Insights Into Curie‐Temperature and Phase Formation of Ferroelectric Hf1−xZrxO2 with Oxygen Defects from a Leveled Energy Landscape

The phase composition of HZO thin films is critical for the ferroelectric and electrical properties of the films and the devices they are integrated into. Optimization is a major challenge since the phase formation depends significantly on many influencing variables that are only partially understood so far. The Curie temperature is identified as an important parameter for understanding the behavior, since it depends sensitively on Zr content, the density of oxygen‐related defects, layer thickness, and external stress. A two‐step process, phase formation by pure kinetic transformation followed by nucleation, is proposed for phase formation. This is necessary because nucleation theory alone cannot explain the experimentally observed dependence on oxygen content. The classical nucleation model is modified at two crucial points. First, the polycrystalline structure is incorporated which allows the size effect to be implemented. Furthermore, the interface energies between the child and parent phase, which result from static ab initio calculations, are rescaled from dynamical effects. The resulting model is used to calculate the phase fractions during thermal processing. The results for the most important influencing variables are discussed and compared with experimental results. The causes of the undesired monoclinic phase are further analyzed.


Introduction
HfO 2 and ZrO 2 and their solid solutions Hf 1−x Zr x O 2 are ferroelectric materials with excellent properties for micro-electronic and other technical applications. [1]These range from dense nonvolatile memories, [2,3] inmemory computing, [4] and neuromorphic devices, [5] negative capacitance transistors, [6] via thin film piezoelectric transducers [7] to thin film pyroelectric devices. [8]The ferroelectric properties of Hf 1−x Zr x O 2 have been associated with the presence of the metastable crystalline orthorhombic (Pca2 1 ) polar o-phase [9] which is formed in a thermal anneal step following the deposition process.To optimize the performance of the devices, the formation of the maximal possible volume fraction of the o-phase is desired because monoclinic (P2 1 /c) m-phase fraction leads to reduced polarization and detrimental depolarization field effects.The tetragonal (P4 2 /nmc) t-phase fraction can lead to similar effects and make an undesired wake-up procedure necessary.A sufficient understanding of the phase formation process is still lacking [10] and is related to the mechanism of polarization reversal, which is still under strong scientific debate. [11]he understanding is further complicated because a number of process parameters have a considerable influence on the phase formation process.As an example, a lot of attention has recently been paid to the oxygen content in thin Hf 1−x Zr x O 2 films, which is important to achieve reliable devices. [12]A controlled amount of oxygen defects seems to be needed to counteract oxygen scavenging from the electrodes during the anneal.Furthermore it has been found that a vast amount of oxygen stabilizes the m-phase. [13]Moreover, oxygen-related defects have been identified as being detrimental to the reliability performance of the films. [14,15]Controlling the oxide content is, therefore, indispensable, and a thorough understanding of the effects is desirable.
The classical nucleation theory has been proposed for the description of o-phase formation in Hf 1−x Zr x O 2 films, [16] which utilizes inhomogeneous interface energy effects [17][18][19] and provides kinetic information in the form of a thermodynamic barrier.
Nucleation theory applied to polarization reversal leads to a specific, measurable switching dynamics, which differs from the Kolmogorov-Avrami-Ishibashi (KAI) model of laterally moving domain walls. [20]Experimentally, for thin polycrystalline films, KAI has been excluded for small electric fields, [21] but nucleation theory and Landau switching models have been found to be compatible with experimental switching data. [11]Meanwhile, the theoretical approaches to describe the phase transformation from t-to o-phase based on phonon modes are being developed, [22,23] which implies a Landau model of homogeneous switching.
