Spin-orbital-entangled J{}_{\mathbf{eff}}=1/2 state in 3d transition metal oxide CuAl{}_{2}O{}_{4}

Spin-orbital-entangled J=1/2 state in 3d transition metal oxide CuAlO

Choong H. Kim chim82@snu.ac.kr Center for Correlated Electron Systems, Institute for Basic Science (IBS), Seoul 08826, Republic of Korea Department of Physics and Astronomy, Seoul National University, Seoul 08826, Republic of Korea    Hwanbeom Cho Center for Correlated Electron Systems, Institute for Basic Science (IBS), Seoul 08826, Republic of Korea Department of Physics and Astronomy, Seoul National University, Seoul 08826, Republic of Korea    Santu Baidya Center for Correlated Electron Systems, Institute for Basic Science (IBS), Seoul 08826, Republic of Korea Department of Physics and Astronomy, Seoul National University, Seoul 08826, Republic of Korea    Vladimir V. Gapontsev M.N. Miheev Institute of Metal Physics of Ural Branch of Russian Academy of Sciences, 620137 Ekaterinburg, Russia    Sergey V. Streltsov M.N. Miheev Institute of Metal Physics of Ural Branch of Russian Academy of Sciences, 620137 Ekaterinburg, Russia Ural Federal University, 620002 Ekaterinburg, Russia    Daniel I. Khomskii II. Physikalisches Institut, Universität zu Köln, D-50937 Köln, Germany    Je-Geun Park jgpark10@snu.ac.kr Center for Correlated Electron Systems, Institute for Basic Science (IBS), Seoul 08826, Republic of Korea Department of Physics and Astronomy, Seoul National University, Seoul 08826, Republic of Korea    Ara Go arago@ibs.re.kr Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS), Daejeon 34126, Republic of Korea    Hosub Jin hsjin@unist.ac.kr Department of Physics, Ulsan National Institute of Science and Technology (UNIST), Ulsan 44919, Republic of Korea

Transition metal oxides exhibit various competing phases and exotic phenomena depending on how their reaction to the rich degeneracy of the d-orbital Imada et al. (1998); Kugel and Khomski (1982); Khomskii2014 . Large spin-orbit coupling (SOC) reduces this degeneracy in a unique way by providing a spin-orbital-entangled ground state for 4d and 5d transition metal compounds Kim et al. (2008, 2009); Plumb et al. (2014); Kim et al. (2014a). In particular, the spin-orbital-entangled Kramers doublet, known as the J=1/2 pseudospin, appears in layered iridates and -RuCl Jackeli and Khaliullin (2009), manifesting a relativistic Mott insulating phase. Such entanglement, however, seems barely attainable in 3d transition metal oxides, where the SOC is small and the orbital angular momentum is easily quenched. From experimental and theoretical evidence, here we report on the CuAlO spinel as the first example of a J=1/2 Mott insulator in 3d transition metal compounds. Based on the experimental study, including synthesis of the cubic CuAlO single crystal, density functional theory and dynamical mean field theory calculations reveal that the J=1/2 state survives the competition with an orbital-momentum-quenched S=1/2 state. The electron-addition spectra probing unoccupied states are well described by the j=1/2 hole state, whereas electron-removal spectra have a rich multiplet structure. The fully relativistic entity found in CuAlO provides new insight into the untapped regime where the spin-orbital-entangled Kramers pair coexists with strong electron correlation.

