Phase diagrams and polarization reversal in nanosized Hf x Zr 1-x O 2-y

To describe the polar properties of the nanosized Hf x Zr 1-x O 2-y , we evolve the “effective” Landau-Ginzburg-Devonshire (LGD) model based on the parametrization of the Landau expansion coefficients for the polar and antipolar orderings. We have shown that the effective LGD model can predict the influence of screening conditions and size effects on phase diagrams, polarization reversal and structural properties of the nanosized Hf x Zr 1-x O 2-y of various shape and sizes. To verify the model, we use available experimental results for Hf x Zr 1-x O 2 thin films and oxygen-deficient HfO 2-y nanoparticles prepared at different annealing conditions. X-ray diffraction, which was used to determine the phase composition of the HfO 2-y nanoparticles, revealed the formation of the ferroelectric orthorhombic phase in them. Micro-Raman spectroscopy was used to explore the correlation of lattice dynamics and structural changes appearing in dependence on the oxygen vacancies concentration in the HfO 2-y nanoparticles. Since our approach allows to determine the conditions (


I. Introduction
Ferroelectric memory elements like FeRAMs and FETs offer fast-switching and low-power consumption benefits, but face integration challenges with modern silicon-based CMOS technology [1,2].Lead-free binary hafnium (HfO2) and zirconium (ZrO2) oxides emerge as promising candidates for FeRAMs and FETs due to the discovery of ferroelectricity and antiferroelectricity in their thin films, which are Si-compatible.The binary oxide Hf1-xZrxO2 is recognized as next-generation Siintegrable materials according to the International Roadmap for Devices and Systems (IRDS™ 2021: Beyond CMOS) [1].
Bulk HfO2 and ZrO2 are high-k dielectrics without ferroelectric properties in a wide range of temperatures (below 1200 K) and pressures (below 12 GPa) that is confirmed by Raman spectroscopy [3,4].However, nanomaterials based on their solid solutions exhibit complex behavior influenced by various structural transitions.In particular, the ferroelectric properties observed in Hf1-xZrxO2 (0  x  1) thin films stem from the polar orthorhombic phase, which results in ferroelectric phase metastability disregarding its higher energy compared to the bulk nonpolar monoclinic phase.The properties of the Hf1-xZrxO2 thin films vary significantly depending on the factors like the substrate material, annealing conditions, deposition methods, film thickness, and dopants concentration [5,6,7].Depending on these factors and x variation from 0 to 1, Hf1-xZrxO2 thin films can manifest as dielectric, ferroelectric, or antiferroelectric materials [8,9].
Theoretical [10,11,12] and experimental [13,14,15] evidences underscore the crucial role of surface and grain boundary energies, as well as oxygen vacancies, for optimizing the Hf1-xZrxO2 nanomaterials for practical applications in advanced FeRAMs and FETs technologies.However, analytical research and optimization efforts are needed to fully exploit the potential of these binary oxides.In particular, the role of the Zr doping, oxygen vacancies, size, screening and surface effects in HfxZr1-xO2-y nanoparticles are very poorly described theoretically, despite several experimental studies reveal their high potential for controllable synthesis [16,17], nanoelectronics and capacitor technology [18], bio-safety and bio-medical applications [19,20].
Here we consider the phase diagrams and polarization reversal in the nanosized HfxZr1-xO2-y using the "effective" Landau-Ginzburg-Devonshire (LGD) model [21].This approach is based on the parametrization of the Landau expansion coefficients for the polar (FE) and antipolar (AFE) orderings in HfO2-based compounds from a limited number of polarization-field curves and hysteresis loops.To verify the model, we use available experimental results for HfxZr1-xO2 thin films to determine the Landau expansion coefficients.Using the coefficients, we calculate the polarization hysteresis and phase diagrams in HfxZr1-xO2 and oxygen-deficient HfO2-y nanoparticles, assuming that the oxygen vacancies can stabilize the polar orthorhombic phase.X-ray diffraction was used to determine the phase composition of the HfO2-y nanoparticles prepared at different annealing conditions.Micro-Raman spectroscopy was used to explore the correlation of lattice dynamics and structural changes appearing in the HfO2-y nanoparticles.

