We report the first case of the successful measurements of a localized spin antiferromagnetic transition in delafossite-type PdCrO2 by angle-resolved photoemission spectroscopy (ARPES). This demonstrates how to circumvent the shortcomings of ARPES for investigation of magnetism involved with localized spins in limited size of two-dimensional crystals or multi-layer thin films that neutron scattering can hardly study due to lack of bulk compared to surface. Also, our observations give direct evidence for the spin ordering pattern of Cr3+ ions in PdCrO2 suggested by neutron diffraction and quantum oscillation measurements, and provide a strong constraint that has to be satisfied by a microscopic mechanism for the unconventional anomalous Hall effect recently reported in this system.
Angle-resolved photoemission spectroscopy (ARPES) has been one of the most powerful tools to investigate the electronic structure of crystalline solids. With the help of the third generation synchrotron facilities and the electron spectrometers with angle resolved mode, ARPES now can be utilized for the study of the magnetic structure as well as the electronic one1. In particular, ARPES has revealed various kinds of magnetic fluctuations in high-temperature superconductors and magnetic thin films, providing key information to understand the underlying physics2,3. However, there is the last piece of the puzzle left in ARPES for magnetic structure study: observing a long range ordering of localized magnetic moments. Actually, Fermi surface (FS) folding induced by many kinds of order other than localized magnetic moments has been observed by ARPES. The most probable reason for this exception is that the potential difference by localized moment ordering is so small that the band electrons can hardly feel the potential change, resulting in negligible or non-detectable changes in the photoemission signals. Here, at long last, we report the first case of the successful ARPES measurements of a localized spin antiferromagnetic transition in delafossite-type PdCrO2. Our observations give direct evidence for the spin ordering pattern of Cr3+ ions in PdCrO2 suggested by neutron diffraction4,5 and quantum oscillation measurements6, and provide a strong constraint that has to be satisfied by a microscopic mechanism for the unconventional anomalous Hall effect (UHAE) recently reported in this system7. Also, this demonstrates a new pathway for investigation of magnetism in limited size of two-dimensional crystals or multi-layer thin films that neutron scattering can hardly study due to lack of bulk compared to surface.
Metallic delafossite-type oxides (Fig. 1a) ABO2 (A = Pd and Pt; B = Cr, Co, and Rh) are famous for their good conductivity8. In particular, the PdCoO2 exhibits the best conductivity in the normal state oxides. The conductivity is even better than that of Pd metal, and so this case has been scrutinized at length by ARPES study9, x-ray absorption spectroscopy study10, ab-initio band structure calculations11,12, and thermo-transport study13. The large two-dimensional FS consisting of the fast conduction electrons with a long life time was confirmed to be the origin of the anisotropic high conductivity in the previous studies. The interesting and noticeable features for PdCoO2 are two-fold; the metallic Pd layers with the large sp-like band dispersions and the insulating triangular lattice CoO2 with the low-spin (S = 0, 3d6) configuration. When the Co3+ ion in the trigonally distorted octahedron MO6 (M = transition metal ion) is replaced by Cr3+ ions in the same crystal structure, the only difference with respect to the electronic structure lies in the spin quantum number of the t2g manifolds in the transition-metal ions. It simply changes the quantum number from S = 0 (full filling) to S = 3/2 (half filling), i.e. from non-magnetic CoO2 to magnetic CrO2 triangular lattice. In this case, since the Cr ions form a two-dimensional triangular lattice (2DTL) with the antiferromagnetically interacting local spins, the system becomes much more attractive. As is well known, this kind of spin geometry would produce the magnetic frustration and the possible quantum liquid ground state14,15. Further, this system has highly metallic layers, and so it provides a rare opportunity to investigate the interactions between the frustrated local magnetic moments in a 2DTL and the itinerant electrons in an adjacent layer. Indeed, recent transport studies have reported that an unconventional anomalous Hall effect (UAHE) was observed in this system, revealing an exotic behavior of an antiferromagnetic 2DTL with metallic layers6,7. However, since the UAHE reported in this system requires an incompatible spin ordering to the known antiferromagnetic ordering, more refined experimental and theoretical characterization is essential to understand the underlying physics.
