Effects of shear strain on HZO ferroelectric orthorhombic phases

Hafnium zirconium oxide (Hf_xZr_1–xO₂, HZO) and doped HfO₂ become intensively investigated materials nowadays for the promising applications in ferroelectric field-effect transistor (FeFET), ferroelectric RAM (FeRAM), and ferroelectric tunneling junction (FTJ) because of its high ferroelectricity showing in the thin film (<20 nm) and also good compatibility in the fabrication of complementary metal–oxide–semiconductor (CMOS).^1,2 Since the discovery of ferroelectricity in the Si-doped HfO₂ thin film at 2011,³ various strategies, such as doping,^4–11 oxygen defects control,^12–15 anneal conditions,^16–19 growth on different substrates,^20–22 and Hf/Zr superlattice,^15,23–26 were extensively explored to enhance the most essential property, remanent polarization (P_r), in the HZO thin film. After years of studies, about half Zr substitution in HfO₂ (i.e., Hf_0.5Zr_0.5O₂) with using titanium nitride (TiN) or tungsten (W) as electrodes in a metal–ferroelectric–metal (MFM) structure^{8,15,16,18–24} and using silicon as substrates in the metal–ferroelectric–semiconductor (MFS) structure^14,25,26 under annealing temperature of 400–700 °C could obtain good ferroelectricity in the HZO thin film even down to a thickness of 2 nm.²⁵ On the other hand, studies via theoretical modeling had also investigated many aspects of HfO₂/ZrO₂ crystalline phases, such as relative stability, remanent polarization, polarization switching mechanism, as well as the effects of strain, doping, oxygen vacancies, and interfacial energies on ferroelectricity.^27–42 The origin of ferroelectricity in the HZO thin film is most commonly considered to be the formation of ferroelectric orthorhombic (Pca2₁, No. 29, oIII-phase) phase.^1,2 While some recent works also proposed the rhombohedral (R3/R3m, No. 146/160) phases^31,39,43,44 and the other ferroelectric orthorhombic (Pmn2₁, No. 31, oIV-phase) phase^{27,31,33,38,41,45} could be another possible sources of ferroelectricity in HZO films. In addition to the ferroelectric phases, nonpolar phases with lower bulk energies like monoclinic (P2₁/c, No. 14, m-phase), tetragonal (P4₂/nmc, No. 137, t-phase), cubic (⁠ $Fm 3 ¯m$⁠, No. 225, c-phase), and nonpolar orthorhombic (Pbca, No. 61, oI-phase) phases are also frequently discussed and analyzed in experimental and theoretical studies. Furthermore, theoretical models of phase diagrams were also developed/simulated based on calculated surface/interfacial energies to explain the observed phenomena and to predict possible tactics for increasing the ferroelectricity in HZO thin films.^35,36,42,46

To identify the existence of ferroelectric oIII-phase, grazing incidence x-ray diffraction (GIXRD) analyses, high-resolution transmission electron microscopy (HRTEM), and high-angle annular dark-field STEM (HAADF-STEM) images are frequently used techniques in experiments. However, the nature of polycrystalline forming in the fabricated HZO thin film usually causes the diffraction patterns of many phases coexisting in GIXRD.^{3–8,12–19,23} Together with the close positions of oIII-phase (111) and t-phase (110) peaks around 30°, it is hard to quantitatively measure the content of oIII-phase via GIXRD analyses. On the other hand, comparing theoretical models with HRTEM and HAADF-STEM images could identify the crystalline phases of local structures in the HZO film;^45,47–52 however, regions with distorted structures may not be well identified. More sophisticated techniques such as nanobeam mapping of x-ray absorption spectroscopy (XAS), were used to analyze the content of oIII-phase, t-phase, and m-phase in the HZO thin film with a resolution of 100 nm.⁵³ Due to the inevitable strain induced structural distortion caused by doping, defects, and interface conditions, the phases of the local structure could not perfectly be defined in the above-mentioned experimental analyses.

Some theoretical works had studied the effects of in plane strain on the relative stability among HZO phases to inspect/support how the choice of substrate will affect the ferroelectricity of the HZO thin film.^30,33,37,39 As inferred in experiments, oIII-phase is more stable than t-phase under in-plane tensile stress to cause the enhancing of ferroelectricity.^20–22 However, the effects of shear strain have not been theoretically studied, which also occurs in distorted structures of the HZO film. In this theoretical study, it shows shear strain (by varying α angle) could dramatically alter the relative stability among HfO₂/ZrO₂ oIII-, oIV-phase, and m-phase. Meanwhile, shear strain also has very different effects on the remanent polarizations and coercive fields of oIII- and oIV-phases. The results indicate the existence of oIV-phase makes the HZO film have strong tolerance to structural distortion while keeping ferroelectricity.

Here, $P$ is the polarization, $U$ is the bulk energy, and $E$ is the electric field. $α$⁠, $β$⁠, $γ$⁠, and $U 0$ are fitting parameters.