The homogeneous switching of a whole domain or grain avoids the contribution of interface effects.However, the large amount of activation energy to possibly switch domains larger than a few nm -the typical size of critical nuclei in classical, nucleation theory -is only conceivable if soft modes exist.In this context the discrepancy between static barrier heights of the energy landscape, calculated at T = 0 K [24] from ab initio, and values obtained from experiments becomes important.Hoffmann et al. [25] finds a height of the double well -which can be associated with a switching barrier -of about 1meV for (HZO), compared to an ab initio calculated barrier by Qi et al. [26] of about 30 meV.Lomenzo et al. [27] reconstructed the energy landscape between the t-phase and the o-phase for ZrO 2 and found about 6meV compared to an ab initio calculated barrier by Reyes-Lillo et al. [28] of about 40meV.The difference between the measured and the static elevation in the energy landscape calculated at T = 0 K is consistent with a further significant discrepancy between theory and simulation, the discrepancy between measured and calculated coercive field.The measured coercive field in Hf 1−x Zr x O 2 is about 1-2 MeV cm −1 , and the values calculated from ab initio are about one order of magnitude larger. [26]owever, a remedy for the calculated energy landscape is possible through a dynamic approach provided by molecular dynamics calculation.First simulations with machine-learned potentials with almost ab initio quality show a clear flattening of the energy landscape due to thermal motion and is detected with an applied external field, which comes surprisingly close to the experimental one. [29,30]Additionally the thermodynamic quantities, like the total energy maintain values, consistent with static calculations.The conclusion is that calculated thermodynamic state variables like total energy may be used without dynamic correction to describe the driving force of phase transitions.However, when kinetic barriers are used in the model, a rescaling to smaller values or leveling needs to be applied.Results from calculations of the minimum energy path of Landau-type phase transitions have been obtained (see Supporting Information SI 1.3).Interphase (coherent interface) energies, which appear at the phase boundary in a nucleation process, have been calculated previously [31] at T = 0 K, and a relation to the kinetic barrier height of homogeneous transition was suggested.A leveling of the values is also plausible here.In our model, we will reduce the size of the barrier-related quantities by a factor of five, according to the previously observed discrepancy in the coercive field between theory and experiment.
Recently, classical nucleation theory has been explored for experimental data and simulations [32,33] to check the range of applicability.In some material systems for the crystallization process from the amorphous phase, molecular dynamics simula-tions have found nonclassical nucleation pathways that bypass the high barrier for homogeneous nucleation in a first step. [34,35]n such cases, kinetic processes dominate over thermodynamic processes, which are the basis of the classical nucleation theory, and the energy contribution of the interface between the core and the amorphous medium is weakened.In a second step, after crystallization is complete, phases are transformed according to classical nucleation theory including the effects of interface energy.
In this paper, a two-step phase formation model is proposed.The first step assumes that the nuclei from the amorphous deposit in a highly symmetric phase, namely the t-phase.This is justified by Ostwald's rule.With continued thermal activation by annealing, it is assumed that these nuclei grow and coalesce without affecting the free energy from the interface to the amorphous environment, and the stability of the nuclei is controlled by the kinetic barrier to competing crystal phases.Thereby phase transformations to more favorable phases such as o-phase or mphase during growth as Landau-type transformation are allowed.In the second step, after crystallization is completed, a transformation to more favorable phases is possible and is described as controlled by the classical nucleation theory with thermodynamic barriers.The separation of the processes into two steps is an idealization, since the growing grains, which are already too large to transform according to Landau could already start transformation according to the nucleation theory.
This model investigates the phase composition depending on Zr-content, temperature, and oxygen content for Hf 1−x Zr x O 2 .It turns out that both steps of the model are necessary to describe the experimentally observed phenomena.The conditions for ideal phase composition are found and even the conditions for ZrO 2 with a substantial o-phase fraction are identified.The strong increase of the m-phase fraction with increasing film thickness becomes explained.In addition temperature effects are investigated, and finally, in an overview the impact of the investigated process parameters in simulation and experiment are discussed.