The spin-orbital-entangled Kramers doublet has emerged in the 4 and 5 transition metal compounds with the configuration due to a large atomic spin-orbit coupling (SOC) assisted by moderate electron correlation. It has also brought about a variety of novel phenomena, including a 5d analogue to high cuprate in a square lattice Wang and Senthil (2011); Kim et al. (2014), topological insulators Shitade et al. (2009); Kim et al. (2012), the Kitaev model Jackeli and Khaliullin (2009); Chaloupka et al. (2010); Kitagawa et al. (2018); Plumb et al. (2014); Winter2017 , Weyl semi-metals Wan et al. (2011), axion insulators Go et al. (2012), and so on Rau et al. (2016). It is interesting to ask how the spin-orbital-entangled state behaves under strong electron correlation Witczak-Krempa et al. (2013). However, this question remains hypothetical, simply because no transition metals can possibly possess both large SOC and strong electron correlation simultaneously. If we take large SOC strength as a prerequisite for the spin-orbital entanglement in the configuration Martins et al. (2016), the intriguing strongly correlated =1/2 state in real materials seems impractical. Recently, a spin-orbital-entangled state has been proposed for Co environments, which is yet to be confirmed experimentally Liu and Khaliullin (2018); Sano et al. (2018).

A simple atomic model, in which the five electrons occupying the triply degenerate -orbital are under strong Coulomb interactions, can shed light on how to realise the strongly correlated =1/2 state, even with small SOC. A nonzero SOC within the atomic model favours the =1/2 doublet as its ground state Abragam and Bleaney (1970). Therefore, strong electron correlation and the narrow bandwidth of -orbitals with preserved cubic symmetry are a simple recipe for the crystalline realisation of the atomic model, and thus, for the strongly correlated =1/2 state. Given the single hole in the atomic model, the system is represented by a simple non-interacting Hamiltonian that reads , where is the atomic SOC and is the tetragonal crystal field induced by Jahn-Teller distortion. The lowest eigenstate of the single hole is Kramers doublet, written as


where , , and denotes the spin-1/2 spinor Jackeli and Khaliullin (2009). Once Jahn-Teller distortion is dominant (), the orbital degeneracy is lifted, and the orbital angular momentum is quenched; thus, we end up with the spin-only =1/2 state (=1) accompanied by the symmetry-lowering tetragonal distortion, which frequently occurs in 3 transition metal oxides. In the strong SOC limit, the spin-orbital-entangled =1/2 state (=1/3) arises while preserving the cubic symmetry. When the atomic is embedded in a crystal, two limiting solutions are possible due to the competition between the Jahn-Teller distortions and SOC (Fig. 1a).

In this work, we demonstrate that the single crystal CuAlO with Cu in tetrahedral sites retains the cubic symmetry, and represents the strongly correlated =1/2 Mott phase by hosting the crystalline version of the atomic model. Spin-orbital entanglement in this weak SOC limit is ascribed to the tetrahedrally coordinated in the isolated CuO. Disconnected tetrahedra reduce the bandwidth of 3-orbitals, approaching the atomic limit. Because -orbitals are not directed to the ligands in tetrahedra, the weak - hybridisation in CuO reduces the energy gain from the Jahn-Teller distortions and makes its quenching of the orbital angular momentum unlikely. Cooperating with electron correlation, the =1/2 ground state from the =1 orbital and =1/2 spin angular momenta of the configuration are stabilised even with the otherwise very small strength of the bare SOC 50 meV) of Cu atoms. In the strongly correlated =1/2 state, many-body multiplets and a one-particle state appear concurrently in the hole and electron excitation spectra of CuAlO, respectively.

CuAlO is among the rare normal spinel cuprates with Cu at the tetrahedral site (Fig. 1b). We succeeded in growing CuAlO single crystal of mm size by a flux method (see Methods). Using a high-resolution single-crystal diffractometer, we carried out Rietveld refinement and confirmed that our single crystal CuAlO sample forms in the cubic spinel structure with the space group of (#227) and =8.083 Å, which is consistent with our earlier results from powder samples Nirmala et al. (2017). The square of the structural factor of each reflection point in Fig. 1c shows good agreement between the observed and calculated intensities. From X-ray diffraction data, we confirmed that there is no tetragonal distortion in CuAlO. As shown in Fig. 1b, the well-isolated CuO tetrahedra form a diamond lattice. In the cubic crystal field of ligand tetrahedra, the electron in the Cu ion fully occupy -orbitals, leaving a single hole in the subshell. There is no common oxygen shared by the neighbouring CuO tetrahedra. This drives the system close to the atomic limit, with a small -orbital bandwidth and strong electron correlations. The absence of the Jahn-Teller distortion in its single crystal sample makes CuAlO a promising candidate to host the =1/2 state in 3 transition metal oxides.

