Itinerant ferromagnetism of the Pd-terminated polar surface of PdCoO₂

${\mathrm{P}\mathrm{d}\mathrm{C}\mathrm{o}\mathrm{O}}_2$ is comprised of triangular-lattice metallic Pd planes separated by insulating ${\mathrm{C}\mathrm{o}\mathrm{O}}_2$ layers. This crystal structure has natural cleavage planes above and below each Pd layer: Inequivalent surface terminations would therefore be expected (Fig. 1A). Consistent with this, we find a strong variation in the relative intensity of Co $3p$ and Pd $4p$ core levels measured by X-ray photoelectron spectroscopy (XPS) at different patches of the cleaved sample surface (Fig. 1B). This is correlated with a marked difference in the electronic structure measured by ARPES (Fig. 1 C–E). Across the sample, we find a steeply dispersive state that we attribute as the Pd-derived bulk band (36). In Fig. 1C, our measurements additionally show a pair of massive hole-like bands (Fig. 1C, labeled “S-${\mathrm{C}\mathrm{o}\mathrm{O}}_2$”). These have been assigned previously as surface states from the ${\mathrm{C}\mathrm{o}\mathrm{O}}_2$ termination (36, 37), consistent with our XPS measurements from the same sample region, which yield the greatest ratio of Co $3p$:Pd $4p$ core-level spectral weight.

Measurements from a different patch of the same sample (Fig. 1E) reveal a completely different surface electronic structure, which we describe in detail below. Our XPS measurements exhibit a much larger spectral weight of the Pd than the Co-derived core-level peak for this sample region (Fig. 1B), and we thus attribute these as surface states originating from the Pd-terminated surface (“S-Pd”). We note that for most regions where such states are visible, we observe a superposition of these spectral features with those of the ${\mathrm{C}\mathrm{o}\mathrm{O}}_2$-terminated surface (e.g., Fig. 1D). This indicates a rather limited spatial extent of typical Pd-terminated regions, with a mixed surface termination within our probing light spot area ($\sim \hspace{0.17em}50\hspace{0.17em}\mathrm{\mu }$m diameter). In the following, we show our highest quality ARPES data obtained from a sample that exhibits such a mixed surface terminations. Similar results were obtained from multiple samples.

The measured Fermi surfaces and dispersions from such a mixed surface termination are shown in Fig. 2. The ${\mathrm{C}\mathrm{o}\mathrm{O}}_2$-terminated surface states form hexagonal and nearly circular hole-like Fermi surfaces about the Brillouin zone center, split-off by a surprisingly large Rashba-like interaction arising due to a large energy scale of inversion symmetry breaking at this surface (37). We do not consider these further here.

Two larger electron-like Fermi surfaces with sharp linewidths are also visible. From comparison with previous experiments (32, 36), we attribute the largest of these as the Pd-derived bulk Fermi surface. The other has a similar topography but slightly smaller average $k_F$ and has greater spectral weight when measured in the second Brillouin zone (Fig. 2A). This band is also evident as a steeply dispersing state in our measured dispersions (Fig. 2B) and appears to be an approximate replica of the bulk state but shifted toward the Fermi level by $\sim \hspace{0.17em}430$ meV. In principle, it is not surprising for a polar charge to change the binding energy of states localized at a polar surface. Indeed, for the ${\mathrm{C}\mathrm{o}\mathrm{O}}_2$-terminated surface, the absence of Pd atoms above the topmost ${\mathrm{C}\mathrm{o}\mathrm{O}}_2$ block leads to a formal valence of $3.5+$ rather than $3+$ for the surface Co layer, corresponding to an experimentally observed (35, 37) effective hole-doping of $\approx \hspace{0.17em}0.5$ holes per unit cell. Similar electrostatic arguments for the Pd-terminated surface (35) would suggest a formal valence of Pd^0.5+ at the surface, compared with Pd¹⁺ in the bulk. This, however, should manifest as a surface doping of $\approx \hspace{0.17em}0.5$ electrons per unit cell, seemingly in contrast to the hole-doped copy of the bulk band observed experimentally here.

