Antiferroelectricity in ZrO 2 and Ferroelectricity in Zr, Al, La Doped HfO 2 Nanoparticles

The dependence of the polarization P in Hf 1-x Zr x O 2 nanoparticles on electric field, dopant concentration x, size and temperature are studied using the transverse Ising model and the Green’s function method. Pure ZrO 2 shows at high electric fields an antiferroelectric behavior. Pure HfO 2 is a linear dielectric in the monoclinic phase. With increasing ZrO 2 content the ( ) P E of HZO shows a ferroelectric behavior. The composition dependence x of the remanent polarization ( ) r P x has a maximum for x = 0.5. For x = 0, pure HfO 2 , and x = 1, pure ZrO 2 , 0 r P = . P increases with decreasing HZO nanoparticle size. The influence of Al and La doping on r P in HfO 2 nanoparticles is also studied. The exhibiting of the ferroelectricity in ion doped HfO 2 is due to a phase transformation and to an internal strain effect. The observed results are in good qualitative agreement with the experimental data.

Below a critical size of 30 nm pure ZrO 2 is stabilized in the tetragonal phase at room temperature which is considered as a crystallite size effect [23]. There are also reports for critical sizes for the tetragonal to monoclinic transformation between 15 -20 nm [24] [25] [26]. The tetragonal phase of HfO 2 is stabilized for d < 3.6 -3.8 nm [27].
The phase stability and the ferroelectricity of orthorhombic HZO ferroelectric material are theoretically investigated by Chen et al. [28] with density functional theory (DFT) computations. Oxygen defect impacts on ferroelectricity in HZO are studied using first-principles calculations by Wei et al. [29]. Also with the DFT Materlik et al. [30] have studied the ferroelectric phase of HfO 2 , ZrO 2 and HZO. Batra et al. [31] revealed later that the results of Ref. [30] might not be correct. The experimentally observed stress in HZO films is tensile [32] whereas Batra et al. [31] reported a compressive stress.
The physical origin of the AFE hysteresis in ZrO 2 NPs and the ferroelectricity in HZO and Al, La doped HfO 2 NPs is still under debate. The aim of the present paper is to investigate theoretically these problems using a microscopic model and the Green's function technique.

Model and Green's Function
The properties of Zr doped HfO 2 , Hf 1−x Zr x O 2 , NPs can be described by the transverse Ising model [33]: The pseudo-spin operator z i S characterizes the two positions of the ferroelectric unit at the lattice point i. ij J is the pseudo-spin interaction between the pseudo-spins at sites i and j which is positive or negative in the ferroelectric or We assume that The factor x gives the concentration of the Zr ions which substitute the Hf ions, whereas ( ) 1 x − is the concentration of the Hf ions.
The retarded Green's function is defined as: The operator i B stands for the set The polarization P of a HZO NP is obtained as: The mixed transverse pseudo-spin-wave excitations ij  in a given shell n are calculated from the poles of the Green's function (4) Materials Physics and Chemistry   22   1 tanh , 4 2 where N ′ is the number of lattice sites.