Our resistivity measurements on CuAlO single crystals show insulating behaviour with an activation energy of approximately 0.56 eV (see Supplementary Information). In Fig. 1d, we also measured the magnetic susceptibility and found Curie-Weiss behaviour with =-149 K. An effective moment value is approximately 2.29 , which is larger than that of =1/2 (1.73 ) and very close to the value measured in -RuCl (2.20 ), where the =1/2 states have been established Plumb et al. (2014). In Fig. 1e, we show the magnetic contribution of the heat capacity () by subtracting the phonon contribution using the Debye model. data exclude any possible long-range order down to 0.5 K other than a broad hump around 2 K. Additionally, as shown in the inset, no anomaly was observed in . Our extensive measurements using several microscopic techniques support that there is no long-range order down to low temperature. The large frustration factor, 75, and the broad hump in make CuAlO a highly frustrated system with the diamond lattice. The origin of the frustration can be related to the interplay of nearest- and next-nearest exchange interactions in a diamond lattice Bergman et al. (2007).

To understand the absence of Jahn-Teller distortion and its physical consequences, we investigated the total energy landscape using first-principles density functional theory (DFT) for the Coulomb interaction (=)=7 eV and SOC =1 as a function of volume and tetragonal distortion . As shown in Fig. 2a, the only stable (and thus global) minimum solution occurs at =1.0025 and =1, whose structural properties are consistent with single crystal data. The electronic structure and projected density-of-states (PDOS) of this solution are shown in Fig. 2b. In cubic CuAlO, the unoccupied band above the Fermi level can be perfectly projected onto the =1/2 doublet with =0.32 in Eq. (1). Since the unoccupied state in the configuration basically represents a single hole, the electron-addition spectra are well described by the spin-orbital-entangled doublet. On the other hand, the electron-removal spectra form a many-body multiplet structure, resulting in the mixture of =1/2 and 3/2 components in the PDOS plot. This differs from the common expectation for the weakly correlated =1/2 state, realised, for example, in SrIrO. The multiplet effects appearing in the electron spectrum of CuAlO become clear in the dynamical mean field theory (DMFT) calculations shown later.

Even though there is no other stable solution in Fig. 2a, interestingly, the total energy landscape suggests that a possible Jahn-Teller distorted =1/2 state may be stabilised under high pressure. At higher pressure, the Cu-O bond length gets shorter, giving rise to larger crystal field splittings induced by Jahn-Teller distortions. By constraining the volume decrease by 3%, the =1/2 state at =0.93 has a lower energy than the =1/2 state at =1. Therefore, two distinct =1/2 and =1/2 phases can be realised by applying pressure values in the experimentally accessible range. The electronic structure of the Jahn-Teller distorted =1/2 state is shown in Fig. 2c. Due to the large tetragonal distortions, the unoccupied state mostly has the character with =0.89.

