Observation of Giant Spin Splitting and d-wave Spin Texture in Room Temperature Altermagnet RuO2

doi:10.21203/rs.3.rs-3966669/v1

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Observation of Giant Spin Splitting and d-wave Spin Texture in Room Temperature Altermagnet RuO2

https://doi.org/10.21203/rs.3.rs-3966669/v1

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Recently, a new magnetic phase called altermagnetism has been proposed, ushering in a third distinct magnetic phase beyond ferromagnetism and antiferromagnetism. It is expected that this groundbreaking phase exhibits unique physical properties such as C-paired spin-valley locking, anomalous Hall effect, nontrivial Berry phase, and giant magnetoresistance. Among all the predicted candidates, several room temperature altermagnets are suggested to host significant potential applications. Nevertheless, direct evidence about the spin pattern of the room temperature altermagnet is still unrevealed. RuO₂ is identified as the most promising candidate for room temperature d-wave altermagnetism exhibiting a substantial spin splitting of up to 1.4 eV in previous research. In this study, utilizing angle-resolved photoemission spectroscopy (ARPES), we report experimental observation of the giant spin splitting in RuO₂. Furthermore, employing spin-ARPES, we directly observed the d-wave spin pattern. Our results unequivocally show that RuO₂ is a perfect d-wave altermagnet with great potential for upcoming spintronic applications.

Physical sciences/Physics/Condensed-matter physics/Electronic properties and materials

Physical sciences/Physics/Condensed-matter physics/Magnetic properties and materials

Magnetism, a foundational domain in condensed matter physics, conventionally classifies magnetic materials into two primary phases: ferromagnets and antiferromagnets (although unconventional forms like ferrimagnetism and chiralmagnetism also recognized). Ferromagnets exhibit macroscopic magnetization in spatial space and spin splitting in reciprocal momentum space (Fig.1a). Conversely, typical antiferromagnets show no net macroscopic magnetization in spatial space and no spin splitting in reciprocal momentum space when time reversal symmetry combined with inversion or translation operation is preserved (Fig.1b)^1–3. Recent theoretical investigations have identified materials with substantial spin-split band structures but no net macroscopic magnetization (Fig. 1c)^2–18. These materials challenge easy classification as either ferromagnets or antiferromagnets. This has led to the proposal of altermagnetism, a third magnetic phase that appropriately describes such materials^2,3. In contrast to conventional antiferromagnets, which have spin sublattices that are linked to one another through straightforward translation operations, altermagnetism involves linking sublattices with opposite spins using rotation operations^2,3,19. Applying the same rotational angle in momentum space causes to a spin flip between up and down. As outcome of this phenomenon, even-order wave symmetries of the spin pattern, such as the d-, g-, i-waves, and others^2–4,20.

The s-wave Cooper pairs in conventional superconductors find analogous counterparts in ferromagnetism. Meanwhile, the investigation of unconventional d-wave superconductivity has been going on for more than two decades, which has led scientists to ask if there is an equivalent unconventional d-wave counterpart in magnetic materials²¹. Efforts have been made to investigate d-wave magnetism within the context of strong electronic correlations^22–24. On the other hand, at the fundamental level of an effective single-particle description of magnetism, altermagnetism presents the prospect of unconventional d-wave or high even-parity wave magnetism^2–16,18,25. This suggests that altermagnets can inherently possess robust d-wave properties^2,3, which would broaden the category of spin quantum phases.

The investigation of altermagnetism greatly expands the symmetry category in the field of magnetism. Due to their unique physical properties – such as stability under perturbing magnetic fields, strong noncollinear spin currents, anomalous Hall effect, C-paired spin-valley locking, nontrivial Berry phase, giant magnetoresistance - altermagnets appear to be promising candidates for promoting the next generation of information technology,^{5,8,11–17,25–32}. Despite these encouraging prospects, it's important to note that the experimental exploration of the electronic structures of altermagnets is still in its early stages.