Figure 1b shows a top view of an upper Cr (big yellow balls) layer, a Pd (red balls) layer, and a lower Cr (small yellow balls) layer in the delafossite structure. When the system is antiferromagnetic below TN = 37.5 K, the neutron powder diffraction study suggested that the spin directions of the Cr ions are ordered into a non-collinear 120° structure. In this spin structure, the Cr ions are classified into three kinds (denoted as A, B, and C in Fig. 1b) with respect to their spin directions. The intensity analysis of the magnetic Bragg peaks suggested that the three spin vectors lie in a plane containing the c-axis4,5. The adjacent Cr layer has the same structure as denoted as A', B', and C' in Fig. 1b, but the exact phase difference between the layers is not clear. The crystal structure and the suggested magnetic phase for our samples were checked by the X-ray diffraction (XRD) and neutron diffraction (ND) measurements above and below TN as shown in Figs. 1(c) and 1(d). The superstructure peaks around 26° in the low temperature (T = 20 K) ND pattern are not seen at the paramagnetic phase (T = 60 K), indicating that the origin of the peaks is spin ordering. Meanwhile, there are neither superstructure peaks at the corresponding angles () nor differences between the XRD patterns above and below TN except for the normal peak position shifts due to thermal lattice expansion. This shows that the symmetry of the crystal structure does not change across the magnetic transition.
The S = 3/2 spin quantum number of the Cr3+ ions and the metallic behavior of the Pd layers are revealed in the Cr 2p → 3d x-ray absorption spectra (XAS) and the Pd 3d x-ray photoemission spectrum (XPS) as shown in Fig. 2. The XAS spectra drawn with red line (T = 60 K) and with blue line (T = 28 K) in Fig. 2a were obtained with the photon polarization vector normal to the ab-plane of the single crystalline samples. The electron configuration and the spin state can be determined by comparing the XAS spectrum with that of a reference system since a transition metal 2p → 3d absorption spectrum is a kind of fingerprint for the electronic energy structure of the transition metal ions16. If we compare our XAS spectra with that of Cr2O3 and of CrO2, each of which represents a trivalent S = 3/2 Cr ion and a tetravalent S = 1 Cr ion, respectively17, the valence and the spin state of the Cr ions in PdCrO2 are clearly determined to be trivalent and S = 3/2 irrespective of the magnetic phase. The metallic Pd layers can be checked by analyzing the Pd 3d XPS spectrum as shown in Fig. 2b. As is the case of iso-structural PdCoO2, the asymmetric line shape is prominent, indicating that the photoholes are well screened by the conduction electrons through the creation of electron-hole pairs across the Fermi level9. The quantitative line shape analysis with the Doniach-Šunjić (DS) model18 gave the asymmetric parameter value of α = 0.26. This value is a little larger than that of Pd metal but equal to that of PdCoO2, and lies in a reasonable region9,19. These spectroscopic observations support the idea that PdCrO2 is an ideal system to investigate the detailed balance between the itinerant electrons and the adjacent localized spins in the 2DTL.
A phase transition involving the superstructure was directly observed by our ARPES measurements as shown in Fig. 3a. At the temperature of 100 K (bottom), we observed only one large electron-like hexagonal FS in a Brillouin zone (BZ). Meanwhile, at T = 20 K (top), it is clearly seen that the extra FS features other than the original FS appear in our constant energy ARPES intensity map at the Fermi level (EF). Since PdCrO2 is known to have an antiferromagnetic phase below TN = 37.5 K, these new FSs are naturally attributed to the folded FS's arising from the reduced BZ of an antiferromagnetic phase. If this is the case, it is a milestone in ARPES because no ARPES study has reported a successful observation on a long range ordering of localized spins in a magnetic system. However, to justify this conclusion, the other possible reasons must be checked. First, a structural transition accompanied by the antiferromagnetic transition is suspected, but this possibility can be excluded by our temperature-dependent ND/XRD as explained in Figs. 1(c) and 1(d). The temperature-dependent diffractions show only a normal thermal expansion behavior and no abrupt peak shift around TN. Second, charge ordering or charge density wave is a candidate, but this can be also ruled out by the temperature-dependent X-ray diffraction study that shows no extra-peaks below TN. Furthermore, we did not observe any signal indicating mixed valency of the cations or charge disproportionation in our temperature-dependent core-level XPS study, as partly shown in Fig. 2. Last, a temperature-dependent surface reconstruction should be checked for a possible origin since ARPES is a surface-sensitive probe. As to this possibility, however, a recent ARPES study on PdCrO2 has reported totally different and temperature-independent surface FSs from ours as well as almost identical bulk FSs to ours20. This clearly shows that our temperature-dependent FS folding is not relevant to a surface reconstruction. Thus, the band folding in our ARPES data cannot but be relevant to the magnetic transition.
