Three-dimensional atomic scale electron density reconstruction of octahedral tilt epitaxy in functional perovskites

Octahedral tilts are the most ubiquitous distortions in perovskite-related structures that can dramatically influence ferroelectric, magnetic, and electronic properties; yet the paradigm of tilt epitaxy in thin films is barely explored. Non-destructively characterizing such epitaxy in three-dimensions for low symmetry complex tilt systems composed of light anions is a formidable challenge. Here we demonstrate that the interfacial tilt epitaxy can transform ultrathin calcium titanate, a non-polar earth-abundant mineral, into high-temperature polar oxides that last above 900 K. The comprehensive picture of octahedral tilts and polar distortions is revealed by reconstructing the three-dimensional electron density maps across film-substrate interfaces with atomic resolution using coherent Bragg rod analysis. The results are complemented with aberration-corrected transmission electron microscopy, film superstructure reflections, and are in excellent agreement with density functional theory. The study could serve as a broader template for non-destructive, three-dimensional atomic resolution probing of complex low symmetry functional interfaces.

C omplex oxides interfaces have become a vibrant research focus in condensed matter physics and material science [1][2][3][4][5] , since they are a fertile playground for emergent phenomena such as, magnetism 6 , ferroelectricity 7 , interface charge transfer 8 , two-dimensional free electron gases 9 , superconductivity 10 , and topological states 11 through strategies in modern materials design, including strain tuning [12][13][14] , artificial layering 15,16 , spatial confinement 17 , and interfacial coupling [18][19][20][21][22][23][24][25] . Control of octahedra tilts in complex oxides via film-substrate interface design, or tilt epitaxy, has been predicted to be a powerful knob for tuning various functional properties, including inversion symmetry breaking [26][27][28] , magnetism 18,22,29 , and electronic orders 30 . Although the tilt epitaxy promises a potentially wonderful route for designing these functionalities in various materials, the experimental reports on realizing tilt epitaxy are still very limited. So far, the tilt epitaxy has been used to stabilize polar distortions in metallic ultrathin nickelates films 20 and to manipulate magnetic anisotropy in SrRuO 3 18,29 and La 2/3 Sr 1/ 3 MnO 3 22 . However, in these works, only in-phase octahedra tilt along one of the three crystallographic axes are experimentally resolvable. Moreover, strain and substrate termination effects can convolute with tilt epitaxy, which remain unexplored.
In general, there are three outstanding challenges in implementing tilt epitaxy. The first is that substrate tilts can transfer into the film only to within~10 unit cells, thus necessitating ultrathin films to observe these dramatic effects. Secondly, experimentally determining the complete three-dimensional (3D) structure of such tilt epitaxy interfaces with atomic resolution is quite a formidable task. Direct aberration-corrected scanning transmission electron microscopy (STEM) is now routinely used for probing atomic structures with picometers metrology; however, they probe the potential of two-dimensional projections of atomic columns, and deconvolving the information along depth direction is a challenge 22 , as we illustrate in this work. Coherent Bragg rods analysis (COBRA) [31][32][33][34][35][36][37][38][39][40][41] , which reconstructs 3D electron density with atomic resolution based on a phase retrieval algorithm taking advantage of the interference between the diffracted X-ray beams from the thin film and the substrate, is a promising technique for such purpose. COBRA requires no special sample preparation (such as in STEM) and is readily applicable to any epitaxial system with film thickness <20 nm. However, previous COBRA studies have mostly focused on systems with high symmetry, e.g., 4mm point group, and heavy cations. The complete 3D analysis of oxygen octahedra for a low symmetry system is still an outstanding challenge. Other emerging 3D imaging techniques (see Supplementary Table 1) include coherent diffraction imaging 42 , tomography 43 , topography 44 , ankylogaphy 45 using X-ray or electrons, each with its own merits and drawbacks. The third challenge is to be able to deconvolve the influence of the tilt epitaxy from that of strain and surface termination effects that may coexist.
