opposite contributions of antiparallel layers of Eu spins. However, any such measurement averages the absorption spectrum over several layers, whose contributions are weighted by their proximity to the surface; a small XMCD signal is still detectable for that reason, whose shape is in excellent agreement with an Eu²⁺ contribution (27). The measured XMCD in Fig. 2B is roughly two orders of magnitude smaller than the value expected for an isolated, fully saturated Eu moment if the atom is in a 4f⁷(6s²) configuration (Eu²⁺, with a total spin S = 7/2) (21, 28). The small yet clear magnetic signal, which vanishes above T_Néel (SI Appendix, Fig. S3), is measured without applying any external magnetic field, and testifies the presence of long-range magnetization in the bulk. Thus, these results confirm that layered antiferromagnetism (A-type) sets in spontaneously in EuSn₂P₂ when crossing T_Néel.

The small residual XMCD signal is compatible with both out-of-plane and in-plane magnetization of a single Eu layer due to the geometry of the experiment; insights into the layer-dependent electronic and magnetic properties are given in Fig. 2 C and D by vectorial spin-resolved ARPES results (29). ARPES probing depth is in the range of two Eu layers, i.e., much more sensitive to the magnetic signal of the topmost layer, taking into account the photon energy used and the large c lattice constant of the unit cell (the periodicity of the AFM order is 2c/3, as sketched in Fig. 1A) (30). Indeed, a strong (up to 19%) spin polarization of Eu 4f states along the x direction is observed below T_Néel, while along both the y direction and the z direction (see SI Appendix, Fig. S4) no magnetic signal is found. The in-plane magnetization appears below and disappears above T_Néel, in a reversible way. These results confirm that the long-range magnetization is fully Eu-derived and aligns spontaneously to exhibit 2D ferromagnetism (in-plane ferromagnetic intrarlayer coupling). We stress that the presence of a robust magnetic signal in zero applied magnetic field is a prerequisite for axion-driven effects and antiferromagnetic spintronics application (15).

Having ascertained the nature of the long-range magnetization and the presence of strong local magnetic moment at the Eu-sites with pure Eu²⁺ character, we now unravel the topological properties of EuSn₂P₂ in the presence of the in-plane AFM order. We have computed the Z₄ invariant, the Z₂ invariant, and the mirror Chern number (MCN) n_M. The parity-based higher-order Z₄ invariant is given by $Z_4={\displaystyle {\sum }_{i=1}^8{\displaystyle {\sum }_{n=1}^{n=\mathit{occ}}\frac{1+{\xi }_n({\text{Γ}}_i)}{2}}}mod 4$, where ${\xi }_n({\text{Γ}}_i)$ is the parity eigenvalue (+1 or −1) of the nth band at the ith time-reversal invariant point ${\text{Γ}}_i$ and n = occ is the number of occupied bands (31). The Z₄ invariant is well defined for an inversion symmetric system such as EuSn₂P₂, even in the absence of TRS. Our calculations show Z₄ = 2 with in-plane AFM, suggesting that EuSn₂P₂ is a 3D AFM axion insulator, with a quantized topological magnetoelectric effect (axion coupling $\theta = \pi$) (32, 33). Based on this result, one would expect that EuSn₂P₂ displays gapped surface states on all faces and gapless chiral hinge states between two gapped surfaces. Although generally undefined in a system with broken TRS, Z₂ can be expanded to classify topological states in magnetic systems if the system possesses a specific magnetic configuration, for instance an AFI state, in which one defines the combined symmetry $\text{S}=\text{Θ}{T}_{1/2}$, where $\text{Θ}$ is the time-reversal operator and $T_{1/2}$ is the lattice translational symmetry of the crystal that is broken by the AFI order (12, 34). Under this definition, we calculated the Z₂ invariant by the Wilson loop method, resulting in a nontrivial Z₂ = 1 (see SI Appendix, Fig. S10A) (34). Next, we consider that EuSn₂P₂ in its AFI phase retains a mirror plane that is perpendicular to the in-plane magnetic moment, corresponding to a nontrivial MCN n_M = 1 on the k_x = 0 plane, indicating that EuSn₂P₂ is also a topological crystalline insulator.

The presence of both a nontrivial Z₂ and the MCN n_M = 1 has important consequences on the topology of the system. The extra mirror symmetry and Z₂ index induced by the in-plane magnetic order constitutes an additional and robust protection of the Dirac fermions. Such a symmetry preserves a gapless surface state on the side surface (010) around $\overline{\mathrm{\Gamma }}$ and on the (001) surface along the $\overline{\mathrm{\Gamma }}-\overline{M}$ direction (see SI Appendix, Figs. S9C and S10D) (19).

