# Probing Ferroic States in Oxide Thin Films Using Optical Second Harmonic Generation
## 1 Introduction
### Ferroic Domain Engineering in Single Layers and Superlattices  
The combination of epitaxial strain, surface termination and electrostatics in ferroic thin films and heterostructures can be used for domain engineering. Deterministic control of the domain formation  
Figure 1: Thickness dependence of the polar state in ferroelectric thin films. (**a**) The polarization \(P\) and depolarizing field \(D\) have a thickness dependence, resulting in the minor influence of the depolarizing field on polarization in the high thickness range and a suppression of polarization at low thicknesses; (**b**) When the bound charges can be screened by the environment, the depolarizing field is minimized and a single domain state can be stabilized over a wide thickness range; (**c**) In the absence of charge screening, however, the depolarizing field favors a multi-domain state. Flux closure or vortices can also form in a specific thickness or strain regime. process toward the design of complex ferroic states in three dimensions is a necessary means to realize next-generation oxide-based devices [48; 49; 50; 51]. One instructive example of strain engineering the ferroic domain structure in single layers is the case of multiferroic BFO thin films. In this system, [PERSON] and coworkers [52] demonstrated the ability to modify the ferroelectric domain state by using relevant substrate orientations and surface states. Starting from the eight symmetry-allowed polar axes along the \(<\)111\(>\) crystallographic directions of the pseudo-cubic unit cell, the number of as-grown ferroelectric variants could be tuned to either a four ferroelectric variants or a two ferroelectric variants configuration, when grown on the cubic perovskite (001) or (110) orientation of SrTiO\({}_{3}\) (STO) substrates, respectively. A single domain state could be achieved when using (111) oriented STO (see Figure 2a-c). Further tuning of the domain architecture was achieved with the use of scandate substrates [53]. DyScO\({}_{3}\) (DSO) crystallizes in the orthorhombic space group with a pseudo-cubic unit cell exhibiting a slight anisotropy between the in-plane lattice parameters [54]. This results in an in-plane anisotropic strain which can further lift the degeneracy of polar domains of perovskite ferroelectrics [53]. Scandate substrates have also been used to induce multiferroicity in STO thin films [55; 56] and EuTiO\({}_{3}\) layers [57; 58]. Beyond epitaxial strain considerations alone, the macroscopic out-of-plane polarization orientation and the domain state in thin film ferroelectrics also drastically dep  
Figure 2: Domain engineering in ferroic thin films. (**a–c**) Control of the number of ferroelectric variants in BFO using STO substrate orientation. (001), (110) and (111) orientations correspond to (**a–c**), respectively. Reproduced with permission from [52], Copyright Wiley-VCH Verlag GmbH and Co. KGaA, 2007); (**d**) The insertion of a dielectric ultrathin lanthanum-doped BFO layer between BFO and the metallic SrRuO\({}_{3}\) (SRO) buffer induces a transition from a 71\({}^{\circ}\) stripe domain architecture to a 109\({}^{\circ}\) stripe domain pattern. Reprinted with permission from [62], Copyright American Chemical Society, 2016; (**e**) The insertion of a dielectric STO layer between the SRO and PbTiO\({}_{3}\) (PTO) ferroelectric film stabilizes nanoscale 180\({}^{\circ}\) domains. Reprinted with permission from [64], Copyright American Chemical Society, 2014.  
environment [59] and the surface termination of the substrate [60; 61]. Screening conditions affect the stability of domain walls and are a key parameter in the control of complex domain structures. By inserting a dielectric layer between the ferroelectric thin film and the metallic buffer, the 109\({}^{\circ}\) stripe domain state could be favored in BFO [62] and 180\({}^{\circ}\) domains were stabilized in PbTiO\({}_{3}\) (PTO) [63; 64]; see Figure 2d,e, respectively.