In this work, we tackle all three of these outstanding challenges. We study ultrathin films of a prototypical perovskite with a complex tilt pattern, namely calcium titanate on various substrates that provide a range of tilt and strain states. We report the atomic scale 3D reconstruction of the electron density across these low symmetry epitaxial complex oxides interfaces by COBRA, the first such feat where both substrate and film possess three octahedral tilts in addition to polar distortions. The reconstruction requires high quality mapping of diffractions in a large reciprocal space volume and generalized computer routines for handling the large experimental data set. Specifically, we present COBRA reconstructed electron densities (EDs) of ultrathin epitaxial CaTiO 3 films on NdGaO 3 (110), DyScO 3 (110) and La 0.29 Sr 0.71 Al 0.65 Ta 0.35 O 3 (001) (LSAT) substrates, each offering a unique combination of strain and octahedral tilt patterns across the interface. Combining COBRA studies with complementary scanning transmission electron microscopy (STEM) and density functional theory (DFT) reveals the distinct roles of tilt epitaxy, strain and surface termination. We find that, in addition to epitaxial strain effect inducing polar distortion in the film, the tilt epitaxy monoclinically distorts the film and clamps the in-plane oxygen octahedral tilts of CaTiO 3 on LSAT substrate, giving rise to significantly higher polar transition temperatures (>900 K) in ultrathin CaTiO 3 films (8 u.c. or~3.0 nm thick) than previously reported for thicker films (>10 nm) [46][47][48][49] . Moreover, an unexpected out-of-plane polarization is observed in tensile strained CaTiO 3 thin films with directions dictated by the interfacial valence mismatch. These tilt epitaxy as well as valence mismatch effects should be present in all epitaxial complex oxides systems and strongly mediate the properties of ultrathin epitaxial films, which provide new routes to artificially control materials functionalities.

Results
Interplay of strain and octahedral tilts at the interface. CaTiO 3 has a centrosymmetric Pnma space group and is comprised of corner-shared oxygen octahedral network with interstices filled by calcium and titanium atoms. In bulk form, the oxygen octahedra exhibit out-of-phase oxygen octahedra tilts of 9.1°(a − ) about the [100] pc and the [001] pc axes (pc: pseudocubic), and an in-phase (b + ) tilt of 9.2°about the [010] pc axis, denoted as a − b + a − using the Glazer notation 50,51 . Previous literatures using STEM imaging and dielectric measurements on 10 nm or thicker biaxially strained epitaxial CaTiO 3 films on different substrates have shown that a paraelectric to ferroelectric phase transition occurs with a tensile strain of >1.1%, leading to an in-plane polarization [46][47][48][49] . However, we find that the story changes dramatically for ultrathin (8 u.c. or 3.0 nm) films used in this study, where out-of-plane polarization also arises, Curie temperatures are significantly higher, the effect of chemical termination at the interface and geometric oxygen octahedral tilt mismatch between substrate and CaTiO 3 becomes important. To ascertain the above effects, an 8 u.c. of CaTiO 3 was epitaxially grown on NdGaO 3 (110) or (or: orthorhombic), DyS-cO 3 (110) or and LSAT(001) pc substrates, with a tensile strain of 1.1%, 3.3% and 1.2%, respectively, by using Molecular Beam Epitaxy (see Methods). Similar to CaTiO 3 , NdGaO 3 and DyScO 3 possess a − b + a − tilt pattern, with out-of-phase tilts (a − ) of 10.3°and 15.0°, respectively, and in-phase tilts (b + ) of 9.8°and 13.0°, respectively. LSAT adopts a simple cubic structure with 0°tilts (or a 0 a 0 a 0 under Glazer notation). The abrupt tilt mismatch in octahedral tilt angles across the interfaces are schematically displayed in Fig. 1a, b, where NdGaO 3 and DyScO 3 prefer enhanced angles along [100] pc and [010] pc axes (Fig. 1a), while LSAT tends to suppress those angles (Fig. 1b).