The electronic structure of PdCrO2 measured by ARPES in the paramagnetic phase is very similar to that of iso-structural non-magnetic PdCoO2 except for the absence of surface states9,12. Both systems commonly have a two-dimensional FS with a rounded hexagonal cross section in a BZ. In Fig. 3b, we show the ARPES image cut along the vertical dotted line in Fig. 3a bottom and the calculated band structure along the Γ-K line using the linearized augmented plane wave method with local orbitals (LAPW + LO) in the generalized gradient approximation (GGA). For simplicity, the calculation was set in the rhombohedral unit cell assuming a ferromagnetic ordering, which produces the over-split Fermi points, P1 and P2. Except for the spin-split feature, the calculated band structures are quite consistent with the ARPES results. The conduction band is very dispersive, indicating that its characters are mostly hybridized 4d-5s orbitals of Pd, as in the case of PdCoO2. The carrier velocity given by at point P0 estimated in the ARPES data is , while the velocities at point P1 and P2 in the calculation are 3.4 and , respectively. The faster carrier velocity in the ARPES data than that in the band calculation probably originates from the underestimation of the Pd 5s-4d orbital mixing in the density functional theory by neglecting the correlation effect of Cr 3d electrons. The inverse mean free path (1/lMFP ~ Δk = 0.021(1) Å−1) and the relaxation time (τ = 1/vkΔk = 5.7(4) × 10−15 s) are comparable to those of PdCoO2.
The electronic structure of the antiferromagnetic phase does not look much different from that of the paramagnetic phase except for the folded bands. Also, the folded bands are almost identical to the original ones at least near the Fermi level as can be compared in Fig. 3c, where they are displayed along the Γ′-ΓAFM line in the antiferromagnetic phase of Fig. 3a top in three different manners, namely the ARPES intensity image (bottom), its waterfall plot derived from the momentum distribution curves near the Fermi level (middle), and the fitted curve at the Fermi level (top), respectively. For example, if we compare the velocity vx of the folded conduction band at Pfold in Fig. 3c to that of PAFM and of PPM in Fig. 3a, all values fall within . The change of the FS topology in the antiferromagnetic phase can be also readily checked in Fig. 3c. The dispersion of the folded conduction bands (the red dots and the yellow inverse triangles in each panel) along the Γ′-ΓAFM line produces one hole-like hexagonal FS and two electron-like triangular FSs in a BZ, as shown in Fig. 4.
The detailed analysis on the cross-sectional FS areas in the antiferromagnetic phase is represented in Fig. 4. Based on the ARPES-measured FS image, we determined the size of a rounded hexagonal FS δ of paramagnetic phase and translated it by the reciprocal lattice vectors to reproduce the folded FSs. The diagonal length and the height in Fig. 4 are 1.94(5) and 1.76(4) Å−1, respectively, which are ~5% reduced in comparison with those of PdCoO2. The cross-sectional area of each FS branch is α = 0.083(9), β = 0.35(2), = 1.05(4), and δ = 2.66(6) Å−2, respectively. These are remarkably consistent with the recent quantum oscillation measurements, where the frequency for each FS branch is α ~ 0.8, β ~ 3.3, ~ 10.5, and δ ~ 27.5 kT, respectively6.