To understand how electron correlation and SOC affect the ground state of the system, we explored the phase diagram in CuAlO by plotting as a function of and (Fig. 3a). For the given value of and , we calculated the total energy by varying and to find a global minimum solution. The phase diagram is divided into blue and green regions that correspond to the spin-orbital-entangled =1/2 (1/3, 1) and the Jahn-Teller distorted =1/2 (1, 1) states, respectively. The competition between SOC and Jahn-Teller distortion results in the separation of two distinct solutions. As correlation strength increases, the phase boundary shifts toward the smaller , demonstrating that the effective SOC is enhanced by electron correlation Liu et al. (2008); Pesin and Balents (2010), and the cubic =1/2 state is stabilised. In Fig. 3b, the total energy curves are depicted with a fixed value of (=7 eV) and varying . For small SOC, two local minima appear in the total energy curves at and , in which the solutions become =1/2 and =1/2 states, respectively. For nominal SOC strength (=0.4), the =1/2 state at =0.93 has the lowest energy. Increasing stabilises the local minimum at and simultaneously destabilises the one at , leading to a discontinuous transition of the energy minimum from tetragonal =1/2 to cubic =1/2 states. Similar behaviour occurs in the total energy curves with a fixed (=) and varying ; increasing also tends to make the =1/2 state more stable than the =1/2 state (Fig. 3c). The strong electron correlation helps the small SOC of the Cu -orbital to overcome the Jahn-Teller distortion, enabling the spin-orbital-entangled ground state.

We also conducted DMFT calculations on top of the DFT-based Wannier Hamiltonian to clarify how robust the -ness is under quantum fluctuations. All correlated copper 3 and uncorrelated oxygen 2-orbitals were included; the multiplets in the -orbitals near the Fermi level were treated by the DMFT, while fully occupied -orbitals were considered at the Hartree-Fock level. We present the self-consistently obtained DMFT spectral function and PDOS in Fig. 4a for a realistic parameter set (=8 eV, =1 eV, and =50 meV). First of all, we note that the strong =1/2 hole character is manifested in the DMFT calculation, indicating that the =1/2 state is stable with respect to local quantum fluctuations. The states below the Fermi level, however, exhibit additional dynamic weight transfer originating from multiplets. Focusing on the manifold just below the Fermi level, the lowest excitation spectra show a mixture of =1/2 and 3/2 characters. This reveals a unique signature of the strongly correlated =1/2 state obeying the -coupling scheme, which is distinct from the weakly correlated counterpart.

The weight distribution can be understood by the atomic model with dominating Hund’s coupling in Fig. 4b. The lowest peak below the Fermi level in the atomic model is composed of three overlapping sub-peak structures, denoted by , and . Each sub-peak is categorised by either =1/2 () or =3/2 (); the mixture of the =1/2 and 3/2 components in the lowest hole excitation shows the close correspondence between the DMFT spectral function and the atomic multiplet description. (This behaviour becomes even clearer in an independent -only DMFT calculation, excluding the and oxygen contribution as shown in Supplementary Information.) The uniqueness of the excitation spectra is further highlighted by comparison with the case of iridates. We investigated the atomic model with a strong SOC regime () that can be compared to the =1/2 state in 5 iridates Kim et al. (2008). (See Supplementary Information for the hole excitation spectrum of the atomic model over the whole parameter range.) In this strong SOC regime closer to the -coupling scheme, electron removal spectra exhibit two prominent peaks, clearly separated by the large SOC and categorised by =1/2 and =3/2 character, respectively (Fig. 4c). This feature is reflected in previous experimental and theoretical reports in SrIrO Kim et al. (2008); Martins et al. (2016); Arita et al. (2012); Zhang et al. (2013); Pärschke et al. (2017), where the DFT single-particle band structure provides a reasonable description given that multiplet effects are less important in this parameter range.

The DMFT calculations show a genuine Mott insulator without breaking the time-reversal symmetry, whereas the DFT solution requires symmetry breaking to open a gap. Although the copper network in this system has a bipartite structure, the paramagnetic ground state persists in the DMFT results. The hole weights are equally distributed in the Kramers pair in Eq. (1) for the entire parameter range considered in this study. Even if we apply a small staggered magnetic field to stabilise an antiferromagnetic order, the magnetic moment quickly disappears as soon as the staggered field is turned off. The suppression of magnetic order may arise from frustration effects, stemming from larger second-neighbour hopping amplitudes than nearest-neighbour ones Bergman et al. (2007).(See Supplementary Information.) The origin and nature of the nonmagnetic Mott phase of CuAlO are beyond the scope of the present work. Given the possibility of being extended to the =1/2 spin liquid, however, the lack of long-range magnetic order is of great interest, requiring further study.

