Theoretical investigation of the electrocaloric effect of a ferroelectric nanocube
Introduction
Ferroelectric nanomaterials have aroused strong interest in the industry because of their novel physical properties, which may have great potential for future device applications. The potential applications of ferroelectric nanomaterials are truly impressive and broad, including multilayer capacitors [1], energy storage [2], and memory nanodevices [[3], [4], [5]].
The electrocaloric effect (ECE) corresponds to a temperature variation induced by an electric field in adiabatic conditions. In the presence of thermal exchange with the environment, this defines the heat exchanged as a function of the electric field (isothermal condition). In the case of ferroelectric materials, the ECE is closely related to the pyroelectric effect. Electrocaloric materials can provide an environmentally friendly and efficient solution for solid-state cooling devices [6]. Many theoretical [7,8] and experimental [9,10] studies dealing with the ECE have been performed on various materials, and new compositions with very high electrocaloric activity have been studied. For instance, Zhu et al. [11] studied the ECE in Pb(Zn 1/3Nb2/3)O3-PbTiO3. They reported a giant ECE in single Pb(Zn1/3Nb2/3)O3-PbTiO3 crystals. Lu et al. [12] measured the ECE in ferroelectric and antiferroelectric lanthanum-doped Pb(Zr,Ti)O3 relaxor. They obtained a large ECE comparable to that predicted by the phenomenological theory. A giant ECE was successively demonstrated in several ferroelectric relaxors [[13], [14], [15], [16]], single crystals [17], polymer-ceramic composites [18,19], and bilayer films [20]. On the other hand, the ECE has been theoretically studied by atomistic first-principles-based simulations [21,22], Landau's phenomenological thermodynamic model [23], the phase field method [24], and Monte Carlo simulation [25]. Furthermore, state-of-the-art work has shown that microstructure design is an effective method to push the ECE closer to what is required for practical applications [26,27]. Chen and Fang [28] used the core-shell structure to study the effect of grain size on the ECE of BaTiO3 nanoparticles. It was observed that the adiabatic temperature change moves to lower temperatures as the grain size decreases. Recently, Ye et al. [29] obtained a giant ECE in PbTiO3 nanoparticles with the phase field method. They found that the ECE can be enhanced by compressive misfit strain and reduced by tensile misfit strain. Ma et al. [30] studied the ECE and state transition in BaTiO3 doped with Zr using combined canonical and microcanonical Monte Carlo simulation and experimental measurements for comparison. They suggested that a small concentration of Zr leads to a high ECE. In this regard, insight from microscopic theory is beneficial to provide additional support and guidance, leading to an enhanced electrocaloric performance.
In this article we investigate in detail the electrocaloric performance of a ferroelectric nanocube with core-shell morphology using the effective field theory based on the probability distribution technique; this has never been studied before. In Section 2, we define the model and briefly give the formulation of the polarizations at each site of the nanocube using the theoretical framework of the effective field theory. In Section 3 we present the numerical results, while Section 4 provides a brief conclusion.
We consider a ferroelectric Ising nanocube with core-shell structure containing sites, and each site is occupied by a spin-1/2 Ising atom (see Fig. 1). The numbers of the core and shell spins are 27 and 98, respectively. So the Hamiltonian of the system can be expressed as
The exchange interaction between two nearest-neighbor sites of the core (shell) is denoted by (), while stands for the exchange interaction between two nearest-neighbor sites of the shell and the core. and denote the z and x components of a quantum spin at each site i of the Ising nanocube, μ is the effective dipole moment at site i, and E is the external electric field.
The above Hamiltonian can be written as follows:where includes the contributions associated with site i, is the remainder, , and . From the statistics of our system, the starting point of the single-site cluster approximation for the operator at site i is given bywhere , where T is the temperature and is the Boltzmann constant, and p is the number of order parameters for spin 1/2. The evaluation of the traces over selected spins in the former equations yieldswhere
The longitudinal polarization of the system is given by
In the current case, we follow the approach described in Ref. [31,32], and then we use the integral representation of the Dirac delta distribution. We find that the longitudinal polarization for a fixed configuration of neighboring spins of site i is given by
To perform the averaging on the right-hand side of the previous equation, we introduce the probability distribution of spin 1/2, and then we obtainwhere the distribution function is given by
Details of the calculations regarding the longitudinal polarizations , , , and at the core and , , , , , and at the shell are given in the Appendix.
The longitudinal polarizations of the shell and the core and the total longitudinal polarization per site are defined as follows:
And
We also calculate some thermodynamic quantities (internal energy U, specific heat , and entropy S). The internal energy per site is given by
The expressions for , , , , , , , , , and can be obtained by substitution of the function with [33] in the expressions for the longitudinal polarizations , , , , , , , , , and at the core and the shell.
The specific heat can be determined from the following relation:
Accordingly, we can get the entropy:
On the basis of the Maxwell relation , the reversible adiabatic changes in the entropy and temperature due to an applied electric field E are given by the following relations:
Andwhere is the specific heat and is the difference in the applied electric fields.
Section snippets
Results and discussion
In this section, we discuss the influence of various physical parameters, such as the external electric field and interfacial exchange coupling, on the thermal and electrocaloric properties of a ferroelectric nanocube with a core-shell structure. We have fixed and .
The thermal variations of the specific heat , internal energy , and entropy are calculated by solving Eqs. (16), (16)17) numerically; the results obtained are depicted in Fig. 2. Fig. 2 (a)–(c) displays the
Conclusion
In summary, the effective field theory method is used to study the ECE and thermal properties of a ferroelectric Ising nanocube in the presence of an external electric field. The entropy change and the adiabatic temperature change as a function of temperature were investigated for different physical parameters. Our calculation shows that the adiabatic temperature change () strongly depends on the external electric field, and in this case , although for a weak electric field.
CRediT authorship contribution statement
Mustapha Tarnaoui: Conceptualization, Formal analysis, Writing - original draft. Ahmed Zaim: Investigation, Supervision, Validation. Mohamed Kerouad: Investigation.
Acknowledgments
This work was performed with the support of URAC: 08 and the project PPR2: (MESRSFC-CNRST).
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