## 2 Theoretical Description  
The theoretical model of describing an AFE phase is a simple two-sublattice model by [PERSON] [37]. [PERSON] and [PERSON] developed a phenomenological model using Gibbs free energy as a function of two dependent FE and AFE order parameters [38; 39]. [PERSON] et al. proposed a three-dimensional phenomenological model to successfully describe the temperature-, pressure-, and composition-induced ferroelectric to antiferroelectric phase transitions, which are consistent with the experimental phase diagrams [32]. Therefore, we adopted the same model and parameters to investigate the electrocaloric effects of Sm-doping BFO. To distinguish the FE and AFE phases, AFE and FE phases were divided into two sublattices, \(a\) and \(b\), associated with polarization \(\overrightarrow{P_{a}}\) and \(\overrightarrow{P_{b}}\) respectively. If \(\overrightarrow{P_{a}}\) and \(\overrightarrow{P_{b}}\) are parallel to each other, the system exhibits the FE phase, as sketched in Figure 1a, otherwise, if \(\overrightarrow{P_{a}}\) and \(\overrightarrow{P_{b}}\) are antiparallel, it exhibits the AFE phase, as illustrated in Figure 1b. When \(\overrightarrow{P_{a}}\) and \(\overrightarrow{P_{b}}\) are both zero, it exhibits the PE phase. The FE and AFE order parameters, \(\overrightarrow{p}\) and \(\overrightarrow{q}\), are defined as follows [40]:  
\[\overrightarrow{p}=\frac{1}{2}\bigg{(}\overrightarrow{P_{a}}+\overrightarrow {P_{b}}\bigg{)},\ \overrightarrow{q}=\frac{1}{2}\bigg{(}\overrightarrow{P_{a}}^{\prime}- \overrightarrow{P_{b}}^{\prime}\bigg{)} \tag{1}\]  
Figure 1: Schematic of the (**a**) ferroelectric (FE) and (**b**) antiferroelectric (AFE) phases. In Equation (1), \(\overrightarrow{P_{a}}^{\prime}\) and \(\overrightarrow{P_{b}}^{\prime}\) were used instead of \(\overrightarrow{P_{a}}\) and \(\overrightarrow{P_{b}}\) to represent the AFE order parameter, because the polarization associated with sublattices of the AFE phase are sometimes different from that of the FE phase. Taking the unpolarized and unstressed PE phase as the reference state, the total free energy density of the Sm-doping BFO system under an electric field can be expressed as [32]:  
\[ \begin{array}{ll}f_{total}=a_{i}p_{i}p_{j}+&a_{ijkl}p_{i}p_{j}p_{k}p_{l}+a_{ ijklmn}p_{l}p_{j}p_{k}p_{l}p_{m}p_{n}+b_{ij}q_{i}q_{j}+b_{ijkl}q_{i}q_{j}q_{k}q_{l} \\ &+b_{ijklmn}q_{i}q_{j}q_{k}q_{l}q_{m}q_{n}+t_{ijkl}p_{l}p_{j}q_{k}q_{l}+\frac{ 1}{2}s_{ijkl}q_{ij}q_{kl}-Q_{ijkl}p_{i}p_{j}q_{kl}\\ &-\Lambda_{ijkl}q_{i}q_{j}\sigma_{kl}-\frac{1}{2}\varepsilon_{0}E_{i}^{2}-p_{ i}E_{i}\end{array}  \tag{2}\]  
where \(p_{i}\) and \(q_{i}\) are the \(i\)th components of the FE and AFE order parameters; \(a_{ij}\), \(a_{ijkl}\), \(a_{ijklmn}\) and \(b_{ij}\), \(b_{ijkl}\), \(b_{ijklmn}\) are the FE and AFE dielectric stiffnesses, respectively. \(t_{ijkl}\) are coupling coefficients between the FE and AFE order parameters. \(s_{ijkl}\) are the elastic compliance constants; \(Q_{ijkl}\) and \(\Lambda_{ijkl}\) are corresponding electrostrictive coefficients; \(\sigma\) is \(i\)th component of applied stress in Voigt notation; \(\varepsilon_{0}\) is the vacuum permittivity; \(E_{i}\) is the external applied electric field. Among all the coefficients, only \(a_{11}\) and \(b_{11}\) are assumed to be dependent on composition and temperature. All the coefficients used in this work are taken from reference [32]. The applied hydrostatic pressure tensors satisfy \(\sigma_{1}=\sigma_{2}=\sigma_{3}=-\sigma\) and \(\sigma_{4}=\sigma_{5}=\sigma_{6}=0\). The EC effect was investigated based on indirect measurements using the Maxwell relation \(\left(\partial P/\partial T\right)_{E}=\left(\partial S/\partial E\right)_{T}\), where the total polarization \(P\) includes \(p_{i}\) and \(q_{i}\).