The Curie-Temperature
Important for the polar o-phase formation from the t-phase is the value of the Curie-temperature T C as the temperature where the Gibbs energy density vanishes The difference in internal energy Δu and entropy Δs depends on the Zr-content, the oxygen content, and has been calculated from ab initio, see SI 1.2 (Supporting Information).Concerning the enthalpy, we found it reasonable to ignore the stressstrain contribution when the grain of the o-phase grows in a favorable orientation, which is a reasonable assumption because the barrier is lowest.The contribution of the grain size depends on the surface to volume ratio A/V of a typical geometry weighted with the grain interface energy .The typical geometry has A/V = 3/R = 6/d for both spherical (then 2R = d) or cylindrical  [38] (symbols) with the model (lines) for A/V = 6/d for typical geometries where film thickness d is varied from to 30 nm.The oxygen density varies from 1 at% vacancies to 1 at% interstitial.b) Remanent polarization comparison for a 10 nm HZO film [38] deposited with 1s O 2 pulses compared to the model prediction as a function of the process temperature.
geometry with cylinder raidus R and height d = 2R, where d is the film thickness.A/V does not deviate much from this value for bodies with aspect ratios close to 1. Materlik et al. [17] have assumed the difference of experimental surface energies as interface energy, which amounted to a few 100 mJ m −2 for a t/o interface.Batra et al. [36] calculated the surface energies from ab initio and obtained similar values for the difference depending on orientation.Both authors used their values as contribution to the free energy to explain phase stabilization in a purely thermodynamical model.Park et al. [16] developed a phenomenological classic nucleation theory and assumed the interface energy to be only 36 mJ m −2 to be consistent with experimental data.His goal was the calculation of the thermodynamic barrier.The phases were allowed to be in a meta-stable state with a large barrier toward ground state instead of being at the absolute minimum of the free energy.Finally, a coherent interface energy was calculated directly ab initio from Falkowski et al. [31] They obtained values starting from 117 -200 mJ m −2 for the t/o interface for ZrO 2 and HfO 2 , respectively, and 300 -450 mJ m −2 for t/m and o/m interfaces.
In our present model, we calculate the thermodynamic barrier for the second step.In contrast to the previously used classical nucleation theory (see SI 2, Supporting Information.),however, we distinguish between the grain interface energy Γ, relevant for the size effect, and the interface energy  0 , relevant for calculating the barrier.The justification is that the grain interface is defined by a structural inhomogeneity (pinning at electrode or differently oriented grain) and that dynamical corrections to statically calculated values are less effective.According to our argument of a leveled energy landscape consistent with molecular dynamics calculation, the value of  0 is reduced by a factor of five compared to Γ.The interpolation between  0 and Γ happens with increasing nucleus radius r modeled with the phenomenological expression.The expression describes the value of the interface energy  (r) when the nucleus in the child phase starts to grow, increasing slowly from  0 with radius r, and finally rising steeply to Γ while approaching the grain radius R closer than the transition scale .
The motivation for this polycrystalline extended nucleation model (see SI III, Supporting Information.)results from two major observations.If  0 is not small, the activation energy barrier for nucleation is unrealistically large.Furthermore, if Γ is small, the size effect on the Curie temperature would be too small.The parameters used in our model are summarized in the SI III C (Supporting Information).
For the size effect on the Curie-temperature, an interface energy of 36mJ m −2 like from Park's work leads to a temperature shift of about 100 °C.The observations -see the discussion of Figure 1 below -hints to a larger effect, which is more consistent with a value for Γ of about 100 mJ m −2 .For example in ZrO 2 T C is predicted to be about 650 K [37] from ab initio calculations, but is found to be below room temperature for 10 nm thin films because these films are tetragonal at room temperature.
In a recent paper [38] values for T C have been measured for different Zr and oxygen content in 10 nm Atomic Layer Deposition (ALD) films.For comparison in Figure 1a of our T C -model with these data, we assume that 1 at% oxygen vacancies (vac) and 1 at% interstitial (int) in the simulation (see SI 1., Supporting Information) correspond to an oxygen pulse time of 0.1 and 5s, and that a pulse time of 1s produces a stoichiometric film.The relation between oxygen supply and defect concentration is based on the following observations: Alcala et al. [12] have found the oxygen vacancy concentration of HZO close to oxidizing electrodes with XPS photoelectron spectroscopy to be 0.1-0.8%,which seems to be typical for a moderately oxygen deficient film.Mittmann et al. [39] measured the unit cell volume in PVD HZO-films with different Zr content and oxygen flow of 0-5 sccm, and found a variation consistent with DFT calculated volumes from ± 1% oxygen variation, with an uncertainty of about a factor two.That low oxygen flow leads to oxygen deficiency (vacancies) is accepted.That large oxygen flows lead to oxygen excess (interstitial) was argued with delaminated top electrodes.Materano et al. [13] measured ALD Hf 1−x Zr x O 2 films and found the phase composition comparable to PVD films when they varied the ozone dose time between 0.1-5 s.Finally, the data for Figure 1 were measured on the same equipment with the same conditions.The sensitivity of T C to grain size can be compared to the results from Hoffmann et al. [40] where the authors found a 200 °C increase for a grain radius from 5 to 25 nm where the grains had a fixed height of 9 nm (implying A/V from 0.62 -0.3).We used A/V from 0.6 -0.2 for our 2R = d spheres/cylinders for d = ranging from 10 to 30 nm and obtained a shift of 300 °C.The T C model shows an excellent agreement with the data and indicates a stable region of the ophase below the respective lines as shown in Figure 1a.Whereas for HfO 2 the o-phase can be stable even for a thickness d of 5 nm, for ZrO 2 the o-phase can only be expected for oxygen-rich films or films thicker than about 10 nm.Films on TiN electrodes are expected to crystallize in the t-phase because an oxygen deficiency is created.The influence of oxygen and Zr content will later be discussed in detail within the discussion of the nucleation rates.In addition, the thickness was varied, which greatly influenced the T C .This aspect will also be discussed within the nucleation theory model.There have also been reports about ferroelectricty below 2 nm, which are highly debated and its origin is not yet unraveled. [41,42]Possible reasons include stress effects or effects from electric fields, caused by fixed charges, which are known to contribute to o-phase stabilization [43] Figure 1b shows the polarization as a function of temperature in a 10 nm HZO film which reduces while approaching the simulated T C of 490 °C = 763 K.The polarization can be taken as an indicator for the o-phase fraction.The remaining material may crystallize into the t-phase or m-phase.Note that in the experiment [38] Figure 1b the data show a decrease of the polarization with increasing temperature, but the polarization was recovered after reducing the temperature again.The reversibility persisted if the Curie-temperature was not exceeded.In the polycrystalline nucleation model the phase fraction results from partially transformed grains with stable radius r < R. A stable interface at a r larger than the critical radius r* does not exist in classical nucleation theory (see Figure S5, Supporting Information) and is the result of the polycrystalline modification of (r) Equation (2).

