## 1 Introduction  
### Structural Phase Transition in Films of Multiferroic Bismuth Ferrite  
Multiferroic materials, where different ferroic orders coexist, are of paramount importance for the next generation of multifunctional devices for spintronics and memory technologies, since the material state in one of the ferroic orders could be controlled by the others. Bismuth ferrite BiFeO\({}_{3}\) (symbolized as BFO) is the most paradigmatic member of the multiferroics family [1]. This lead-free piezoelectric is the most famous due to the coexistence of the ferroelectricity and magnetism at room temperature [1, 2]. However, in recent years the high sensitivity of BFO state to strain and light is attracting increasing attention [2, 3, 4, 5, 6, 7, 8, 9, 10]. In particular, not only the unit cell parameters but BFO crystal structure can be controlled by strain in the epitaxial growth of thin BFO films on the substrates: the morphotropic rhombohedral-tetragonal phase transition can be induced [2, 3, 4]. Since piezoelectric, ferroelectric, optical, and elastic properties of materials are expected to exhibit anomalies near morphotropic phase boundaries, the control of the BFO by epitaxial strain could engineer thin films with enhanced interactions between different ferroic orders, which is highly sought for future information and communication technologies.  
Besides the strain-controlled epitaxial growth technique, and among other possible methods, a possible way to approach the morphotropic boundary between the rhombohedral and the tetragonal phases of the material can be the modification of its composition via doping [4, 11, 12, 13]. In particular, the effects of Praseodymium (Pr) substitution on the structure, magnetism, and electrical properties of mixed compounds (Bi\({}_{1-x}\)Pr\({}_{x}\))(Fe\({}_{0.95}\)Mn\({}_{0.05}\))O\({}_{3}\) (symbolized as BPFMO) thin films were systematically investigated [12, 13]. X-ray diffraction and Raman spectra results confirmed that a phase transition from rhombohedral to tetragonal structure occurs at about 15% Pr substitution. These results demonstrate that doped BFO films with compositions near a phase boundary may be promising candidates for multifunctional applications due to the simultaneous possession of the maxima in ferromagnetic and ferroelectric properties. The other signatures of the phase transition in condensed matter could be elastic anomalies [14]. In nanometers-thick BFO films, the softening of elastic modulus at sub-MHz frequencies by over 30% in the vicinity of the electrically induced rhombohedral-tetragonal phase transition has been reported [15]. The elastic anomaly has been accompanied by two- to three-fold enhancement of local piezoresponse, suggesting potential applications of BiFeO\({}_{3}\) in next-generation frequency-agile electroacoustic devices based on the utilization of the soft modes underlying successive ferroelectric phase transitions [15]. The combination of the experimental results [12, 13, 15] of high applicative significance motivates the studies of the elastic, piezoelectric, optical, and other, yet unmeasured, properties of BFO films across the structural phase transition induced by doping. The objective of our research reported in the following Sections 2-4 was to employ a picosecond laser ultrasonics technique, namely the time-domain Brillouin scattering (TDBS) technique, in combination with optical methods, atomic force microscopy, and scanning electron microscopy, to reveal the anomalies in the acoustical and optical parameters of the BPFMO films at GHz frequencies induced by approaching the morphological phase boundary via Pr substitution.