ZrO2 is a wide-band insulating material with a high dielectric constant. With increasing temperature in ZrO2 exist monoclinic, tetragonal, orthorhombic and cubic phases. Antiferroelectric (AFE)-like double-hysteresis loops are observed in ZrO2 thin films   where the structure is tetragonal at room temperature    . Using density functional calculations Reyes-Lillo et al.  have studied the experimentally reported field induced phase transition in ZrO2 thin film   which corresponds to an intrinsic effect.
Furthermore, ferroelectricity was found in HfO2 thin films doped with Zr (HZO)        as well as with Si, Y, Al, Gd, La  - . It must be noted that pure HfO2 and ZrO2 are not ferroelectric. HfO2 exists with increasing temperature in monoclinic, tetragonal and cubic phases . In nano-materials the tetragonal phase extends to lower temperatures . For different Zr content x the HZO thin films show dielectric (x = 0), ferroelectric (for example x = 0.5) and AFE (for example x = 0.7) properties, which are due to the involvement of monoclinic (m-phase, P21/c-dielectric), orthorhombic (o-phase, Pca21-ferroelectric) and tetragonal (t-phase, P42/nmc-AFE) phases depending on the Hf:Zr ratio . Wei et al.  reported that the polarization P in HZO increases with decreasing nanoparticle (NP) size. In HZO thin films P also increases significantly when the film thickness decreases .
Below a critical size of 30 nm pure ZrO2 is stabilized in the tetragonal phase at room temperature which is considered as a crystallite size effect . There are also reports for critical sizes for the tetragonal to monoclinic transformation between 15 - 20 nm   . The tetragonal phase of HfO2 is stabilized for d < 3.6 - 3.8 nm .
The phase stability and the ferroelectricity of orthorhombic HZO ferroelectric material are theoretically investigated by Chen et al.  with density functional theory (DFT) computations. Oxygen defect impacts on ferroelectricity in HZO are studied using first-principles calculations by Wei et al. . Also with the DFT Materlik et al.  have studied the ferroelectric phase of HfO2, ZrO2 and HZO. Batra et al.  revealed later that the results of Ref.  might not be correct. The experimentally observed stress in HZO films is tensile  whereas Batra et al.  reported a compressive stress.
The physical origin of the AFE hysteresis in ZrO2 NPs and the ferroelectricity in HZO and Al, La doped HfO2 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.
2. Model and Green’s Function
The properties of Zr doped HfO2, Hf1−xZrxO2, NPs can be described by the transverse Ising model  :
The pseudo-spin operator characterizes the two positions of the ferroelectric unit at the lattice point i. is the pseudo-spin interaction between the pseudo-spins at sites i and j which is positive or negative in the ferroelectric or AFE case, respectively. The dynamics of the model with strength is determined by the operator . E is an external electric field. Here mean Zr (or Al, La) or Hf. , for pure ZrO2, and , for pure HfO2. Thus, . has two values— and . The interaction term has three different values— , and .
The Hamiltonian (1) can be written in explicit form as ( ):
We assume that
where . The factor x gives the concentration of the Zr ions which substitute the Hf ions, whereas is the concentration of the Hf ions.
The retarded Green’s function is defined as:
The operator stands for the set , , , , where , are Pauli operators (S = 1/2, ).
The polarization P of a HZO NP is obtained as:
The mixed transverse pseudo-spin-wave excitations in a given shell n are calculated from the poles of the Green’s function (4) using the method proposed by Tserkovnikov  :
where is the number of lattice sites.
3. 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 ZrO2 we will firstly consider the electric field dependence of the polarization in the tetragonal phase of a ZrO2 NP with N = 3 shells for T = 300 K. Materlik et al.  showed that AFE behavior of pure ZrO2 thin films is observed after stabilization of the tetragonal phase for d < 35 nm. Using the lattice parameters for ZrO2 from Ref.  in the tetragonal phase a = 5.06, b = 5.18, c = 5.06 (Å) we obtain the following model parameters: , , , . The tetragonal structure is PbZrO3 (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 . The results are presented in Figure 1, curve 1. This AFE behaviour is in agreement with the experimental data of Ref.     . The polar AFE phase exists under a certain magnitude of the external electric field. When T increases, above a critical temperature only paraelectric properties can be observed. So, we can conclude, that one explanation of the origin of the AFE-ty in ZrO2 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. .
Now we will study the electric behaviour for different electric field, temperature, crystal phase and size of Hf1-xZrxO2 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 in the defect sizes (denoted as ) compared to the undoped samples. The radius of the tetravalent Zr ion (86 pm) is a little larger than that of the Hf ion (85 pm), i.e. there is a small tensile strain ( ), in agreement with the experimental data
Figure 1. (Color online) Electric field dependence of the polarization in Hf1-xZrxO2 NPs for N = 3 shells, , and different x values: (1) 1; (2) 0.5; (3) 0.
of Shiraishi et al.  for HZO thin films, whereas Batra et al.  reported a compressive stress.