The 3D structures of the above systems were investigated using the COBRA method by modeling the interference between diffractions from ultrathin epitaxial CaTiO 3 films and the three substrates. As shown in Fig. 1c, the synchrotron X-ray diffraction from such epitaxial system form crystal truncation rods (CTRs) at integer H, K values (in-plane directions of the film) with a continuous distribution along the L (thickness direction of the film) in the reciprocal space. By rotating the sample about its surface normal (L) axis, the CTRs intersect with the Ewald sphere at different L positions, and the diffraction in the full reciprocal space can be mapped out. Measurements on CaTiO 3 /NdGaO 3 , CaTiO 3 /DyScO 3 and CaTiO 3 /LSAT were performed at both room temperature and 30 K (at which all films are in polar state). A phase-retrieval algorithm (COBRA) is then employed to reconstruct the 3D electron density in real space 31   All the atomic positions, including oxygen atoms, are clearly visible as discrete peaks in the electron density maps. The high quality of the electron densities suggests the films are epitaxial and of high crystallinity. We first focus on the domain states of the three systems. Since both NdGaO 3 and DyScO 3 have the same space group (Pnma) as CaTiO 3 , the epitaxially grown 8 u.c. CaTiO 3 is expected to follow the crystallographic orientation of substrates to minimize the interfacial energy. Indeed, a monodomain of 8 u.c. CaTiO 3 on NdGaO 3 and DyScO 3 is confirmed by the symmetry exhibited in CTRs (See Supplementary Note 2), as well as the consistent oxygen octahedral tilt pattern (a − b + c − ) across the interfaces, as seen in Fig. 2a, b. However, since LSAT possesses an effectively higher (cubic) symmetry (Fm3m) than CaTiO 3 , four symmetry equivalent domains exist within the X-ray probe area (~500 µm) with equivalent fractions, as evidenced by the symmetry of the measured CTRs (See Supplementary Note 2). Therefore, the reconstructed electron densities of CaTiO 3 /LSAT contains folded structural information, as shown in Fig. 2c, which is the result of spatially translating CaTiO 3 into a single pseudocubic unit cell 52  respectively displayed in Fig. 2d-f. The interconnected oxygen octahedra networks can be clearly seen in all three slices, as sketched by black squares. The tilt angles of the oxygen octahedra, α, β, and γ, are indicated on each slice. Here we focus on the impact of tilt mismatch on those angles across the interface. The β angle of the rightmost GaO 2 layer in ac slice is about 11 ± 1°, while its neighboring TiO 2 layer next to the interface developed a tilt of 10 ± 1°, which is close to the value of NdGaO 3 substrate and slightly larger than that of bulk CaTiO 3 . Similar behavior is also observed in bc slice, where α angles of 12 ± 1°and 11 ± 1°are measured for neighboring GaO 2 and TiO 2 layers across the interface. This suggests that the interconnected octahedra network in ac and bc slices can effectively propagate the in-plane tilts (α and β) of substrate into the epitaxial film. However, with this picture of tilt epitaxy, one would expect the out-of-plane γ tilt to be relatively decoupled across the interface. Indeed, as shown in Fig. 2f, the GaO 2 and TiO 2 layers near the interface give γ angles of 11 ± 1°and 8 ± 1°, showing a relatively large change. CaTiO 3 on DyScO 3 possesses very similar structural characteristics as on NdGaO 3 , where the exact same behavior can be observed; thus it is not shown here. The slices of folded 3D electron density for CaTiO 3 /LSAT are shown in Fig. 2g-i. As discussed in Supplementary Note 3, the tilts of the oxygen octahedra will give rise to broadened oxygen peaks, representing multiple equivalent oxygen atom positions at corresponding oxygen sites. The broadening of oxygen peaks is indicated by the solid ellipses, which mark the contours of oxygen peaks. In this case, the tilt angles can be extracted by fitting the splitting of the oxygen atoms at each site. The ac slice in Fig. 2g shows an increase of ellipticity of oxygen peaks from the interface to the surface of the film, indicating an increase of tilt angles. The bc slice (Fig. 2h) shows the same behavior due to the symmetry equivalency of a and b axes of the folded electron density. The in-plane tilt angles (α and β) for the neighboring Al 0.65 Ta 0.35 O 2 and TiO 2 layers are respectively 0 ± 2°a nd 4 ± 2°, yielding significantly smaller in-plane tilt magnitudes in CaTiO 3 film compared to its bulk values. This again agrees with the tilt epitaxy effect through the interface. Similarly, the γ angles are 4 ± 3°for Al 0.65 Ta 0.35 O 2 and 8 ± 3°for TiO 2 layers near the interface, confirming that the γ tilt of the CaTiO 3 film is decoupled from the substrate.
Quantitative analysis of the 3D electron densities is performed as follows. By fitting each peak in the electron density with 3D Gaussian functions, the 3D coordinates of atoms in each unit cell from 5 u.c. beneath the substrate to the surface of the 8 u.c. thick film are extracted. Electrical polarization vectors are calculated by using cations displacements relative to anions (oxygen) and their nominal charges. The projections of polarization vectors on ac and bc planes are plotted as black arrows on the corresponding faces of the electron density maps (Fig. 2a-c), depicting a polar phase at 30 K in films on all three substrates. For convenience of comparison, one of the four equivalent domains on LSAT is plotted.