The effect of magnetic order is outlined in Fig. 3 A and B, which compare the spectral weights calculated for the (001) surface along the $\overline{\mathrm{\Gamma }}-\overline{M}$ direction in both the nonmagnetic and AFM configurations. The overall surface band structure does not change significantly, with the exception of the strong intensity from the Eu-derived states, absent in nonmagnetic calculation and centered around −1.5-eV binding energy (BE) in the AFM configuration, in good agreement with the ARPES characterization. The topological Dirac surface state ($S_{\mathit{topo}}$) is visible in the bulk gap at $\overline{\mathrm{\Gamma }}$ in the nonmagnetic case at 0.28 eV above the Fermi level (light blue inset of Fig. 3A). The onset of magnetic order shifts the locus of the Dirac point in momentum space without opening a gap (Fig. 3C); the Dirac crossing position above the Fermi level curtails its direct observation by ARPES (14, 18). Other surface states around the bulk band gap are found both in nonmagnetic and AFM cases, labeled $P$ since they correspond to the $P$ band observed in ARPES. In the nonmagnetic case, these surface states are degenerate at $\overline{M}$ (Fig. 3A, red inset) and merge into the bulk valence band around $\overline{\mathrm{\Gamma }}$ above the Fermi level. The broken TRS in the AFM phase opens a gap at $\overline{M}$ (Fig. 3B, red inset), not visible in the experiment due to the vanishing intensity at higher BE and to the overlap with other bulk bands. By applying to this calculation an on-site energy of 0.2 eV on the surface atoms to push up the energy position of the surface states (Fig. 3D), we find that the connectivity of the topological surface state changes dramatically (35, 36): $S_{\mathit{topo}}$ is no longer merged into the projected bulk band around the $\overline{\mathrm{\Gamma }}$ point as in Fig. 3C but connects with one of the $P$ bands and forms a new surface band, labeled $S_{\mathit{topo}}+P$, which starts from bulk conduction band around $\overline{\mathrm{\Gamma }}$, crosses the boundary of the BZ, and connects to the bulk valence band. As the surface on-site energy does not affect the bulk topology, the connectivity of surface state maintains its character. Therefore, the $P$ band is part of a topological surface state.

Spin-ARPES results shown in Fig. 3 F and G provide clear spectroscopic evidence of the topological character of the P-derived states. Spin-ARPES energy distribution curve (EDC) spectra measured at T =15 K (i.e., below T_Néel) at two opposite momentum values display clear spin-momentum locking of the $P$ band (as well as the surface resonance $\sigma$) with polarization inversion of the y component of the spin (sketch in Fig. 3E). The same spin inversion is also observed above T_Néel (SI Appendix, Fig. S7). These results indicate that 1) the observed spin polarization has a nonmagnetic origin, 2) TRS is preserved as expected for a topological surface state (2, 23, 37, 38), and 3) the topological character is strong and not influenced by in-plane layered ferromagnetism, as opposed to what has been observed in MnBi₂Te₄ for the out-of-plane direction of the magnetization (14). As outlined above, calculations predict the presence of two split P bands, which should carry opposite spins. The splitting is so small that it cannot be resolved by the energy and momentum resolutions of our system (30 meV and 0.01 Å⁻¹, respectively), yet the imbalance in the spin-projected weight of the two bands results in a net polarization of the photoemission signal, revealing their topological character.

Our theoretical and experimental observations provide clear evidence that EuSn₂P₂ has a termination-dependent surface electronic structure with multiple topology linked to the alignment of the Eu magnetic moments. In particular, our theoretical results predict the existence of gapless surface states protected by the combination of different nontrivial topological invariants, i.e., the inversion symmetry plus the presence of a mirror plane perpendicular to the magnetic moment. Our experimental results confirm this scenario, with the observation of spin-momentum locking of the P-derived surface states, coexisting with in-plane ferromagnetism. The observed extra topological protections in EuSn₂P₂ are expected to vanish when the mirror in-plane symmetry is broken, as for example in presence of an out-of-plane spin component. This symmetry-derived and termination-dependent link between topology and magnetism may open new routes in the realization of topological spintronics devices and in topological antiferromagnetic heterostructures.

This work has been partly performed in the framework of the Nanoscience Foundry and Fine Analysis (NFFA-MUR Italy) facility. The work at Princeton was supported by the NSF (NSF Materials Research Science and Engineering Centers grant DMR 2011750). We thank D. di Sante and S. Picozzi for valuable discussion.