The electric field dependence of the polarization in Hf0.5Zr0.5O2 NPs is shown in Figure 1, curves 1-3. ZrO2 and HfO2 have almost equivalent crystal phases, with almost identical lattice parameters. It is seen that pure HfO2 (Figure 1, curve 3) in the monoclinic phase is a linear dielectric with no notable nonlinear response of the polarization curve. As the ZrO2 content increases, the curve reaches its maximum value for doping concentration x = 0.5 (Figure 1, curve 2). ZrO2 displays an AFE-behavior at high fields, where the polarization response becomes non-linear with hysteresis (Figure 1, curve 1). In the non-polar state where the polarization P = 0 we obtain a linear dependence in the polarization below the Curie-Weiss temperature (curve 1). Above with increasing temperature, when the temperature is between and (the ferroelectric phase transition Curie temperature), , there is a polar state, and the hysteresis loop is similar to the ferroelectric one (curve 2, x = 0.5). In this temperature region the crystal is in the orthorhombic phase where the electric dipoles are aligned parallel to their nearest neighbors, i.e. . Using the lattice parameters for HZO from  a = 5.06, b = 5.14, c = 5.27 Å we have calculated the following model parameters: , . We assume , . The begin of the polar ferroelectric state corresponds to the monoclinic to orthorhombic phase transformation ( ). A similar ferroelectric hysteresis curve is obtained also for x = 0.4. This ferroelectric behaviour of HZO nanostructures is reported in Ref.       . Above in the temperature interval the polar phase becomes to be metastable. Because of this the hysteresis curve shows a ferroelectric behavior. For (the AFE phase transition temperature), we observe the AFE-like state ( ) (Figure 1, curve 1), typical for pure ZrO2 NP (x = 1), the crystal phase is tetragonal. The polar phase cannot be induced when the temperature T is around even under an external electric field. For temperatures higher than the AFE transition temperature in the cubic phase remain only paraelectric properties. The monoclinic phase decreases with increasing the ZrO2 content. It can be seen from Figure 1 that the remanent polarization is zero for pure HfO2 and ZrO2. reaches at doping concentration x = 0.5 its maximum value.
In Figure 2 is shown the composition dependence x of the remanent polarization in HZO NPs. For x = 0, for pure HfO2, . With increasing of x increases, reaches at x = 0.5 its maximum value and then in pure ZrO2, x = 1, is again zero. The experimentally reported maximum value of the remanent polarization is in the interval x = 0.5 - 0.6    . Mueller et al.  have shown that for x = 0.5 the ferroelectric phase is stable between 100 - 400 K. In this temperature interval HZO thin films for x = 0.7 show a transition to a double-loop hysteresis, whereas pure ZrO2 thin films remain in this double-loop hysteresis starting from low temperatures.
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    . 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 . Park et al.  reported also that the o-phase increases with decrease thickness in HZO film. Clima et al.  show that oxygen vacancies can reduce drastically the polarization reversal barriers.
Finally, we will consider the effect of different ion doping on the electric properties of HfO2 NPs. Variations of Al and La doping concentration influences the crystallographic structure of the NP and therefore the polarization. The insertion of a 3+ (Al) or 4+ (La) cation in the HfO2 lattice leads to the appearance of oxygen vacancies to keep the charge balance. The radius of the Al ion (67.5 pm) is smaller compared to the ionic radius of the Hf ion (85 pm) (i.e. in our model we have ). Figure 4 shows the remanent polarization of the HfO2 NP as a function of the Al-concentration (Figure 4, curve 1). The 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 HfO2 NP behaves as a paraelectric material. Mueller et al.  showed that the ferroelectricity is related to the non-centrosymmetric orthorhombic phase which is stabilized at low Al doping concentration.
Figure 2. The remanent polarization of Hf1-x ZrxO2 NPs for , , , , and different Zr conzentration x.
Figure 3. Size dependence of the polarization of HZO NPs for , .
Figure 4. Doping concentration dependence of the remanent polarization of a HfO2 NP for doping with: (1) Al ( ); (2) La ( ) ions.
A similar behavior for the Al concentration dependence of the dielectric constant in HfO2 thin films is reported by Yoo et al. .
The electric properties of La doped HfO2 NPs are also studied. The radius of the La ion (117.2 pm) is larger compared to the ionic radius of Hf (85 pm) (this means ). Batra et al.  have shown that La doping stabilizes the orthorhombic phase. It can be seen from Figure 4, curve 2, that compared to the Al doping, the ferroelectric region for the La doped HfO2 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 is larger for the La doping than that for the Al doping (Figure 4, curves 2 and 1). The maximum value of is observed for x = 0.14. Schroeder et al.  reported also that La shows the highest remanent polarization values of all ion doped HfO2 thin films. Our results confirm the experimental data of Ref.   for Al and La doped HfO2 thin films. It must be noted that the observed here maximum values of the ion doped HfO2 NPs are comparable to the values reported for Al-doped (x = 0.025 - 0.03   and for La-doped (x = 0.12 ) HfO2 epitaxial thin films.
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 ZrO2 NPs and the ferroelectricity in HZO and Al, La doped HfO2 NPs which is still under debate. Therefore, we have investigated the dependence of the polarization P in ion doped HfO2 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 . It can be concluded that the change in the polarization with respect to the doping concentration in HfO2 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 ZrO2 displays in the tetragonal phase an AFE-behavior ( ) at high fields inducing a t-o phase transformation. Pure HfO2 is a linear dielectric in the monoclinic phase. With increasing the ZrO2 content in HZO the hysteresis loop is consistent with that for ferroelectric materials ( ). shows a maximum for x = 0.5. For x = 0 and x = 1 Pr = 0. It is shown that the properties of these three compounds—ZrO2, HfO2 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 .
The influence of Al and La doping on in HfO2 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 HfO2 nanostructures. Both remanent polarizations have a maximum value at x ≈ 0.03 and 0.14, respectively. The curve for La doping is shifted to higher doping concentrations and is broader due to the larger radius of the La ion. Moreover, is larger for La-doped compared with that of Al-doped HfO2 NPs.
There are some differences in the electric properties of ion doped HfO2 and ZrO2 nanostructures   . For example Yoo et al.  observed that the dielectric constant in Al doped HfO2 thin films undergoes a maximum whereas in Al doped ZrO2 thin films it decreases. The electric properties of ion doped HZO and ZrO2 NPs will be considered in the next paper.
One of us (A. A.) acknowledges financial support by the Bulgarian National Fund “Scientific Studies” (contract number KP-06-OPR 03/9).
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