# Probing Ferroic States in Oxide Thin Films Using Optical Second Harmonic Generation
## 1 Introduction
### Current Challenges in Probing Ferroic Domains in Thin Films  
Currently, state-of-the-art observation of ferroelectric domain distribution in the volume is provided primarily by scanning transmission electron microscopy (STEM). This imaging technique allows spatial mapping of local polarization but remains a destructive analysis and is highly dependent on the sample preparation. Other experimental techniques such as synchrotron X-ray diffraction-based techniques [21; 26; 68] can identify polarization orientation and additional periodicities originating from domain formation, but the involvement of large-scale facilities renders their generalization towards day-to-day operation impractical. For a complete understanding of the role of epitaxial strain and electrostatics at the created oxide interface, noninvasive access to the polarization state is essential.  
When it comes to probing several ferroic orders simultaneously, as is desirable in multiferroics, additional experimental constraints need to be considered. Most ferroelectric multiferroic materials exhibit antiferromagnetic ordering and therefore do not possess a net magnetic moment. Hence, antiferromagnetic domain visualization remains challenging. An efficient way to observe antiferromagnetic domains is using the principle of ferroic domain imprint transfer [69; 70; 71; 72; 73]. A thin ferromagnetic layer is deposited in direct contact with the multiferroic layer. The interfacial exchange  
Figure 3: Scanning transmission electron microscopy on complex polar domains. (**a**) Observation of ferroelectric vortices in PTO/STO superlattice. Reprinted by permission from [48], Copyright Macmillan Publishers Ltd., 2016. (**b**) Close-up into a single vortex. (**c**) Flux-closure domain patterns in a PTO/STO multilayer system. From. Reprinted with permission from [65], Copyright AAAS, 2015.  
coupling with the underlying antiferromagnetic layer results in the formation of ferromagnetic domains in the ferromagnetic film that can be resolved using scanning electron microscopy with polarization analysis (SEMPA) [72] as shown in Figure 4a,b, or magnetic force microscopy (MFM) [72; 74]. The underlying ferroelectric domain state is however no longer reachable using conventional scanning probe experiments, i.e., piezoresponse force microscopy (PFM), after the ferromagnet has been deposited.  
Recently, a single spin magnetometer technique was developed based on point-like impurity nitrogen-vacancy (NV) defect in diamond [75; 76; 77]. This scanning-probe technique was used to spatially resolve the local magnetic order in multiferroic antiferromagnetic BFO thin films [78]. A periodic modulation in the magnetic response corresponding to a spin cycloid was measured within a single ferroelectric domain; see Figure 4c,d. This spatially resolved characterization of antiferromagnetic domains does not prevent the subsequent ferroelectric analysis in the same surface area and signifies important progress towards the analysis of multiferroic domains.  
However, the direct access to multiferroic domains in buried configurations, i.e., in a device architecture or in a superlattice, remains impossible using the scanning probe approach.