Magnetization density distribution of Sr$_2$IrO$_4$: Deviation from a local $j_\text{eff}=1/2$ picture

TL;DR

This study reveals that the magnetization density in Sr$_2$IrO$_4$ deviates significantly from the expected local $j_ ext{eff}=1/2$ model, highlighting a dominant $xy$ orbital contribution and challenging previous assumptions about its electronic structure.

Contribution

The paper provides direct experimental evidence of deviation from the local $j_ ext{eff}=1/2$ picture in Sr$_2$IrO$_4$, using polarized neutron diffraction and advanced density reconstruction techniques.

Findings

Magnetization density is anisotropic and aspherical.

$xy$ orbital contribution dominates the magnetization density.

Deviation from the local $j_ ext{eff}=1/2$ model is significant.

Abstract

$5d$ iridium oxides are of huge interest due to the potential for new quantum states driven by strong spin-orbit coupling. The strontium iridate Sr$_2$IrO$_4$ is particularly in the spotlight because of the so-called $j_\text{eff}=1/2$ state consisting of a quantum superposition of the three local $t_{2g}$ orbitals with -- in its most simple version -- nearly equal population, which stabilizes an unconventional Mott insulating state. Here, we report an anisotropic and aspherical magnetization density distribution measured by polarized neutron diffraction in a magnetic field up to 5~T at 4~K, which strongly deviates from a local \jeffHalf picture even when distortion-induced deviations from the equal weights of the orbital populations are taken into account. Once reconstructed by the maximum entropy method and multipole expansion model refinement, the magnetization density shows cross-shaped positive four lobes along the crystallographic tetragonal axes with a large spatial extent, showing that the $xy$ orbital contribution is dominant. The analogy to the superconducting copper oxide systems might then be weaker than commonly thought.

Figures

a compares the electron and spin density distributions corresponding to the perfectly cubic and the distorted cases.

Equations

$$|j_{\text{eff}}=\frac{1}{2},\pm \frac{1}{2}⟩=\frac{1}{\sqrt{3}}\left(|xy,\pm \sigma ⟩\pm |yz,\mp \sigma ⟩+i|xz,\mp \sigma ⟩\right).$$

$$R_{\text{PND}}=\frac{I_{\uparrow }}{I_{\downarrow }}=\frac{F_N^2+2p{\sin }^2\alpha F_N{F}_M+{\sin }^2\alpha {F_M}^2}{F_N^2-2pe{\sin }^2\alpha F_N{F}_M+{\sin }^2\alpha {F_M}^2},$$

Full text

Sr2IrO4 possesses a tetragonal structure with $I4_{1}/acd$ space group, in which the IrO6 octahedra are rotated by $\approx$11∘ around the $c$-axis with an opposite phase for the neighboring Ir ions and it orders antiferromagnetically below $T_{\text{N}}\approx 230$ K Huang et al. (1994); Kim et al. (2008); Ye et al. (2013); Dhital et al. (2013); Sung et al. (2016). Strong spin-orbit coupling (SOC) stabilizes an unconventional Mott insulating ground state, which is commonly described by a spin-orbital product state within a so-called $j_{\text{eff}}=1/2$ model Kim et al. (2008, 2009); Jackeli and Khaliullin (2009); Wang and Senthil (2011). In the most simple version of this model, $5d$ electrons at the Ir4+ ($5d^{5}$) ions occupy the $t_{2g}$ states with an effective angular momentum $l_{\text{eff}}=1$, which are split by the relatively large SOC into a $j_{\text{eff}}=1/2$ doublet and a $j_{\text{eff}}=3/2$ quartet. The Coulomb repulsion induces a gap in the narrow half-filled $j_{\text{eff}}=1/2$ band, and stabilizes the Mott insulating state with the pseudospin $j_{\text{eff}}=1/2$ Kim et al. (2008); Jackeli and Khaliullin (2009), which consists of three equally populated spin-orbital components in the $t_{2g}$ band (Fig. 1a):

[TABLE]

where $\pm\sigma$ denotes the spin of the electrons.

While resonant and inelastic X-ray scattering Kim et al. (2009, 2012) gave credit to a description in terms of these $j_{\text{eff}}=1/2$ states Bertinshaw et al. (2019), the simple description with equal weights for the orbital populations has been questioned owing to the tetragonal distortion that is not negligible Chapon and Lovesey (2011); Haskel et al. (2012); Moretti Sala et al. (2014). Indeed, the most simple model with equal weights is realized only for a perfect cubic symmetry, while any lattice distortions (compression, elongation or tilting of the octahedra around the $c$-axis) split the $t_{2g}$ orbitals into three non-degenerate Kramers doublets Abragam and Bleaney (2012); Jackeli and Khaliullin (2009); Chapon and Lovesey (2011); Perkins et al. (2014); Moretti Sala et al. (2014); Lenz et al. (2019), driving the lowest energy state away from the equal weight case of Eq. (1). In the following we adopt the nomenclature common to the field and still refer to the states that diagonalize the local Hamiltonian in the presence of the distortions present in Sr2IrO4 as $j_{\text{eff}}=1/2$ and $j_{\text{eff}}=3/2$ states. Fig. 1a compares the electron and spin density distributions corresponding to the perfectly cubic and the distorted cases.