Calculations have predicted dozens of potential altermagnet candidates^2,3, and recent research has reported band spin splitting in materials with gaps of about 0.3 eV in MnTe₂ and MnTe, about 0.8 eV for CrSb^2,3,33–36. Energy splitting and critical temperature are two criteria that are important for the candidates’ potential of application. To be applied in spintronics, the critical temperature must be higher than room temperature; also, to increase transport robustness, the energy splitting should be three to five times larger than the room temperature induced energy broadening, that is to say greater than 300-400 meV. Remarkably, of these candidates, only three (RuO₂, MnTe, and CrSb) are in the promising region for potential application, as shown in Fig.1e³. However, till now, there hasn’t been any reporting of the direct experimental observation of the spin pattern of these intriguing room temperature candidates (while plaid-like spin splitting has been reported in MnTe₂ very recently with critical temperature ~80K³⁵). RuO₂ stands out among the promising candidates, exhibiting the largest spin-splitting at around 1.4 eV and high critical temperature more than room temperature^3,6,37–39. This substantial spin splitting in RuO₂ is of great scientific interest and has a wide range of potential uses. Novel observations in RuO₂ include the reporting of anomalous Hall effect and in-plane Hall effect^40–43, along with the prediction of giant and tunneling magnetoresistance^44–46, the identification of titled spin currents and crystal thermal transport^47,48, and the documentation of terahertz emission induced by spin splitting effect^49,50. Additionally, superconductivity has been observed in thin-film RuO₂^51–53. Topological superconductors, such as p-wave superconductivity, are suggested to arise from the introduction of superconductivity into altermagnet, where the spin degeneracy is lifted and spin is consistent between k and -k^54,55. Evidence of time reversal symmetry breaking has been reported through the difference between circular plus and minus light-induced photoemission spectra⁵⁶. Moreover, the presence of nodal lines in the electronic structure has also been studied⁵⁷.

Despite the several novel findings in RuO₂, direct spectroscopic evidence proving the giant spin splitting and d-wave spin pattern has been lacking until now, but they are essential for confirming RuO₂ as an ideal altermagnet with substantial application potential. In the current study, we bridge this gap by employing synchrotron-based ultraviolet (UV) ARPES, soft X-ray ARPES, and spin-ARPES. Using these advanced techniques, we provide strong direct evidence that giant spin splitting exists in RuO₂, the most optimal room temperature altermagnet candidate. Moreover, our study directly captures the d-wave spin pattern in the momentum space, which represents a significant advance in the experimental verification of RuO₂'s unique altermagnetic properties.

The crystal structure of RuO₂, shown in the inset of Fig. 1d, hosts the space group denoted as P42/mnm (SG136). The Ru atom is surrounded by six O atoms, forming an octahedron with 2-fold rotation symmetry. The spin configuration of Ru within the two sublattices can be connected through a 90-degree rotation operation combined with a translation operation, which breaks parity time symmetry. The crystal and spin configuration distinctly characterize the d-wave altermagnetism that is anticipated in RuO₂. X-ray photoemission spectroscopy shows both O and Ru peaks in Fig. 1d. Fig. 1f and g, respectively, show the band structures in the paramagnetic phase and altermagnetic phase. Double-degenerate bands are shown in the paramagnetic phase, while in the altermagnetic phase, distinct spin splitting is visible along the GM and AZ directions. In the graphical representation, the red color bands indicate spin-up states, whereas the blue color bands indicate spin-down states. The significant spin-splitting along the GM direction is especially noteworthy. Our predicted calculation shows energy splitting as high as 1.54 eV as indicated by the double arrows along GM direction in Fig.1g, which is similar to the previously reported 1.4 eV³. This makes it the most substantial among the candidates proposed in the recent predictions (Fig.1e).