The Two-Step Model
It is assumed that during film deposition, seeds already crystallize in the t-phase, as supported by experimental observations. [39] separate thermal anneal step realizes the growth of the seeds and the thin film's complete crystallization.The proposed phase formation model based on a two-step process is visualized in Figure 2 as an example the results are shown in Figure 3 in its different evaluation stages for HZO.The first step contains the growth of the seed and possibly a homogeneous phase transformation until the crystallization temperature is reached.This step is considered to be Landau-type with a kinetic barrier (kb), where the activation energy density e a has been calculated as minimal energy path (MEP) (see SI 1. and SI 1.3, Supporting Information) and has the unit energy per volume.In the second step, after the crystallization has been completed, the phase composition is governed by the thermodynamic barriers (tb) calculated based on the classic nucleation theory.To compare the different barriers in an energy landscape, both need to be brought to a common denominator.The explicit derivation can be found in the SI.II (Supporting Information).
The thermal barrier free energy density is in fact nearly independent from the interface energy  and results in g* = Δh/2 (Δh = Helmholtz energy difference), which gives a qualitative picture of the energy landscape for the first step and the second step in Figure 3a.
The densities must be multiplied with relevant volumes to obtain the energy barrier.For the kb this volume is the seed size V, which varies and is in the order of a few nm.In our model we choose a cube of size V = (2 nm) 3 based on experimental findings before annealing. [44]For the tb this volume is the critical volume V*, which depends on composition, oxygen content, as well as the Curie-temperature.V* is calculated from the maximum of the free energy w.r.t. the radius at r = r*, including the sizedependent (r), and results in the thermodynamic barrier (see SI III.A., Supporting Information for details).
The reaction rate has two contributions active in step 1 and step 2, the length of step 1 depends on the crystallization temperature. [18]The time depends on the ramp-up of the thermal anneal equipment.The prefactors are calculated as the secondary nucleation attempt frequency according to ref. [45] and globally adjusted to A 1 =0.1 m −3 s −1 and A 2 =0.8 m −3 s −1 to compare better with the data.The trend of larger seeds with higher Zr content decreases the contribution of the first step to the transformation.This is taken into account with the crystallization temperature, which is smaller for higher Zr content and makes the first step less important.
Finally, the phase fractions are calculated from a first order balance equation, taking the reaction rates for transformation into account.We did not use the Kolmogoroff-Johnson-Mehl-Avrami equation for grain growth, because the polycrystals formed after step 1 do not change in size after step 1 and only change composition in step 2 with a probability, but a balance equation Note that the annealing process is varied depending on the experimental circumstances and is changing accordingly between a rapid temperature anneal (RTA) and a fast furnace anneal.