In addition, a strong hybridization between Ir $5d$ and O $2p$ orbitals, which seems to be natural for a large spatial extent of $5d$ orbitals, has been proposed to account for a large reduction of the ordered magnetic moment Kim et al. (2008) as well as for antiferromagnetic (AFM) exchange interactions between the nearest-neighboring Ir ions and for the canted magnetic moments following the octahedral rotations Jackeli and Khaliullin (2009); Perkins et al. (2014). The strong hybridization of the $d$-orbitals with the $p$-orbitals of the ligand oxygen is reminiscent of K2IrCl6 Lynn et al. (1976) and the isostructural ruthenate Ca1.5Sr0.5RuO4 Gukasov et al. (2002), where similar covalency effects have been reported. In Sr2IrO4, recent muon spin relaxation measurements have suggested the formation of oxygen moments Miyazaki et al. (2015), and charge redistribution between adjacent IrO2 and SrO layers has been revealed using electron spin resonance measurement Bogdanov et al. (2015). Further, unusual magnetic multipoles have been proposed to be observed by neutron diffraction Lovesey and Khalyavin (2014) and recently a hidden magnetic order having the same symmetry as a loop-current state has been observed by polarized neutron diffraction Jeong et al. (2017).

The magnetic moments of Ir ions are confined in the $ab$-plane and track the staggered octahedral rotation in an $-++-$ sequence along the $c$-axis in the unit cell Ye et al. (2013). Owing to this canted AFM structure, each IrO2 layer has a weak ferromagnetic (WFM) moment along the principal crystallographic axis in the $ab$-plane at zero magnetic field. This WFM is compensated due to the $-++-$ stacking sequence whereas, in a magnetic field higher than $H_{c}\approx 0.3$ T applied in the $ab$-plane Kim et al. (2008); Ye et al. (2013), a net homogeneous WFM moment appears in the plane (inset of Fig. 1b) above the metamagnetic transition. Remarkably, this WFM moment follows the direction of applied magnetic field in the $ab$-plane Fruchter et al. (2016); Nauman et al. (2017); Porras et al. (2019) and attains a saturation value of $\approx$0.08$\mu_{\text{B}}$/Ir in the field of 1 T Fruchter et al. (2016). In the current experiment, a uniform magnetic field ($H$) up to 5 T has been applied along the vertical direction (Fig. 2a). The IrO6 octahedral rotation generates two additional magnetic terms in the simple Heisenberg-type Hamiltonian Porras et al. (2019): $J_{z}$ and Dzyaloshinskii-Moriya terms, which restrict the angle between adjacent pseudospins to $\pi+2\alpha$ with the octahedral rotation angle $\alpha$ Jackeli and Khaliullin (2009). However, it does not break the in-plane rotational symmetry as the pseudospins are free to rotate in the plane while keeping the same canting angle between them (the situation is shown in the inset of Fig. 1b for a field applied along the $[110]$ direction). Therefore, under the applied magnetic field, the WFM moment does not interlock with the rotation of IrO6 octahedra in contrast with the AFM staggered moment at zero field.

The existence of this WFM allows us to probe the magnetization density distribution in crystals by polarized neutron diffraction (PND). This technique is unique because it provides direct information about the 3-dimensional distribution of the magnetization throughout the unit cell, which in turn allows for a determination of the symmetry of occupied orbitals. This method has been successfully used in the study of ferromagnetic ruthenate Ca1.5Sr0.5RuO4, isostructural to Sr2IrO4, where an anomalously high spin density at the oxygen site and the $xy$ character of the Ru $d$-orbitals have been reported Gukasov et al. (2002).

