Compass Impurity Model of Tb Substitution in Sr₂IrO₄

Introduction.—The layered iridate compound Sr₂IrO₄ has attracted much attention recently partly due to its close resemblance to the cuprate superconductors Kim et al. (2008); Jackeli and Khaliullin (2009); Watanabe et al. (2010); Wang and Senthil (2011). It is described by a one-band (pseudo)spin-$1/2$ Hubbard model like the isostructural cuprate parent compound La₂CuO₄ Wang and Senthil (2011). Therefore, many interesting phenomena common to the cuprates are also found in Sr₂IrO₄. For example antiferromagnetic (AF) order exists in the parent compound Kim et al. (2009); Dhital et al. (2013); Ye et al. (2013), and upon doping Fermi arcs are seen in angle-resolved photoemission spectroscopy (ARPES) Kim et al. (2014). Moreover both scanning tunneling spectroscopy Yan et al. (2015) and ARPES Kim et al. (2015) suggest a low temperature nodal gap. Whether the latter is due to superconductivity is currently actively investigated.

The microscopic origins of the effective one-band description in Sr₂IrO₄ and cuprates, however, are quite different. For example in the parent compound the half filled band in cuprates has mixed copper $d_{x^{2}-y^{2}}$ and oxygen $p_{x,y}$ characters Zhang and Rice (1988). For most purposes spin-orbit coupling (SOC) is negligible. In contrast, the relevant band of Sr₂IrO₄ derives from an effective total angular momentum (pseudospin) $1/2$ spin orbit coupled crystal field orbital. Due to the narrow bandwidth even the relatively weak on-site Coulomb correlation can render it Mott insulating Kim et al. (2008). At low temperatures the magnetic moments associated with this band become AF long range ordered Kim et al. (2009); Dhital et al. (2013); Ye et al. (2013).

In a recent experiment Wang et al. (2015) Tb⁴⁺ impurities are substituted into Sr₂IrO₄ to replace the Ir⁴⁺ ions. The Néel temperature is fully suppressed by less than $3\%$ Tb substitution. At low temperatures the hysteretic magnetic susceptibility and the linear-$T$ specific heat behaviors suggest the formation of a spin glass state. These phenomena are reminiscent of the insulating lightly hole-doped cuprates, in which the Néel order is also replaced by a spin glass state at low temperatures Keimer et al. (1992); Chou et al. (1993); Niedermayer et al. (1998); Coneri et al. (2010). Theoretically it has been shown the doped holes create nonlocal dipolar distortions, i.e., magnetic vortex-antivortex pairs around the holes, which quickly destroy the AF order Aharony et al. (1988); Glazman and Ioselevich (1989, 1990); Cherepanov et al. (1999); Shraiman and Siggia (1988); Kou and Weng (2003a, b); Mei (2013). However, we expect the microscopic mechanism for the Tb substituted Sr₂IrO₄ to be different because the isovalent Tb⁴⁺ substitution does not introduce extra charge carriers in the IrO₂ plane.

In this work, we first show that the magnetic interaction between the Tb⁴⁺ impurity and its surrounding Ir⁴⁺ ions is given to a good approximation by

in which the summation runs over the nearest neighboring Ir⁴⁺ sites ($i=\pm\hat{x},\pm\hat{y}$) of the Tb site (see Fig. 1, left panel). $\tilde{S}_{\gamma_{i}}$ is the spin operator of the Tb⁴⁺ ion (slightly rotated due to the lattice distortion as will be discussed later) and $J^{\mathrm{eff}}_{i,\gamma_{i}}$ is the pseudospin operator of the Ir⁴⁺ ion at site $i$. $\gamma_{i}=x$ ($y$) for $i=\pm\hat{x}$ ($\pm\hat{y}$). The Ising-like interaction on each bond favors the magnetic moments aligning along the bond direction like the compass model Nussinov et al. (2004); Nussinov and Fradkin (2005); Douçot et al. (2005), so we call it the compass impurity model. As we shall show such highly anisotropic magnetic interaction is rooted in the spin-orbital coupled nature of the Ir⁴⁺ pseudospin-$1/2$ atomic levels. In previous studies such Ising-like magnetic interaction can arise only from edge-sharing IrO₆ octahedra Jackeli and Khaliullin (2009). Our result opens a new route to strong exchange anisotropy in iridates with the corner-sharing IrO₆ octahedron structure.

