Pressure-induced spin-state ordering in Sr$_2$CoO$_3$F

TL;DR

This paper theoretically investigates how pressure influences the low-temperature phases of Sr$_2$CoO$_3$F, revealing a novel spin-state ordered phase characterized by a checkerboard pattern of cobalt spin states.

Contribution

It introduces a combined LDA+DMFT and strong-coupling model approach to identify a new spin-state ordered phase under pressure in Sr$_2$CoO$_3$F.

Findings

Identification of a spin-state ordered phase under pressure

Checkerboard arrangement of cobalt spin states

Presence of paramagnetic and antiferromagnetic regimes

Abstract

We study theoretically low-temperature phases of a recently synthesized compound Sr$_2$CoO$_3$F under pressure. The analysis combining LDA+DMFT and a strong-coupling effective model points to the existence of not only normal paramagnetic and antiferromagnetic regimes, but also a spin-state ordered phase in a certain range of applied pressure and low temperature. This order is characterized by a checkerboard arrangement of different spin states of cobalt atoms in the lattice.

Equations

$$\hat{\mathcal{H}}=\sum _i{\hat{\mathcal{H}}}_{\text{at}}^{(i)}+\sum _{ij}{\hat{\mathcal{H}}}_t^{(ij)},$$

$${\hat{\mathcal{H}}}_{\text{at}}^{(i)}=\sum _{\kappa \lambda }{h}_{\kappa \lambda }^{ii}{\hat{c}}_{i\kappa }^{†}{\hat{c}}_{i\lambda }^{}+\sum _{\kappa \lambda \mu \nu }{U}_{\kappa \lambda \mu \nu }{\hat{c}}_{i\kappa }^{†}{\hat{c}}_{i\lambda }^{†}{\hat{c}}_{i\nu }^{}{\hat{c}}_{i\mu }^{},$$

$${\hat{\mathcal{H}}}_t^{(ij)}=\sum _{\kappa \lambda }{h}_{\kappa \lambda }^{ij}{\hat{c}}_{i\kappa }^{†}{\hat{c}}_{j\lambda }^{},i\ne j.$$

$${\hat{\mathcal{H}}}_{\mathrm{eff}}=\sum _{\mathbf{i}}\sum _{\mathbf{e}=\pm \mathbf{a},\pm \mathbf{b}}\sum _{\alpha \beta \gamma \delta \in \mathcal{L}}{\epsilon }_{\alpha \beta \gamma \delta }^{(\mathbf{e})}\left({\hat{d}}_{\mathbf{i}\alpha }^{†}{\hat{d}}_{\mathbf{i}\beta }^{}\right)\left({\hat{d}}_{\mathbf{i}+\mathbf{e}\gamma }^{†}{\hat{d}}_{\mathbf{i}+\mathbf{e}\delta }^{}\right),$$

$$M=\sum _{i=1,2}(-1{)}^i\sum _{m=1}^5(n_{i,m\uparrow }-n_{i,m\downarrow }),$$

$$D=\sum _{i=1,2}(-1{)}^i[n_{i,\mathrm{H}{\mathrm{S}}_1}-(n_{i,\mathrm{LS}}+n_{i,\mathrm{I}{\mathrm{S}}_1})],$$

Full text

I Introduction

Trivalent cobalt oxides with perovskite structure host a number of unusual physical phenomena resulting from the presence of several low-energy multiplets in the atomic spectrum of octahedrally coordinated Co3+ Tanabe and Sugano (1954). The parent compound LaCoO3 has been attracting attention since the 1950s due to its thermally driven spin-state crossover and insulator-metal transition Goodenough (1971). Hole doping as in La1-xSrxCoO3 leads to formation of clusters with large magnetic moments and eventually to the appearance of a ferromagnetic metal with increasing hole concentration Wu and Leighton (2003); Phelan et al. (2006). Ferromagnetism is observed also in strained films of LaCoO3 Fuchs et al. (2007); Fujioka et al. (2013), although its insulating character suggests that a different mechanism is at play. In the layered perovskite LaSrCoO4 low-temperature spin glass behavior was reported Guo et al. (2016). Compounds from the (Pr1-yYy)xCa1-xCoO3 family exhibit a ’hidden’ order at temperatures as high as 130 K, which breaks time-reversal symmetry, but does not exhibit ordered moments Tsubouchi et al. (2004); Hejtmánek et al. (2010). Understanding coupling between Co ions is essential to capture these diverse phenomena. Recent resonant inelastic x-ray scattering (RIXS) experiments on LaCoO3 Wang et al. (2018) showed that not only spin exchange, but also mobile spinful excitons are important for the low-energy physics of cobaltites. Studies on simplified models Kuneš and Augustinský (2014); Hoshino and Werner (2016); Nasu et al. (2016) as well as material specific mean-field calculations on cobaltites Afonso and Kuneš (2017); Hsu et al. (2012); Yamaguchi et al. (2017) uncovered a number of possible ordered phases in the vicinity of spin-state crossover including antiferromagnet (AFM), spin-state order (SSO) characterized by static checkerboard arrangement of atoms in distinct spin states, or an exciton condensate. In particular, the last one giving rise to a peculiar magnetism has been the subject of intense research recently Khaliullin (2013); Cao et al. (2014); Jain et al. (2017).