Single crystal synthesis  Single crystals of CuAlO were grown using a flux method. Initial raw materials of polycrystalline CuO (99.995 %, Alfa Aesar) and AlO (99.995 %, Alfa Aesar) were mixed in a 1:1 molar ratio with a sufficient amount of flux, anhydrous sodium tetraborate (99.998 %, Alfa Aesar) Fregola et al. (2012). The mixture in a platinum crucible was annealed using a ceramic tube furnace flowing under O gas. After annealing up to 1350 C, where it remained for 24 h, the mixture was cooled slowly to 750 C. After removing the flux by applying diluted HCl ( 17 %), we obtained dark-brown single crystals of CuAlO (inset in Fig. 1c). The crystal size was about 1 mm with a typical octahedral shape of spinel-type crystals.
Single-crystal X-ray diffraction  We subsequently carried out high-resolution single-crystal diffraction experiments (XtaLAB P200, Rigaku with a Mo source, =0.710747 Å) to confirm the (#227) cubic phase with a lattice parameter of =8.083 Å. The number of total independent reflections was 148. The observed intensity, and the calculated intensity, or square of the structural factor, of each reflection point were well matched. This led to sufficiently good agreement factors: , , and . Detailed structural information is provided in TABLE 1 of Supplementary Information.
Bulk properties characterisation  The magnetic property of CuAlO single crystal was verified through a direct current (DC) magnetisation measurement using an MPMS3 system (Quantum Design). The temperature-dependent magnetic susceptibility was measured under 500 Oe, which was parallel to the crystal axis of (001), using a zero-field-cooled (ZFC) method from 2 to 350 K. The The heat capacity () of the CuAlO single crystal was measured down to 0.5 K using a PPMS-9ECII system (Quantum Design) equipped with a 3He option. To obtain the magnetic contribution of the heat capacity (), we subtracted the phonon contributions using two methods: a Debye model was fit with the two Debye temperatures of 484 and 1,089 K, and the other model used experimental data measured on ZnAlO, the nonmagnetic homologue with the same spinel structure. Both methods provided similar results.
First-principles calculations  Our total energy and electronic structure calculations were based on DFT within the PBEsol functionals Perdew et al. (2008), as implemented in Elk code http://elk.sourceforge.net . Brillouin zone integrations were performed using 666 grid sampling; the basis size was determined by =9.0. We fully optimised the structure with the force criterion of 510 eV/Å. The simplified rotationally invariant DFT+ formalism by Dudarev et al. Dudarev et al. (1998) was adopted in the DFT++SOC calculations. It should be mentioned that the calculation details, such as choice of the double counting term, value of , etc., may affect the set of the exact parameters that stabilises =1/2 and =1/2 solutions. (See, e.g. Fig. 3.) However, for the cubic symmetry with , as found in our experiment, the =1/2 state is the ground state. We chose Dudarev’s formalism with the most reasonable and accepted parameter values (=7 eV) for our calculations, because on the one hand this approach is well proven, and on the other hand it best described the experiment. More details of theoretical calculations will be discussed in a separate publication. For the magnetic structure, we employed a collinear Néel antiferromagnetic order in which the moments were aligned along the -axis.
Dynamical mean-field theory calculations  We performed DMFT calculations to investigate multiplet effects in the single-particle excitation spectra. Maximally localised Wannier functions Marzari and Vanderbilt (1997) were obtained from the DFT full Cu 3+oxygen 2 bands in the absence of and SOC. As such, SOC and the rotationally invariant local Coulomb interaction at each Cu ion were treated by the dynamical mean-field theory (DMFT), where the double counting correction was applied using the fully localised limit scheme Solovyev et al. (1994). The correlations involving -orbitals were calculated by the Hartree-Fock approximation and the oxygen orbitals were assumed to be noninteracting. We employed the exact diagonalisation Caffarel and Krauth (1994) (ED) as an impurity solver for the zero-temperature DMFT calculations. From the DFT-based maximally localised Wannier Hamiltonian, each Cu site was mapped onto an effective impurity Hamiltonian with 6 correlated -orbitals and 18 bath orbitals. After the two impurity Hamiltonians were solved independently by the ED, the local Green function as a 6868 matrix was updated with two 1010 self-energy matrices. We used the conjugate gradient method to obtain new bath parameters that minimised the distance function between the inverse Green functions defined using fictitious Matsubara frequencies with inverse temperature =128 eV. The DMFT self-consistency loop was terminated if the bath parameters remained unchanged after the bath fitting.
Code Availability  The codes used for DMFT+ED calculations are available from AG upon reasonable request.
Data Availability  The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