This realization leads us to propose that the (uniform) compass model, which has topologically protected double degeneracy and may serve in quantum computation as qubits protected against decoherence Douçot et al. (2005), can be realized in ordered double perovskites composed of spin-orbital-coupled $d^{5}$ ions (Ir⁴⁺, Rh⁴⁺, Ru³⁺, etc.) and half-closed-shell $f^{7}$ ions (Tb⁴⁺, Gd³⁺, Eu²⁺, etc.).

In an antiferromagnet with easy-plane anisotropy, e.g., Sr₂IrO₄, the “compass impurity” induces a distortion of the AF order parameter which decays as $r^{-2}$ away from the impurity. In the dual Coulomb gas picture of the XY model the impurities induce quenched vortex quadrupoles. In the following we shall show the thermal vortices triggered by the impurity quadruple potential causes a low temperature reentrant transition to a disordered phase for arbitrarily small impurity concentration. Moreover, the Néel order is fully suppressed by only a few percent substitutions consistent with the experiment. Further experimental predictions shall also be discussed.

Compass impurity model.—In the dilute impurity limit we first consider a single Tb⁴⁺ ion embedded in the IrO₂ as shown in the left panel of Fig. 1. The Tb⁴⁺ ion has electronic configuration $4f^{7}$. Because the $4f$ orbitals are very localized the crystal field effects is negligible and all $f$ orbitals are nearly degenerate. The Hund’s rule coupling leads to a large spin $S=7/2$ and an orbital singlet state on each Tb⁴⁺ ion.

The Tb-O-Ir bonds are slightly distorted due to the rotation of the IrO₆ and the TbO₆ octahedra around the $z$ axis. The rotation angle $\theta_{i}=\pm\theta$ ($\theta\simeq 11^{\circ}$) for $i\in A$ and $B$ sublattices respectively. Because the electric field perpendicular to the bond (which spoils the spin conservation in the hopping process) and the wavefunction overlap are only affected to order $\sin\theta$, as a good approximation we consider the bond to be straight and perform symmetry analysis in the usual (orbital, spin) basis.

In the undistorted case the Tb-O-Ir bond along $x$ direction has reflection symmetries along $y$ and $z$ axes, which allows the following nearest neighbor hopping parameters to be non-zero:

Here $|d_{a}\rangle_{\hat{x}}$ and $|f_{b}\rangle_{o}$ denote the Ir⁴⁺ orbitals at $\hat{x}$ and Tb⁴⁺ orbitals at the origin respectively. We note that $t_{3}={}_{o}\langle f_{xyz}|H_{t}|d_{yz}\rangle_{\hat{x}}$ is also allowed by symmetry, but it is much smaller than those in Eq. (2). The reasons are two fold: (1) the direct wavefunction overlap of the $d_{yz}$ and the $f_{xyz}$ orbitals is much smaller due to the $d_{yz}$ orbital orientation, (2) all possible oxygen $p$ orbitals mediated hopping processes are prohibited. Therefore we neglect the $t_{3}$ term in the rest of this work.

The $90^{\circ}$ rotation around the $x$ axis is also an approximate symmetry of the Tb-O-Ir bond if the electrons are well localized on the Ir and Tb ions. It relates the hopping parameters in Eq. (2) such that $t^{\prime}_{1}\simeq t_{1}$ and $t^{\prime}_{2}\simeq t_{2}$. This symmetry is well respected in the Ir-O-Ir bond of Sr₂IrO₄: the nearest-neighbor hopping parameters of the $d_{xy}$ and the $d_{zx}$ bands, which are also related by this rotation, are nearly equal: $t_{xy}=0.36~\mathrm{eV}$ and $t_{zx}=0.37~\mathrm{eV}$ Watanabe et al. (2010). The Tb⁴⁺ $4f$ orbitals are more localized, so this approximate symmetry should also be respected. Therefore, the nearest neighbor hopping between the Tb⁴⁺ and the Ir⁴⁺ ions along the $\hat{x}$ bond is described by

Here $\sigma$ is the spin component along the $z$ direction.

