Magnetic ground state of ${\mathrm{Sr}}_2{\mathrm{IrO}}_4$ and implications for second-harmonic generation

The currently accepted magnetic ground state of ${\mathrm{Sr}}_2{\mathrm{IrO}}_4$ (the $-++-$ state) preserves inversion symmetry. This is at odds, though, with recent experiments that indicate a magnetoelectric ground state, leading to the speculation that orbital currents or more exotic magnetic multipoles might exist in this material. Here, we analyze various magnetic configurations and demonstrate that two of them, the magnetoelectric $-+-+$ state and the nonmagnetoelectric $++++$ state, can explain these recent second-harmonic generation (SHG) experiments, obviating the need to invoke orbital currents. The SHG-probed magnetic order parameter has the symmetry of a parity-breaking multipole in the $-+-+$ state and of a parity-preserving multipole in the $++++$ state. We speculate that either might have been created by the laser pump used in the experiments. An alternative is that the observed magnetic SHG signal is a surface effect. We suggest experiments that could be performed to test these various possibilities and also address the important issue of the suppression of the RXS intensity at the $L_2$ edge.

Figure 1

Crystal and magnetic symmetries of ${\mathrm{Sr}}_2{\mathrm{IrO}}_4$ in the $4{\mathrm{IrO}}_2$ planes of the unit cell: (a) $z=\frac{1}{8}$; (b) $z=\frac{3}{8}$; (c) $z=\frac{5}{8}$; (d) $z=\frac{7}{8}$. The three magnetic patterns of interest, $-++-$ (black arrows), $++++$ (blue arrows) and $-+-+$ (red arrows), are shown. They differ by the order of the ferromagnetic in-plane component along the $b$ axis. Iridium atoms are labeled as in Table 1.

Figure 2

The alternating intensities (at the RXS maximum for $L_3)$ for magnetic pattern $-+-+$ as opposed to $++++$, of (a) the $\left(1,0,4n-1\right)$ and $\left(1,0,4n+1\right)$ reflections, and (b) the $\left(0,1,4n-1\right)$ and $\left(0,1,4n+1\right)$ reflections. These simulations were performed with the FDMNES code for a cluster radius of 6.5 Å and a Hubbard $U$ on the Ir sites of 2 eV. For $\left(10L\right)$, the azimuthal angle was $0^{\circ }$ and for $\left(01L\right)$, it was ${90}^{\circ }$. In all cases, the polarization geometry was SP, with an assumed core-hole lifetime of 5.25 eV. Similar oscillations are found for a cluster radius of 3 Å, though they are less pronounced.

Figure 3

Schematic representation of the SHG process. E1 transitions are represented in red and M1 transitions in blue. The three processes leading to ${\chi }^{\left(e\right)}$ are shown in (a), (b), (c). The nine processes leading to ${\chi }^{\left(m\right)}$ are shown in (d), (e), (f). All these processes are expressed in Appendix pp1 through Eqs. (A21) to (A32). Each label ${\chi }^{(m,j,i)}$ is explicitly defined in one of these equations. The nonresonant processes are highlighted by an energy level represented by a dotted line. The denominators ${\mathrm{\Delta }}^{\left(i\right)}$, defined in the text, correspond to doubly-resonant, singly-resonant, and nonresonant processes.

Figure 4

Second harmonic signal in SS geometry at (a) 295 K and (b) 175 K [5]. The zero of the angle is along the $a$ axis. The solid curves are fits to angular functions as tabulated in Ref. [17]. For (a), the curve is the square of $0.91-0.42sin\left(4\psi \right)-1.69{sin}^2\left(2\psi \right)$. Note that the smallest coefficient is for the only allowed term in $4/mmm$ (the rest only occur for $4/m)$. For (b), the curve is the square of $0.86-0.40sin\left(4\psi \right)-1.70{sin}^2\left(2\psi \right)-0.03{cos}^3\left(\psi \right)-0.25{cos}^2\left(\psi \right)sin\left(\psi \right)-0.11cos\left(\psi \right){sin}^2\left(\psi \right)+0.005{sin}^3\left(\psi \right)$. Note the small changes in the first three coefficients relative to (a), and the fact that the last four coefficients indicate the significance of the octupole terms.

Figure 5

Sketch of the density of states of ${\mathrm{Sr}}_2{\mathrm{IrO}}_4$, with the possible transitions leading to the SHG signal: a doubly resonant one, involving $J_{\mathrm{eff}}=3/2$ electrons, and a nonresonant one, involving O $2p$ electrons.

Figure 6

A schematic representation of the Kramers doublet formation. When $jj$ coupling is considered (first spin-orbit (SO) and then octahedral crystal field (OCF), right-to-center path) it appears that the Kramers doublet is the only purely $j=5/2$ state (in red). All other states are a mixing of $j=3/2$ and $j=5/2$ (blue and red lines).

Figure 7

Ir $L_2$ edge XAS for two different core-hole widths. The core-hole width of total fluorescence yield, $\mathrm{\Gamma }=5.69$ eV, does not allow one to separate the underlying peak structure. The A peak should be absent in the octahedral limit. The $-++-$ state was assumed (the $-++-$ and $++++$ states give indistinguishable results at this scale). The Fermi level sets the zero of the energy scale.

Figure 8

Iridium $L_1$ XAS, highlighting E2 transitions below 8 eV and the positions of the $t_{2g}$ and $e_g$ states (see the main text for explanations). The $-+-+$ state was assumed, and the Fermi level sets the zero of the energy scale.