# Probing Ferroic States in Oxide Thin Films Using Optical Second Harmonic Generation
## 1 Introduction
### Ferroelectric Thin Films  
While tremendous advances have been made in understanding ferroelectrics on their own, fundamental aspects of their behavior in the ultrathin regime or once inserted in multilayers or superlattice architectures are still under intense investigation [19; 20; 21]. The buildup of a ferroelectric polarization \(P\) in a material is accompanied by the accumulation of bound charges at its surfaces. As a result, the system hosts an internal field, induced by these bound charges, which is oppositely oriented to the polarization direction; see Figure 1. This depolarizing field \(D\) depends on the thickness of the ferroelectric layer and its electrostatic environment, such as the presence of metallic electrodes or the surrounding gas partial pressure and their respective charge screening efficiencies [22; 23; 24]. The role of the depolarizing field becomes critical in ultrathin ferroelectric single layers and in superlattices, where interface effects may become predominant [19; 21; 25; 26]. As illustrated in Figure 1, a small change in film thickness and charge environment will have a drastic influence on the manifestation of a ferroelectric state and its domain configuration. For instance, in superlattices, polarization can be stabilized at ultra-low thickness [27; 28] or a global enhancement of ferroelectricity can be achieved [29]. In a charge compensating environment, a single domain state is favored in a wide range of thicknesses; see Figure 1b. In the case of absent or incomplete charge screening, the presence of the depolarizing field will result in splitting of the single-domain state into a 180\({}^{\circ}\) multi-domain pattern, the formation of flux-closure domains or the emergence of ferroelectric vortices depending on the thickness and strain state considered; see Figure 1c.  
In ferroelectric-based heterostructures, the intricate character of the interactions between polar and non-polar, metallic and non-metallic layers and the difficulty to access buried ferroelectric states renders the control of polarization challenging. Furthermore, probing the emergence of ferroelectricity or the domain state in thin films as function of thickness, strain and temperature is hindered by the lack of contactless probing techniques. Leakage currents [30] in the ultrathin regime and at elevated temperatures prevent the understanding of ferroelectricity in the low thickness regime.  
On the structural level, pioneering theory and experimental work dealing with ferroelectric thin film behavior under epitaxial stress evidenced the potential of epitaxial strain as a new degree of freedom for controlling the ferroelectric state in single layers [7]. In situ oxide growth diagnostic tools such as reflection high energy electron diffraction (RHEED) [31] together with the development of single crystal substrate manufacture, allow precise engineering of the strain state in high-quality thin films. Using a combination of phase field simulations and experimental techniques, it was further shown that strain can be used to tune the ferroelectric polarization [32], enhance the Curie temperature [33; 34] and induce a selectivity to specific ferroelectric switching events [35]. Epitaxial strain can also be used to induce a ferroic state. For instance, ferromagnetism was strain induced in orthorhombic LuMnO\({}_{3}\) thin films [36] while a polar state was stabilized in tensile strained SrMnO\({}_{3}\) thin films [37]. However, experimental evidence of such a control remains inaccessible in multilayer architectures due to the increasing role of interfaces and the lack of a direct probe of ferroelectricity in a buried configuration.