Robust long-range magnetic correlation across antiphase domain boundaries in ${\mathrm{Sr}}_2{\mathrm{CrReO}}_6$

We report a resonant elastic x-ray scattering study of a thin-film sample of ${\mathrm{Sr}}_2{\mathrm{CrReO}}_6$, which has one of the highest ferrimagnetic transition temperatures among ordered double perovskites. We found resonantly enhanced magnetic Bragg peaks at $\mathbf{Q}=$ (odd, odd, odd) at both the rhenium $L_2$ and $L_3$ edges, which coincide with the structural Bragg peaks of ${\mathrm{Sr}}_2{\mathrm{CrReO}}_6$. By analyzing the widths of these Bragg peaks, we extracted very different structural and magnetic correlation lengths. The former is about 15 nm, while the latter is constrained by the instrumental resolution to be at least 90 nm. We argue that a finite structural correlation length is consistent with the existence of antiphase nanodomains in our sample. On the other hand, a much larger magnetic correlation length indicates that the magnetic correlation extends far beyond the boundaries of individual domains and is consistent with strong antiferromagnetic coupling between different antiphase domains. Last, from the azimuthal dependence of the magnetic intensity, we show that the magnetic moments lie perpendicular to the $c$ axis, which explains the earlier bulk magnetization data.

Figure 1

(a) Unit cell of an ideal double perovskite. Throughout the paper, we use cubic notation with $a\approx 7.8\AA$. (b) Experimental setup for resonant elastic x-ray scattering. The angle $\theta$ denotes a rotation within the scattering plane, and the azimuthal angle $\psi$ denotes a rotation of the sample around the momentum transfer $\mathbf{Q}={\mathbf{k}}_f-{\mathbf{k}}_i$. $\psi =0$ is defined as the sample orientation where the reference wave vector ${\mathbf{Q}}_{\mathrm{ref}}=(1,1,0)$ is in the scattering plane, as shown. An x-ray can be polarized either parallel or perpendicular to the scattering plane, denoted by $\sigma$ and $\pi$ polarizations, respectively. Polarizations of outgoing x rays are denoted with a prime. (c) Energy dependence of the $\sigma -{\pi }^{\prime }$ channel elastic intensity at $\mathbf{Q}=(1,1,5)$ at the Re $L_2$ edge. The energy dependence is obtained for different $\theta$'s. The solid black circles are the energy dependence at the peak $\theta$ value, or ${\theta }_{\mathrm{peak}}$, while open triangles are for the background scan obtained $0.{05}^{\circ }$ away from ${\theta }_{\mathrm{peak}}$. The energy dependence of x-ray florescence is shown by the blue solid line.

Figure 2

Rocking curve, or $\theta$ scans, at (a) $\mathbf{Q}=(1,1,5)$, (b) $\mathbf{Q}=(1,1,3)$, and (c) $\mathbf{Q}=(1,1,7)$ in the $\sigma -{\pi }^{\prime }$ channel. The two incident energies, $E_r=11.959\mathrm{keV}$ and $E_{\mathrm{nr}}=11.94\mathrm{keV}$, are indicated by vertical lines in Fig. 1. (d) Temperature dependence of the rocking curves at $\mathbf{Q}=(1,1,7)$ obtained with the resonant incident energy $E_r$ in the $\sigma -{\pi }^{\prime }$ channel. All curves have been shifted horizontally to match the peak positions. The shaded region denotes the nonresonant background. Solid lines are fit to the data using a sum of two Gaussian peaks. The temperature dependence of the integrated intensity extracted from the fit is shown as black circles and red squares in the inset for the broad background and the sharp peak, respectively. Temperature readings were strongly fluctuating during the measurement depending on the position of the tip of the thermocouple. Values given in the legend and the inset should therefore be taken as a rough estimation of the sample temperature. (a)–(d) were obtained at 4-ID at NSLS-II. However, the rocking curve in (a) was obtained in a different experiment from the rest. $\mathbf{Q}$ scans along (e) $H$ (f), $K$ (g), and $L$ at $\mathbf{Q}=(1,1,7)$ in the $\sigma -{\sigma }^{\prime }$ channel and off resonance in the $\sigma -{\pi }^{\prime }$ channel. The intensity is normalized with respect to the peak intensity. The $H,K,L$ values of the peak position are designated to be zero in all plots.

Figure 3

(a) $\theta$ scans of the $\mathbf{Q}=(1,1,5)$ magnetic Bragg peak at different azimuthal angles $\psi$. The legend also shows the corresponding sample angle ${\theta }_s$, which measures the sample rotation in the scattering plane from perfect grazing geometry. All $\theta$ scans are first normalized with respect to the diffuse tail with $\theta \ge 0.{05}^{\circ }$ before a separately measured nonresonant background is scaled and subtracted to obtain the magnetic signal. The inset shows the raw $\theta$ scans before normalization and background subtraction. (b) Azimuthal dependence of the $\theta$-integrated intensity. The solid (dashed) line is the calculated azimuthal dependence at (1,1,5) for an ordered moment perpendicular (parallel) to the $c$ axis. The reference wave vector is ${\mathbf{Q}}_{\mathrm{ref}}=(1,1,0)$.

Figure 4

(a) Energy dependence of the $\sigma -{\pi }^{\prime }$ channel elastic intensity at $\mathbf{Q}=(1,1,5)$ at the Re $L_3$ edge. The energy dependence is obtained for different $\theta$'s. Data shown by solid black circles are obtained for $\theta$ that maximizes the Bragg peak intensity, or ${\theta }_{\mathrm{peak}}$, while those shown by open triangles are obtained $0.{05}^{\circ }$ away from ${\theta }_{\mathrm{peak}}$. The energy dependence of the x-ray florescence is shown by the blue solid line. (b) $\theta$ scan at $\mathbf{Q}=(1,1,5)$ near the $L_3$ edge, obtained with both the resonant ($E_r=10.536\mathrm{keV}$) and nonresonant ($E_{\text{nr}}=10.524\mathrm{keV}$) energies. Data shown by red diamonds give the $\theta$ scan of the resonant magnetic signal after scaling and subtracting the nonresonant background.

Figure 5

(a) Comparison between the $L_2$ edge rocking curves at even $\mathbf{Q}$'s (open symbols) and $\mathbf{Q}=(1,1,7)$ (solid symbols). For $\mathbf{Q}=(1,1,7)$, data from both the $\sigma -{\pi }^{\prime }$ and $\sigma -{\sigma }^{\prime }$ polarization channels are shown. The $\sigma -{\pi }^{\prime }$ data at $\mathbf{Q}=(1,1,7)$ are the magnetic intensity obtained by subtracting the nonresonant diffuse background. The peak intensities of all rocking curves have been scaled to 1 for comparison. (b) Schematic drawing of APDs separated by APBs (thick lines) in our sample. Open and solid circles denote Cr and Re atoms, respectively. The average domain size is indicated by $\xi$. Red and black denote the two types of domains where positions of Cr and Re ions are exchanged. (c) One-dimensional model of the APB (thick vertical line in the middle). The two domains are colored black and red as in (b). Re magnetic moments $m_n^{Re}$ are shown by black vertical arrows above each Re atoms. The phase factor $exp(i\vec{Q}·{\vec{x}}_n)$ used to calculate the magnetic Bragg peak intensity at odd $\vec{Q}$'s is shown below each Re atom.