The polarizations vector evolution along the growth direction is discussed next. As illustrated in Fig. 3a-c, CaTiO 3 films on DyScO 3 , NdGaO 3 , and LSAT, respectively exhibit average inplane polarizations of −20.6 ± 2.1, −14.9 ± 1.8, and 13.5 ± 2.5μCcm −2 along the a-axis (green circles). There is no measurable polarization along the b-axis (blue circles). The magnitudes of the in-plane polarizations qualitatively agree with the larger tensile strain states on DyScO 3 (3.3%) and smaller tensile strain on NdGaO 3 (1.1%) and LSAT (1.2%); the numbers however deviate from previous theory predictions on bulk state 47 , which will be addressed in detail further on. Most interestingly, unexpected out-of-plane polarization components are observed in CaTiO 3 on all three substrates (red circles in Fig. 3a-c).
Remarkably, COBRA indicates that the directions of the outof-plane polarizations appear to be dictated by the substrate terminations, due to a valence mismatch effect 21 . The NdO layer termination of NdGaO 3 substrate, as indicated by the black line in Fig. 2a, with a valence mismatch of +1, prefers an out-of-plane polarization towards the +c direction. In contrast, electron density in Fig. 2b, c (black lines) indicate that an ScO 2 and Al 0.65 Ta 0.35 O 2 (BO 2 ) termination with a valence mismatch of −1 and −0.3, respectively, result in an out-of-plane polarization direction of −c. These COBRA reconstructed surface terminations are confirmed experimentally using transmission electron microscopy as shown in Supplementary Figure 5.
The combination of in-plane and out-of-plane polarization components determine the polarization vectors as shown, which lie in a single mirror plane, indicating the monoclinic symmetry with m in the ac plane. Note that to reveal this low symmetry distortion by COBRA, one has to collect enough truncation rods; in this particular case, up to 47 CTRs for each system were collected. Further, in contrast to previous COBRA algorithm, the current phase retrieval algorithm was generalized to include all possible crystallographic symmetries.
Quantitative structural analysis of the oxygen octahedral tilt angles α, β, and γ, respectively about the a, b and c axes, are performed by analyzing oxygen atom peaks in each BO 6 octahedron and are depicted as pie charts along the edges of the electron density maps (Fig. 2a-c) as well as plotted in Fig. 3d-f as green (α), blue (β), and red (γ) circles. COBRA reveals a gradual change in the α and β values and a relatively drastic change in the γ values across the interfaces for all three systems, as suggested by the 2D slices in Fig. 2d-i. This is expected due to the tilt epitaxy effect on the α and β angles between the substrates and their corresponding CaTiO 3 films through the shared interfacial oxygen atoms (as illustrated in Fig. 1a, b), while the γ angles are not affected, allowing for a drastic change across the interface.
The above reconstructed 3D oxygen tilts and polar displacements in the film suggest the role of substrate strain, substrate oxygen octahedral tilts, and substrate terminations. To understand and deconvolve these effects, density functional theory (DFT) calculations were performed (see Methods). The DFT results for the three components of the polarization and the octahedral tilts for each film system are plotted using solid lines in Fig. 3a-f, showing excellent agreement between the calculations and COBRA experiments. To uncouple the changes in the octahedral tilts induced by tilt epitaxy from the changes due to a pure biaxial strain, bulk calculations on strained CaTiO 3 (with no interfaces) were performed. The differences in the magnitude of in-plane octahedral tilts, |Δα| + |Δβ|, between the film and the substrate were smaller in the case where the tilt epitaxy effect is present versus when the substrate simply imparts a biaxial strain. (See Supplementary Table 2 for summary of tilt angles) For example, for the CaTiO 3 /NdGaO 3 film system, |Δα| + |Δβ| = 0.63°from DFT which agrees well with the measured 0.7 ± 0.5°f rom the COBRA reconstruction. In contrast, it is 1.21°from the strained bulk calculation where there is no interface tilt epitaxy effect, thus indicating the important role of tilt epitaxy in minimizing the in-plane tilts difference (|Δα| + |Δβ|) across the interface. A similar trend for |Δα| + |Δβ| is seen for the other two film systems: 6.33°(DFT including tilt epitaxy) and 3.7 ± 1.3°( COBRA) versus 8.5°(DFT bulk without tilt epitaxy) for the CaTiO 3 /DyScO 3 ; and respectively, 8.58°and 12.7 ± 1.0°versus 18.88°for the CaTiO 3 /LSAT system. These qualitatively excellent and quantitatively good comparisons between DFT and COBRA confirm the tilt epitaxy and valence mismatch effects on the inplane and out-of-plane polarization components of epitaxial CaTiO 3 thin films.
