Evidence of Slater-type mechanism as origin of insulating state in Sr2IrO4

TL;DR

This study demonstrates that the insulating state in Sr2IrO4 arises from a Slater-type antiferromagnetic ordering rather than spin-orbit driven Mott physics, by including extended neighbor interactions in first-principles calculations.

Contribution

It provides evidence that the insulating behavior in Sr2IrO4 is due to unconventional magnetic ordering, challenging the previous Jeff=1/2 Mott insulator model.

Findings

Magnetic structure with up-up-down-down order matches experiments.

Electronic gap and optical spectra are consistent with this magnetic ordering.

Extended neighbor interactions are crucial for understanding the ground state.

Abstract

For iridates with large spatially extended 5d orbitals, it may be anticipated that distant neighbor interactions would play a crucial role in their ground state properties. From this perspective, we investigate the magnetic structure of Sr2IrO4 by including interactions beyond first and second neighbors, via super cell modeling. Adopting to first-principles scalar relativistic methods, it is found that the minimum in total energy among various magnetic structures correspond to a up-up-down-down type antiferromagnetic ordering of the Ir ions for which the magnitude of the electronic gap, that of the Ir local moments and, the facsimile of the two-peaked structure in the optical conductivity spectra of Sr2IrO4 were found to be in good agreement with the experiments. The results unequivocally evidence that the origin of the electronic gap in Sr2IrO4 is due to an unconventional antiferromagnetic ordering of Ir ions, thereby classifying the system as a Slater magnet, rather than the spin-orbit coupling driven Jeff = 1/2 Mott insulator.

Figures

Tables

Table 1. Table 1: : Table showing the spin ordering and the space group of the magnetic structures, generated using 2 × 2 × 1 2 2 1 2\times 2\times 1 super-cell framework. Here, d N ​ N ( i ) superscript subscript 𝑑 𝑁 𝑁 𝑖 d_{NN}^{(i)} represents the i t ​ h superscript 𝑖 𝑡 ℎ i^{th} near neighboring shell with respect to a central Ir spin ( ↑ ↑ \uparrow ) ion. The first four near neighboring distances are also shown.

	Space group	$d_{NN}^{\left(1\right)}$ ($3.88$Å)	$d_{NN}^{\left(2\right)}$ ($5.48$Å)	$d_{NN}^{\left(3\right)}$ ($7.01$ Å)	$d_{NN}^{\left(4\right)}$ ($7.75$ Å)
AF1	$I$-$4_{2}d$	$4$($\downarrow$)	$4$($\uparrow$)	$4$($\uparrow$) $4$($\downarrow$)	$4$($\uparrow$)
AF2	$P{4}_1{2}_{1}2$	$2$($\uparrow$) $2$($\downarrow$)	4($\downarrow$)	$4$($\uparrow$) $4$($\downarrow )$	$4$($\uparrow$)
AF3	$Pnna$	$2$($\uparrow$) $2$($\downarrow$)	$2$($\uparrow$) $2$($\downarrow$)	$4$($\uparrow$) $4$($\downarrow$)	4($\downarrow$)
FM	$I{4}_1/acd$	$4$($\uparrow$)	$4$($\uparrow$)	$8$($\uparrow$)	$4$($\uparrow$)

Full text

Sr2IrO4 is an insulator at all temperatures Kim2 ; Kim1 ; Jin ; Rao ; Fisher ; Cao ; Castaneda ; Kini ; Klein and undergoes an antiferromagnetic transition below $240$ K Cao ; Castaneda ; Kini ; Klein ; Chikara ; Crawford ; Cava ; Shimura . Assuming that the strength of spin-orbit coupling (SOC) is comparable with that of crystal field interaction, Coulomb correlation and Hund’s coupling, a new quantum paradigm has been proposed Kim2 . In this model, the crystal field split Ir $t_{2g}$ states are further split by SOC into a four-fold degenerate $J_{eff}$ $=$ $\frac{3}{2}$ quartet and a two-fold $J_{eff}$ $=$ $\frac{1}{2}$ doublet states. With Ir in its $+4$ formal valence state, the low energy $J_{eff}$ $=$ $\frac{3}{2}$ states are fully filled with two electrons each, leaving the $J_{eff}$ $=$ $\frac{1}{2}$ doublet singly occupied. Furthermore, since the bandwidth of the J_{eff}$$=$$\frac{1}{2} doublets are significantly narrow, Coulomb correlation splits the doublets into an upper and lower Hubbard band, thereby rendering the system an insulating ground state Kim2 ; Kim3 . The model successfully accounts for both electron localization and insulating state on equal footing and derive consistent support from resistivity measurements, photo-emission spectroscopy, optical conductivity, absorption spectroscopy and model Hamiltonian based calculations Kim2 ; Kim1 ; Chikara ; Takayama .