The typical experimental setup for PND, shown in Fig. 2a, consists of a neutron polarizer, a flipping device that reverses the incident neutron polarization, a magnet and a detector. The sample is magnetized by a magnetic field applied along the vertical axis and scattering intensities of Bragg reflections for the two opposite states (spin-up and spin-down) of the incident polarization are measured. They are used to calculate the so-called flipping ratio, allowing access to the Fourier components of the magnetization density, as

[TABLE]

where $F_{N}$ is the nuclear structure factor and $F_{M}$ is the magnetic structure factor. $p$ and $e$ are the polarization efficiency of the polarizer and flipper, respectively, and $\alpha$ is the angle between the scattering vector and the magnetization sup . The flipping ratios $R_{\text{PND}}$ of more than 280 $(hkl)$ reflections were measured in the weakly ferromagnetic state above the metamagnetic transition at 2 K for two magnetic field orientations, $H\|[010]$ and $H\|[\bar{1}10]$ (well above the critical field $H_{c}\approx 0.3$ T Kim et al. (2009); Porras et al. (2019)). The measured intensities for two orientations were averaged sup . As shown in Fig. 2b, the magnetic structure factors $F_{M}$ were directly obtained from the measured flipping ratios by using Eq. (2) and known nuclear structure factors $F_{N}$. For convenience, the amplitudes are given in Bohr magnetons, normalized by the number of Ir atoms (8) in the unit cell, and taken in absolute values to remove alternating signs of the phase factor. The amplitude, $F_{M}(0)$, is imposed in agreement with the saturation moment (0.08$\mu_{\text{B}}$/Ir) given by the uniform magnetization measurement Fruchter et al. (2016).

In the dipole approximation, i.e. at low momentum transfer, $F_{M}(Q)$ is usually described by a smooth decreasing function of $Q$, called the magnetic form factor, corresponding to a linear combination of radial integrals calculated from the electronic radial wave function. Instead in Fig. 2b, one observes a large distribution of the measured structure factor indicating unusually large anisotropy. That large anisotropy is explained by a predominance of $xy$-orbital as shown below using the reconstruction of the magnetization density in real space. The theoretical radial integrals $\left<j_{n}\right>$ for an isolated Ir4+ ion Kobayashi et al. (2011) are also shown in Fig. 2b for comparison. We recall that $\left<j_{2}\right>$,$\left<j_{4}\right>$ and higher-order integrals are needed to describe the departures from spherical symmetry. As seen from Fig. 2 except for the $(0,0,l)$ reflections, decreasing gradually with increasing $Q$, the majority of reflections strongly deviate from any expected smooth curve. Moreover, while the $(0,0,4n)$, $(2,0,4n)$ and $(2,2,4n)$ reflections are close to the $\left<j_{0}\right>$ curve in a small $Q$ region, the $(1,1,4n+2)$ and $(2,1,2n+1)$ reflections deviate from it quite strongly. This indicates an aspherical magnetization density, which is typical of ions with one or two unpaired electrons in the $d$-orbitals Lynn et al. (1976); Shamoto et al. (1993); Zaliznyak and Lee (2004). In addition, one can see that high-$Q$ reflections like the $(4,0,0)$, $(4,2,0)$ and $(4,4,0)$ ones show anomalously large values.

Next, a real space visualization has been performed by a reconstruction of the magnetization density, using two different very well-established and widely used approaches; a model-free maximum entropy method (MEM) Papoular and Gillon (1990) and a quantitative refinement using the multipole expansion of the density function Coppens (1997). Both techniques have advantages and limits and should be employed where they are the most efficient. Typically, no assumption is made for the initial magnetization distribution in MEM whereas the $d$-orbitals shape is constrained in the multipole expansion.

Since the crystal structure is centrosymmetric, the magnetization density can be directly reconstructed from the measured magnetic structure factors by MEM Papoular and Gillon (1990). Fig. 3a-d, shows the 3-dimensional magnetization density reconstructed by using a conventional flat density prior. A positive magnetization density in red color denotes a magnetic moment density parallel to the applied magnetic field and a negative one in blue is antiparallel. There are three key features to be noted in the figure. First, the magnetization density at Ir sites has four positive density lobes directed along the $a$, $b$ axes, corresponding to a dominant positive magnetization density of $d_{xy}$ orbital symmetry (Fig. 3b). The two other components of the effective $j_{\text{eff}}=1/2$ state model, $d_{yz}$ and $d_{xz}$, which would form an axially symmetric doughnut-shaped density above and below the $xy$ plane (see Fig. 1a), does not appear as seen in Fig. 3c,d. Thus the WFM density originates predominantly from the $xy$ orbital (a schematic illustration of the magnetic components in this situation is given in the Supplemental file (Fig. S4) sup , in contrast with the local j_{\mathrm{eff}}=1/2\picture). Second, positive density lobes are very strongly elongated, in such a way that some magnetization density is delocalized well beyond of the IrO6 octahedra. Third, contrary to the expectation of strong iridium oxygen ligand hybridization, no visible induced magnetization density appears at the oxygen sites. Actually, no significative polarization dependence has been found in any of dozens measured $(2,1,2n+1)$ reflections where oxygen atoms contribute. This is in contrast with the isostructural $4d$ compound Ca1.5Sr0.5RuO4, where $\sim$20% of the magnetic moment is transferred to the in-plane O sites Gukasov et al. (2002). However, one can notice the presence of a negative magnetic density mostly along the Ir-O direction existing between the large positive lobes. In fact, a significant negative magnetization density as large as half of the net moment is essential for a better description in the MEM analysis sup .