Moreover, rather than a simple rigid shift of the bulk Pd-derived valence band, our measured dispersions (Fig. 2 B and C) reveal a much richer surface electronic structure. As well as the steep band discussed above, Fig. 2B shows a flat-topped band located $\sim \hspace{0.17em}100$ meV below the Fermi level and a pair of electron-like bands crossing $E_F$ near the Brillouin zone center (all labeled S-Pd in Fig. 2B). The outer of these electron pockets can also be seen in our Fermi surface measurements (Fig. 2A), while the innermost band is not clearly observed at the photon energy used. Additionally, multiple fully occupied bands are found at the Brillouin zone face $\overline{\text{M}}$-point (Fig. 2C).

We demonstrate below that such a rich multiband electronic structure of the Pd-terminated surface of ${\mathrm{P}\mathrm{d}\mathrm{C}\mathrm{o}\mathrm{O}}_2$ is the result of an instability to itinerant surface ferromagnetism. Fig. 3 shows the dispersions of the bulk and Pd-derived surface states extracted from the measurements shown in Fig. 2B as well as from equivalent measurements performed using a different photon energy and/or light polarization. These different experimental conditions lead to varying transition matrix elements, selectively enhancing the visibility of different features (see, for example, SI Appendix, Fig. S1), thus ensuring that we capture the dispersions of all bands accurately. If we assume that these surface states are spin-degenerate and extract their Luttinger count from our experimentally measured Fermi surfaces, we find an electron count that is unphysically high. In contrast, if we treat them as spin-polarized, we find a surface carrier density that is $0.55\pm 0.03$ electrons per unit cell higher than that of the bulk, which is close to the additional 0.5 electrons per unit cell that would be expected from the polar surface charge. We note that spin-polarized states are generically allowed at surfaces, where the breaking of inversion symmetry can lead to a momentum-dependent spin-splitting of the Rashba-type (38). If this was the origin of the surface state spin polarization here, however, then the spin-splitting should vanish at the zero-momentum $\overline{\mathrm{\Gamma }}$-point, where time-reversal symmetry enforces a Kramers degeneracy. In contrast, the clear splitting of spin-polarized bands that we observe at the $\overline{\mathrm{\Gamma }}$ point (Figs. 2B and 3) indicates that time-reversal symmetry is broken here [i.e., $E\left(k=0,\uparrow \right)\ne E\left(k=0,\downarrow \right)$]. Moreover, a Rashba-like origin of the observed spin-polarized bands would be incompatible with the large ($\approx \hspace{0.17em}430$ meV) energy splitting of the steep bands that we observe, which is more than three times the atomic spin-orbit coupling of Pd.* Together, these observations therefore rule out a Rashba origin of our observed spin-polarized surface states; instead, they provide conclusive empirical evidence that the Pd-terminated surface of ${\mathrm{P}\mathrm{d}\mathrm{C}\mathrm{o}\mathrm{O}}_2$ is ferromagnetic.

Although the above conclusion is independent of theory, it is also reinforced by explicit calculations (Fig. 4; see Materials and Methods), which further demonstrate how the surface ferromagnetism arises as a result of an intrinsic Stoner-like instability driven by the polar surface charge. This mechanism is shown in Fig. 4. The band that forms the bulk Fermi surface, while steep at the Fermi level (34, 39), becomes much flatter above $E_F$ (evident, for example, along $\overline{\mathrm{\Gamma }}-\overline{K}$ in the bulk $k_z$-projected calculations shown in Fig. 4A). Compensating the surface polarity should lead to electron doping of the surface layer, as discussed above. This will create a surface copy of the bulk band. Its flat band top and the high DOS associated with it are pushed downward toward the Fermi level, as evident in our nonmagnetic supercell calculations shown in Fig. 4B. From a spin-polarized calculation (Fig. 4C), we find that the DOS at $E_F$ is sufficiently high to trigger a Stoner transition. The band at the Fermi level therefore exchange splits into a pair of spin-polarized bands (labeled $\alpha$ and $\beta$), which can be identified as the experimentally observed flat ($\alpha$ in Fig. 3) and steep ($\beta$) states. Consistent with previous DFT calculations (35), we also find how additional near-$E_F$ surface states inherit a similar exchange splitting, forming the majority-spin and minority-spin $\gamma$ and $\delta$ bands observed at the Brillouin zone center in our ARPES measurements (Fig. 3).