This work was supported by Institute for Basic Science (IBS) in Korea (Grant No. IBS-R009-D1 (CHK, SB), IBS-R024-D1 (AG), IBS-R009-G1 (HC, JGP)), the Basic Science Research Program of the National Research Foundation (NRF) of Korea under Grant No. 2016R1D1A1B03933255 and 2017M3D1A1040828 (HJ). The work of GVV and SVS was supported by the Russian science foundation (grant 17-12-01207), while DIKh thanks the Deutsche Forschungsgemeinschaft (SFB 1238) and German Excellence Initiative.

Author contributions
JGP initiated the project. HC synthesized the single crystal and performed measurements. CHK, SB, VVG, SVS performed DFT, while AG carried out DMFT calculations. All authors contributed to the discussion and writing of the manuscript.


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Figure 1 Crystal structure and energy diagram of CuAlO.
, Two possible ground states from the competition between Jahn-Teller distortion () and spin orbit coupling (), resulting in =1/2 and =1/2 states, respectively. b, The crystal structure of CuAlO. The grey, light blue, and black spheres represent Cu, Al, and O atoms, respectively. The Cu atoms surrounded by the O tetrahedron form a diamond lattice. c, A Rietveld refinement result of single-crystal X-ray diffraction data of CuAlO and (inset) a picture of a CuAlO single-crystal. d, Inverse susceptibility () versus temperature of CuAlO taken with an applied field of 500 Oe: the Curie-Weiss fit (line) with the Curie-Weiss temperature () of -149 K and the effective moment of 2.29 /Cu. e, (main panel) and (inset) as a function of down to 0.5 K, indicating no signature of magnetic ordering. The Debye fit of CuAlO and heat capacity of ZnAlO to estimate the phonon contribution are depicted in the inset.

Figure 2 Total energy landscape and two competing phases.
, Total energy landscape as a function of and with =7 eV and =1. b-c, Band structure and projected density-of-states (PDOS) for =1.0025 and =1.00 (b) and =0.970 and =0.93 (c), corresponding to =1/2 and =1/2 states, respectively.

Figure 3 Phase diagram of CuAlO from density functional theory calculations.
, The phase diagram as a function of Coulomb interaction () and spin-orbit coupling (SOC) (). b-c, Total energy curve versus with varying , fixed (b) and varying , fixed (c). Different symbols of each energy curve indicate the corresponding parameters set in the phase diagram (a). Colour schemes denote values for given solutions.

Figure 4 Multiplets in dynamical mean field theory calculations.
, Spectral weights and PDOS from DMFT calculations for =8 eV, =1 eV, =0.05 eV. While the spectral gap is roughly proportional to , the splitting of the hole spectra below the Fermi level depends on and . Schematic illustration of the single-electron/hole excitation spectra from b, the strongly correlated () and c, the weakly correlated () =1/2 ground state. In c, is used for simplicity.

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