To confirm the symmetry found by MEM, multipole expansion was performed for an alternative refinement of the WFM density Hansen and Coppens (1978); Coppens (1997). It is composed of radial and angular parts: Slater-type radial wave functions and real spherical harmonic density functions sup . In Fig. 3e-h, the magnetization density distribution with the best refinement is shown. The main positive magnetization density lobes located between the local $x$- and $y$-axis appear clearly, which corresponds to the $d_{xy}$ symmetry. Therefore, the multipole expansion model fully confirms the $d_{xy}$ symmetry found by MEM. A benefit of the multipole method is to determine the contribution of all five $d$-orbitals to the magnetization. Using the orbital-multipole relations Coppens (1997), the magnetic moments on each orbitals were obtained as: $+0.48$, $-0.051$, $-0.035$ and $-0.314\mu_{\text{B}}$/Ir for $d_{xy}$, $d_{yz/xz}$, $d_{z^{2}}$ and $d_{x^{2}-y^{2}}$, respectively. Thus a positive $d_{xy}$ and to a lesser extent a negative $d_{x^{2}-y^{2}}$ orbital are dominant in the refinement (the latter effect is minor in the MEM method), while the $d_{yz/xz}$ orbitals are barely populated. Interestingly, the admixture of $d_{x^{2}-y^{2}}$ character to the j_{\mathrm{eff}}=1/2\orbital also has been found in first principles simulations Martins et al. (2017).

It is obvious that the refinement of multipoles with a single radial exponent cannot fit the widely delocalized density. Therefore, we introduce in the refinement a second radial exponent to describe the delocalized Ir density. Such a model shows a considerably better agreement factor ($R_{w}\sim 0.09$) compared to the model with a single radial exponent ($R_{w}\sim 0.18$) (see Supplemental section Vsup ). It confirms the anomalously large spatial extent of the magnetization density of Ir found by the MEM analysis. To appreciate the relevance of the obtained magnetization maps, we calculate the magnetic structure factors from the optimized MEM and multipoles results. By plotting them along with the measured ones in Fig. 2c (for MEM) and Fig. 2d (for multipoles), one sees that the calculated densities with MEM reproduce better the experimental data.

While the standard modeling of layered iridates by means of an anisotropic super-exchange Hamiltonian within the effective local $j_{\mathrm{eff}}=1/2$ picture correctly captures the WFM moment of the ground state and explains the most salient magnetic properties of Sr2IrO4 Jackeli and Khaliullin (2009); Perkins et al. (2014); Martins et al. (2018); Bertinshaw et al. (2019); Lenz et al. (2019), this local $j_{\mathrm{eff}}=1/2$ model is at odds with the present findings: its simplified version with equal orbital weights would suggest a homogeneous magnetization density, while taking into account distortions would – even worse – enhance the $xz$ and $yz$ orbital weight of the hole and thus of the WFM (see Fig. 1a).

These considerations give an additional twist to the exotic properties of Sr2IrO4 and the possibilities of modeling them as well as to the relationship to superconducting copper oxides. Recently, our PND results has been interpreted in terms of spin anapole Khalyavin and Lovesey (2019), pointing towards the existence of multipole correlations that goes beyond the local $j_{\mathrm{eff}}=1/2$ picture. An alternative interpretation based on a momentum-dependent composition of the orbital carrying the hole in terms of atomic $t_{2g}$ states will be published elsewhere Lenz et al. (2020). In this type of model, the hole resides in an orbital that results from a non-local, that is a $\mathbf{k}$-dependent superposition of Wannier functions of $t_{2g}$ character. In this light, it is less surprising that neither the simplified j_{\mathrm{eff}}=1/2\picture discussed above nor the state that takes into account the structural distortions but remains restricted to a local superposition of atomic orbitals describes our present experimental findings.

In summary, using PND we have evidenced a magnetization density distribution in Sr2IrO4 that is inconsistent with the local j_{\mathrm{eff}}=1/2\picture. The measured magnetic structure factor shows a strong axial anisotropy and anomalous values at large $Q$, which indicate an aspherical magnetization density distribution with a significant orbital contribution. Real space visualization exhibits a dominant $d_{xy}$ orbital character with highly elongated lobes of Ir magnetization densities towards the next Ir atoms. Although a strong $d$-$p$ hybridization is expected in Sr2IrO4, the magnetization density at the ligand oxygen sites is barely present. Our results elucidate that the ground state of Sr2IrO4 substantially deviates from the commonly accepted local j_{\mathrm{eff}}=1/2\state.

I Supplementary Material

See pages - of Sr2IrO4-MD-supp-PRL-aug2020-arxiv.pdf