We note that the surface magnetism observed here is thus qualitatively different to the recent observations of ferromagnetism arising due to uncompensated moments at the surface of, for example, antiferromagnetic (AF) ${\mathrm{E}\mathrm{u}\mathrm{R}\mathrm{h}}_2{\mathrm{S}\mathrm{i}}_2$ (40). Instead, it reflects an intrinsic instability of the underlying electronic structure that can be triggered by a pronounced self-doping of the system in response to its polar surface charge. From our extracted Luttinger areas, we estimate a total magnetization, $M=\left(N^{\uparrow }-N^{\downarrow }\right){\mu }_B=\left(0.59\pm 0.03\right){\mu }_B$ per unit cell. As the wavefunctions of the surface states are mostly localized on the surface Pd layer (see SI Appendix, Fig. S2), this layer also hosts the largest moment (found to be 0.46 ${\mu }_B$ in our calculations; see Materials and Methods). Nonetheless, finite orbital mixing with the subsurface ${\mathrm{C}\mathrm{o}\mathrm{O}}_2$ block means that the near-surface Co also inherits a finite, but much smaller, moment, found to be 0.12 ${\mu }_B$ in our calculations.

Finally, we show our measurements of the Pd-terminated surface of the sister compound, ${\mathrm{P}\mathrm{d}\mathrm{C}\mathrm{r}\mathrm{O}}_2$ (Fig. 5). As in ${\mathrm{P}\mathrm{d}\mathrm{C}\mathrm{o}\mathrm{O}}_2$, the bulk band, which is also responsible for the high bulk conductivity of ${\mathrm{P}\mathrm{d}\mathrm{C}\mathrm{r}\mathrm{O}}_2$ (41), is evident as a steeply dispersing state with the largest $k_F$. We again observe a copy of this state shifted toward the Fermi level (labeled “$\beta$” in analogy with the ${\mathrm{P}\mathrm{d}\mathrm{C}\mathrm{o}\mathrm{O}}_2$ case discussed above), an $\alpha$-band with its flat top at $\approx \hspace{0.17em}150$ meV below the Fermi level and, in this case, a single electron-like ($\gamma$) band crossing $E_F$ near the Brillouin zone center. This electronic structure is consistent with previous measurements of the chromate (42). Moreover, there is an almost one-to-one correspondence with the surface states that we observe for the Pd-terminated surface of ${\mathrm{P}\mathrm{d}\mathrm{C}\mathrm{o}\mathrm{O}}_2$, allowing us to again assign the $\alpha$ and $\beta$ bands as an exchange-split pair in ${\mathrm{P}\mathrm{d}\mathrm{C}\mathrm{r}\mathrm{O}}_2$, and the $\gamma$-band as a spin-majority band, with its spin-minority partner being unoccupied. Excitingly, while the ${\mathrm{C}\mathrm{o}\mathrm{O}}_2$ layer in ${\mathrm{P}\mathrm{d}\mathrm{C}\mathrm{o}\mathrm{O}}_2$ is band insulating and nonmagnetic, the ${\mathrm{C}\mathrm{r}\mathrm{O}}_2$ layer in bulk ${\mathrm{P}\mathrm{d}\mathrm{C}\mathrm{r}\mathrm{O}}_2$ is Mott insulating and hosts local-moment AF order on the Cr sites (41). Combined with the analysis of ${\mathrm{P}\mathrm{d}\mathrm{C}\mathrm{o}\mathrm{O}}_2$ presented above, however, our measurements indicate that ${\mathrm{P}\mathrm{d}\mathrm{C}\mathrm{r}\mathrm{O}}_2$ nonetheless also supports ferromagnetism at its Pd-terminated surface.

