# Probing Ferroic States in Oxide Thin Films Using Optical Second Harmonic Generation
## 3 Probing Ferroic States in Thin Films Using SHG
### Optical Second Harmonic in Domain Engineered Ferroic Thin Films  
The SHG technique is a powerful technique for the analysis of polar states in thin films possessing a net ferroelectric polarization, i.e., in a single domain state or in domain-engineered systems [88]. Uniaxial tetragonal ferroelectric PbZr\({}_{0.2}\)Ti\({}_{0.8}\)O\({}_{3}\) (PZT), the most technologically relevant ferroelectric compound, shows a thickness-dependent domain distribution [91]. A fully strained film can exhibit a single domain state with a polar axis pointing out of plane, along the long tetragonal \(c\)-axis. Above a threshold thickness (of about 60 nm in the case of PZT grown on LaAlO\({}_{3}\) substrates [91]), strain relaxation will induce a coexistence of nanosized in-plane oriented domains, i.e., \(a\)-domains with a polarization pointing along the in-plane oriented long axis, together with the out-of-plane \(c\)-domains [92; 93]. Figure 7: Substrate- and surface-related SHG. (**a**) Polarizer measurements performed on DSO in normal incidence (0\({}^{\circ}\) tilt angle) transmission. (**b**) Surface-related SHG response in anisotropy configuration at 45\({}^{\circ}\) incidence in reflection from an SRO thin film grown on STO substrate. The \(a\)-domains nucleate from either the top or the lower interfaces and their distribution in the volume can have major implication on the ferroelectric switching dynamics and thus on the targeted applications [94; 95; 96]. It is, however, non-trivial to access the ferroelectric domain distribution in the thin film volume when using scanning probe techniques due to their limited depth sensitivity. As discussed in the previous section, the optical SHG technique only probes polar distortions projected on a plane perpendicular to the k-vector of the incoming light wave [16; 17; 18]. It is, therefore, possible to disentangle the contribution of \(a\)- and \(c\)-domains by changing the orientation of the ferroelectric polar axis with respect to the incident light beam [97]. The single-domain state of a thin film (15 nm thick) with a polarization pointing out of plane can be confirmed by PFM as shown in Figure 8a. For a fully \(c\)-oriented tetragonal film with point group \(4 mm\), three independent \(\chi_{ijk}\) components (\(\chi_{zzz}\), \(\chi_{zxx}\) and \(\chi_{zxx}\)) are non-zero [80; 97]. At normal incidence (\(\theta=0^{\circ}\)), no second harmonic response is expected since the out-of-plane directed ferroelectric polarization then points along the k-vector. When tilting the ferroelectric system with respect to the incident light beam, the increment of the incidence angle will result in an increase of the associated SHG signal as shown in Figure 8b. The specific light polarization configuration is described in the inset of Figure 8b. The angular dependence of the SHG light in the polarizer measurement identifies the projection of the polar distortion to be along the direction of the double-lobe shape, as seen in Figure 8c. Here, the [001]\({}_{\text{DSO}}\) direction of the substrate is orthogonal to the tilt-axis ([1; 2; 3; 4; 5; 6; 7; 8; 9; 10]\({}_{\text{DSO}}\)) and represents the projection of an out-of-plane polarization. For a thicker layer, the epitaxial strain relaxation promotes the growth of nanosized stripe-domains with an in-plane polarization orientation, as evidenced by the PFM scan performed on the 75 nm thick film in Figure 8d. Here, specific light polarization configurations allow discriminating between the in-plane (in black) and the out-of-plane (in blue) contributions to the nonlinear optical response, as shown in Figure 8e. It is worth emphasizing that the SHG intensity associated with the in-plane  
Figure 8: Optical detection of \(a\)-domains in PZT. **(a)** PFM scan of a single domain, \(c\)-oriented film. The scale bar is 1 \(\upmu\)m. **(b)** SHG tilt scan showing the increased contribution from \(c\)-domains with increased tilting angle.