Spin-orbit-enhanced magnetic surface second-harmonic generation in ${\mathrm{Sr}}_2\mathrm{Ir}{\mathrm{O}}_4$

An anomalous optical second-harmonic generation (SHG) signal was previously reported in ${\mathrm{Sr}}_2\mathrm{Ir}{\mathrm{O}}_4$ and attributed to a hidden odd-parity bulk magnetic state. Here we investigate the origin of this SHG signal using a combination of bulk magnetic susceptibility, magnetic-field-dependent SHG rotational anisotropy, and overlapping wide-field SHG imaging and atomic force microscopy measurements. We find that the anomalous SHG signal exhibits a twofold rotational symmetry as a function of in-plane magnetic field orientation that is associated with a crystallographic distortion. We also show a change in SHG signal across step edges that tracks the bulk antiferromagnetic stacking pattern. While we do not rule out the existence of hidden order in ${\mathrm{Sr}}_2\mathrm{Ir}{\mathrm{O}}_4$, our results altogether show that the anomalous SHG signal in parent ${\mathrm{Sr}}_2\mathrm{Ir}{\mathrm{O}}_4$ originates instead from a surface-magnetization-induced electric-dipole process that is enhanced by strong spin-orbit coupling.

Figure 1

(a) Intralayer Néel order of the $J_{\text{eff}}=1/2$ pseudospins and the $c$-axis stacking pattern of the net in-plane moments (purple arrows) at $H=0$ ($-++-$) and $H>H_c$ ($++++$) below $T_N$. (b) Schematic of RA-SHG setup. The angle of incidence ($\theta$) is fixed to ${10}^{\circ }$ while the scattering plane angle ($\varphi$) varied. The fundamental input ($\omega$) and SH output ($2\omega$) beams can be selected as $P$ or $S$ polarized. An in-plane magnetic field $H$ (green arrow) can be applied at varying angles $\beta$ relative to [100]. Temperature dependence of $P_{\mathrm{in}}-S_{\mathrm{out}}$ SHG intensities acquired in a field-cooled warming scheme for (c) $H=0\mathrm{mT}$, (d) $75\mathrm{mT}$, and (e) $370\mathrm{mT}$, with $\varphi$ fixed at the lobe of maximum intensity. The curves are normalized to the room-temperature value of the zero field data. Insets: (c) RA data for $P_{\mathrm{in}}-S_{\mathrm{out}}$ polarization geometry measured at $T=80\mathrm{K}$ and $H=0\mathrm{T}$. As reported in Ref. [7], this pattern is produced by the interference between a $C_4$ crystallographic EQ term with a $C_1$ magnetic ED term as illustrated on the right. Filled and white lobes denote opposite optical phase. The relative lobe intensities are highly sensitive to small changes in the EQ and ED tensor elements, which are found to vary slightly between this work and Ref. [7]. (d) Magnetization curves for $30\mathrm{mT}$ and (e) $370\mathrm{mT}$ in units of ${10}^{-3}{\mu }_{\text{B}}$/Ir.

Figure 2

(a) RA-SHG patterns acquired in $P_{\mathrm{in}}-S_{\mathrm{out}}$ geometry at $80\mathrm{K}$ for different applied magnetic field angles ($\beta$) with a fixed field strength of $370\mathrm{mT}$. Solid black lines are fits to the bulk EQ $+$ MD model described in the main text using a fixed set of susceptibility tensor elements and only varying $\beta$ [23]. (b) Summation of the field-dependent RA data over all $\beta$, where $\beta$ runs from $0^{\circ }$ to ${345}^{\circ }$ in ${15}^{\circ }$ increments. Solid lines show fits to a model that includes both the bulk EQ $+$ MD terms and the anomalous $C_1$ term, where the latter either rotates with $\beta$ (black) or is independent of $\beta$ (red).

Figure 3

(a) White light microscopy and (b) wide-field SHG image of (001) cleaved ${\mathrm{Sr}}_2\mathrm{Ir}{\mathrm{O}}_4$ acquired at $295\mathrm{K}$ in zero magnetic field. $P_{\mathrm{in}}-P_{\mathrm{out}}$ geometry was used with $\varphi \sim {185}^{\circ }$. Scale bars, $50\mu \mathrm{m}$. (c) SHG image acquired after first and (d) second cool down to $80\mathrm{K}$ starting from $295\mathrm{K}$. (e) Zoomed-in SHG image acquired after third cool down. (f) Composite image of several atomic force microscopy height maps. The yellow solid (magenta dashed) lines in (e) mark where a monolayer (bilayer) step was observed in the height maps.

Figure 4

(a) Zoomed-in SHG image acquired in $P_{\mathrm{in}}-P_{\mathrm{out}}$ geometry with $\varphi \sim {185}^{\circ }$ at 80 K and zero magnetic field. The local orientations of the $C_1$ order parameter (purple arrows) were determined by scanning RA measurements. (b) Atomic force microscopy height profile along a line cut shown by the dashed line in (a). Insets show the local $P_{\mathrm{in}}-S_{\mathrm{out}}$ RA pattern in each region. Black lines are fits to the EQ $+$ ED model described in the main text. (c) Order parameter orientation at $80\mathrm{K}$ in each of the four regions after successive thermal cycles from $80\mathrm{K}\to 295\mathrm{K}\to 80\mathrm{K}$.