Now that the observed FS folding clearly originates from the localized spin antiferromagnetic ordering, it is important to understand why and how this observation was realized in our ARPES measurements. We think that the key factor is a considerable hybridization between the localized Cr 3d orbitals and the itinerant Pd 4d orbitals through the oxygen anions. This feature has been also revealed in the recent quantum oscillation study6. In the antiferromagnetic phase, in addition to the normal hopping between Pd ions, the conduction electron in the O-Pd-O dumbbells has spin-dependent hopping paths through three kinds of Cr ions A(A'), B(B'), and C(C') in the adjacent upper(lower) layer, as shown in Fig. 1b. Taking into account both the normal hopping and the magnetic hopping, each strength of which is parameterized by t0 and t1 for the magnetic super-cell, we considered a single layer model crystal to set up the tight-binding Hamiltonian described in detail below.
Figure 5 shows the band structure of our model crystal. When the magnetic hopping is absent (t0 = 1, t1 = 0), the model crystal has one rounded hexagonal FS that resembles the ARPES-measured FS in the paramagnetic phase as shown in Fig. 5a. When turning on the magnetic hopping, the folded bands get their intensities and the antiferromagnetic order of the Cr layers is reflected in the FS map of Fig. 5b and c. The optimal resemblance is obtained at t1 ~0.4 in this model. Here, we point out that the absolute value of t1 can be different from the true value to some degree because of the simplification of our TB model, but it shows a trend consistent with the essential features observed in the ARPES measurements. The success of this model strongly implies that the hybridization between the conduction electrons and the localized spins plays a key role in observing the ordering of the localized spins in ARPES. Actually, this also explains why the surface states observed by Sobota et al. do not appear in our ARPES data20. The surface states appear in the ARPES data obtained with photons, but are absent in our ARPES data. The photoionization cross section ratio of Cr 3d to Pd 4d is for 55 eV photons and for 120 eV photons, respectively21. Thus, the surface states with mostly Pd 4d characters appear much intensely in the 55 eV ARPES data. By the same argument, the folded bands in the antiferromagnetic phase are strongly hybridized with the Cr 3d orbitals, so they appear only in our 120 eV ARPES data.
The FS folding due to the antiferromagnetic superstructure observed in ARPES provides a strong constraint on the microscopic origin of the UAHE recently reported in this system. Until now, two studies have reported the UAHE, but the details are quite diverse yet6,7. Takatsu et al. found an unusual nonlinear field dependence of the Hall resistivity curve, ρxy(H), with a hump under a magnetic field of 1–3 T at temperatures below T* ~ 20 K, and interpreted it as an UAHE. In order to explain the nonlinearity of ρxy(H) in the frame of the scalar spin chirality mechanism22,23, they proposed another magnetic structure with a broken periodicity below T*. Meanwhile, Ok et al. measured the Hall resistivity up to H = 32 T, and found nonlinearity both at H ~ 2 T, and near the Néel temperature TN. The latter was interpreted as an UAHE possibly originating from the finite scalar spin chirality induced by an external high magnetic field at the temperature region with antiferromagnetic spin-fluctuations, but the former as a magnetic breakdown effect. Our ARPES measurements directly evidenced that the periodicity is sustained at least down to T = 20 K. Here, it is worth addressing that the ARPES data was obtained over the whole momentum space, so that the confidence on the new ordering pattern is incomparable to that from other data measured at single momentum point such as diffraction patterns. Combined our ARPES result with the absence of any abrupt behavior in the magnetization curve M(H) as a function of applied magnetic field for PdCrO27, the UAHE at the high magnetic field region near TN is more consistent.
The successful observation of the local spin order by ARPES also suggests a novel method for magnetic structure study of a small two-dimensional crystal such as AgNiO2, Ag2MO2 (M = Cr, Mn, Ni), Fe1.3Sb or an epitaxial thin film with localized magnetic moments for which neutron scattering is usually not applicable due to the low efficiency. Our experimental results demonstrate that ARPES can be more suitable for these cases if a few metallic layers are covered on the surface with balanced hybridization to the local magnetic moments.
