Their results show that spin-splitting in delafossite oxides reaches as much as the full strength of the spin-orbit coupling of the relevant orbitals. This highly surprising result is a consequence of unusually strong symmetry breaking at the surface, enabled by the crystal structure. This insight could be used to design new materials exhibiting record-breaking spin-splitting.
Delafossites are layered oxide compounds of the general formula ABO2, in which triangularly coordinated noble metal (A, A=Pt,Pd) layers are separated by transition metal oxide (BO2, B=Co, Rh, Cr) blocks (Fig. 1a). The platinum and palladium based delafossite metals have a single, fast Pt/Pd based band crossing the Fermi level, and making a hexagonal Fermi surface. Their room temperature conductivities are among the highest of any metal, and higher per carrier than that of elemental copper1. Their low temperature mean free paths are extremely long, reaching as much as 20 μm (~105 lattice spacings) in as-grown PdCoO2. The long mean free path has led to the observation of a number of anomalous transport properties, including hydrodynamic electron flow.
While this fascinating transport is taking place in the noble metal, the transition metal oxide layer of PtCoO2 and PdCoO2 is insulating. This is because the Co3+ is in the d6 configuration, with the t2g-like orbitals completely occupied and eg-like orbitals empty. The bulk compound can thus be thought of as a natural heterostructure, with alternating metallic and insulating layers.
The qualitative observation of the spin-split states on a surface is thus not unexpected, however the size of the splitting observed is surprising. To see this, one needs to compare the size of the splitting in the delafossites with other well-known systems. The splitting in momentum in the delafossite reaches as much as 0.13 Å-1, comparable to 0.1 Å-1 in BiTeI and 0.26 Å-1 in Bi/Ag surface alloys4. Those are some of the largest momentum splittings known, and it is no accident they are typically found in bismuth containing compounds, as the atomic spin-orbit coupling (SOC) of bismuth 6p orbitals is as large as 1500 meV. The spin-orbit coupling of Co is only 70 meV, 20 times weaker! The observation of such large splitting in the delafossites is thus highly surprising, and requires further explanation.
To understand the large splitting it is necessary to realise that the atomic spin-orbit coupling is not the only relevant energy scale; rather the size of the splitting is determined by the relative magnitude of the spin-orbit coupling and inversion symmetry breaking (ISB) energy scales, with the smaller scale limiting the splitting. In the vast majority of spin-split states the ISB scale is small, limiting the splitting to a moderate fraction of the atomic spin-orbit coupling. In delafossites the splitting at some points of the zone becomes comparable to the atomic spin-orbit coupling (Fig. 3), indicating that the inversion symmetry breaking energy scale is larger than the spin-orbit one, enabling the maximum possible spin splittings to be achieved. Understanding the mechanism of this symmetry breaking is highly relevant for design of future spintronic materials.
The key point is the structure of the top CoOp2 player (Fig. 1b). Co electrons can hop from one Co to another either via the oxygen in the layer above (O1), or via the oxygen in the layer below (O2). In the bulk these two paths are equivalent, however the top oxygen has no noble metal above it. This causes an on-site energy shift between the two oxygens, and makes the hopping path through the surface oxygen much more likely. The inversion symmetry breaking is no longer a small perturbation of the dominant kinetic hamiltonian; rather the kinetic hamiltonian itself becomes asymmetric. This asymmetry of hopping integrals is estimated to be about 40%, or about 150 meV. This is more than twice the atomic spin-orbit coupling of cobalt, allowing the spin splitting to assume the full atomic spin-orbit coupling strength, as observed.
What is more, this kinetic mechanism of inversion symmetry breaking has a potential for achieving truly large spin splitting scales in compounds containing heavier ions with larger spin-orbit coupling. Usually increasing the atomic spin orbit coupling causes the splitting to be limited by the smaller inversion symmetry breaking scale, however the kinetic inversion symmetry breaking is proportional to the bandwitdh, and will also grow as heavier atoms with larger orbitals are used. This suggests a route to stay in the 'strong inversion symmetry breaking' limit even when the absolute size of spin orbit coupling is large. A proof that this is indeed possible is found by measuring the surface electronic structure of previously unstudied PdRhO2. Similar surface states are observed to the cobalt-based compound, but the energetic spin splitting in the rhodate reaches a ~2.5 times larger value of 150 meV, a record for an oxide, reflecting the 2.5 times larger atomic spin-orbit coupling. This therefore confirms the kinetic energy-coupled inversion symmetry breaking mechanism.
To conclude, it has been shown here that a 'recipe' for achieving large spin-splitting is to use heavy elements with large atomic spin orbit coupling and large orbital overlaps in a structure where the preferred hopping paths are out of plane. Due to the loss of bonding at a surface such kinetic energy will become asymmetric, and introduce a large inversion symmetry breaking energy scale, allowing the full spin-orbit coupling energy to be utilised in spin-splitting surface states.
Related publication: Sunko V, Rosner H, Kushwaha P, Khim S, Mazzola F, Bawden L, Clark OJ, Riley JM, Kasinathan D, Haverkort MW, Kim TK, Hoesch M, Fujii J, Vobornik I, Mackenzie AP, King PDC. Maximal Rashba-like spin splitting via kinetic- energy-coupled inversion-symmetry breaking. Nature 549, 492–496, doi:10.1038/nature23898 (2017).
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