External pressure allows experimental control of the crystal-field splitting and relative energies of multiplets on the Co site. It eventually converts a high-spin (HS) to a low-spin (LS) atomic ground state. Trivalent RCoO3 (R$=$La, Pr, Y) perovskites host the LS Co3+ already at ambient pressure and thus the spin-state crossover cannot be induced by its increase. In contrast, Sr2CoO3F, a recently synthesized compound Tsujimoto et al. (2011), contains HS Co3+ ions that order antiferromagnetically below $T_{\rm N}=323$ K at ambient pressure Tsujimoto et al. (2012). Unlike its layered analog LaSrCoO4 Guo et al. (2016), Sr2CoO3F is free from structural disorder. Under pressure, the ground-state configuration changes from HS to LS. At room temperature, the full conversion to LS is concluded at around 12 GPa Tsujimoto et al. (2016).

In this paper, we study the pressure-driven spin-state crossover in Sr2CoO3F and explore possible ordered phases by combining ab initio and mean-field numerical approaches. Our results reproduce the experimental observations of HS AFM at low pressure and LS state at high pressure. In addition, at intermediate pressure we predict the appearance of the SSO phase. We compare the electronic structure of Sr2CoO3F with isoelectronic LaCoO3 and its layered analog LaSrCoO4 Guo et al. (2016); Afonso et al. (2018).

II Computational method

We use the multiband Hubbard model to capture the electronic correlation in the studied material. The calculations are carried out in several steps. Starting from density functional theory (DFT) calculations, the bilinear part of the Hubbard Hamiltonian is obtained by projecting the DFT Hamiltonian onto the Co-$3d$ Wannier orbitals. To incorporate the electronic correlation effects we add the local Coulomb interaction parametrized by the Slater integrals $F_{0}$, $F_{2}$, and $F_{4}$. Two theoretical approaches are applied for the same set of input parameters: strong coupling expansion followed by the mean-field treatment of the effective model (SC-MF) and dynamical mean-field theory (DMFT).

II.1 LDA and tight-binding parameters

The electronic structure calculations are performed in the framework of the DFT with local density approximation (LDA) Kohn and Sham (1965); Hohenberg and Kohn (1964) to the exchange correlation potential. Sr2CoO3F consists of layers of distorted CoO3F corner-sharing octahedra separated by Sr atoms; see Fig. 1(a).

The crystal parameters are taken from Ref. Tsujimoto et al. (2011), amounting to $a=3.83145$ Å and $c=13.3201$ Å at ambient pressure. Following Ref. Tsujimoto et al. (2016), the space group $I4/mmm$ is kept unchanged throughout the studied pressure range. The calculations are carried out with the wien2k package Blaha et al. (2001). The muffin-tin radii are set 2.22 for Sr, 1.90 for Co, 1.63 for O, and 2.18 for F in atomic units. The Brillouin zone was sampled with a $k$-mesh grid of the size $10\times 10\times 10$.

The tight-binding parameters for the Co-$3d$ bands are obtained by projecting the LDA Hamiltonian to Wannier orbital basis Kuneš et al. (2010); Mostofi et al. (2014). With these, the system is described by the five-orbital Hubbard Hamiltonian

[TABLE]

where

[TABLE]

Here, $\hat{c}^{\dagger}_{{i}\kappa}$ ($\hat{c}^{\phantom{{\dagger}}}_{{i}\kappa}$) is a fermionic creation (annihilation) operator, $i$ and $j$ refer to the site, while $\kappa,\lambda,\mu,\nu$ are the combined orbital and spin state indices. The matrix $h_{\kappa\lambda}^{ii}$ describes the crystal-field splitting and spin-orbit coupling (SOC) 111The employed amplitude of SOC $\zeta=56$ meV is based on the electronic-structure analysis in LaSrCoO4 Afonso et al. (2018).. The crystal-field splitting increases with pressure $P$, as summarized in Fig. 2(a). To determine the local interaction $U_{\kappa\lambda\mu\nu}{(F_{0},F_{2},F_{4})}$, we fix $F_{0}=3.0$ eV Karolak et al. (2015), $F_{4}/F_{2}=0.625$ Pavarini et al. (2011); Pavarini (2014), and treat the Hund’s coupling ${J}=(F_{2}+F_{4})/14$ as a tunable parameter.