We then project Eq. (3) onto the Ir⁴⁺ pseudospin-$1/2$ atomic levels with the following replacement Wang and Senthil (2011): $c_{j,xy,\sigma}^{{\dagger}}\rightarrow-i\sigma\sqrt{1/3}e^{i\theta_{j}\sigma/2}d_{j,\sigma}^{{\dagger}}$ and $c_{j,zx,\sigma}^{{\dagger}}\rightarrow\sigma\sqrt{1/3}e^{i\theta_{j}\sigma/2}d_{j,-\sigma}^{{\dagger}}$ and find

In the second term the pseudospin is not conserved in the hopping process. The reason is that while the spin is conserved [see Eq. (3)] the spin of the $d_{zx}$ component in the pseudospin-$1/2$ states is antiparallel to the pseudospin due to the SOC. This leads to the anisotropic magnetic interaction in Eq. (1) as we shall see below.

Taking into account the onsite Coulomb repulsion on the Ir⁴⁺ and the Tb⁴⁺ ions we derive the second order perturbation Hamiltonian and project it onto the $S=7/2$ subspace of the Tb⁴⁺ ion. The effective magnetic interaction on the $\hat{x}$ bond is found to be $H_{\hat{x}}=-\alpha\tilde{S}_{x}J^{\mathrm{eff}}_{x,\hat{x}}$, in which $\tilde{S}_{x}=e^{i\theta S_{z}}S_{x}e^{-i\theta S_{z}}$ is the slightly rotated spin operator of the Tb⁴⁺ ion. The interaction strength $\alpha=\frac{4}{21}(U_{d}^{-1}+U_{f}^{-1})(t_{1}^{2}+t_{2}^{2})$, in which $U_{d}$ [$U_{f}$] is the energy difference between the $(d^{6},f^{6})$ [$(d^{4},f^{8})$] and the $(d^{5},f^{7})$ electron configurations due to the onsite Coulomb repulsion. The magnetic interaction on the Tb-O-Ir bond along the $y$ direction is derived in the same fashion, $H_{\hat{y}}=-\alpha\tilde{S}_{y}J^{\mathrm{eff}}_{y,\hat{y}}$ with $\tilde{S}_{y}=e^{i\theta S_{z}}S_{y}e^{-i\theta S_{z}}$. Combining $H_{\hat{x}}$ and $H_{\hat{y}}$ gives the compass impurity model, Eq. (1). The Hund’s rule coupling on the Ir⁴⁺ gives an AF Heisenberg-type correction, which is smaller than the compass-type interaction by one order of magnitude.

If Ir⁴⁺ (or other ions with $d^{5}$ electronic configuration and strong SOC) and Tb⁴⁺ (or other $f^{7}$ ions) form ordered double perovskites with the chemical formula A₄BB^′O₈ (layered quasi-two dimensional structure) or A₂BB^′O₆ (three dimensional structure), in which $\mathrm{B}=d^{5}$ ions and $\mathrm{B}^{\prime}=f^{7}$ ions occupy different sublattices with unequal (pseudo)spin sizes in the 2D square or 3D cubic lattice, the effective magnetic interaction is given by the compass model,

in which $\gamma_{ij}=x,y$ (and $z$ in the 3D case) for the bond $\langle ij\rangle$ along the $x,y$ (and $z$) directions respectively.