II. The "effective" Landau-Ginzburg-Devonshire model for nanosized HfxZr1-xO2-y
To determine the spatial-temporal evolution of polarization in the nanosized HfxZr1-xO2-y we use the Kittel-type model [22] incorporating polar and antipolar modes [23,24,25] combined with the LGD approach [21].Corresponding LGD free energy functional  additively includes a bulk partan expansion on the 2-th and 4-th powers of the polar (  ) and antipolar (  ) order parameters,   ; a polarization gradient energy contribution,   ; an electrostatic contribution,   ; and a surface energy,   . has the form [15,21]: where the constituent parts are Here   is the volume and  is the surface of the nanosized HfxZr1-xO2-y.Polarization vector is  ⃗ = ( 1 ,  2 ,   ).The "effective" LGD expansion coefficients   ,   ,   ,   and  are the functions dependent on Zr content "x" and oxygen deficiency "y";   is the polarization gradient tensor,   and   are the surface energy coefficients;  0 is a universal dielectric constant,   is a background permittivity;   are the electric field components (,  =1, 2 and 3).For classical ferroelectric films with a pronounced temperature-dependent and strain-dependent soft mode, the coefficients   and   linearly depend on the temperature and strains (see e.g., Refs.[26,27]).However, this is not the case for the HfxZr1-xO2-y.
where ℎ is the thickness of HfxZr1-xO2-y film, λ is the effective Debye-Hukkel screening length in the semiconducting electrodes, which is usually very small (much less than 1 nm),   is the lattice dielectric permittivity of the electrodes,  is the voltage applied between the electrodes (see Fig. 1(a)).
For the single-domain spherical nanoparticle the analytical expressions for the electric field components have the form: where  is the radius of the HfxZr1-xO2-y nanoparticle, λ is the effective screening length in the screening shell with the relative dielectric permittivity   (see Fig. 1(b)).Here, λ also can be rather small (less than 0.1 -1 nm) due to free charges and surface band bending in the shell.Only if λ ≫  and   ~ , the field in the nanoparticle core is of the same order as the applied field  3 0 .The derivation of Eqs.( 4) is given in Ref. [29].are, which physical range is (0.5 -5) nm [31].
To apply the analytical expressions (4) we should assume that Λ , → ∞ and λ is very small (much less than 1 nm), and thus the free charges in the electrodes (or in the shell) provide an effective screening of the HfxZr1-xO2-y spontaneous polarization and prevent the domain formation, so that the assumption of the single-domain state in HfxZr1-xO2-y is self-consistent.For higher λ one should use the finite element modeling (FEM) to account for the possible domain formation.
As a rule, the polar order parameter is observable (i.e., measurable), and the antipolar order parameter cannot be directly measured.However, the nonlinear coupling between   and   changes the field dependence   ( 3 ).

A. Thin films of HfxZr1-xO2
To verify the effective LGD model, we use available experimental results for HfxZr1-xO2 thin films at room temperature [8].The geometry of the considered heterostructure, consisting of a HfxZr1-xO2 film of thickness ℎ placed between conductive TiN electrodes is shown in Fig. 1(a).The temperature dependence of the polar properties of Zr-doped HfO2 is considered elsewhere [32].
Examples of how the LGD model works quantitatively are shown in Figs.A1 and A2 in Supplement [33].Here polarization hysteresis loops, measured experimentally in HfxZr1-xO2 thin films by Park et al. [8], are shown.The films thickness was 9.2 nm and the Zr content "x" varied from 100 % to 50 %.The films were covered with conducting TiN electrodes.Fits of dielectric, paraelectric, antiferroelectric, and ferroelectric loops show that the proposed effective LGD free energy qualitatively and semi-quantitatively describes the experimental results for the case of perfect screening (λ = 0).The LGD-model parameters, determined from fitting of experimental results from Park et al. [8], are listed in the last four columns of Table AI in Appendix A [33].
The composition dependences   (),   (),   (),   (), and (), determined from the fitting of experimental results [8] and further interpolated entire the range 0 ≤  ≤ 1, are shown in Notably that Eqs.( 6)-( 9) are valid in the narrow range of the HfxZr1-xO2 sizes, e.g., for the film thickness range 5 nm ≤ ℎ ≤ 15 nm, because thinner films lose their ferroelectric properties and thicker films are described by another set of LGD coefficients (see other experimental results in Ref. [8] and their fitting in Ref. [21]).
Using the free energy (1), expressions for the electric field (4a) and x-dependences of the LGD coefficients ( 6)-( 9) we calculated the phase diagram of the HfxZr1-xO2 thin films in dependence on the Zr content  and the ratio ℎ  ⁄ (see Fig. 2(c)).It is seen from the diagram that the increase of  from 0 to 0.17 leads to the transition of the dielectric (DE) state to the antiferroelectric (AFE) phase, then to the mixed ferrielectric (FEI) state for x  0.5, and then to the ferroelectric (FE) phase for >0.5.The further increase of  from 0.6 to 0.7 leads to the gradual disappearance of ferroelectricity, and to the appearance of the paraelectric (PE) phase at >0.7, which continuously transforms to the reentrant DE state for >0.8.At the same time the diagram become ℎ-dependent for ℎ  ⁄ > 10, since the contribution of the depolarization field becomes negligibly small with the increase of the ratio ℎ  ⁄ (see Eq.(4a)).The condition of "weak" screening, 5 < ℎ  ⁄ < 10, is the actual range of the film thickness effect manifestation, as well as the domain formation is possible exactly in the range of ℎ  ⁄ .
It may seem that the diagram in Fig. 2(c) is applicable for arbitrary film thickness, and the phase boundaries depend on the ratio ℎ  ⁄ and Zr fraction .However, we need to remind readers that the -dependence of LGD coefficients given by Eqs.( 6)-( 9) are valid for 5 nm ≤ ℎ ≤ 15 nm.Thus,   To study the screening and size effects, the ratio   ⁄ should be changed (see e.g., Eq.(4b)).
Phase It may seem that the diagrams are applicable for arbitrary radius  and the phase boundaries depend on the   ⁄ and .However, the x-dependence of LGD coefficients given by Eqs.( 6)-( 9) is valid in the narrow range of sizes, i.e., for 2.5 nm ≤  ≤ 7.