Breaking of parity-time symmetry results in band degeneracy lifting of altermagnet with respect to the paramagnetic phase. Experimentally, the observation of band splitting provides strong evidence for the establishment of long-range magnetic order. The specific cut along the GM direction provides an excellent point of reference for verifying this claim. Both UV and soft X-ray photon energy-dependent mappings were conducted in order to precisely determine the correct photon energies and locations in the momentum space, as shown in Fig. 2b and c. The natural cleavage of the sample along the (110) direction, causes the central point to alter between G and M with varying photon energy. The soft X-ray ARPES data in Fig. 2c distinctly reveals a clear periodic structure. In order to define the photon energies associated with high symmetry planes that align well with the lattice parameters and set the inner potential, vertical cuts at the center of the Brillouin Zone (BZ), as shown in Fig. 2d, offer insights into the band structure along the GM direction (see more details in Supplementary Materials section 1 and Fig. S1). On the other hand, owing to the k_z broadening effect, the out-of-plane dispersion in UV ARPES data is weaker but still noticeable, as shown in Fig. 2b. Since the inner potential remains consistent for both soft X-ray and UV measurements, we double confirmed the k_z relationship with photon energy by deriving the UV photon energies associated to high symmetry planes from the soft X-ray data. This derivation, as shown in Fig. S1c, exhibits excellent agreement with the UV data. To investigate the in-plane Fermi surface, two photon energies, 115 eV and 74 eV, were chosen to measure the electronic structure associated with the G and M planes, respectively, as plotted in Fig. 2e and f.

We designate the direction parallel to GM as k_x and the direction parallel to GZ as k_y. Plots of the horizontal cuts crossing the BZ center are shown in Figs. 2g and h, which both match to the GM spectra but have different central locations. Within these ARPES spectra, we identify four bands near Fermi level labeled as α - δ. The α band has a flat characteristic that almost exactly on the Fermi level. The adjacent β and γ bands cross the Fermi level between G and M. Along the perpendicular GZ direction, these two bands degenerate, as illustrated in Fig. 2e, where we also identify surface states (SS) forming a quasi-1-D Fermi surface along with one small pocket. The δ band is broad in energy and it bends back at the binding energy of approximately 0.4 eV. This band dispersion is consistent with the results of DFT calculations, as shown in Fig. 2j. Soft X-ray Fermi surface maps at the G plane is presented in Fig. 2k, without the presence of surface bands. The cut along the GM direction is plotted in Fig. 2j, with the DFT calculation superimposed. Apart from the flat α band at the Fermi level, bands β to δ are clearly identified. The double-peak in momentum distribution curve (MDC) in Fig. 2i clearly shows the separation of the β and γ bands. To verify if the α band is a surface state, a surface band structure projection calculation is given in Fig. S2. The results show no discernible evidence of surface bands along the GM direction on the Fermi level near α band, but one dispersive hole surface band located at around -0.25 eV above the δ band (see more details in Supplementary Materials section 2). In a previous study, a flat surface band was linked to the bulk nodal line⁵⁷. However, even if such surface states exist, our computational analysis reveals that they are topologically unprotected and overlap with the bulk band projection, basically extending the wavefunction of the same bulk flat band onto the top layer. This observation suggests that bulk spectra are included in the α band. We explain the vanishing of the α band at soft X-ray area by a slight chemical potential shift between the surface and bulk regions caused by the band bending effects near the surface. Upon detailed comparison, we found that a calculated band structure with energy shift upward of 0.36 eV agrees well with the UV-APRES data. Meanwhile, a 0.41 eV upward shift of the calculated bands is necessary to align with the soft X-ray ARPES data, where the flat α band is just above the Fermi level. On the other hand, detailed analysis about the polarization dependent data (see Supplementary Materials section 2, and Fig. S3) shows that along GM direction, one surface band is strengthened under LV polarization while bulk bands are apparent under LH polarization. Circular polarization, on the other hand, has both surface and bulk states along the same cut.

In the paramagnetic phase, the electron bands crossing the Fermi level (E_F) and centered around G ( and bands) are initially double degenerate. The double degeneracy does not lift in the whole BZ as illustrated in Fig. S4. On the other hand, this degeneracy is lifted when the long-range magnetic order begins to emerge. In Fig. 2g,h,i, we experimentally observed band splitting between and bands near the E_F along the GM direction. It indicates the formation of long-range magnetic order and provides evidence for the requirement of altermagnetism in RuO₂.