Single crystals of ${\mathrm{P}\mathrm{d}\mathrm{C}\mathrm{o}\mathrm{O}}_2$ and ${\mathrm{P}\mathrm{d}\mathrm{C}\mathrm{r}\mathrm{O}}_2$ were grown by a flux method in sealed quartz tubes (26, 48). These were cleaved in situ at the measurement temperature of $T\sim \hspace{0.17em}10$ K. ARPES measurements were performed at the I05 beamline of Diamond Light Source using a Scienta R4000 electron analyzer and variable light polarizations with photon energies between 60 and 120 eV (49).

DFT calculations were performed using the full-potential local-orbital (FPLO) code (www.fplo.de, version fplo14.00-47) (50–52), using the Perdew–Burke–Ernzerhof (52) formalism and including spin–orbit coupling. The surface electronic structure was calculated using a symmetric slab containing 9 Pd layers, with a vacuum gap of 15 Å. For the inner layers of the slab, the bulk experimental crystal structure (34) was used, while the surface atomic positions were relaxed. We note that, for the unrelaxed surface, we do not stabilize a magnetic state. By fully relaxing the topmost Pd layer to minimize forces on the surface Pd atoms, we obtain a sizable increase of the Pd–O distance and find a stable ferromagnetic state with a magnetic moment of 0.55 ${\mu }_B$ per unit cell. The moment is predominantly carried by the surface Pd (0.40 ${\mu }_B$) with minor contributions from the subsurface Co (0.10 ${\mu }_B$) and O (0.04 ${\mu }_B$) just below the surface. As shown in SI Appendix, Fig. S3, the peak at the Fermi level deriving from the Pd-related surface state becomes considerably narrower for the relaxed surface with the increased Pd–O distance compared with the unrelaxed one. This leads to a strong increase (by about 50%) of the DOS ($E_F$) and the respective Pd 4$d$ DOS, strongly favoring a magnetic splitting of these states in a Stoner picture. The stability of the ferromagnetic surface state slightly increases if not only the surface Pd but also the ${\mathrm{C}\mathrm{o}\mathrm{O}}_2$ layer nearest the surface is relaxed, increasing the magnetic moment to 0.62 ${\mu }_B$ per unit cell (Pd: 0.46 ${\mu }_B$; Co: 0.12 ${\mu }_B$; O: 0.05 ${\mu }_B$). These structural relaxations are used for the calculations presented in the main text. For the DOS shown in Fig. 4, a boxcar smoothing of 25 meV has been used to remove unphysical spikes appearing in the calculated DOS for the supercell.

Experimentally, we find that the surface states at $\overline{\mathrm{M}}$ are all located below $E_F$ (Fig. 2C), unlike for our supercell calculations. In general, we note that the exact ordering of bulk and different surface features in our calculations is sensitive to application of a Hubbard U term on the Co sites as well as an intrinsic off-stoichiometry of the slab due to the polar nature of the structure. Taking into account the strong Coulomb repulsion at the Co-site in a mean field way, applying DFT + $U$ [$U$ = 4 eV (37)], the total moment remains almost unchanged, but the contribution of Pd increases sizeably (Pd: 0.54 ${\mu }_B$; Co: 0.05 ${\mu }_B$; O: 0.04 ${\mu }_B$). As a consequence, this leads to a larger splitting in energy of the Pd-dominated surface states, as shown in SI Appendix, Fig. S4, but does not qualitatively influence any of the conclusions presented here.