III. X-ray diffraction and Raman spectra of HfO2-y nanoparticles
It is known that oxygen vacancies in oxides cause the effect of the so-called "chemical pressure" (another name is the Vegard stresses) of the crystal lattice [34].The stresses affect the formation of structural states unstable under normal conditions in HfO2 and ZrO2 oxides, which have from 3 to 4 polymorphic forms.Thus, the Vegard stresses induced by the oxygen vacancies can shift the stability conditions and lead to the metastability of the orthorhombic polar phase in oxygen- The nanopowder samples were obtained in two ways of synthesis.
(1) By organonitrate synthesis from mixtures of as-prepared Hf hydroxide, ammonium nitrate and dextrin.The heating temperature of the mixtures was maintained in the range of (400 -700)°C.To create reducing conditions for the synthesis, an excess of the organic additive relative to the NO2 oxidant was used.Below we discuss results of experimental studies for the group of white-colored samples, named as the "Sample 1", which were prepared by the way (1), namely annealed at 700°C for 6 hours in air.
(2) By pyrogenic synthesis from the same mixtures of hydroxides washed from nitrates with the addition of dextrin in the CO+CO2 environment heated in the temperature range of (400 -700)°C for 6-20 hours.Excess of carbon from the powders obtained in this way was removed by the short-term exposure (10-15 minutes) in air at the temperature (450 -500)°C.Below we discuss results of experimental studies for the 3 groups of grey-colored samples, named as the "Sample 1, 2, 3", which were prepared by the way (2).Note that the Sample 2, annealed at 700°С for 6 hours in CO+CO2 atmosphere, has a light grey color; the Sample 3, annealed at 650°С for 16 hours in CO2 atmosphere, has a grey color; and the Sample 4, annealed at 600°С for 16 hours in CO+CO2 atmosphere, has a dark-grey color.
X-ray diffraction (XRD) was used to determine the phase composition of the HfO2-y nanoparticles.We use an XRD-6000 diffractometer with Cu-Kα1 radiation, and the measured angle (2θ) was from 5° to 70°.To identify the crystallographic phases in the studied system we used the database of the International Committee for Powder Diffraction Standards (JCPDS PDF-2).
Micro-Raman spectroscopy was used to explore the correlation of lattice dynamics and structural changes appearing in dependence on the oxygen vacancies concentration.Raman spectra were measured using a Renishaw InVia (England) micro-Raman spectrometer equipped with a DM2500 Leica confocal optical microscope.A laser operating at a wavelength of λ = 633 nm was used to measure the Raman scattering spectra.Processing of Raman spectra was performed using the WiRE 5.2 program, which was used to determine the peaks and decompose the bands into components.All measurements were performed at room temperature.
The XRD spectra of the Sample 1, shown in Fig. 5(a), reveals the pure monoclinic phase ("m"), as anticipated for a stoichiometric HfO2 nanopowder.The average size of coherent scattering regions with the monoclinic symmetry is 13 nm.The Raman spectra of the Sample 1, shown in Fig. 5(b), has many sharp peaks (not less than 8) located below the 800 cm -1 , and then the Raman signal intensity strongly increases and saturates above the 800 cm -1 .The saturation can be related with the significant amount of Raman-active luminescence centers.The XRD spectra, shown in Fig. 6(a)-(c), correspond to the oxygen-deficient HfO2-y Samples 2, 3 and 4, respectively.The XRD spectra demonstrate the gradual increase of the orthorhombic phase ("o61"), and the gradual decrease of the monoclinic ("m") occurring under the change of annealing conditions.In particular, the fraction of non-FE monoclinic phase gradually decreases from 68.2 % to 27.3%, and the fraction of FE orthorhombic phases gradually increases from 32.0% to 72.7% for the Samples 2, 3 and 4, respectively.We relate the change in the phase composition with the gradual increase of the oxygen vacancies concentration, which appear due to the change of the annealing conditions.Simultaneously with the composition change the average size of coherent scattering regions, which have the orthorhombic symmetry, varies from 10 nm to 14 nm, and the average size of coherent scattering regions, which have the monoclinic symmetry, varies from 9 nm to 23 nm (see