Results
As already discussed, the formation process starts with the tphase and its seed will grow out of the amorphous phase governed by the kinetic barriers (kb) till the crystallization process is completed.At this time, depending on the kinetic barrier, the nucleus may transform into the o-phase or the m-phase.The finally crystallized t-phase, o-phase, and m-phase continue transforming governed by thermodynamic barriers.
We illustrate the results first by the example of HZO in Figure 3.The free energy landscape in (a) illustrates that at 20 °C the t-phase can transform into the o-phase and the m-phase, by overcoming the barriers.Due to the free energy reduction of the t-phase at 600 °C only a transformation into the m-phase is possible, for both the path via tb and kb.The transformation rates in (b) show that the t-to o-transformation is possible via the kb (green symbols) up to a temperature of about T C .Above T C the reverse o-to t-transformation will be favored.The t-to m-(black symbols) is simultaneously possible in the complete temperature range.All kb-related kinetic transformation rates show an increase with temperature.
After the crystallization time  the kinetic rates drop to zero and the nucleation rates are relevant.Since the t-phase energy depends on temperature according to Δh = Δu − TΔs relative to the m-phase, the t-to o-nucleation barrier (green line for its transformation rate) increases (the rate nearly vanishes) with temperature according to Equation (3) and shows a steep maximum close to T C , when the o-phase energy is reached.The singularity is not exactly at T C but is close to it and depends on the size effect (see SI 3., Supporting Information).Heating above T C does not increase the o-phase fraction.Therefore, heating up to T C seems to be the best condition for achieving the optimal o-phase fraction.Furthermore, the process time should be kept short, since o-to m-and t-to m-transformation processes (blue and black lines) are not suppressed and lead to the undesired m-phase.Most important for the final phase composition is the cooling process.Reducing the temperature leads to a massive increase of the tto o-(green line) transformation.The o-phase formation during the cooling has been pointed out by Park et al. using a conventional classical nucleation theory model [16] without the size effect.The transient simulation in Figure 3c shows again that the phase composition depends sensitively on the cooling phase of the fast furnace process, which is typically not specified in the process recipes.The transient evolution of the phase composition is calculated by Equation (5).During the crystallization time , the kinetic processes result in a small amount of the m-phase.During the nucleation time , the m-phases increase mainly from the tto m-transition.Finally, in the cooling phase, most of the o-phase fraction is created from the t-phase, but some fraction of o-phase is lost to the m-phase.This phenomenon, will be investigated in subsection 3.5.
Next, we investigate the dependence of the model on the Zr content and the density of oxygen defects, as shown in Figure 4.
Regarding the oxygen content, several publications have recently documented the effects of the oxygen supply experimentally. [13,39,44,46]We directly compare our simulations with the results of Materano et al. [13] who processed 10 nm films of Hf 1−x Zr x O 2 over a range of Zr content x and different amounts of oxygen supply followed by the same 600 °C anneal in a fast furnace equipment.
The first look at the t-to m-and o-to m-nucleation rates shows that interstitials suppress the m-phase formation.This is a contradiction to the experiments.In the classical nucleation theory model, this is simply caused by the decrease of the free energy of the t-phase by interstitials, the thermodynamic barrier has the size g* = Δh/2.This observation can be transferred to a possible nucleation from the amorphous phase.Nucleation has been suggested [10] as the crystallization mechanism.However, interstitials lower the free energy (see SI I.A and C, Supporting Information) and increase the difference Δh to the amorphous phase, increasing the barrier height.Again this contradicts the experimental results.The observation of the wrong trend for the oxygen dependence inherent to the nucleation model, which results both in amorphization as well as annealing, is the reason for proposing the two-step mechanism in this paper.
The kinetic barrier, however, strongly favors the m-phase for interstitial rich films, because the interstitials destabilize the tetragonal phase.In summary, large amounts of interstitials lead to a significant transformation into the m-phase in step 1 from the kinetic barrier.In step 2, the m-phase may be formed due to further process conditions.The observation can be confirmed by looking at the transient evolution of the phase composition in Figure 5.The figure contains a comparison of the phase fraction of Materano et al. and our simulation as a histogram.Depending on the crystallization time  in the interstitial rich samples, the m-phase is formed from the beginning.For ZrO 2 the time  is much shorter than for HfO 2 because of the lower crystallization temperature.Experimental data and simulation compare well for some cases, but there are also large discrepancies.Some of the discrepancies may be due to imprecise assumptions.For "ZrO 2 int" we assumed 1% interstitial in the simulation.Reducing this value, the simulation result comes closer to "ZrO 2 ", which fits much better.Furthermore, experimentally, the m-phase fraction is known to rapidly increase with O 2 -pulse from Xu et al. [46] For The histograms compare the results with experimental results from Materano et al. [13] the three HfO 2 cases, the simulation indicates a larger o-phase fraction than the ALD data.But for physical vapor deposition films of the same thickness and similar anneal temperature, see Mittmann et al.SI Figure S15 (Supporting Information) [44] the data are very comparable.It could be that ALD data are affected by carbon incorporation, which is not part of our model.Severe, however, is the discrepancy for HZO where the simulation consistently predicts a larger o-phase fraction for HfO 2 .
Regarding the Zr content dependence, the major effect is the lower free energy of the t-phase relative to the o-phase for increasing Zr.This moves T C significantly from 760 °C for HfO 2 to 165 °C for ZrO 2 .If the process temperature is adapted to T C , the transformation to o-phase happens during the cooling phase.During the process time, undesired transformation to the mphase may happen.These t-to m-phase and o-to m-phase transition rates depend on further conditions, as discussed later.
The case of ZrO 2 is special because T C is below the crystallization temperature and the o-phase cannot be formed but this situation can be reversed with increased oxygen content.