However, few observations had also hinted to itinerant characteristics in Sr2IrO4 Kini ; Arita ; Li ; Yamasaki ; Piovera . Scanning tunneling microscopy finds that the electronic gap emerges in the close vicinity of the magnetic transition Li , whereas optical conductivity measurement deduce a strong reduction in the optical gap with increasing temperature Moon . Also, a metal-insulator transition is observed in the ultrafast dynamics of photo-excited carriers which indicate to a underlying Slater-type mechanism in Sr2IrO4 Hsieh . Magnetic susceptibility and isothermal magnetization measurements find the effective paramagnetic moment and the saturation moment as $0.5$ $\mu_{B}$ and $0.14$ $\mu_{B}$, respectively which is far less than the expected spin-only value of 1$$\mu_{B} for localized spin of $S$ $=$ $\frac{1}{2}$ Cao ; Kini ; Ye . The reduction in the magnitude of the Ir moment indicates to strong hybridization between Ir $5d$ and O $2p$ orbitals. In addition, Sr2IrO4 displays weak ferromagnetism which is attributed to spin canting Kim1 ; Crawford ; Kim3 ; Ye . It has been addressed in terms of nontrivial exchange interactions accounting for the strong coupling of orbital magnetization to the lattice Jackeli ; Liu1 . The weak ferromagnetism although vanishes with increasing pressure, system retains the insulating ground state Haskel . The effect is attributed to an increased tetragonal crystal field thereby substantiating the interplay of structural distortions and SOC, which affects the balance between the isotropic magnetic coupling and the Dzyaloshinskii- Moriya anisotropic interaction. It is also highlighted that distorted in-plane bond angle in Sr2IrO4 can be tuned through magnetic field Ge and epitaxial strain Serrao . Besides, the in-plane anisotropic nature and inter-layer coupling are also seen to play an important role in the low field magnetic nature of Sr2IrO4 Gim . Therefore, in the view of these experimental findings, it is clear that there is a subtle interplay of SOC, crystal field, Coulomb correlations, magnetic exchange interactions, and the local chemistry of the underlying IrO6 motifs in Sr2IrO4.

Significantly, what appears less emphasized in Sr2IrO4 is the effect of distant near neighbor interactions on the magnetism and its electronic structure properties. The magnetic structure as refined from neutron diffraction associates a non-collinear Neel type ordering of the Ir spins in the crystallographic $a-b$ plane, with the spin orientation rigidly tracking the staggered rotation of the IrO6 along the $c$-axis Ye . However, the Ir $5d$ orbitals being much extended in space and that they strongly hybridize with the near neighboring O $2p$ orbitals, it may be anticipated that the magnetic interactions in the $a-b$ plane would significantly extend over distant neighbors than those along the $c-$axis. The antiferromagnetic ordering temperature as high as $240$ K, can be well thought of one such consequence of distant neighbor magnetic exchange interactions.

Here, we present a comprehensive investigation of the electronic and magnetic structure of Sr2IrO4, by means of first principles density functional theory. To include interactions beyond first nearest neighbors, we model few antiferromagnetic structures on an underlying super-cell of dimension 2a$$\times$$2a$$\times$$c, where $a$ and $c$ are the tetragonal lattice parameters of Sr2IrO4. Consistent with the previous works, we find that the local approximations to the exchange correlation potential, such as local density approximation (LDA) LDA-PW and generalized gradient approximation (GGA-PBE) GGA-PBE fail to capture the antiferromagnetic insulating ground state of Sr2IrO4. However, using the modified Becke- Johnson potential (mBJ) Tran , we find that the equilibrium corresponds to an unconventional \uparrow$$\uparrow$$\downarrow$$\downarrow type antiferromagnetic ordering of the Ir ions in the $a$-$b$ plane. The predictive powers of the calculation are substantiated by the consistency it yields with the experiments. The magnitude of the insulating gap and that of the Ir local moment and, the double peak structure in the materials optical absorption spectra are found to be in good agreement with the experiments. These findings suggest that the underlying mechanism that drives Sr2IrO4 as an antiferromagnetic insulator is Slater-type, which is in stark contrast with the widely discussed SOC driven $J_{eff}$ $=$ $\frac{1}{2}$ Mott model.