II.2 Strong coupling expansion and mean-field approximation

We diagonalize the local Hamiltonian $\hat{\cal H}^{(i)}_{\text{at}}$ to obtain the set $\cal{L}$ of low-energy atomic states of the system. The multiplets with lowest energies are shown in Fig. 2(b). An important feature of the present Slater-Condon term of the local Coulomb interaction is that IS1 and HS1 states have lower energies than IS2 and HS2, respectively. This contrasts with simpler (density-density or Slater-Kanamori) parametrization of interactions, or with the non-interacting limit, where IS2 and HS2 become lower due to the tetragonal crystal-field splitting.

Next, we apply the Schrieffer-Wolff transformation to second order in the hopping $\hat{\cal H}^{(ij)}_{t}$ and keep the nearest-neighbor terms only; see also Ref. Afonso et al. (2018) for details. For each pressure $P$, the above procedure leads to an effective Hamiltonian that spans a finite set of the lowest atomic states [see, e.g., Fig. 2(b)]. Using the Schwinger boson representation, the effective Hamiltonian can be written as follows

[TABLE]

where $\hat{d}^{\dagger}_{\bf{i}\alpha}$ ($\hat{d}^{\phantom{{\dagger}}}_{\bf{i}\alpha}$) is a creation (annihilation) operator of boson representing the atomic state $\alpha$ on the site $\bf{i}$. Hard-core constraint $\sum_{\alpha}\hat{d}^{{\dagger}}_{i\alpha}\hat{d}^{\phantom{{\dagger}}}_{i\alpha}=\hat{1}$ is imposed on the physical states.

Due to the complexity of the model (4) for analysis with exact methods, we apply the mean-field decoupling in the local basis. We restrict the SC-MF calculations to a two-site unit cell, which allows us to account for both the SSO and AFM ordering. The access to local observables of the $\braket{\hat{d}^{\dagger}_{i\alpha}\hat{d}^{\phantom{\dagger}}_{i\beta}}$ type allows us to probe orders involving local superpositions of different multiplets, e.g., excitonic condensates Kuneš (2015).

The excitation spectrum for the model (4) can be obtained by the generalized spin-wave approach Sommer et al. (2001); Afonso et al. (2018). With the intent to search for an incipient excitonic instability, we apply it in the LS phase. Here, one component, $\alpha={\rm LS}$, is the vacuum state. The terms in Eq. (4), which contain three or four operators with $\alpha\neq{\rm LS}$ describing interactions between excitations, are omitted. Therefore, the spin-wave analysis is restricted to the low-temperature region, where the thermal population of excited states can be neglected.

II.3 LDA+DMFT approach

The LDA+DMFT calculations are performed using the w2dynamics Wallerberger et al. (2019) implementation of the strong-coupling continuous-time quantum Monte Carlo solver for the Anderson impurity model with full Coulomb vertex $U_{\kappa\lambda\mu\nu}$. The superstate-sampling algorithm is used to boost the efficiency of the calculations Kowalski et al. (2019). The small off-diagonal elements in the hybridization function due to the distortion of the Co atom are neglected. After the DMFT self-consistent calculation is converged, the spectral functions along the real-frequency axis are obtained by the analytic continuation using the maximum entropy method Wallerberger et al. (2019); Geffroy et al. (2019). To study SSO and AFM ordering, we employ a unit cell containing two Co atoms. The calculations are performed at $T=290$ K.

III Results and discussion

At high pressure (large crystal-field splitting), the global ground state of Sr2CoO3F corresponds to all atoms in the LS state. The dominant effect of pressure is to change the crystal-field splitting; see also Fig. 2(a). As pressure decreases, the energy difference between LS and other [intermediate-spin (IS) and HS] states reduces. When the energy of atomic excitations nears the energy of LS states, the global ground state changes. This is the so-called “spin state crossover” regime, where interactions between excitations and their mobility become important.

The HS excitations interact repulsively on nearest-neighbor (NN) bonds. The reason for that is the superexchange mechanism. A HS state surrounded by LS sites can lower its energy by a number of virtual hopping processes. In contrast, when two HS states with the same orbital and spin character occupy NN sites, the virtual hopping is blocked by the Pauli principle. The interaction is thus strongly repulsive (in Sr2CoO3F, it is calculated to be around $0.2$ eV per bond). For HS states with different spin/orbital character, Pauli blocking is not complete and the NN repulsion is thus weaker. Note that even in AFM with antiferroorbital (AFO) arrangement, two HS states still interact repulsively on NN bonds (typical values are around 30 meV per bond). The interactions involving IS states have more diverse structure ranging from weakly repulsive to weakly attractive depending on the spin, orbital and bond orientations.