Ferroelectric ZrO 2
Stoichiometric ZrO 2 is mostly antiferroelectric due to the low T C .With an increased oxygen content a higher T C can be achieved.An interstital concentration of 1 at% already destabilizes the tphase, but besides the o-phase the m-phase may form.Therefore, a varying oxygen surplus is investigated, and the simulation results are shown in Figure 6.Additionally, the thickness is increased since ZrO 2 was found to be partially ferroelectric [46] at a thickness of 45 nm.The higher thickness increases the T C as well (Figure 1).An oxygen surplus of 0.6at% at 45 nm pushes T C beyond the crystallization temperature.This opens up a window for the annealing process to achieve a higher o-phase formation and results in a significant remanent polarization P r .As a general explanation of making ZrO 2 ferroelectric, we can state that thickness and oxygen interstitial increase the T C above the crystallization temperature.As a result, the material can crystallize into the o-phase already during deposition.Moreover, the o-phase content could further increase after an additional annealing step.

Thickness Effect
The influence of the thickness on the phase formation rates has already been discussed for the case of ZrO 2 .With increasing thickness d, the grain size is enhanced.In a broader context, the fraction of t-phase in HfO 2 and HZO is reducing with increasing thickness.To further investigate this effect, the phase transformation rates in HZO for different thicknesses were calculated (see SI 3.3, Supporting Information) and the resulting phase fractions are shown in Figure 7.At a small thickness below 5 nm the possibility of achieving a ferroelectric film is very low since T C is below the crystallization temperature.As a result, the film stays in the t-phase as indicated by a high dielectric constant. [47]The increase of the m-phase fraction with larger HZO film thickness is related to an increase of the thermodynamic nucleation rates towards the o-to m-and to t-to m-transformation as well as an increase of T C .This has been observed in various experiments. [44,48,49]

Temperature Effect
Recently, a systematic study of the effect of the annealing temperatures has been performed on HZO. [50]A 15% increase of the m-phase by enhancing the annealing temperature from 600 to 1000 °C was found in experiment.In simulation, however, by raising the temperature while keeping the same cooling rates in the RTA processes, similar concentration changes can be seen, as illustrated in Figure 8.It has been suggested by Han et al. [51] that stress effects interact in processes with large temperature variations.Such effects are not included in this temperature simulation.

Quenching Effect
Quenching describes the rapid cooling after annealing.In the case of HfO 2 this resulted in an increased o-phase fraction compared to processes with slower cooling. [52,53]In our simulation, the cooling period in the RTA process is reduced by a factor of 10, as seen in Figure 9.As a result, a significant phase transformation is prevented, and only a small m-phase fraction is generated, so that a larger wake-up effect should occur.But this results in a larger o-phase fraction, since less m-phase is generated overall.The reduced m-phase fraction in the simulated quenching process results from a shortened process time.