The single crystals of PdCrO2 were grown by the NaCl flux method using powder samples described in the literature24. The powder samples were synthesized by the following metathetical reaction: Pd + PdCl2 + 2LiCrO2 → 2PdCrO2 + 2LiCl. In this step, the precursor LiCrO2 was prepared by the solid state reaction method, starting from the stoichiometric mixture of Li2CO3 and Cr2O3 at 850°C for 24 hours. The obtained crystals are silvery hexagonal plates, and the typical size is 1.5 × 1.5 × 0.1 mm3. The XRD/ND experiments were performed on the powder samples to check the crystallographic/magnetic phases. The XRD patterns were obtained at the 3A beamline of the Pohang Light Source (PLS) with . The ND patterns were obtained at the high-resolution powder-diffraction neutron beamline of HANARO research reactor, Korea. The neutron beam was monochromatized to 1.833 Å by using single-crystal Ge (331) plane.
The XAS measurements were performed at the 2A beamline of the PLS. The chamber pressure, resolving power of the photon beam, and the sample temperature were kept at ~10−9 Torr, 2500, and below 20 K, respectively. After the single crystalline samples were cleaved in situ, the absorption spectra were recorded in the total electron yield mode and were normalized by the photon flux.
The photoemission experiments were performed at the 4A1 beamline of the PLS with a Scienta SES-2002 electron spectrometer25. Linearly polarized light was incident on the samples with polarization vector parallel to the kx-direction in the ARPES data. The photon energy was set to for ARPES, and to for XPS. The total energy resolution is ~60 meV, and the momentum resolution was set to be ~0.01 Å−1. The crystals were cleaved in situ by the top post method at the temperature of 100 K under the pressure of ~7.0 × 10−11 Torr. Due to the quasi two-dimensional structure of the crystals, the cleaved surface was shiny and well oriented. After cleaving the samples, we collected the ARPES data in the paramagnetic phase at 100 K, and then lowered the sample temperature to 20 K for the antiferromagnetic measurements. After collecting the low temperature ARPES data in a full range of k-space, we raised the sample temperature to 100 K again to check the disappearance of the folded band in a partial range of k-space. Unlike the PdCoO2 case9, we did not apply the thermal cycling process to PdCrO2 since the surface states reported in Ref. 20 were not observed in our ARPES measurements with photon energy.
Tight-binding model hamiltonian
As in the case of the PdCoO2 band structure, the bands near Fermi level mainly consist of non-magnetic Pd and Pd 5 s hybridized orbitals10,12. The Cr moments order as 120° super cell making six Cr spin-dependent hopping paths (see Fig. 1b), but the total moments of Cr around a given O-Pd-O dumbell sum up to zero. Therefore, the magnetic moment of the Pd layer is constraint to zero by symmetry. Here, we ignored all the higher order hopping to make the situation simple, and only consider non-magnetic hopping term t0 (>0), and magnetic hopping term t1 (>0) in our tight-binding Hamiltonian, where is the annihilation(creation) operator for a conduction electron, and i, j, σ, σ′ are the indices for lattice sites and spin directions, respectively. The magnetic hopping matrix is determined by the direction of Cr moments (overlap of electronic states) in the hopping path, as in the super-exchange. In this calculation, has been used, where Sk is the Cr moment in the hopping path k.
To extract information of band unfolding spectral intensities, we use the following formalism26: where K(k) and J(n) denote the super-cell (normal cell) crystal momentum and the band index, respectively. The effect of super-cell potential can be projected to the original BZ, giving spectral intensities. In the tight binding form, the evaluation of |〈kn|K J〉|2 is trivial, i.e. they are just delta functions which can be determined by spatial relations. To set the Fermi surface, we integrate the states over BZ, and set the number of electron as 1 (half filled) to describe one hole in Pd nominally.
This work was supported by the National Research Foundation (NRF) of Korea Grant funded by the Korean Government (MEST) (Nos. 2010-0010771 and 2012M2B2A4029607). K.K. and B.I.M. acknowledge the support of NRF (Nos. 2009-0079947 and 2011-0025237) and KISTI (No. KSC-2012-C2-27). H.D.K. was supported by NRF funded by MEST (No. 2009-0090561).