With decreasing pressure, the energy gap between LS and exited states shrinks. If the lowest excitations are localized but strongly interact on NN bonds, a checkerboard arrangement of these and LS states is expected Kuneš and Křápek (2011); Karolak et al. (2015). However, as soon as some low-lying excitations are mobile, a competing excitonic condensation (EC) instability becomes important Kuneš (2015); Sotnikov and Kuneš (2016); Afonso et al. (2018).

We start our presentation in the high-pressure LS phase. To examine the EC scenario in Sr2CoO3F, we employed the spin-wave analysis. It reveals flat bands for the lowest excitations; see Fig. 3(b).

Only the IS4 excitations have a substantial dispersion. The flat dispersion of the IS1 and IS2 excitations originates from small $d_{z^{2}}$–$d_{z^{2}}$ hopping amplitude between the nearest neighbors. For comparison, the $d_{z^{2}}$–$d_{z^{2}}$ hopping amplitude is 147 meV Wang et al. (2018) and 133 meV Afonso et al. (2018) in LaCoO3 and LaSrCoO4, respectively, while in Sr2CoO3F it averages 14 meV in the studied range of the applied pressure, i.e., an order of magnitude smaller. This can be understood by considering the shape of the $d_{z^{2}}$ Wannier orbitals shown in Fig. 3(a). Asymmetric hybridization with the apical O and F atoms causes tilting of the Wannier tails on the in-plane O sites. As a result, the overlapping tails of the neighboring Wannier $d_{z^{2}}$ orbitals are almost orthogonal. In contrast, the $d_{x^{2}-y^{2}}\otimes d_{xy}$ IS4 excitons possess a dispersion with a large band width, similar to LaCoO3 Wang et al. (2018). Nevertheless, the large crystal-field energy of the $d_{x^{2}-y^{2}}$ state gives rise to a 0.7-eV gap, which excludes any excitonic instability.

We use the SC-MF analysis to study phase diagram of Sr2CoO3F under pressure, shown in Fig. 4.

There are three phases: (i) nonmagnetic (NM) or thermally induced paramagnetic (PM) LS phase; (ii) paramagnetic SSO phase; (iii) AFM HS (with an admixture of IS) phase. SOC only slightly affects the overall structure of phase diagrams. At low temperatures, see Figs. 4(a) and 4(b), SOC removes additional antiferro-orbital (AFO) ordering of the AFM (HS) phase. It also stabilizes SSO at low temperatures leading to wider regions in the $P$-$J$ planes. With the temperature increase, SSO order becomes more susceptible to thermal fluctuations in the presence of SOC, see Figs. 4(c)–4(f).

A noticeable broadening of the SSO phase with temperature in Figs. 4(a), 4(c), and 4(e), as well as the re-entrant PM-SSO-PM features in Fig. 4(d) at fixed $J\in[0.6,0.75]$ eV, resemble the Blume-Emery-Griffiths (BEG) model Blume et al. (1971). Indeed, the $J$-$T$ phase diagrams in Figs. 5(a) and 5(b) have the same shape as those of the BEG model Hoston and Berker (1991); Kuneš (2015), where the Hund’s coupling $J$ controls the splitting between the LS and HS states, i.e., it plays the same role as the single-ion anisotropy term in the BEG model. The shrinking of the area occupied by the SSO phase both at low and high temperatures arises from the dual role played by temperature. On one hand, temperature generates the HS sites necessary for the SSO phase to form. On the other hand, thermal fluctuations destroy the SSO order when the temperature is too high. Note that the PM-SSO and PM-AFM transitions are continuous, while the SSO-AFM transition is of the first order, both in Sr2CoO3F and in the BEG model.

The broadening of the SSO phase in the presence of SOC at low temperature can be attributed to the splitting of the HS and IS energy multiplets. The lower critical temperatures for both SSO and AFM phases in the presence of SOC are due to renormalization of the coupling constants in the effective model (4).