Interface controlled polarization state. The influence of interfacial tilt epitaxy and valence mismatch effects is expected to diminish as the epitaxial film thickness increases and should be much more prominent in ultrathin films. Fig. 3g, h show the polarization comparison between literature values on thick CaTiO 3 films (>10 nm) [46][47][48][49] and values on ultrathin films studied in this work, where theoretical phase field simulations (green lines), DFT (yellow lines) on strained bulk CaTiO 3 , dielectric measurement results on >10 nm thick CaTiO 3 films (green squares, blue triangles), as well as COBRA (open stars) and DFT (closed stars) results on ultrathin (8 and 6 u.c., respectively) epitaxial CaTiO 3 films are presented. The comparison of in-plane polarization (Fig. 3g) shows a perfect match between experimentally measured polarization on thick CaTiO 3 films (green squares and blue triangles) and theoretical calculation (green and yellow lines) on strained bulk CaTiO 3 , where no interface is present. DFT calculations (closed stars) on ultrathin epitaxial CaTiO 3 films agree well with COBRA results (open stars). The slight increase in the in-plane polarization at 3.3% strain for ultrathin films is consistent with the fact that higher tensile strain favors larger in-plane polarization. The ultrathin films give significantly higher in-plane polarization under 1.1-1.2% strain (NdGaO 3 and LSAT) and lower polarization under 3.3% strain (DyScO 3 ) as compared to thick films, which suggests that the interfacial tilt epitaxy effect can dominate the properties of these films. Similarly, in Fig. 3h, tensile strained bulk CaTiO 3 exhibits zero out-of-plane polarization components, while ultrathin films show a clear non-zero polarization with decreased magnitude at 3.3% tensile strain. This non-zero polarization in ultrathin films again display the effect of interfacial tilt epitaxy and its competition with strain. We also notice that the tilt angles of these ultrathin films change significantly over the first few unit cells, and then tend to relax on approaching the surfaces; however, they do not fully relax to the bulk value within the 8 u.c. This explains the reduced out-of-plane polarization on approaching the film surface. (The outermost u.c. has surface effects and is not included in the discussion here.) With the tilt epitaxy being the driving force, the long-range electrostatic interaction also plays a role in stabilizing this out-ofplane polarization state. The larger out-of-plane polarization in the first few layers favors the polarization with the same direction in the rest of the film, while a depolarization field leads to the relaxation of the out-of-plane polarization from the interface to the film surface.
Probing tilt epitaxy by STEM. Aberration-corrected scanning transmission electron microscopy (AC-STEM) was also employed to confirm the structure determined from COBRA reconstruction at room temperature, to the extent possible by STEM. Atomic resolution annular bright field scanning transmission electron microscopy (ABF-STEM) of the above three epitaxial systems along [010] pc (b-axis) zone axis reveal high quality CaTiO 3 thin films that are epitaxially grown on three different substrates, as  Probing the polar state using SHG method. To confirm the point group symmetry resolved by COBRA, and to investigate the polar transition Curie temperature (T C ), we employ optical second harmonic generation (SHG), schematically shown in Fig. 5a, where a linearly polarized light λ = 800 nm is incident onto the sample at an angle θ, and the second harmonic signal at λ = 400 nm is measured. In Fig. 5b Figure 6). The large enhancement of polar transition temperature, T C , in the 8 u.c. CaTiO 3 films in this work as compared to thicker films is a direct result of the interfacial tilt epitaxy effect, which stabilizes the polar phonon soft mode against its competing centric oxygen octahedral modes through interfacial coupling 53 . For the same reason, 8 u.c. CaTiO 3 on LSAT exhibits the highest T C (>900 K) among all three systems, due to its smallest tilt angles arising from the tilt epitaxy on a substrate with no tilts. Symmetry of CaTiO 3 is determined by SHG polarimetry, where s-/p-polarized SHG signal components, I 2ω,s and I 2ω,p , are measured as a function of incident polarization angle φ (see Methods). Theoretical modeling (see Fig. 5c and Supplementary Fig. 10) indicates a single domain point group of m for the films on NdGaO 3 and DyScO 3 and four equivalent m domains with each domain fraction of 0.25 for the film on LSAT. These results are consistent with the polar states extracted from the COBRA electron density maps.