Impurity-induced quadrupolar distortion.—The magnetic interaction of the pure Sr₂IrO₄ compound has an easy-plane anisotropy induced by the Hund’s rule coupling on the Ir⁴⁺ ions Jackeli and Khaliullin (2009); Watanabe et al. (2010), so the magnetic moments form inplane AF order at low temperature Kim et al. (2009); Dhital et al. (2013); Ye et al. (2013). With Tb substitution such an anisotropy is strengthened by the compass-type interaction. For example, the magnetic moments of Heisenberg model with a single compass impurity align in the $xy$ plane in its classical ground state, which is obtained numerically. In experiments Wang et al. (2015), the uniform susceptibility along $c$ axis is larger than the inplane susceptibility at $x=0.03$, which also shows that the inplane AF correlation is stronger. Therefore, we shall study the impact of the compass impurities on the antiferromagnetic XY model.

The compass impurity induces local frustration to the AF order. We numerically calculate the classical ground state of a single compass impurity embedded in the XY antiferromagnet described by

in which $\langle ij\rangle^{\prime}$ excludes the bonds connected to the impurity site. The result is shown in Fig. 2, left panel, where the arrows indicate the direction of the staggered magnetic moments.

Due to the ferromagnetic Ising-like interaction around the impurity, the AF moment on the impurity site lies antiparallel to the total AF moment of the system. The nearby AF moments are also distorted to gain the anisotropic interaction energy. This creates vortices and antivortices in the plaquettes adjacent to the impurity, i.e., the AF moment orientation changes by $\pm 2\pi$ as one encircles the plaquette as shown in Fig. 2, left panel ¹¹1The vorticity in a single plaquette is counted by adding up the phase differences along the bonds $\phi_{ij}=\phi_{i}-\phi_{j}$, in which $\phi_{ij}$ are restricted to the interval $(-\pi,\pi]$ Jelić and Cugliandolo (2011).. Therefore, the compass impurity induces a vortex quadrupolar distortion of the AF order. The long range behavior of the AF moment orientation $\phi(\mathbf{r})$ is given by the solution of the following equation,

in which $Q_{ij}$ is a normalized ($\det Q=-1$) traceless symmetric tensor indicating the orientation of the quadrupole moment and $A$ is the quadrupole strength.

The solution of Eq. (7) is $\phi(\mathbf{r})=\bar{\phi}+\delta\phi(\mathbf{r})$ with the distortion $\delta\phi(\mathbf{r})$ given by

which fits the numerical results perfectly as shown in Fig. 2, upper right panel. Therefore, the impurity-induced distortion to the AF order extends nonlocally and decays as $r^{-2}$.

The quadrupole strength $A$ is obtained by fitting Eq. (8) for different compass impurity strength $\alpha$ (Fig. 2, lower right panel). It is of $O(1)$ order and increases monotonically with $\alpha$.

Suppression of AF order.—The classical XY model without impurities has the famous Kosterlitz-Thouless (KT) transition at finite temperature, which is driven by the unbinding of thermally activated vortex-antivortex pairs Kosterlitz and Thouless (1973). In the presence of quenched vortex dipoles a reentrant transition to a disordered phase occurs at low temperatures Rubinstein et al. (1983).

The compass impurities with quenched vortex quadrupoles turn out to have a similar impact on the AF order. In the presence of many quadrupolar impurities the continuum Hamiltonian is given by

in which $\rho_{s}$ is the spin stiffness. The second term describes the interaction of the AF moment field with the quadrupolar impurity potential,

In the dilute impurity limit, the quadrupole-quadrupole interaction decays as $r^{-4}$ and can be neglected. Therefore, both the position $\mathbf{r}^{\mathrm{q}}_{l}$ and the orientation $Q_{ij}(l)$ of each quadrupole are treated as quenched random variables without spatial correlation. Upon disorder average (d.a.) we have

in which $x$ is the impurity concentration.