In the calculations presented in Fig. 1g, considerable spin splitting is shown in GMAZ plane. We investigate the band structure obtained at of 115 eV photon energy to examine the band splitting within this plane. The corresponding Fermi surface and 3D electronic structure are shown in Fig. 3c and d. Figures 3e-j plot experimental results for a set of bands parallel to the GM direction, which are designated as cut 1-6. While Fig. 3k-p show calculated results for the same bands. Along the GM direction, the spin-polarized α band, which is located exactly on the Fermi level, is most isolated in energy space close to the middle of the GM. The Kramer pair band with opposite spin is located at around 1 eV above the Fermi level. Together, the δ band and the α band share the same spin orientation, forming a large, purely spin-polarized energy range spanning from -0.7 eV till above E_F. This property renders RuO₂ an ideal and robust candidate for spintronic applications. Experimental data in Fig. 3e-j show detailed band splitting along cut 2-6, which agree well with the calculations (Fig. 3k-p). Meanwhile, the occupied band exhibits the largest spin splitting up to 1.54 eV in the energy range of -0.7 to -2.5 eV. Fig. S6 shows the experimental data for a wide energy range, revealing the signature of this huge spin splitting. Additionally, Fig. S7 presents a comparison between computations and experimental results along cut 1-6 collected with soft X-rays. The general band dispersion in the experimental data is in agreement with that obtained with UV light, except the bands are broader and the energy resolution is lower. This comprehensive comparison provides solid evidence that the experimental observations and theoretical predictions under the altermagnetic phase are consistent.

Although the band splitting characterized the altermagnetic phase, direct confirmation of the d-wave altermagnetism in RuO₂ requires the measurement of spin-resolved band structure and the associated d-wave spin pattern. To achieve this, we conducted spin-ARPES measurements using a VLEED detector connected to the DA30 Scienta Omicron detector. The data are acquired with LH polarized light to remove the effect of surface bands. Given the unique characteristic of a d-wave spin pattern, wherein the spin direction undergoes a flip after a 90-degree rotation, we carefully chose four points symmetrically around the M point along the GM direction (Fig. 4a-c) in the out-of-plane momentum space within the GMX plane, as indicated in Fig.4r, where the spin-splitting is prominent in the calculation (Fig. 4b,c). The spin-up and spin-down energy distribution curves (EDCs) For the left k point are plotted in Fig. 4f and the zoomed-in view is shown in Fig. 4g. It is shown that the density of states for spin-up is larger than that for spin-down between -0.7 eV and the Fermi level. Fig.4h displays the intensity difference between spin up and down, indicating a spin-up polarization for the and bands that is in good agreement with the prediction as depicted in Fig. 4b (EDC1). Meanwhile, the right point exhibits similar results, as seen in Fig. 4o,p,q. Furthermore, we varied the photon energy to tune the momentum space to the lower points (54 eV) and upper point (95 eV). Fig. 4i,j and k present the corresponding EDCs for spin-up and spin-down of the lower point. As expected, the density of states for spin-down is larger than that for spin-up from -0.7 eV up to the Fermi level. Concurrently, the upper point exhibits a comparable spin polarization, as seen in Fig. 4l,m,n, indicating the spin down polarization. Fig. 4d displays the spin polarization of the four EDCs, which reaches a maximum of around 10%. Notably, the EDCs exhibit a considerable background, leading us to mitigate its impact. To address this, we applied a fitting procedure to the four EDCs, including a linear background and Gaussian peaks multiplied by Fermi Dirac function, as demonstrated in Fig. S8. For both spin-up and spin-down EDCs, the same linear background is subtracted at the same location in order to eliminate the impact of background on the spin differential. The spin polarization is greatly increased by this background subtraction, with the largest polarization up to 20% at both the upper and lower points. By aggregating the spin polarizations from all four points, we obtained compelling and direct evidence supporting the d-wave spin pattern of altermagnetism in RuO₂.

In conclusion, we have successfully conducted a systematic study of RuO₂ by employing synchrotron-based UV ARPES, soft X-ray ARPES, and spin ARPES. Not only have we observed the spin-splitting, but we have also directly witnessed the d-wave spin pattern in the ideal altermagnetic material, RuO₂. As emphasized earlier, altermagnets exhibit great scientific promise and have the potential for diverse applications. Meanwhile, the lack of experimental investigations into the spin-polarized electronic structures of altermagnet RuO₂ underscores the significance of our findings. As the experimentally validated ideal candidate with large spin splitting and with critical temperature much higher than room temperature, RuO₂ not only exhibits novel physical properties but also presents itself as a promising candidate for the development of next-generation spintronic devices in the near future^{41,42,44–46}.

