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**DFT study of lead-free environmentally friendly ferroelectric materials.**

In recent years, metal oxide based materials have been widely studied because of their interesting properties such as high-temperature superconductivity, colossal magnetoresistance and ferroelectricity. Among those oxides, the ferroelectric compounds are subject of a special attention due to their used in the Nonvolatile Ferroelectric Random Access Memories (NvFRAM), the electrooptical devices and the Micro/Nano-Electro-Mechanical Systems (M/NEMSs) such as sensors and actuators, etc…^{1-3}. So far, the majority of ferroelectric materials employed in the micro-electronics industry are made of lead. The PZT (PbZr_{x}Ti_{1-x}O_{3}) is particularly used because of these remarkable physical properties^{4}; depending on *x*, di-, piezo-, ferro-electrical properties can be modulated on a large range. However, the European 2002/95/CE directive, also known as RoHs, provides for the prohibition and limitation of use of dangerous substances for health and the environment within electric and electronic equipment. In this context a research towards new ferroelectric lead-free phases proves to be essential.

The main lead-free materials being investigated around the world are BaTiO_{3}, BaSr_{x}Ti_{1-x}O_{3}, LiNbO_{3}, BaZrO_{3}, Sr_{x}Ba_{1-x}Nb_{2}O_{6}, KTa_{1-x}NbO_{3}, Na_{x}Bi_{1-x}TiO_{3}, Bi_{4}Ti_{3}O_{12}, CaCu_{3}Ti_{4}O_{12}, other compounds are also studied such as Inx(GeSb_{2}Te_{5})_{1-x}, Bi_{4-x}La_{x}Ti_{3}O_{12}, BaNd_{2}Ti_{5}O_{14}, (Sm,Sr)Bi_{2}Ta_{2}O. However, none of these compounds presents physical properties as interesting as the compounds with lead, namely PZT. Thus, we have been interested by lead-free compounds where their ferroelectric properties remain very little studied. These compounds are based on rare earth and transition metal oxides and they belong to the A_{2}B_{2}O_{7} family layered structures with A= (La, Ce, Pr, Nd) and B=(Ti, Zr).

A_{2}B_{2}O_{7} structures can exist in two forms, the cubic pyrochlore structure and the layered perovskite structure, also known as pseudo-pyrochlore structure. The pyrochlore, A_{2}B_{2}O_{7}, structure (Fd3m, Z=8), is an anion-deficient derivative of fluorite, AX_{2} (Fm3m, Z=1), with two types of cations ordered on the A- and B-sites and one eighth of the anions removed. The structure can be envisioned as interpenetrating networks of BO_{6} octahedra and A_{2}O chains of distorted cubes (Fig. 1)^{5-6}. Empirically, the pyrochlore structure is stable when the radius ratio R_{A}/R_{B}=1.46–1.80^{6}. When the A-site in A_{2}Ti_{2}O_{7} materials is occupied by La, Ce, Pr, or Nd, (R_{A}/R_{B} >1.80), a monoclinic layered perovskite-type structure is adopted^{7}. Layered perovskite-type structures can also be obtained by subjecting A_{2}Ti_{2}O_{7} pyrochlores to high pressure, e.g. Eu_{2}Ti_{2}O_{7} ^{8}.

Figure 1. The crystal structure of pyrochlore.

The layered A_{2}B_{2}O_{7} structures are the n=4 members of the homologous series having the general formula A_{n}B_{n}O_{3n+2}, with ^{9-16}. With respect to the parent perovskite (n=∞), the A_{2}B_{2}O_{7 }layered structure has an extra layer of O_{2} inserted along the perovskite [110] direction after every four (n) distorted perovskite units^{9}(Fig. 2). These layered oxides have attracted great interest for their ferroelectric and photocatalytic properties. Among them we can mention lanthanum dititanate, La_{2}Ti_{2}O_{7} (LTO), which is characterized by an extremely high Curie temperature T_{C} > 1500 °C, spontaneous polarization Ps = 5 μC/cm^{2}, high permittivity ε= 42–52 and highpiezoelectric properties ^{17}. Thermal stability and low dielectric losses open the way for a wide use of LTO as a key component in the fabrication of microwave ceramics and nanocomposites ^{18-20}. High refractive indices n_{a} = 2.28, n_{b} = 2.32 and n_{c} = 2.30 (λ= 632.8nm) and a reasonable level of nonlinear optical susceptibility were measured for this compound^{17}. LTO has been also reported to have good photocatalytic activity in the water-splitting reaction^{21-22}and in the oxidative decomposition of CH_{3}Cl_{4}^{23}when combined with nickel oxide.

Figure 1. La_{2}Ti_{2}O_{7 }layered structure projected on the (100) and (010) plane.

In parallel with advances in laboratory synthesis, the past decades has seen a revolution in the atomic-scale theoretical understanding of ferroelectricity, especially in perovskite oxides, through first-principles density functional theory investigations. The central result of a density-functional-theory (DFT)^{24-25}calculation is the ground-state energy computed within the Born-Oppenheimer approximation; from this the predicted ground-state crystal structure and electronic properties are directly accessible. For the physics of ferroelectrics, the electric polarization and its derivatives, such as the Born effective charges and the dielectric and piezoelectric tensors, are as central as the structural energetics, yet proper formulation in a first-principles context long proved to be quite elusive. Expressions for derivatives of the polarization corresponding to physically measurable quantities were presented and applied in density-functional perturbation theory calculations in the late 1980s^{26}. A key conceptual advance was establishing the correct definition of the electric polarization as a bulk property through the Berry-phase formalism of King-Smith, Vanderbilt, and Resta^{27-28}. With this and the related Wannier function expression^{29}, the spontaneous polarization (P_{S}) and its derivatives can be computed in a post-processing phase of a conventional total-energy calculation, greatly facilitating studies of polarization-related properties.

In the present project we have chosen to study the ferroelectric properties of the A_{2}B_{2}O_{7} compounds, with A= (La, Ce, Pr, Nd) and B=(Ti, Zr), by mean of DFT and the modern theory of polarization. For perovskite oxides, the presence of oxygen and lanthanide metals significantly increases the computational demands of density-functional total energy calculations compared to those for typical semiconductors. Hence, we have chosen to use ultrasoft pseudopotentials^{30}, and projector-augmented wavefunction potentials (PAW)^{31}methods as implemented in the Vienna Ab-Initio Simulation Package (VASP)^{32-34}. In these methods, the core electrons around each ion are replaced by an effective (pseudo) potential. This allows a relatively small plane wave Fourier expansion to be used to represent the single particle wavefunctions since the remaining valence electrons typically have smooth wavefunctions.

As a first step, to predict ground-state crystal structures, the usual method is to minimize the total energy with respect to free structural parameters in a chosen space group. The space group is usually implicitly specified by a starting guess for the structure. For efficient optimization, the calculation of forces on atoms and stresses on the unit cell is essential and is included now in every standard first-principles implementation following the formalism of Hellmann and Feynman^{35}for the forces and Nielsen and Martin^{36}for stresses. Calculations on the La_{2}Ti_{2}O_{7 }and_{ }Nd_{2}Ti_{2}O_{7 }prototype structures showed a good agreement with experimental values both at the local density approximation (LDA) and generalized-gradient approximation (GGA) for the exchange-correlation functional ^{37}. Spontaneous polarizations of these compunds was found to be around 8 μC/cm^{2 }which is in good agreement with the electrical measurements reported in literature^{17, 38}.

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