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Zirconia nanoparticles and nanostructured systems

Stability of rhombic (tetragonal-like) and cubic-like zirconia nanoparticles is investigated using the density functional pseudopotential method in a general gradient approximation. The binding energy (Eb) increases with increasing of rhombic particles, but the cubic-like particle is more favorable than rhombic ones. All studied nanoparticles have nonzero energy distances (ΔE) between occupied and non-occupied states. A simple model of nanostructured zirconia has been constructed using a Zr16O32 cluster as a unit element for a three dimensional superlattice with cubic symmetry. This work was supported by the Russian Foundation of Basic Researches, the grant # 04-02-9700.


Introduction
Zirconia (ZrO 2 ) is interesting as a ceramic material with useful optical, electrical, thermal, hardness and other characteristics. A monoclinic (m) phase of pure ZrO 2 is stable below 1170°С. From 1170°С up to 2370°С pure ZrO 2 has a tetragonal structure, but at temperatures between 2370°С and 2706°С (the melting point) it exists as cubic [1,2]. The cubic phase can be stabilized (e.g. the temperature of the "с → t" transition may be decreased), by inducing such additives as MgO, CaO, Y 2 O 3 and so on.
One of the ways to construct stable ceramics on the base of high temperature phases of zirconia is the using of nanoparticles. There are few works on the formation of stable thin films and nanoparticles from pure (undoped) zirconia of tetragonal and cubic structure [3][4][5][6][7][8][9][10][11]. It has been shown [11] experimentally and theoretically that the tetragonal-cubic transition takes place (at low temperatures) at a particle size of about two nanometers. However, theoretical results of that investigation were obtained without taking into account of the relaxation of the atomic structures of nanoparticles. The goal of this work is to study equilibrium atomic and electronic structures of cubic and tetragonal zirconia nanoparticles. Besides, we have studied a model of a three-dimensional net constructed from cubic zirconia nanoparticles.
using the FHI98pp code [15]; they were tested for the ability to reproduce the basic characteristics of the bulk materials: namely, equilimrium lattice constants and bulk modules have been calculated.
The plane wave energy cutoff of 40 Ry was implemented. The maximal sizes of the investigated particles were about 1.5 nanometer; the distances between them were about 0.8 nm. A single k-point (0;0;0) of the Brillouin zone was used to study nanoparticles, while for modeling of bulk zirconia two special points (0.25;0.25;0.25) and (0.25;0.25;0.75) have been taken with the weights of 0.25 0.75, 0.75, respectively.

Nanoparticles
In contrast with Ref. 11 we concentrated our opinion on studying of stoichiometric particles of zirconia. To analyze the equilibrium geometry and energetics of nanoparticles we studied two types of particles: namely, the Zr 6    Initial positions of atoms in the Zr 6 O 12 , Zr 10 O 20 and Zr 14 O 28 particles corresponded cubic phase, but symmetry of particles was not cubic. Symmetry of the given particles is symmetry of a rectangular parallelepiped which lateral surfaces are perpendicular to its bases.
Our calculations have shown that parameters a and b are equal to 0.46 nm in all studied particle. The c parameter is equal to 0.489 nm for Zr 6 O 12 and growths up to 0.503 nm for the Zr 14 O 28 . Thus, the c/a relation is growing from 1.063 up to 1.094. Perhaps, the reason is a special geometry of those particles: they have the same sizes at two directions corresponded to the a and b parameters and are different in the third direction that corresponds to the c parameter (see fig.2). The bond-bond angle α in the basis of the parallelepiped cell is not equal to 90º as is characteristic for a cubic phase. It is equal to 100º for the smallest particle (Zr 6 O 12 ) and 92º for the largest one (Zr 14 O 28  and Zr 14 O 28 particles have been averaged because they are different inside of particles and at their borders. It is necessary to note, that in all cases lateral surfaces of cells-parallelepipeds remain orthogonal to their bases, i.e. there is no tendency to occurrence of monoclinic symmetry. In order to speak about features of transitions between cubic and not cubic phases, it is necessary to study a relaxation of nanoparticles which starting geometry is cubic.  . 3) as an example of such particle. This particle has been constructed by a cutting up of the Zr 16 O 32 cluster from the lattice of cubic zirconia with addition of three zirconium atoms and six oxygen atoms.
The geometry optimization has shown that symmetry of the Zr 19 O 38 particle remains cubic-like after its full relaxation. Parameter a (the cube edge shown by a dotted line in fig. 2) was found equal to 0.505 nm. Table II contains the values of the binding energy E b and the energy "gap" ∆E for zirconia particles investigated by us. Comparison of these data shows that the particle with cubic geometry is energetically more favorable, than particles which geometry is close to tetragonal. It must be noted that positions of oxygen atoms in the Zr 19 O 38 particle do not precisely correspond to the perfect cubic positions, therefore their deviation from equilibrium places do not generate a tendency of the oxygen subsystem to distortion. In other words, a small disordering of cubic symmetry caused by dimensional effect stabilizes a cubiclike symmetry of the particle without it's doping.
Calculated values of the energy "gap" (even for bulk zirconia) are rather less than the experimental value of 6 eV [16]. This is a common problem for calculations based on the one-particle approach (see Ref. 17).
In order to compare the electronic structures of nanoparticles and bulk zirconia in details we constructed densities of states (DOS) using 0.2 eV wide Gaussian broadening functions centered on each electron level. These DOS are plotted in fig. 3. One can see that DOS plots for the Zr 6 O 12 , Zr 10 O 20 and Zr 14 O 28 particles arrive to the DOS for bulk zirconia with increasing of a particle size.

A model of nanostructured zirconia
A simple model of nanostructured zirconia can be constructed using the Zr 16 O 32 cluster as a unit element of the three dimensional superlattice with cubic symmetry. Figure 4 demonstrates the planar projection of this superlattice. The equilibrium binding energy is -24.61 eV per ZrO 2 unit that lays between -24.15 eV obtained for the Zr 19 O 38 cubic-like nanoparticle and -25.91 eV calculated for the bulk cubic ZrO 2 . Thus, a stable nanostructured porous zirconia can be formed from cubic-like nanoparticles with the energy gain of 0.5 eV per ZrO 2 unit.   The plot of DOS for the porous nanostructured zirconia is shown at figure 5. It demonstrates a rather wide energy gap of 2.6 eV between occupied and no occupied states.