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The chemical ordering, electrical, optical, and magnetic properties of$Na_{0.5}Bi_{0.5}TiO_{3}$ (NBT) and 25$\%$ dysprosium doped NBT (DyNBT) wereinvestigated in the framework of first-principles calculations using the fullpotential linearized augmented plane wave (FP-LAPW) method based onspin-polarized density functional theory implemented in the WIEN2k code. Wedemonstrated that NBT structure is stable in the 001 A-site configuration,while DyNBT presents an A-site disorder perceived by the minimal energydifference between the different A-site configurations. A significant magneticmoment of $5{\mu}B$ emerges in DyNBT system, while NBT is known to benon-magnetic. Dysprosium in NBT matrix seems to form an ionic bonding withoxygen atoms whereas Bi-O forms covalent bonding which is responsible for thedecrease of the polarization value from 42.3 ${\mu}C/cm^{2}$ for NBT to 22.08${\mu}C/cm^{2}$ for the doped compound. In the second part, the transitiontemperature and the hysteresis loops of $Na_{0.5}(Bi_{1-x}Dy_{x})_{0.5}TiO_{3}$system x = 0 - $25\%$ were investigated using the Monte Carlo simulation. Weobserved a decrease in the transition temperature as a function of dysprosiumintroduction. We pointed out from the hysteresis loops, an apparent decrease ofthe coercive field together with the remanent polarization as a function ofdoping and also as a function of temperature. Our proposed model was seen toapproach the values of experimental studies.

Electrical and structural properties of pure and dysprosium doped $Na_{0.5}Bi_{0.5}TiO_{3}$ system: DFT and Monte Carlo simulation