Table A3
in Appendix A [33]).The Raman spectra of the Samples 2-4, shown in Fig. 6(d), do not have any sharp peaks located below 3500 cm -1 .Instead, the Raman signal intensity reveal a very diffuse maxima located between (1000 -2500) cm -1 .The maxima height gradually decreases from the Sample 2 to the Samples 3 and 4, respectively.The behavior is characteristic for the very high amount of Raman-active luminescence centers, which concentration increases due to the increase of oxygen vacancies concentration in the samples.

Raman shift (cm
Hence, the X-ray diffraction, complemented by the Raman spectroscopy, revealed the formation of the ferroelectric orthorhombic phase under the increase of oxygen vacancies amount in the HfO2-y nanoparticles prepared at different annealing conditions.The determined fractions of the monoclinic and orthorhombic phases along with the sizes of the coherent scattering regions allow us to use the information for the application of effective LGD model.Obtained results are discussed in the next section.

IV. Polar properties of the oxygen-deficient HfO2-y nanoparticles
Using the fractions of the monoclinic and orthorhombic phases along with the sizes of the coherent scattering regions, which were determined from the XRD data, we apply the effective LGD model to the stoichiometric and oxygen-deficient HfO2-y nanoparticles.We regard that the concentration of oxygen vacancies is maximal at the surface of quasi-spherical nanoparticle and monotonically decreases towards its center (see Fig. 7(a)).The vacancies, being elastic dipoles, create elastic Vegard strains [35,36], which increase the stability of the FE orthorhombic phase due to electrostriction coupling [11,15,21].

V. Conclusions
We have shown that the effective LGD model can predict the influence of screening conditions and size effects on phase diagrams, polarization reversal and structural properties of the nanosized HfxZr1-xO2-y of various shape and sizes.The effective LGD parameters was determined from the available experimental results for HfxZr1-xO2 thin films.
We prepare oxygen-deficient HfO2-y nanoparticles, where the XRD revealed the formation of the ferroelectric orthorhombic phase.Micro-Raman spectroscopy was used to explore the correlation of lattice dynamics and structural changes appearing in the nanoparticles.We apply the effective LGD model to the HfO2-y nanoparticles and explain the appearance of the ferroelectric orthorhombic phase in dependence on the oxygen vacancies concentration.
Since our approach allows to determine the conditions (shape, sizes, Zr content and/or oxygen vacancies amount) for which the nanosized HfxZr1-xO2-y are ferroelectrics or antiferroelectrics, we hope that obtained results are useful for creation of next generation of Si-compatible ferroelectric gate oxide nanomaterials.      .

Figure 1 .
Figure 1.(a) The geometry of the considered heterostructure, consisting of a HfxZr1-xO2 film of thickness ℎ placed between conductive TiN electrodes.(b) The radial cross-section of the HfxZr1-xO2 nanoparticle covered with the shell of screening charge with the effective screening length .

