鐵電材料因其獨特的電疇翻轉動力學（domain switching dynamics），對下一代奈米電子學與能量儲存領域至關重要。為了改善微觀密度泛函數理論或巨觀模型的缺失，本研究建立了一個基於時變金茲堡-朗道方程的介觀相場框架，其中納入四種自由能包含朗道-戴馮賽爾（Landau-Devonshire）能、梯度能、彈性能及靜電能，用以觀察電疇隨時空的演化，為消除傳統有限差分法 (FDM)常見的時間步長限制與空間各向異性，本研究使用Python語言自編程式引入了半隱式傅立葉譜方法（SIFSM）將實空間中較為複雜的偏微分方程轉化為倒空間中的代數運算，並結合動態低通濾波算子確保了較高的計算效率與數值穩定性，最後透過功率譜密度（PSD）分析與貝索函數加權根據朗道展開係數（α,β,γ）的隨機模擬，評估了基於三種晶格扭曲的鈣鈦礦結構（即Pnma、R3c與P4bm）的弛豫鐵電體（RFs）數個重要物理性質，結果顯示，給定初始的電極化率（χ_0）為30.94時，由27% Pnma、52% R3c 及21% P4bm組成的混合相在能量儲存應用方面展現出最佳的弛豫鐵電行為，該弛豫鐵電體的奈米極化區域（PNRs）平均尺寸為5nm，計算出電滯曲線中的可回收儲能密度達到13.4 J/cm³和回收效率高達90.2%，此外本研究也計算了能量對極化強度的關係圖，結果顯示與典型鐵電體相比，弛豫鐵電體在二階導數為負的極化區域更為寬廣，因此在負電容應用上具有更高的潛力。

Ferroelectric materials are vital for next-generation nanoelectronics and energy storage due to their unique domain switching dynamics. To bridge the gap between microscopic Density Functional Theory (DFT) and macroscopic models, this study establishes a mesoscopic phase-field framework based on the Time-Dependent Ginzburg-Landau (TDGL) equation, incorporating Landau-Devonshire, gradient, elastic, and electrostatic energies to capture spatiotemporal domain evolution. To eliminate time-step constraints and spatial anisotropy typical of Finite Differential Method (FDM), a Semi-Implicit Fourier Spectral Method (SIFSM) was implemented in Python, transforming spatial derivatives into frequency-domain algebraic operations with a dynamic low-pass filtering operator that ensured high computational efficiency and numerical stability. Finally, with Power Spectral Density (PSD) analysis and Bessel function weighting, a few important properties of the relaxor ferroelectrics (RFs) based on three distorted perovskite structures (i.e., Pnma, R3c & P4bm) were evaluated by random simulation of Landau Expansion Coefficients (α,β and γ). The results indicated that, with an initial electrical susceptibility (P⁄(ε_0 E)) of 30.94, a mixed phase with 27% Pnma, 52% R3c & 21% P4bm exhibited an optimal relaxor-ferroelectric (RF) behavior in terms of energy storage application. The average size of polar nanoregions (PNR) of such a RF was 5 nm, and the recoverable energy storage density was calculated to be 13.4 J/cm3 with a recovery efficiency of 90.2%. Furthermore, the energy vs. polarization plot was calculated. The result indicated that the polarization region for negative second-order derivative was wider for RFs compared to the typical ferroelectrics. So, they have a higher potential for negative capacitance applications.

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