Calculations are based on the full potential linearized augmented plane-wave (FP-LAPW) method as implemented in the Wien2k code Blaha . The lattice parameters were adopted to the experimental values, with a$$=5.48 Å, and $c=25.83$ Å Crawford , and the position coordinates of the Sr and O ions were fully relaxed. The ground state properties were obtained using well-converged basis sets using the Wien2k parameters; R_{MT}$$K_{max}$$= $7$, G_{max}$$=$$24 a.u.-1and $l_{max}$ =$$7 Blaha . Additional local orbitals were also used to account for the semi-core Ir $5p$ states. The exchange correlation potential to the crystal Hamiltonian was considered in mBJ formalism Tran .

Few collinear magnetic structures with different initial Ir spin alignment were considered in the study. These are shown in Table 1, which are described in terms of the Ir spin alignment in the first, second, third and fourth near neighbors designated as $d_{NN}^{(i)}$; $i$ $=$ $1$, $4$. Neglecting non-collinearity, AF1 then represents the experimentally determined structure and FM represents ferromagnetic ordering. In LDA spin polarized calculations, all structures converged to a paramagnetic metallic solution. However, in GGA the AF3 spin configuration converged to an antiferromagnetic metallic solution with an Ir moment of $0.2$ $\mu_{B}$, while all other structures converged to a nonmagnetic solution. The AF3 structure was $-1.4$ meV/f.u lower in energy in comparison to its non-magnetic counterpart. A schematic representation of the AF3 structure is shown in Fig.1. The AF3 unit cell consists of $16$ formula units, with an underlying Pnna symmetry of crystal lattice dimensions $a$ $=$ $5.48$Å, $b=$ $25.83$Å and $c$ $=$ $10.96$Å.

It is well known that the electron density representation of the Coulomb potential in both LDA and GGA leads to an unphysical self interaction. As a result, these approximations tend to reduce the self-repulsion of electrons thereby stabilizing artificially delocalized electronic states Cohen ; Mori . Among various correction schemes that have been proposed Heyd ; Bechstedt ; Georges , we adopt to the mBJ formalism. With $t$ and $\rho$ representing the kinetic energy density and electron density, respectively, a screening term of the form $\sqrt{\frac{t}{\rho}}$ is introduced in the mBJ exchange potential, the contribution of which is calculated by $\frac{\left|\nabla\rho\right|}{\rho}$ Tran . As a result, regions with low density are associated with higher positive potential thereby increasing the energy of these states Tran ; Koller . The mBJ formalism is applicable for Sr2IrO4 and also for other iridates Vijeta2 ; Vijeta1 since the states in the vicinity of Fermi energy are predominantly anti-bonding in character. It should be noted that the anti-bonding orbitals have less electron density, thus the choice of mBJ exchange potential for iridates is well justified.

In Fig.2, we show the mBJ generated total, atom resolved and Ir $5d$ resolved density of states (DOS) of Sr2IrO4 with AF3 spin configuration in the Ir sub-lattice. The spectra reveal Sr2IrO4 to be an insulator with an electronic gap of $0.47$ eV, consistent with the experimental value of $0.54$ eV Moon . Here, we note that the magnitude of the insulating gap in Sr2IrO4 have been reported between 0.1$$-$$0.6 eV, with the lowest determined from the resistivity data fit using a thermal activation model Shimura ; Ge and also from the earlier GGA+U+SOC calculations Kim2 ; Jin . The highest value of the insulating gap of $0.6$ eV follows from the density of states measurements using the scanning tunneling spectroscopy Dai . Intermediate values of the gap are reported from the optical conductivity, resonant inelastic x-ray scattering Kim2 ; Jin ; Kim3 and also from the photoemission spectroscopy measurements Kim2 ; Wang ; Wojek .