Discussion
This work demonstrates that tilt epitaxy, namely, slight changes in octahedral tilts in perovskites through interfacial tilt control can dramatically influence the functional properties of ultrathin films. The reconstructed 3D electron density in ultrathin films clearly reveal the intertwining roles of tilt epitaxy, substrate strain, and substrate surface terminations. These give rise to unexpected out-of-plane and in-plane polarization components, as well as large enhancement of polar Curie temperatures T C . An important highlight of this work is the development of the Fourier phase retrieval COBRA method to successfully reconstruct the 3D atomic resolution structure of low symmetry complex oxides interfaces with all independent octahedral tilts and polarization vectors in both the film and substrate, the most complex low symmetry interface structure reconstructed to date by this technique. With the development of high-energy surface X-ray diffraction 54 that enables the rapid capture of large portions of 3D Bragg rods in reciprocal space, COBRA measurements will become much more efficient and widespread in their application to oxide thin films and heterostructures. This work will motivate progress in the fledgling field of tilt epitaxy engineering in ultrathin perovskite films, and more broadly, provide a powerful non-destructive tool with atomic resolution for probing the electron density of complex functional interfaces.

Methods
Sample growth by molecular beam epitaxy. Epitaxial CaTiO 3 thin films were grown on DyScO 3 , NdGaO 3 , and LSAT using reactive molecular-beam epitaxy (MBE) in a Veeco GEN10 system equipped with reflection high-energy electron diffraction (RHEED) and utilizing a background partial pressure of~5 × 10 −7 Torr of distilled ozone. Calcium was evaporated from an effusion cell and titanium from a Ti-Ball™ sublimation source. The fluxes of the constituent elements, calcium and titanium, were measured using a quartz crystal monitor (QCM) and typical values for each element were around 2 × 10 13 atoms cm −2 s −1 . The 8 u.c. CaTiO 3 films were grown at a temperature of 650°C by co-depositing CaO and TiO 2 . The starting fluxes of the Ca and Ti molecular beams were initially determined by QCM and then the calcium flux was fine-tuned to match the flux of the titanium using shuttered RHEED intensity oscillations. Due to the imperfect growth control, the exact thickness of these films is determined to be slightly larger than 8 u.c. with incomplete surface layers, as discussed in Supplementary Note 7.
Crystal truncation rods measurements and coherent Bragg rods analysis. Crystal truncation rods (CTRs) were measured using a surface X-ray diffraction geometry with a six-circle diffractometer under X-ray photon energy of 16 and 23.9 keV at sectors 12-ID-D and 33-ID-D at Advanced Photon Source, Argonne National Laboratory. Both beamlines have a similar total flux of~2.0 × 10 12 photons s −1 . At 33-ID-D, the X-ray beam was focused by a pair of Kirkpatrick-Baez mirrors down to a beam profile of 50μm (vertical) × 500μm (horizontal). The twodimensional diffraction images of CTRs at each L step in the reciprocal space were recorded with a pixel array area detector (Dectris PILATUS 100 K). Samples were protected under dry helium gas flow in a concealed sample cell during room temperature measurements. Low temperature measurement was achieved with a closed-cycle-cryocooler system (Advanced Research System Model DE-204). A large set of CTRs in the reciprocal lattice coordinate were measured for all three epitaxial CaTiO 3 films at both room temperature and low temperature (30 K), with H max , K max = 8 r.l.u., L max = 9 r.l.u. under 2 × 2 × 2 pseudocubic notation.
3D electron densities (EDs) for the complete atomic structures were reconstructed from the complete set of CTRs by using an iterative phase retrieval technique, known as coherent Bragg rods analysis (COBRA) 32,34,35,52,55 , through an in-house developed MATLAB code, generalized for systems with symmetry lower than 4mm (or simple four-fold symmetry system). Experimental CTR data were first background subtracted, and then properly corrected for geometric and polarization factors. Initial atomic structural model was constructed based on bulk structures with fitted CaTiO 3 c lattice constant using GenX software 56 . Within each iteration, real space and reciprocal space constraints are alternatively applied to reconstruct phase information from measured CTRs. The structural results yielded by the COBRA iterations are found to be insensitive to the initial model, as illustrated in Supplementary Note 8.
