IntroductionFrequency-dependent dielectric response is one of the important properties of ferroelectrics, reflecting the polarization response to high-frequency electric fields. Polarizations are closely related to ferroelectric domain structures, such as single domain, which represents the region with homogeneous polarizations direction. Ferroelectrics usually possess complex domain structures with domain walls (DWs) separating adjacent homogeneously polarized domains. DWs have attracted much attention during the past two decades due to their properties and potential for device designing. The related issues include DW motion, nonvolatile memory, topological defects, enhanced susceptibility, enhanced quality factor, low dielectric loss, and others. (Ba_0.8,Sr_0.2)TiO₃ (BST_0.8) is a ferroelectric usually with multi-domain structures. Previous work identified two typical types of domain walls (DWs), i.e., 90° DWs and 180° DWs. The enhancement of dielectric response in systems with 90° DWs is now well understood, and the behavior of dielectric response in multi-domain systems with 180° DWs remains unclear. Therefore, gaining insights into how 180° DWs affect the dielectric response can clarify the effects in multidomain systems.MethodsWe performed molecular dynamics simulations using the ALFE-H code with the first-principles-based effective Hamiltonian method to study the BST_0.8 system. All the calculations were performed in the NPT ensemble using the Evans-Hoover thermostat, and periodic boundary condition (PBC) along all three directions. To simulate the substrate, a uniform biaxial strain was fixed to the 1.55% in-plane strain. To analyze the multi-domain with different DWs, the simulations started with a self-constructed initial multi-domain polarization configuration. Subsequent 50 ps MD simulation was performed under chosen strains for structural relaxation. In the initial configuration, the magnitude of non-zero components of soft mode on each site was set to 0.1 Å, atomic occupations (alloying) were randomized, and unless otherwise specified, all other mode variables were set to zero. The trajectory of local mode averaged over the supercell during MD simulations was extracted to calculate the dielectric response. The 8 ns MD simulations were performed to obtain an autocorrelation function for any time t ranging from 0 to 1 ns by one step of 10 fs. The fast Fourier transformation (FFT) was performed to calculate the dielectric response. Then two uncoupled damped harmonic oscillators (DHOs) were used to fit the data of dielectric response.Results and discussionThe dielectric response of single domain at 300 K with the different electric fields along [110] from 0 to 2 MV/cm was computed. The computational results can be well fitted with the model of two uncoupled DHOs. The real and imaginary parts of the predicted dielectric response at each chosen electric field both exhibit two peaks. As the electric field increases, the low-frequency mode with 300 GHz at zero field in the system gradually disappears, and a high-frequency mode of larger than 8 THz appears when electric field is larger than 1 MV/cm. The high frequencies modes of 3 THz at zero filed and 8 THz under 1 MV/cm shift towards higher frequencies as the electric field increases. In other words, the present simulations reveal that it is possible to manipulate the frequency of peaks in dielectric response via changing the magnitude of the external electric field.The dielectric responses of multi-domain with 90° DWs and 180° DWs are further analyzed. According to the experimental PFM results, the multi-domain structures are simulated and the dielectric response through MD simulations is calculated. The analysis of the dielectric response of single domain structure and multi-domain structures shows that the single domain structures exhibit high-frequency peaks at > 300 GHz, whereas the multi-domain structures exhibit low-frequency peaks at 8 GHz and 120 GHz for 180° DWs system and at 10 GHz for 90° DWs system, revealing that there exists a low-frequency mode related to collective oscillation of DWs in multi-domain structures. In addition, the frequencies of peaks in multi-domain with DWs are in a gigahertz range, whereas the single domain structure exhibits peaks in a terahertz range. The contribution of DWs to the dielectric response primarily arises from the timescale of DWs motion, such as sliding or breathing, which differs significantly from the high-frequency vibrations of optical phonon modes. The vibrational frequency of DWs is much lower, with relaxation times in the order of nanoseconds, resulting in a response frequency in GHz range, which is far below the terahertz range of optical phonon modes. Therefore, DWs oscillations dominate the dielectric response at a low frequency. Moreover, multi-domain structure with 180° DWs exhibits a unique low frequency mode at 120 GHz, which is significantly different from single domain and 90° DWs system. In other words, multi-domain structures with 180° DWs and 90° DWs exhibit different dielectric responses. There exists a common low-frequency mode related to the oscillations of DWs in BST_0.8.ConclusionsIt was possible to manipulate the frequency of peaks in dielectric response of single domain through changing the magnitude of the external electric field. Domain walls oscillations dominated the dielectric response in a low frequency gigahertz range, whereas the single domain structures exhibited resonant peaks in a terahertz range. Moreover, multi-domain structures with different domain walls in BST_0.8 had different dielectric responses, but the both have a same low-frequency mode at 10 GHz related to the domain walls oscillations. The results of this study indicated the dielectric response behaviors of ferroelectrics induced in an external electric field and internal multi-domain configurations and provided the potential mechanisms and guidance for optimizing application performance.