# 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 scale bar is 2 \(\upmu\)m. **(c)** Polarizer measurement at a 40\({}^{\circ}\) tilt angle evidencing a single polar axis in the film. **(d–f)** Corresponding PFM, tilt scan and polarizer measurement at normal incidence for a mixed \(a\)-/\(c\)-domain state, respectively. domains is non-vanishing at normal incidence (\(\theta=0^{\circ}\)), where no contribution is expected from \(c\)-axis (out-of-plane) domains. The polarizer measurement shown in Figure 8f highlights the presence of a macroscopic in-plane polarization directed along [1-10]\({}_{\text{DSO}}\). Note that in the presence of a complex sub-optical-resolution domain architecture where two sets of orthogonally oriented \(a\)-domains coexist [93], the SHG signal will not yield the spontaneous polarization of each domain but rather reflect the direction of the macroscopic in-plane net polarization resulting from the sum of all the \(a\)-domain contributions. In the present case, this corresponds to a \(45^{\circ}\) rotation of the polar axis shown in Figure 8f, as further discussed below for the case of BFO films. SHG is therefore sensitive to ferroelectric domains smaller than the optical resolution and the presence of \(a\)-domains throughout the volume of the film can be identified optically [97]. We note however that complementary scanning transmission electron microscopy is necessary to address specific domain morphologies throughout the film thickness [97]. The correspondence between the angular dependence of SHG and the orientation of the polar axis can be exploited further to identify the direction of a net polarization when more complex domain structures are considered [88]. For example, domain-engineered multiferroic BFO films subject to an in-plane anisotropic strain present two ferroelectric variants arranged in a stripe pattern (Figure 9a) [88]. The in-plane projections of the polarization in adjacent domains differ by \(90^{\circ}\) as highlighted by the small white arrows in the PFM scan in Figure 9a. This produces an average net in-plane polarization orthogonal to the domain wall elongation (indicated by the large white arrow in Figure 9a). The angular dependence of the SHG light therefore manifests as a two-fold shape oriented along the macroscopic polar axis direction as shown in Figure 9b. In the absence of such in-plane anisotropy, four equivalent ferroelectric variants nucleate, characterized by the absence of an average macroscopic in-plane polarization (Figure 9c). In this case, the associated SHG signal does not depend on the sample orientation; see Figure 9d. Figure 9: SHG on domain-engineered multiferroic BFO. (**a**) PFM scan of a stripe-like domain BFO film. (**b**) Polarizer measurement evidencing a net in-plane polarization in the film. (**c,d**) Corresponding PFM and polarizer measurement for an isotropic domain distribution, respectively. The signal observation in (**d**) does not depend on the sample crystalline orientation. Reproduced with permission from [88], Copyright Wiley-VCH Verlag GmbH and Co. KGaA, 2015. SHG can also detect subtle structural changes such as strain dependent monoclinic distortions in epitaxial films. In particular, large compressive strains (>4.5%) in BFO films induce a tetragonal-like distortion favoring the emergence of a pseudo-tetragonal phase, i.e., BFO T-phase, while smaller compressive strains are associated with a rhombohedral-like distortion stabilizing the pseudo-rhombohedral phase, i.e., BFO R-phase [98; 99]. These two phases possess the same monoclinic point group \(m\) but are characterized by a rotation of the in-plane polarization of 45\({}^{\circ}\) at the unit-cell level [100]. With SHG, these two phases can be simply distinguished by observing the direction of the double-sided lobe [88; 101; 102; 103] as shown in Figure 10a,b. Note that the orientation of the polar axis is inclined by a few degrees with respect to the crystallographic axes due to the monoclinic distortion. The R- and T-phases can also coexist in the same thin film system [98].