In accordance with the ionic model which associates Ir $5d$ manifold with five electrons, the electronic gap is found to reside well within the Ir $t_{2g}$ manifold which extends over the range $E$(eV) $\subset$ $\left[-1.5,\,0.8\right]$. Four distinct localized features, which are predominantly of Ir $d_{xz}$ / $d_{yz}$ orbital characters are observed in the spectra. Three of them are in the occupied part of the spectra centered at $-1.32$ , $-0.77$ and $-0.31$ eV below $E_{F}$ and, the fourth peak at $0.55$ eV above $E_{F}$. The position of these bands indicate to three Ir $t_{2g}$ inter-band transitions with the first, second and third transition energies being $\simeq$ $0.86$ eV , $1.32$ eV and $1.87$eV, respectively. These energies are reasonably in good agreement with the optical conductivity measurements, where two transition peaks labeled $\alpha$ and $\beta$ were determined to be at $\simeq$ $0.5$ eV and $1$ eV, respectively Moon ; Sohn .

However, unlike the $d_{xz}$ / $d_{yz}$ states, the $d_{xy}$ states appear relatively more widespread on the energy scale. We note that the Ir-O distance in Sr2IrO4 corresponds to $1.98$ Å and $2.04$ $\textrm{\AA},$ for in-plane and apical O ions, respectively. For the in-plane O2 ions, the $2p_{z}$ orbitals hybridize with the Ir $d_{xz}$ and $d_{yz}$ orbitals, while the O $2p_{x}$ and $2p_{y}$ mix with the $d_{xy}$, $d_{z^{2}}$ and $d_{x^{2}-y^{2}}$ orbitals. On the other hand, for apical O1 ions the $2p_{z}$ orbitals hybridize with the Ir $d_{z^{2}}$ and the $2p_{x}$ / $2p_{y}$ mix with the $d_{xz}$ / $d_{yz}$ states. Thus, the crystal chemistry suggests a mixing of the Ir $t_{2g}$ and $e_{g}$ states in Sr2IrO4 primarily due to the rotation of the IrO6 octahedra. The rotation of the octahedra mixes the otherwise orthogonal Ir $d_{xy}$ and $d_{x^{2}-y^{2}}$ orbitals and consequently push the $d_{xy}$ states below the Fermi energy. The $d_{xy}$ and $d_{x^{2}-y^{2}}$ hybridization also results in a pseudo-gap like feature which is manifested \simeq$$-0.6 eV below $E_{F}$. Further, the valence band energy integration of the orbitals states showed that the $d_{xy}$+ $d_{x^{2}-y^{2}}$ orbitals are occupied with $\simeq$ $2$ electrons per Ir ion, while the electron occupation in the $d_{xz}$+$d_{yz}$+$d_{z^{2}}$ sums to $3$ electrons per Ir ion, with $d_{xz}$ and $d_{yz}$ occupancy being $1.15$ and $1.29$ electrons, respectively. Also, the integrated DOS of the $d_{xy}$ / $d_{yz}$ orbitals above $E_{F}$ was determined to be \simeq$$1 e*-per Ir ion. Thus, we find that the scalar relativistic calculations with exchange potential as described in the mBJ formalism predicts Sr2IrO4* to be an antiferromagnetic insulator.

The magnitude of the local magnetic moment at the Ir sites in the AF3 structure was calculated as $\simeq$ $0.57$ $\mu_{B}$. The value is significantly higher than those determined from experiment, the latter which deduce the value as $0.2$ $\mu_{B}$ Ye . The overestimation of the Ir local moment might be due to the PBE-GGA functional in the calculation. However, the Ir magnetic moment is found to be much smaller than spin only value of $1$ $\mu_{B}$ anticipated for a $S$ $=$ $\frac{1}{2}$ system. This may be partly attributed to the strong hybridization of the Ir $5d$ $-$ O $2p$ orbitals. The effects of hybridization are also manifested on the induced moments at the O sites. We note that the AF3 structure has a \uparrow$$\uparrow$$\downarrow$$\downarrow type antiferromagnetic ordering of Ir ions in the $a-b$ plane of the tetragonal unit cell. For those in-plane O ions which bridge the Ir ions in the $a$-$b$ plane with same polarization, *i.e., *\uparrow$$\uparrow or \downarrow$$\downarrow the induced magnetic moment is calculated as $\simeq$ $0.12$ $\mu_{B}$ while for oppositely polarized Ir ions the moment is $\simeq$ $0.06$ $\mu_{B}$ . The induced moments on the apical O ions were found to be $0.03$ $\mu_{B}$.