In the dual Coulomb gas picture vortices and antivortices are mapped to electric charges and the impurity vortex quadruples are mapped to electric quadrupoles. The Hamiltonian is given by

in which $a$ is the short-distance cutoff and $m_{l}$ is the vorticity at $\mathbf{r}^{\mathrm{v}}_{l}$. $E_{c}$ is the vortex core energy. The last term is the Coulomb interaction between the vortices and the quenched quadrupolar impurities.

Following the standard KT renormalization group (RG) procedure José et al. (1977); Chaikin and Lubensky (1995) we define the reduced spin stiffness $K=\rho_{s}/k_{\mathrm{B}}T$ and find, from the dielectric function, the renormalized stiffness $K_{\mathrm{R}}$ as

in which $m(\mathbf{r})=\sum_{l}m_{l}\delta(\mathbf{r}-\mathbf{r}^{\mathrm{v}}_{l})$ is the charge density of the Coulomb gas, and $\langle\cdot\rangle_{T}$ is the thermal average. Define the (thermal) vortex fugacity $y=e^{-E_{c}/k_{\mathrm{B}}T}$ we find, to $O(y^{2})$,

in which

where $G(\mathbf{r})=(1/2\pi)\log(r/a)$.

The disorder average can be evaluated using the cumulant expansion Rubinstein et al. (1983),

in which $\delta_{a}$ is the short-range regularization constant for the $\delta$ function, which comes from the core of the quenched quadruples.

From Eqs. (13)–(16) we find that the renormalized stiffness $K_{\mathrm{R}}$ has exactly the same form as in the standard KT transition if $y$ is replaced by $y_{x}$,

The RG equations are given by José et al. (1977); Chaikin and Lubensky (1995)

The RG flow in the $y_{x}$-$K^{-1}$ parameter plane is shown in Fig. 3, left panel. The shaded region, in which the reduced stiffness $K$ flows to a non-zero value, is the KT phase with quasi-long range order (QLRO). The region outside is disordered because $K$ flows to zero. In the absence of impurities $y_{x}$ reduces to the vortex fugacity $y$. Its variation with the temperature is drawn as the dashed black curve – the pure system has QLRO at low temperature and becomes disordered at the KT transition, namely, the Néel temperature $T_{N}(0)$.

In the presence of quenched impurities the extra factor in $y_{x}$ represents the nucleation of vortices near the quadrupoles. Approaching zero temperature it diverges faster than how $y$ vanishes so the system is disordered at zero temperature for any impurity concentration $x$. The variation of $y_{x}$ with temperature is illustrated as the dashed curves for different impurity concentrations. There is a critical concentration $x_{c}$. Below $x_{c}$ the system shows QLRO at an intermediate temperature range and enters a reentrant disordered phase at low temperatures. Because this low-$T$ disordered regime is driven by the impurity potential we believe it can show the spin glass behavior seen in the experiment Wang et al. (2015); Fischer and Hertz (1991). Above $x_{c}$ the intermediate ordered phase vanishes and the Néel temperature $T_{N}$ abruptly drops to zero. This is schematically illustrated in the phase diagram (Fig. 3, right panel).

The critical concentration $x_{c}$ is not universal. It depends on the vortex core energy $E_{c}$, the quadrupole strength $A$ and the inverse quadrupole core area $\delta_{a}$. If we take all these quantities to be of $O(1)$ order, $x_{c}$ is found to be only a few percent. For example, for $E_{c}=2$ and $A^{2}\delta_{a}=2$ we find $x_{c}=0.027$, which is consistent with the quick suppression of the Néel temperature in the experiment Wang et al. (2015).

Summary.—To summarize, the magnetic interaction near Tb impurities in Sr₂IrO₄ is described by the planar compass impurity model. The strong in-plane anisotropy around the Tb site can be detected with nuclear magnetic resonance. The compass impurity induces a long range quadrupolar distortion to the antiferromagnetic order which drives a reentrant transition to a disordered phase at low temperature and quickly suppresses the Néel temperature. Motivated by this work we propose that the compass model can be realized in ordered double perovskites composed of spin-orbital-coupled $d^{5}$ ions and half-closed-shell $f^{7}$ ions.