Thermodynamic Model
The classical nucleation theory and the stabilization of metastable phases with a large barrier were proposed by Park et al. because in 10 nm HZO films the transformation from o-phase to m-phase could be observed [16] at elevated temperature, but was absent at room temperature.The purely thermodynamic stabilization of the o-phase, however, requires the free energy of the o-phase to be below the m-phase. [17]It is interesting to ask for the stabilization of the o-phase in the present model with its choice of parameters.First, the thermodynamically stable phase according to the  free energy Equation ( 1) has been calculated depending on size d and temperature T, and on Zr composition and oxygen content, the results are shown in Figure S11 (Supporting Information).For Hf rich composition with increasing oxygen content, the size effect does stabilize the o-phase thermodynamically at lower temperatures.But especially for HZO the absolute stabilization is only realized in a narrow region very close to 5 nm.
Second, the Curie-temperature for the same parameter set has been calculated and shown in Figure S12 (Supporting Information).Below the critical film thickness a transformation to the o-phase is not possible.Above the critical thickness a phase mixture is predicted, but not necessarily a pure phase.
The conclusion is that the thermodynamic stabilization criterion is too restrictive.O-phase formation in thermodynamically metastable process conditions may occur and are kinetically stabilized.These kinetically stabilized regions, however, are not limited to a single phase.

Discussion
Finally, the individual models are discussed regarding their predictive power in relation to experimental results.For this purpose, the agreements between simulation and experiments for the different models are analyzed and presented in Table 1.
The influence of the Zr concentration in Hf 1−x Zr x O 2 can be reproduced with the kinetic, nucleation, and thermodynamic model components.With increasing Zr content the kinetic trans-formation rates reduce, leading to a large t-phase fraction.For the polycrystalline nucleation model the o-or m-phase formation is hindered from small T C for large Zr concentration, leaving a large t-phase fraction.Thermodynamically the energy difference between the phases is smaller in ZrO 2 , resulting in an increased t-phase stability.As discussed before the nucleation and thermodynamic model cannot reproduce the phase fraction of experimentally varied oxygen concentration in Hf 1−x Zr x O 2 .Therefore, the kinetic model is introduced before crystallization, especially for large oxygen concentrations.Since the relation between ozone dose/oxygen flow and the actual oxygen concentration in Hf 1−x Zr x O 2 is not defined, only the maxima and minima can be compared, meaning large and low oxygen concentrations.In case of a reduced oxygen concentration in HZO a reduced o-phase fraction is measured. [13,54]In HfO 2 a shift toward increase of ophase fraction can be detected. [39]Usually, after annealing, a reduced oxygen concentration can be measured due to scavenging of the electrodes at a moderate ozone dose time/oxygen flow.It should be noted that at low oxygen supply the incorporation of carbon atoms play a role, which is not included in the model.It has already been demonstrated that the kinetic barrier, therefore, the kinetic transformation is primarily independent of the annealing temperature. [29]Concerning nucleation, the annealing temperature is already included in the calculation of the rate for the activation.In the thermodynamic model, the temperature dependence is included in the free energy.The variation of the Curie temperature is only explainable with the thermodynamic and nucleation model due to the temperature dependency and  the size effect as discussed before.Stress and Strain can not be explained by either model, which can be seen in an example calculation in the SI 3. (Supporting Information).The simulations show the opposite trend due to the ratio change between o-and m-phase transformation rates and their free energies.The stress variations of the simulated results have been compared to other simulations studies [43] and are coherent. [51]The last process parameter is the thickness, at the kinetic level, all rates would drastically be reduced due to the larger volume and following increased barriers.For the nucleation and thermodynamic model, the T C increases with thickness for relevant nucleation rates toward oor m-phase, therefore , the phase formation rates are increasing at larger thicknesses, and a larger m-phase will be formed.In general the new developed model based on kinetic and nucleation limited phase transformation can explain the influence of most process parameters besides the stress, which gives even the opposite trends, which is still a mystery.