To confirm reliability of the SC-MF results, we compare local observables from two approaches, SC-MF and DMFT, in two parameter regimes [marked regions in Fig. 4(c)] in Figs. 5(c)–5(f). The staggered AFM and SSO order parameters are defined as follows:

[TABLE]

where $i$ and $m$ are the sublattice and orbital indices, respectively 222In DMFT, the SSO order parameter $D$ is estimated approximately from the electron densities on different $e_{g}$ and $t_{2g}$ orbitals with restricting to the subspace of three lowest multiplets LS, IS1, and HS1., and $n^{\phantom{\dagger}}_{i\alpha}\equiv\langle\hat{d}^{\dagger}_{i,\alpha}\hat{d}^{\phantom{\dagger}}_{i\alpha}\rangle$.

Our results show that the SC-MF approach can be viewed as a sufficient method to estimate positions of the phase boundaries for the system under study. At the same time, as expected, it overestimates the magnitudes of the order parameters in the symmetry-broken phases (and, most likely, critical temperatures). Nevertheless, according to additional analysis, it is more precise in comparison with the commonly used restriction to the density-density type of interactions. In addition, SC-MF allows us to analyze relevant physical observables in a wide temperature range and include the effect of SOC in a straightforward manner.

Although the main driving mechanism for SSO is the strong NN repulsion between HS states on the lattice, there is a sizable contribution of IS states in this compound. In Fig. 6 we show the atomic weights corresponding to the converged solutions from two approaches.

In the SSO phase, the weight of IS states increases with $J$ due to substitution of LS by IS, so that close to the SSO-AFM transition the state becomes rather of the HS-IS than of the HS-LS type. Using the maximum-entropy method to analytically continue the DMFT data, we can compare one-particle spectral functions of Sr2CoO3F of different phases in Fig. 7. The spectra were obtained at two pressures, $P=6.4$ GPa and $P=15$ GPa, with the Hund’s exchange $J=0.64$ eV, which matches best the experimental observations.

The estimated gaps for the AFM, SSO, and LS phases are, respectively, 0.25, 0.21, and 1.35 eV. The spectral weights above the Fermi energy $E_{\rm F}$ of the $d_{x^{2}-y^{2}}$ and $d_{xz/yz}$ states in the AFM (spin $\uparrow$) and SSO (atom 1) panels, respectively, confirm significant contributions of IS1 states that are also apparent in Fig. 6. In Figs. 7(d) and 7(e) we show the $k$-resolved spectral functions of the $d_{x^{2}-y^{2}}$ and $d_{xy}$ orbitals. The spectral functions for both orbitals reflect the doubling of the unit cell and formation of Hubbard bands on the HS site. Since photoemission measurements are not possible under pressure, the experimental observation of the SSO phase is likely to rely on local probes that are able to detect formation of two distinct Co sites such as Raman or Mössbauer spectroscopy or nuclear magnetic resonance.

IV Conclusions

We have reported DFT+DMFT and DFT+SC-MF study of the layered cobaltite Sr2CoO3F under pressure. Unlike in the isoelectronic LaCoO3 and LaSrCoO4, mobile IS excitations were found to play no role in low-energy physics of Sr2CoO3F. Mobile $d_{x^{2}-y^{2}}\otimes d_{xy}$ IS4 excitations exhibit a gap over 0.5 eV due to tetragonal crystal-field splitting, while the other IS excitations involving $d_{z^{2}}$ orbital have substantially reduced mobility due to asymmetric hybridization between Co and apical O and F atoms. The low-energy physics of Sr2CoO3F is governed by NN interactions between atom-bound multiplet excitations and our SC model reduces to a generalized BEG model. As in the BEG model Blume et al. (1971); Hoston and Berker (1991), we find the SSO phase between LS and AFM phases. The SC-MF results are confirmed by matching DMFT results.

The existence of sublattices with two distinct atomic states of Co in the SSO state renders its detection straightforward. For example, a sizable Co-O bond-length disproportionation Knížek et al. (2009) is expected. The SSO state has been invoked in the context of LaCoO3 Bari and Sivardière (1972); Kuneš and Křápek (2011); Karolak et al. (2015), but never experimentally observed. We suppose mobile low-energy IS excitations Sotnikov and Kuneš (2016); Wang et al. (2018) to be the reason. The absence or presence of these in Sr2CoO3F and LaCoO3, respectively, can explain their different magnetic properties — the AFM ordering of Sr2CoO3F vs ferromagnetic correlations in bulk LaCoO3 Tomiyasu et al. (2018) and ferromagnetic ordering in strained LaCoO3 films Fuchs et al. (2007); Freeland et al. (2008). Sr2CoO3F thus provides an important reference system to understand the physics of Co3+ perovskites.

V Acknowledgments

The authors thank M. Pickem, J. Kaufmann, and A. Hausoel for technical assistance with the w2dynamics package. This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 646807-EXMAG). Access to computing and storage facilities provided by the Vienna Scientific Cluster (VSC) is greatly appreciated.