Fig. 2 (
Fig. 2(a) and 2(b).To interpolate the points in the figures, polynomial x-functions are used for the dependence of LGD expansion coefficients on Zr content :

Fig. 2 (
Fig. 2(c) is the diagram showing the strong influence of the Zr content and weaker influence of the screening effects on the phase state of HfxZr1-xO2 thin films.

Figure 2 . 3 .
Figure 2. The composition dependence of the effective LDG coefficients   and   (in 10 9 F/m) (a), as well as   =   and η (in 10 10 Vm 5 /C 3 ) (b) determined from the fitting of hysteresis loops of HfxZr1-xO2 thin films, shown in Figs.A1-A2.(c) Phase diagram of the HfxZr1-xO2 film calculated in dependence of the Zr content x and the ratio ℎ  ⁄ .Parameters   = 10 and   = 3.The abbreviations "DE", "AFE", "FEI", "SDFE" and "PE" refer to the dielectric, paraelectric, antiferroelectric, mixed ferrielectric, and single-domain ferroelectric phases, respectively.A possible region of the poly-domain ferroelectric (PDFE) state stability is located inside the semitransparent cyan area.

Figure 3 .
Figure 3.The quasi-equilibrium dependences of polar (  ) and antipolar (  ) order parameters on the external field  calculated for the HfxZr1-xO2 nanoparticles, which Zr content  changes from 0 to 1 with the step 0.05

Fig. 4 (
Fig. 4(d)for  = 30).The FEI phase transforms to the FE phase with  increase.The further increase of  from 0.6 to 1 leads to the ferroelectricity disappearance at >0.7, and to the appearance of the PE phase, which continuously transforms to the DE state for >0.8.All diagrams inFig.4 become -independent for   ⁄ ≫ 10, since the contribution of the depolarization field becomes negligibly small with the   ⁄ increase (see Eq.(4b)).The range of the incomplete screening conditions, 5 <   ⁄ < 50, is the actual range of the size effect existence.Notably, the domain formation is possible for the weak and incomplete screening (see the semitransparent cyan area of PDFE states in Figs.4(b) and 4(c)).
5 nm.Thus, one should consider Figs. 4 as the diagrams showing the influence of the Zr content and screening effects on the phase state of the HfxZr1-xO2 nanoparticles.
y nanoparticles.To verify the idea, we prepared several samples of stoichiometric HfO2 nanoparticles and oxygen-deficient HfO2-y nanoparticles, which differ by the annealing conditions.
The phase diagram of the spherical HfO2-y nanoparticles, calculated in dependence of oxygen vacancies amount y and the ratio   ⁄ for   = 5 is shown in Fig. 7(b).It is seen from the diagram, that the increase of  from 3 % to 4.8 % leads to the transition of the AFE state to the single-domain (SDFE) or poly-domain (PDFE) ferroelectric phase.The thin region of FEI states separates the AFE state from the SDFE phase for   ⁄ more than the critical value,   ⁄ ≈ 40 (see the thin region located between the black dashed curves).For   ⁄ less than the critical value, the AFE state transforms into the PDFE with y increase.The further increase of  above 4.8 % leads to the ferroelectricity disappearance, and to the appearance of the PE phase, which continuously transforms to the DE statefor further increase of y above 7 % (not shown in Fig.7(b)).The appearance of the FE phase in the oxygen-deficient HfO2-y nanoparticles agrees with the experimental XRD results presented in the previous section.

Figure 7 .
Figure 7. (a) The radial cross-section of the HfO2-y nanoparticle.(b) Phase diagrams of spherical HfO2-y nanoparticles calculated in dependence of oxygen vacancies amount y and the ratio   ⁄ .The abbreviations "AFE", "FEI", "SDFE" and "PE" refer to the paraelectric, antiferroelectric, mixed ferrielectric, and singledomain ferroelectric phases, respectively.A possible region of the poly-domain ferroelectric (PDFE) state stability is located inside the semi-transparent cyan area;   = 5 and   = 3.
vacancies amount y (%) field dependences and magenta dotted curves show the dynamic polarization-field hysteresis loops calculated using the effective LGD model.Blue symbols represent the experimentally measured polarization-field dependences [38].

Table A2 .
The spontaneous order parameters, free energy densities, stability conditions of the thermodynamically stable spatially homogeneous phases of the free energy (1), and critical field(s).

Table A3 .
Rietveld refinement of crystallographic parameters of HfO2-y nanopowders