Buried moiré supercells through $\mathrm{SrTi}{\mathrm{O}}_3$ nanolayer relaxation

We identified a highly ordered moiré lattice at the buried $\mathrm{SrTi}{\mathrm{O}}_3$ (STO)-(La,Sr)(Al,Ta)-oxide (LSAT) interface by high-resolution x-ray diffraction reciprocal space mapping. We found long-ranged ordered supercells of 106/107 unit cells of unstrained STO-LSAT caused by complete lattice relaxation through high-temperature annealing. Transmission electron microscopy images show that this periodicity is based on line dislocations at the interface region. The presence of such ordered superstructures in such widely used complex oxides sets the ideal conditions for moiré-tuned interfacial electronic modifications and ferroelectric supercrystallinity, opening the possibility for interface functionalities and impacting findings on vortex structured multilayers systems.

Figure 1

Slices through the reciprocal space volume (RSV) obtained by x-ray diffraction (XRD) at the (2, 0, $L$) family peaks. The reciprocal space distance between substrate and surface layer peak increases linearly with $L$ diffraction order, yielding an out-of-plane lattice difference of $1.07\pm 0.10\%$. The scattering intensity is scaled logarithmically according to the color bar.

Figure 2

(a) Out-of-plane and (b) in-plane sliced projections of the reciprocal space volume (RSV) obtained by x-ray scattering on the (2, $K$, 4) family peaks, showing a commensurate relation between atomic lattice reflections and the superstructure scattering rods. Black dashed lines mark the center of the Bragg reflection in the $\mathrm{SrTi}{\mathrm{O}}_3$ (STO) reference frame. The scattering intensity is scaled logarithmically according to the color bar. (c) The intensity of two-dimensional (2D) Fourier transform of a theoretical 106/107 moiré lattice reproduces the same scattering behavior if an induced phase shift is considered.

Figure 3

(a) Structural illustration of the $\mathrm{SrTi}{\mathrm{O}}_3\left(\mathrm{STO}\right)-{\left(\mathrm{LaAl}{\mathrm{O}}_3\right)}_{0.3}{\left({\mathrm{Sr}}_2\mathrm{TaAl}{\mathrm{O}}_6\right)}_{0.7}\left(\mathrm{LSAT}\right)$ interface due to lattice relaxation, shown along the STO (100) direction. Starting from perfect overlap of the corresponding unit cells (uc; see black dashed area at $\mathrm{u}{\mathrm{c}}_{\mathrm{STO}}=0$), the local mismatch of the interfacial oxygen atoms increases until it is maximal at half the moiré periodicity (see red dashed area at $\mathrm{u}{\mathrm{c}}_{\mathrm{STO}}=53$). Eventually, the LSAT and STO unit cells overlap again (see black dashed area at $\mathrm{u}{\mathrm{c}}_{\mathrm{STO}}=106$), effectively forming a two-dimensional (2D) 106/107 moiré pattern. (b) High resolution transmission electron microscopy (HRTEM) images of the sample cross-section, showing a highly ordered arrangement (mean distance of 44.6 nm) of dislocations that form where the local mismatch is largest.

Figure 4

(a) Atomic force microscopy (AFM) image of the $\mathrm{SrTi}{\mathrm{O}}_3$ (STO) surface after annealing, showing a common lateral disordered arrangement of islands. (b) Reciprocal space image of the surface morphology obtained via two-dimensional (2D) fast Fourier transform (FFT) (using Hanning window function) of the AFM image shown in (a). The radial-symmetric intensity distribution confirms no preferred alignment direction of the surface features. (c) Average of 20 radial cuts together with a Gaussian fit to quantify the mean correlation length.

Figure 5

Line cuts through the (2, 0, 2) reciprocal space volume (RSV; see Fig. 2) along (a) (0, 0, l), (b) ($h$, 0, 0), and (c) (0, $k$, 0). The out-of-plane fringe pattern in (a) was fitted (blue line) to obtain the film thickness of 33 ± 2 nm. In (b) and (c), along both directions, clear satellite peaks around the central (2, 0, 2) reflection are evident.

Figure 6

Gaussian fits of the first satellite peaks seen in the horizontal line cuts along (a) ($h$, 0, 0) and (b) (0, $k$, 0), as shown in Figs. 5 and 5.

Figure 7

(a) Real-space interference matrix $I_{AB}$ illustrating occurrence of a moiré lattice from two exemplary $\frac{9}{10}$ base lattices, as shown in green and blue. After sufficient unit cell repetitions, the two lattice base points overlap, causing an increase in the scattering length density (SLD) in our model (lattice overlap points marked by orange arrows). These overlap points reoccur in a square symmetry, which constitutes the “moiré lattice” (see red box). (b) and (c) Intensity of the fast Fourier transforms (FFTs) of the interference matrix $I_{AB}$. (b) FFT where all $I_{AB}$ matrix elements have no complex terms. Only the two reflection families stemming from the base lattices $A$ and $B$ (green and blue) are visible. (c) FFT where an arbitrary complex constant has been added at the overlap points shown in (a). In addition to the two base lattice reflections (green and blue), higher order contributions from the interference superstructure appear. These moiré lattice reflections are consistent with experimental observations (see Fig. 2).

Figure 8

Results of the three-dimensional (3D) model calculations, showing the intensity of the fast Fourier transforms (FFTs) that yield reciprocal space volumes (RSVs) along the (a) out-of-plane and (b) in-plane direction. (c) Illustrative sketches of the underlying geometries. The left column shows results from two lattices of different sizes and no out-of-plane corrugation, yielding only Bragg-like reflections that correspond to the two unit cell parameters. The middle column shows results from two lattices of different sizes and 10% out-of-plane corrugation, and the right column shows results from two lattices of different sizes and 10% in-plane atomic displacements. Here, a strong superstructure scattering contribution is observed, confirming both in- and out-of-plane corrugation as a cause for the experimentally observed scattering features.

Figure 9

(a) In-plane and (b) out-of-plane cuts through the reciprocal space volume (RSV) of the (0, 0, 2) reflection of $\mathrm{SrTi}{\mathrm{O}}_3$ (STO)- $\mathrm{NdGa}{\mathrm{O}}_3$ (NGO) sample system. Also, in this case, we observe the same moiré superstructure scattering motif as in the $\mathrm{STO}-{\left(\mathrm{LaAl}{\mathrm{O}}_3\right)}_{0.3}{\left({\mathrm{Sr}}_2\mathrm{TaAl}{\mathrm{O}}_6\right)}_{0.7}\left(\mathrm{LSAT}\right)$ system but with a spacing of $\mathrm{\Delta }h=0.0106$, relating to a real-space $d$ spacing of 36.8 nm. (c) High resolution transmission electron microscopy (HRTEM) images of a sample cross-section, showing the sample lattice dislocation pattern at the interface as for the STO-LSAT system [see Fig. 3] but with a periodicity of 38.2 ± 1.8 nm.