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Single crystals of RuO₂ were grown by the chemical vapor transport method. RuO₂ powder together with the transport agent TeCl₄ (5 mg/cm³) was sealed in an evacuated quartz tube. The quartz tube was then placed into a two-zone tube furnace with 1100 ℃ at the raw material side and 1000 ℃ at the growth side. The reaction was maintained for one week, and millimeter-sized crystals were obtained.

Conventional ARPES measurements were performed at Bloch beam line of MAX-IV synchrotron, the energy and angular resolutions were set to ~20 meV and 0.1°, respectively, and the temperature was set to around 22K; Soft X-ray ARPES measurements were performed at BL25SU beamline of SPring-8 Synchrotron and the energy and angular resolutions were set to ~60 meV and 0.1°, and the temperature was set to around 77K^58,59, respectively; Spin-ARPES measurements were performed at DREAMLINE beamline of Shanghai Synchrotron Radiation Facility (SSRF) and the energy resolution were set to ~100 meV, and the temperature was set to around 11K. The one-direction spin detector measures spin direction parallel to the sample surface aligning with GM direction during the measurements. The samples for all ARPES measurements were cleaved in situ and measured in a vacuum better than 2×10^-10 Torr.

We have calculated the ground state electronic structure of RuO_2. All DFT calculations were performed within the Perdew-Burke-Ernzernhof (PBE)⁶⁰ generalized gradient approximation and projector augmented wave (PAW) method⁶¹ using QUANTUM ESPRESSO package^62,63with pseudopotential from the PSLibrary⁶⁴. The energy cutoff for the plane wave basis is 80 Ry with a charge density cutoff of 400 Ry. A Monkhorst-Pack⁶⁵121218 k-mesh has been used. The electronic correlations on Ru atoms were accounted for using the DFT+U⁶⁶with Ueff = 3 eV. The Wannier functions have been built up using WANNIER90 code⁶⁷. The surface states have been calculated with WannierTools⁶⁸.

Acknowledgements

We acknowledge Prof. Yaobo Huang, Mr. Gexing Qu for help during the spin-ARPES experiments. Funding: This work was supported by Research Grant Council (RGC) via the early career scheme (ECS) with grant number 21304023, the Research Grant Council (RGC) via the Collaborative Research Fund (grant number C6033-22G), the Collaborative Research Equipment Grant (grant number C1018-22E), the National Natural Science Foundation of China (12104379), and Guangdong Basic and Applied Basic Research Foundation (2021B1515130007). C.F. and D.C. thank the European Research Council (ERC Advanced Grant No.742068 `TOPMAT'), the DFG through SFB 1143 (project ID. 247310070) and the Wurzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter ct.qmat (EXC2147, project ID. 390858490) for support. The SX-ARPES measurement was performed under the approval of BL25SU at SPring-8 (Proposal No.2022A2060, 2022B2106 and 2023B1691) We acknowledge MAX IV Laboratory for time on Beamline Bloch under Proposal 20230361. Research conducted at MAX IV, a Swedish national user facility, is supported by the Swedish Research council under contract 2018-07152, the Swedish Governmental Agency for Innovation Systems under contract 2018-04969, and Formas under contract 2019-02496. Author contributions: JZM supervised this project; ZHL, WLL, XL, SYF and JZM performed ARPES experiments with the help of KY; JO, ML and BT; DC and CF synthesized the single crystals; ZHL performed first-principles calculations of the band structure; JZM and ZHL analyzed the data and plotted the figures; JWL contributed to the discussion of this project; JZM and ZHL wrote the manuscript.

Competing interests

The authors declare that they have no competing interests.

Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Materials and additional data related to this paper may be requested from the authors.

There is NO Competing Interest.

SupplementaryMaterials.docx
Supplementary Materials Supplementary Text Sections 1-5. S1. Determining photon energies versus out-of-plane k_z. S2. The nature of a flat band on the Fermi level. S3. SOC and long-range magnetic order. S4. Bulk band splitting in soft X-ray data. S5. EDC fitting of spin resolved data. Supplementary Figures 1 to 8.

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Observation of Giant Spin Splitting and d-wave Spin Texture in Room Temperature Altermagnet RuO2

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