Numerical Results and Discussion
Our NP has an icosahedral symmetry. A certain Hf-spin is fixed in the center of the particle and all other spins are included into shells n. n = 1 denotes the central spin and n = N represents the surface shell. Strain effects on the surface of the NP change the number of next neighbors on the surface and reduce the symmetry. Therefore the pseudo-spin interaction constants can take different values on the surface and in the bulk, denoted with the index "s" and "b", respectively. Moreover, J is proportional to the inverse of the distance between two nearest spins, i.e. of the lattice parameters.
In order to clarify the AFE behavior in ZrO 2 we will firstly consider the electric field dependence of the polarization in the tetragonal phase of a ZrO 2 NP with N = 3 shells for T = 300 K. Materlik et al. [30] showed that AFE behavior of pure ZrO 2 thin films is observed after stabilization of the tetragonal phase for d < 35 nm. Using the lattice parameters for ZrO 2 from Ref. [30] in the tetragonal phase a = 5.06, b = 5.18, c = 5.06 (Å) we obtain the following model parameters: The tetragonal structure is PbZrO 3 (PZO)-like AFE one, the electric dipoles are aligned antiparallel to their nearest neighbors-analogous to the magnetic moments in antiferromagnetic materials, therefore, we chose 0 J < . The results are presented in Figure 1, curve 1. This AFE behaviour is in agreement with the experimental data of Ref. [2] [7] [30] [35] [36]. The polar AFE phase exists under a certain magnitude of the external electric field. When T increases, above a critical temperature crit T only paraelectric properties can be observed. So, we can conclude, that one explanation of the origin of the AFE-ty in ZrO 2 NPs is a phase transformation from a tetragonal to an orthorhombic phase induced by an external electric field which is an intrinsic behavior. This is confirmed by the ab-initio study of Reyes-Lillo et al. [7]. Now we will study the electric behaviour for different electric field, temperature, crystal phase and size of Hf 1-x Zr x O 2 NPs. By doping of ions with different radius appear different strains which give rise to additive changes (increasing or decreasing) of the pseudo-spin interaction constant Advances in Materials Physics and Chemistry of Shiraishi et al. [32] for HZO thin films, whereas Batra et al. [31] reported a compressive stress.
The electric field dependence of the polarization in Hf 0.5 Zr 0.5 O 2 NPs is shown in Figure 1, curves 1-3. ZrO 2 and HfO 2 have almost equivalent crystal phases, with almost identical lattice parameters. It is seen that pure HfO 2 (Figure 1 To completely explain the ferroelectric-phase stability in HZO NPs, we want to focus now on the size dependence of the polarization P in HZO NPs which is demonstrated in Figure 3. It must be noted, that the distance between the shells is ≈10 Å, i.e. we consider NPs with N = 2 -10, i.e. with size of 2 -50 nm. It can be seen from Figure 3 that P increases with decreasing NP size, i.e. the ferroelectric properties disappear in large NPs, thick films and bulk materials, in agreement with the experimental data [9] [18] [40] [41]. This behaviour shows that the m-phase (non-ferroelectric), which is absent or very rarely found in the smallest NPs, increases with increasing size whereas the ferroelectric rhombohedral phase is stabilized by the existing surface strain. To conclude, we show that strain can be used in very small NPs of HZO to induce a ferroelectric phase, with a large polarization P and remanent polarization r P . Park et al. [18] reported also that the o-phase increases with decrease thickness in HZO film. Clima et al. [42] show that oxygen vacancies can reduce drastically the polarization reversal barriers.  Figure 4 shows the remanent polarization r P of the HfO 2 NP as a function of the Al-concentration (Figure 4, curve 1). The r P value increases firstly by increasing the Al concentration starting at x ≈ 0.01. The maximum ferroelectric polarization is reached at x = 0.03 Al, followed by an AFE region between x = 0.04 -0.06 Al. At higher Al-concentrations the doped HfO 2 NP behaves as a paraelectric material. Mueller et al. [16] showed that the ferroelectricity is related to the non-centrosymmetric orthorhombic phase which is stabilized at low Al doping concentration.
A. T. Apostolov et al.   thorhombic phase. It can be seen from Figure 4, curve 2, that compared to the Al doping, the ferroelectric region for the La doped HfO 2 NP which starts at higher x value, x ≈ 0.05, is shifted to higher doping concentrations and is broader due to the larger ionic radius of the La ion. In addition, the remanent polarization r P is larger for the La doping than that for the Al doping (Figure 4, curves 2 and 1). The maximum value of r P is observed for x = 0.14. Schroeder et al. [44] reported also that La shows the highest remanent polarization values of all ion doped HfO 2 thin films. Our results confirm the experimental data of Ref. [15] [44] for Al and La doped HfO 2 thin films. It must be noted that the observed here maximum values of the ion doped HfO 2 NPs are comparable to the values reported for Al-doped (x = 0.025 -0.03 [43] [44] and for La-doped (x = 0.12 [16]) HfO 2 epitaxial thin films.

Conclusions
The properties of HZO are theoretically investigated till now with DFT computations. In this paper for the first time is used the microscopic transverse Ising model in order to clarify the physical origin of the AFE hysteresis in ZrO 2 NPs and the ferroelectricity in HZO and Al, La doped HfO 2 NPs which is still under debate. Therefore, we have investigated the dependence of the polarization P in ion doped HfO 2 NPs on electric field, dopant concentration x, size and temperature. Different from the DFT we study the behavior of the material at finite temperatures. To that aim we use a Green's function technique for 0 T ≠ . It can be concluded that the change in the polarization r P with respect to the doping concentration in HfO 2 NPs is the result of the transformation of the crystalline phase due to the internal stress, of the appearance of an orthorhombic phase exhibiting ferroelectricity. Moreover, we try to clarify some discrepancies in the literature, for example about the appearing strain in HZO NPs (it is tensile and not compressible).
We obtain that pure ZrO 2 displays in the tetragonal phase an AFE-behavior ( 0 J < ) at high fields inducing a t-o phase transformation. Pure HfO 2 is a linear dielectric in the monoclinic phase. With increasing the ZrO 2 content in HZO the hysteresis loop is consistent with that for ferroelectric materials ( 0 J > ).
( ) r P x shows a maximum for x = 0.5. For x = 0 and x = 1 P r = 0. It is shown that the properties of these three compounds-ZrO 2 , HfO 2 and HZO-are changed with ion doping and size. The polarization P increases with decreasing NP size, i.e. the non-ferroelectric m-phase disappears with decreasing size. We show that strain can be used in very small NPs of HZO to induce a ferroelectric phase with large P and r P .
The influence of Al and La doping on ( ) r P x in HfO 2 NPs is also studied. Stress due to the different ionic radii of the doping ions compared to the host ones (which cause different pseudo-spin interaction constants in the defect states) as well as the distribution of oxygen vacancies play a key role for the phase transformations in doped HfO 2 nanostructures. Both remanent polarizations  [46]. For example Yoo et al. [43] observed that the dielectric constant in Al doped HfO 2 thin films undergoes a maximum whereas in Al doped ZrO 2 thin films it decreases. The electric properties of ion doped HZO and ZrO 2 NPs will be considered in the next paper.