Direct Observation of Localized Spin Antiferromagnetic Transition in PdCrO₂ by Angle-Resolved Photoemission Spectroscopy

Abstract

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.

Introduction

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 physics^2,3. 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 diffraction^4,5 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.

Results

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 calculations^11,12 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 t_2g 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 state^14,15. 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 layers^6,7. 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 1

Crystal structure and corresponding diffraction patterns.

(a), Crystal structure of delafossite-type PdCrO₂ (space group ) with the conventional hexagonal unit cell. Each Pd (red sphere) layer and Cr (yellow sphere) layer forms a triangular lattice. The yellow rods between Cr, O and Pd atoms denote the shortest path from a Pd layer to a Cr layer. (b), Top view along the c-axis. For convenience, the O atoms are not displayed. When the Cr layers have the 120° antiferromagnetic ordering, there are three kinds of Cr atoms with respect to the spin directions, denoted as A(A'), B(B') and C(C'), in the adjacent upper(lower) layer. The small rhombus drawn with the dotted lines represents the paramagnetic unit cell, while the large rhombus with the solid lines represents the super-structured antiferromagnetic unit cell. (c), X-ray diffraction patterns for the powder samples obtained at the temperature above and below T_N = 37.5 K. The difference curve (green line) indicates that there is no structure transition across the magnetic transition. (d), Neutron diffraction patterns obtained above and below T_N. The broad hump around 2θ = 26° in the blue curve is a bunch of superstructure peaks corresponding to (1/3 1/3 n) (n = 0, 1/2, 1, 3/2).

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 T_N = 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-axis^4,5. 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 T_N 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 T_N 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 region^9,19. 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.

Figure 2

X-ray absorption/photoemission spectra.

(a), Cr 2p X-ray absorption spectra of PdCrO₂, Cr₂O₃ and CrO₂. The first three spectra look very similar while the fourth one looks quite different from the others. The similarity in transition metal 2p → 3d XAS indicates that the electron configuration of the ions is the same. The two reference spectra are taken from the literature¹⁷. (b), Pd 3d X-ray photoemission spectrum of PdCrO₂. The blue solid line is the simulated spectrum based on the DS line shape model. The red solid line, black dotted line and the green solid line are the fitting spectrum, the background and the fitting error, respectively.

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 (E_F). Since PdCrO₂ is known to have an antiferromagnetic phase below T_N = 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 T_N. 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 T_N. 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.

Figure 3

Antiferromagnetic transition directly observed by ARPES.

(a), ARPES-measured FSs at the paramagnetic state (lower image) and at the antiferromagnetic state (upper image). The FS foldings corresponding to superstructure are clearly observed in the antiferromagnetic phase at T = 20 K, while only one hexagonal FS is seen in the paramagnetic phase. The black hexagons are the surface BZs for the (001) surface of the hexagonal crystal unit cell and the red hexagons are for the magnetic unit cell. (b), ARPES intensity map of paramagnetic PdCrO₂ along the Γ-K direction corresponding to the vertical dotted line in (a) (lower left) and the momentum distribution curve at the Fermi level (upper left). The peak width in the momentum distribution curve at the Fermi level measures an inverse mean free path of the conduction electrons. The lower right panel shows the band dispersions along the Γ-K line calculated by the LAPW + LO method. The calculation reproduces most of the major features in the band structure. (c), ARPES intensity map (lower panel), its waterfall plot derived from the momentum distribution curves near the Fermi level (middle panel) and the fitted curve at the Fermi level (upper panel) of antiferromagnetic PdCrO₂ along the Γ-K-K-Γ line corresponding to the horizontal dotted line in (a). The folded bands crossing the Fermi level are indicated by the yellow triangles, the red dots and the black triangles in each panel. In (b) and (c), a reverse color scale is used in comparison with the scale bar in (a).

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 states^9,12. 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/l_MFP ~ Δk = 0.021(1) Å⁻¹) and the relaxation time (τ = 1/v_kΔk = 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′-Γ_AFM 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 v_x of the folded conduction band at P_fold in Fig. 3c to that of P_AFM and of P_PM 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′-Γ_AFM line produces one hole-like hexagonal FS and two electron-like triangular FSs in a BZ, as shown in Fig. 4.

Figure 4

Cross-sectional areas of the FSs in the antiferromagnetic phase.

The lower part is the ARPES-measured FSs showing the antiferromagnetic transition. The upper part is the schematic FSs based on the analysis of the cross-sectional areas of the FSs. In the antiferromagnetic phase, there are one hole-like hexagonal FS () and two electron-like triangular FSs (α) in a BZ. Due to the small gap size between the FSs, other shapes of the cross-sectional area, e.g. the lens (β) and large rounded hexagon (δ), are found. Those were detected by the quantum oscillation measurements⁶.

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.

Discussion

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.

Figure 5

Tight-binding calculation results with the minimal model Hamiltonian.

Calculated FS intensity map (upper panel) and the corresponding band dispersions (lower panel) along the Γ-M-K-Γ path (red arrows) of (a) t₁ = 0.0 (paramagnetic phase), (b) t₁ = 0.4 and (c) t₁ = 0.6 (t₀ and t₁: normal and magnetic hopping strength). In the dispersion curves, the line thickness is proportional to the relative spectral intensity of the eigenstates at a crystal momentum k. In the panel (a), the thick line corresponds to an intensity 1, while the thin line to 0. At every momentum point, the sum of all spectral intensities is normalized to 1.

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 yet^6,7. Takatsu et al. found an unusual nonlinear field dependence of the Hall resistivity curve, ρ_xy(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 ρ_xy(H) in the frame of the scalar spin chirality mechanism^22,23, 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 T_N. 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 T_N 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), Fe_1.3Sb 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.

Methods

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 k_x-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 orbitals^10,12. 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 S_k 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.