Conclusion
In conclusion, the metastable ferroelectric o-phase formation in Hf 1−x Zr x O 2 thin films, at varying oxygen content, stress, thickness, and annealing temperature, was theoretically examined based on a two-regime phase formation process.The first regime consists of a homogeneous transformation in kinetic processes till the crystallization is completed.This is followed by a nucleation, which model is modified for polycrystalline materials.The major achievement of the model is to describe the shift of the Curie-temperature with Zr-content, oxygen content, and grain size.The Curie-temperature has a central role in the model.The modification is derived by assuming a leveling of the crystal energy landscape from dynamic motion, reducing the interface energy value for the nucleation process.At the same time the interface energy maintains the statically calculated values, allowing a sufficiently large size effect.The kinetic processes during the crystallization in the ramp-up time are responsible for large mphase fractions in layers processed under oxygen rich conditions.The nucleation models systematically do not result in a large mphase formation in such processes.With the help of the modified nucleation theory becoming effective after completed crystallization, the phase formation in multiple experiments can be explained well.A major contribution to the nucleation comes from the cooling process.During processing the temperature is typically close to the Curie-temperature with only a minimal transformation from the t-phase to the o-phase.The model cannot explain the influence of compressive and tensile stress on the phase formation and needs further investigation, since other studies suggest opposite trends, as in experiments.However, the increased m-phase formation for films with increasing thickness can be explained by the Curie-temperature shift from the size effect.The simulation is so far limited to single parameter variations, including correlated effects like the influence of oxygen content on film stress, which requires further research.The simulation is so far limited to single parameter variations, and the inclusion of correlated effects like the influence of oxygen content on film stress requires further research.But some correlated effects can already be foreseeable.To obtain minimal leakage from minimal defects in the final film, the supply of oxygen in the process has to be estimated in advance to compensate for the effect of oxygen consumption during the anneal, which depends on the kind of electrode.Minimal wake-up requires a film with low t-phase content due to excess oxygen, but these conditions are susceptible to contributions from m-phase.

Figure 1 .
Figure 1.a) Comparison of Hf 1−x Zr x O 2 T C data from Schroeder et al.[38] (symbols) with the model (lines) for A/V = 6/d for typical geometries where film thickness d is varied from to 30 nm.The oxygen density varies from 1 at% vacancies to 1 at% interstitial.b) Remanent polarization comparison for a 10 nm HZO film[38] deposited with 1s O 2 pulses compared to the model prediction as a function of the process temperature.

Figure 2 .
Figure 2. Visual illustration of the two step model.a) In a first step, nuclei of in the amorphous deposit can transform kinetically without interfacial effects.b) In a second step, grains can transform according to a modified nucleation model depending on the interfacial energy.

Figure 3 .
Figure 3. a) Free energy landscape of kb (dashed) for first step and tb (lines) for second step at room-and annealing temperature.b) Decadic logarithm of the transformation rates for all phase transformations ("starting phase" > "final phase") are limited by the kinetic barrier kb (symbols) or nucleation described by the thermal barrier tb (lines with same colors) as a function of the temperature, the yellow shaded region is below the crystallization temperature.c) Phase fractions as a function of time in a 600 °C fast furnace anneal process.

Figure 4 .
Figure 4. Formation rates between the t-, o-, and m-phase of Hf 1−x Zr x O 2 phases for varying Zr content and oxygen defects ranging from 1 at% interstitials via stoichiometric films to 1 at% vacancies for kinetic transformation (kb, symbols) and nucleation (tb, lines with same colors) depending on annealing temperature.Indicated are the crystallization temperature (yellow shaded region) and the Curie-temperature T C .

Figure 5 .
Figure 5. Phase fraction of Hf 1−x Zr x O 2 phases for varying Zr and oxygen after a 600 °C fast furnace anneal process based on nucleations rates of Figure 4.The histograms compare the results with experimental results from Materano et al.[13]

Figure 6 .
Figure 6.Phase fractions of the o-phase in ZrO 2 for increasing values of the oxygen content (compare (a) and (b)) and increasing thicknesses (compare (a) and (c)).

Figure 7 .
Figure 7. Phase fraction of Hf 1−x Zr x O 2 phases for varying the thin film thickness from a) 7 nm to b) 10 nm to c) 30 nm with an RTA process of 600 °C.

Figure 8 .
Figure 8. Phase fraction of Hf 1−x Zr x O 2 phases for varying the RTA process temperature from a) 600 °C to b) 800 °C to c) 1000 °C.

Figure 9 .
Figure 9. Phase fraction of Hf 1−x Zr x O 2 phases for reducing the cooling time from a) 200s to b) 20s with an RTA process of 600 °C process.

Table 1 .
Predictive power of models for different dependencies compared to experimental results, agreements are symbolized as "+" disagreements are "-".