## 1 Introduction  
Nowadays, progresses in materials science and engineering have shown increasing reliance on the advantages of materials properties that originate from multi-field couplings. In ferroelectric materials, these properties include electrical-mechanical coupling (i.e., piezoelectricity [1; 2] and flexoelectricity [3; 4; 5]), electrical-thermal coupling (i.e., pyroelectricity [6]), magneto-electricity [7] coupling, dielectricity [8; 9], and so forth. All of the above properties involve the dynamics of ferroelectric polarization as well as reactions to externally applied fields. Electric-field-induced polarization switching in ferroelectric materials has attracted extensive attention from experimental [10] and theoretical [11; 12; 13; 14] perspectives, owing to their unique features which can potentially provide improvements in data storage [15; 16], spintronics [17], tunnel junctions [18], and non-volatile magnetoelectric devices [19]. As less voltage would be needed to generate the same electric field in thin films than that in thick ones, the basic understanding of underlying physics for thickness effects in switching behaviors is of great importance to the applications of ferroelectric thin films in miniaturized devices. One of the most studied ferroelectric materials, BiFeO\({}_{3}\) (BFO), simultaneously possesses ferroelectric properties and antiferromagnetic properties at room temperature, with Neel temperature \(T_{\rm N}\sim 643\) K and Curie temperature \(T_{\rm C}\sim 1103\) K [20; 21; 22; 23]. The crystal structure of BFO below \(T_{\rm C}\) is a rhombohedrally distorted perovskite structure with its space group being \(R3c\), [20] resulting in eight possible variants for spontaneous polarization in BFO, which can be denoted as \(\rm R_{\rm 1}^{\pm}=\pm[111],~{}R_{\rm 2}^{\pm}=\pm[\overline{1}11],~{}R_{\rm 3}^{ \pm}=\pm[\overline{1}11],~{}R_{\rm 4}^{\pm}=\pm[1\overline{1}1]\). Polarization switching arises when an external electric field is applied to the film. Due to the specialty of rhombohedral structures, only three paths are symmetrically allowed for polarization switching. According to the rotation angle, these switching paths can be defined as 71 degree switching, 109 degree switching, and 180 degree switching, respectively. To discover the optimal conditions for BFO films in applications, [PERSON] et al. successively studied the effect of film thickness, annealing treatments, processing temperatures, deposition substrates, and doping on the electric properties, particularly the polarization and leakage current density, of BFO thin films [24; 25; 26; 27]. [PERSON] et al. studied ferroelectric properties including polarization-electric-field (_P-E_) loop and leakage behavior for BFO films varying from 85 to 280 nm [28]. It was found that the coercive field decreased with increasing thickness. Similar conclusions were drawn by [PERSON] et al. in a study about thickness-dependent multiferroic properties of BFO thin films, with film thicknesses ranging from 210 to 830 nm [29]. Theoretically, a thermodynamic model based on the Landau-Devonshire free energy expression and the Landau-Khalatnikov dynamic equation was employed to analyze the effects of film thickness and surface parameter on switching time and coercive field [30]. Results showed that the sign of surface parameter would determine the trend of coercive field changing with film thickness. Dead-layer effects in polarization switching has also been studied in ultrathin BFO films using both experimental methods and statistical analysis [31]. [PERSON] and [PERSON] [12] proposed a micromechanics model to study polarization switching in tetragonal and rhombohedral structures and found that it is easier to move 90\({}^{\circ}\) (for tetragonal) and 70.5\({}^{\circ}\) (for rhombohedral) domain walls than 180\({}^{\circ}\) domain walls.