The generic approach for uncertainty analysis based on a parameterized model is not applicable to COBRA-generated EDs. A method called noise analysis based on statistical analysis is previously used to determine the uncertainties in COBRA results 32 . This method requires COBRA reconstructed EDs of a large number of groups of CTRs adding afterward with random noise and analyzes the degree of scatters in the interested values extracted from EDs, which is extremely costly for analyzing six different systems (three epitaxial structures at room temperature and 30 K) presented in this work. By taking advantages of ultrathin CaTiO 3 films and well-known substrates used in this work, we estimate the uncertainty by including 11 pseudocubic u.c. of substrates into the reconstructed EDs. The structures of substrate unit cells buried underneath the 5th u.c. away from the interfaces should maintain their bulk structures. Therefore, the deviation between EDs and bulk structures for the first 6 u.c. of substrates is used for estimating the magnitude of deviation between COBRA results and the true values. A comparison between above two methods is detailed in Supplementary Note 9.
Density functional theory calculations. The density functional theory (DFT) calculations use the plane-wave basis and projector augmented wave (PAW) 57 method within the Vienna Ab initio Simulation Package (VASP) 58 . The choice of the exchange-correlational functional is the Perdew-Burke-Ernzerhoff (PBE) 59 generalized gradient functional. Convergence tests indicate that energies are converged to within 1 meV atom −1 with a 560 eV cutoff energy, with 20 Å of vacuum in the direction perpendicular to the interface, and with a 8 × 8 × 1 k-point mesh. Spin-polarization is used in all the calculations. Structure relaxation is iterated until the energy differences are below 10 −6 eV and until all forces on the atoms are below 0.05 eV Å −1 . To minimize the unphysical dipole energy arising from the heterostructure, CaTiO 3 thin films were symmetrically introduced on both side of the substrate. During the calculation, all the substrate atoms are fixed at their initial positions and are not allowed to relax.
Scanning transmission electron microscopy imaging. The scanning transmission electron (STEM) images of CaTiO 3 thin films on NdGaO 3 , DyScO 3 , and LSAT under (100) pc , (010) pc , and (110) pc zone axes were collected on an FEI Titan G2 double aberration-corrected HR-STEM at 300 kV with a probe illumination angle of 28 mrad. High-angle annular dark-field (HAADF) and annular bright field (ABF)-like images were obtained with collection angles of 42-244 and 9-51 mrad, respectively. At each sample location, images were taken with the STEM fast scan direction set to 0°and 90°with respect to the substrate interface direction. These image pairs were then drift corrected, after which the images were superimposed.
Analysis of the STEM images was performed using custom-written MATLAB code. The sub-pixel resolution of the cations and oxygen positions were determined by fitting a seven parameter 2D elliptical Gaussian profile (to account for any ellipticity in the intensity distribution) to the HAADF/ABF intensity distribution.
Optical second harmonic generation measurements. Optical second harmonic generation (SHG) polarimetry and temperature dependent measurements were performed in a far-field transmission setup using femtosecond pulses at λ = 800 nm generated by a Spectra-Physics Empower Q-switched Nd:YLF pumped Solstice Ace Ti:Sapphire laser system (100 fs, 1 kHz). The experimental schematic is shown in Fig. 5a, where a linear polarized fundamental field with polarization direction φ incident on the sample at an incident angle θ.The p-polarized (I 2ω,p ) and spolarized (I 2ω,s ) components of second harmonic field (E 2ω ) was first spectrally filtered and then detected by a photo-multiplier tube, using lock-in method (SR830). For each sample, systematic polarimetry was performed by rotating the incident polarization φ at fixed θ for two different sample orientations (O1 and O2). SHG polarimetry on CaTiO 3 /NdGaO 3 at 30 K was performed at incident angles θ = −30°, 0°, 30°for two sample orientations, O1: (S 1 , S 2 , S 3 ) = ([001] or , [1][2][3][4][5][6][7][8][9][10] Figure 10, and is different from CaTiO 3 on NdGaO 3 and DyScO 3 . Temperature-dependent measurements were performed by monitoring the SHG signal while scanning the sample temperature, which was controlled using helium cooled Janis 300 cryostat (for low temperature) and a heater (for high temperature).
Code availability. The computer codes that support the findings of this study are available from the corresponding author upon reasonable request.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.