Parameter modifications in the phase diagrams of a ferroelectric nanoparticle described by the transverse field Ising model

Study Phase Transition Properties of a Ferroelectric Nanoparticle with the Usual Mean-Field Approximation

“…Figure 3 is the same as Figure 2 but for a different particle size S. Obviously, the larger the particle size S, the larger the crossover value Ω bc . These features in Figure 2 and 3 are similar to the corresponding result of the effective-field theory with correlations [5,7]. Figure 4 shows the dependence of the phase diagram on the surface exchange interaction J s , i.e., the curves of T c~Ωs for different values of J s .…”

“…The pseudo-spins on a ring K (0<K≤S) are on a same sublattice K. The pseudo-spins are connected to the nearest-neighbor pseudo-spins each other with spin exchange interaction. The Hamiltonian is [3,4,7] ij iz jz…”

“…In recent decades, the nano-materials have been paid attention more and more in experiment and theory [1][2][3][4][5][6][7][8][9][10][11]. In the experiment, it has been demonstrated that the magnetic properties of nanoparticle-based systems are deviated from the corresponding bulk nano-materials [1].…”

“…In the theory, the transverse field Ising model has been thought to be a good candidate to describe the phase transitions in the ferroelectric nano-systems. The effective-field theory with correlations and the usual meanfield approximation have been used to inspect various physical properties of the ferroelectric nano-systems [3][4][5][6][7][8][9][10][11]. Kaneyoshi has studied the phase diagrams of a ferroelectric nanoparticle by changing the particle size S as well as two ratios of s and p for the exchange interactions and the transverse fields between the corresponding value at the surface and the corresponding value in the bulk within the two frameworks of the effective-field theory with correlations and the usual mean-field approximation.…”

“…Recently, Kaneyoshi [5,6] discussed the phase diagrams and magnetizations in the ferroelectric nanoparticles with different structures with the effective-field theory with correlations. Very recently, Lu [7] investigated the dependence of the phase diagrams on the exchange interactions and transverse fields of a ferroelectric nanoparticle in the (T c , J s ) coordinates and the (T c , Ω s ) coordinates with the effective-field theory with correlations, however, this work has not been carried out by using the mean-field approximation up to now.…”