One of the well accepted methods to validate the electronic structure is by its optical response. In experiments, a double-peak structure with maxima around $0.5$ eV and $1$ eV have been observed, with the former peak being relatively sharper than the latter Kim3 ; Sohn ; Lee . These peaks are associated with two Ir $d-d$ transitions, which in terms of the $J_{eff}$ model are due to the transitions from occupied $J_{eff}$ $=$ $\frac{3}{2}$ and $\frac{1}{2}$ states to the unoccupied $J_{eff}$ $=$ $\frac{1}{2}$ states. The spectra has been well reproduced by the LDA+U+SO calculations, thereby suggesting the importance and interplay of SOC and Coulomb correlations in Sr2IrO4 Kim2 .

In Fig. 3, we show the optical conductivity calculated for Sr2IrO4 with the underlying AF3 structure. Consistent with the experimental spectra, we obtain two characteristic peaks, centered on the energy scale* at $\simeq$ $0.82$ eV and $1.32$ eV, respectively. We note that the position of the peaks are shifted to higher energies in comparison with experiments Sohn ; Propper which is primarily due to the larger band gap estimated ($0.57$ eV) in our calculations. However, what is very consistent between the experiments and that of our results is the energy difference between the two peaks, which is found to be $0.5$ eV. Our results, therefore show that the origin of electronic gap in Sr2IrO4* is associated with the long ranged antiferromagnetic interactions, and hence a Slater-insulator.

We also investigated the effect of spin-orbit coupling (SOC) on the electronic structure of Sr2IrO4. The GGA-mBJ+SOC density of states are shown in Fig.4. In the calculation the SOC was included for the valence states through the second variational step with the mBJ scalar relativistic basis, where states up to $10$ Ry above $E_{F}$ were included in the basis expansion. While the overall features of bonding states are more or less unaltered, we find noticeable changes in the anti-bonding region. The Ir $5d$ bands are more broadened manifesting an enhanced hybridization of the $t_{2g}$ states with the O $2p$ orbitals. The hybridization not only is found to reduce the band gap to $0.17$ eV, but also decreases the magnitude of the Ir local magnetic moment to $0.47$ $\mu_{B}$.

So to check whether the insulating gap is originally due to the unconventional antiferromagnetic ordering of Ir spins and not pertained to the choice of the exchange functional, we also performed GGA+Ueff calculations, with $U_{eff}$ $=$ $2$ eV. Quite interestingly, the overall features in the density of states (Fig.5) were found very much similar to that obtained with the mBJ-GGA. However, the calculated band gap and Ir local moment was $0.11$ eV and $0.41$ $\mu_{B}$, respectively. In general, our results following a comprehensive set of calculations concisely and convincingly show that SOC is lesser significant interaction in rendering Sr2IrO4 an antiferromagnetic insulating ground state.

In summary, using the first-principles density functional theory based scalar relativistic calculations with exchange potential described in mBJ formalism, we find that Sr2IrO4 is an unconventional Slater-type antiferromagnetic system. The calculated magnitude of the electronic gap, that of the Ir local moment and, the two peak structure in the materials optical conductivity are found to be very consistent with the experimental results. Contrary to the present understanding that Sr2IrO4 is a SOC driven $J_{eff}$ Mott insulator, our calculations show that the role of of SOC in Sr2IrO4 is of lesser significance in rendering the system its insulating ground state. Our results, which are based on density functional theory, are expected to stimulate further experimental works with an objective to unravel the magnetic structure of the system and the nature of Ir magnetism, thereby providing robust understanding of iridates, in general.