Figure 2 shows the energy profile of polarization switching for optimized HfO₂ oIV-phase with L–G fitting and its corresponding P–E curve. The simulation data points are well fitted by L–G equation [Eq. (1)] and the estimated E_c is 14.8 MeV/cm, which is obviously larger than most of observed values around 1–3 MeV/cm.^{3–6,8–12,17,20,23,43,47,51} Please be noted that the negative dP/dE segment of solid P–E curve in Fig. 2(b) is not observed in experimental P–E hysteresis loop. A normal P–E hysteresis loop should follow the guide by arrows in Fig. 2(b). During the electric field sweep, the polarization will abruptly change its direction when it meets the turning points for a single crystal domain. Table I summaries the calculated phase energies, optimized lattice constants of four phases shown in Fig. 1, and the remanent polarizations, coercive fields of two ferroelectric phases. All the lattice parameters are given in conventional unit cell with 12 atoms, which is also the primitive cell for m- and oIII-phase but twice of the primitive cell for oIV- and t-phase. The lattice constants of HfO₂ phases are all smaller than the corresponding ZrO₂ phases. The α angles of m-phase are very close for both HfO₂ and ZrO₂. On the other hand, the α angle of HfO₂ oIV-phase is 2° larger than ZrO₂. For both HfO₂ and ZrO₂, oIII-phase is more stable than t-phase but less stable than m-phase with larger energy difference among HfO₂ phases than ZrO₂. oIV-phase is less stable than oIII-phase for HfO₂ and ZrO₂; however, it is more stable than t-phase for HfO₂ but the other way around for ZrO₂.

The polarization strength of HfO₂ and ZrO₂ is in the same order varying from 46.3 to 60.3 μC/cm². ZrO₂ oIII-phase has stronger polarization than HfO₂ but ZrO₂ oIV-phase is weaker than HfO₂. On the other hand, the energy barriers (E_b) for ZrO₂ polarization switching are always much smaller than HfO₂ (Table I). Via L–G model, the oIII-phase has larger estimated E_c than oIV-phase for both HfO₂ and ZrO₂, while E_c values for ZrO₂ ferroelectric phases are all smaller than HfO₂, which indicates Zr doping in HfO₂ can greatly reduce E_b and E_c as predicted by Qi's theoretical results.³⁸ Among the ferroelectric phases, ZrO₂ oIV-phase has the smallest E_c of 5.42 MV/cm. The E_c values of HfO₂ oIII-phase estimated in this study are larger than other theoretical estimations but in the same order.^28,32,38 Clima et al. estimated the E_c of HfO₂ oIII-phase to be 13.4 MV/cm according to the following equation: $E c= P 2 κ ε 0$⁠.²⁸ Here, $κ$ is the relative permittivity. Maeda et al. assigned Born effective charges on ions in simulations under external electric fields to determine the E_c of HfO₂ oIII-phase to be 15.5 MV/cm³² and Qi et al. got a similar value of 15.0 MV/cm with the same method.³⁸ However, the theoretical E_c values are all obviously larger than most of experimental observations ranging around 1–3 MV/cm.^{3–6,8–12,17,20,23,43,47,51} In this study, we found that shear strain will be an important factor to reduce E_c values of HZO other than Zr doping, which will be shown in the following discussion.

The not 90° α angles for both m-phase and oIV-phase indicate shear strain will stabilize these two phases and destabilize the oIII-phase and t-phase with right angle of α. Figures 3(a) and 3(b) show the structural transformations from oIV-like phase to m-like phase of HfO₂ and ZrO₂ with different α angles. It is found that oIII-like structure is the intermediate state in these transformations. In the modeling, the lattice constants are optimized for oIV-like phase at different α angles in order to have easier analysis of the remanent polarizations and coercive fields. The oIII-like and m-like phases are optimized oIII- and m-phase structures constrained with oIV-like phase lattice constants at different α angles except for the case of ZrO₂ oIII-like phase at 84° which could not be optimized in calculation but obtained in cNEB. Figures 3(c) and 3(d) show the variation of relative energies for oIII-like, oIV-like, and m-like phases with respect to α angle. For both HfO₂ and ZrO₂, oIII-like phase becomes more stable when α angle becomes larger, while m-like phase becomes less stable as expected. The HfO₂ oIV-like phase also becomes less stable when α angle becomes larger as expected but the energy variation is much smaller than that for oIII-like and m-like phases. On the other hand, the stability of ZrO₂ oIV-like phase is almost stationary within 84°–90° of α angle [Fig. 3(d)]. The oIV-like phase actually degenerates into t-phase in structure for both HfO₂ and ZrO₂ when α angle is 90°, which implies that the polarization of oIV-like phase will gradually become zero when α angle gets larger and larger.

The results shown in Figs. 3(a) and 3(b) indicate that there is a barrier for HfO₂/ZrO₂ oIV-like phase transforming into m-like phase when α angle varies from 84° to 90°. The smallest barrier is 19.8 meV/atom for HfO₂ and 14.9 meV/atom for ZrO₂ at α angle around 86°. The transformation curves, hence, suggest that once the ferroelectric orthorhombic phases (oIII- and oIV-like phase) have formed in HZO, the phase transition into m-phase by thermal agitation needs to overcome a barrier at least 14.9 meV/atom. The most possible path is transformation at α angle around 86°.

Shear strain will not only alter the relative stability among HZO phases but also affect the ferroelectric properties of oIII-like and oIV-like phases. Figure 4 shows the variations of P_r, E_b, and E_c values for oIII-like and oIV-like phases with varying α angle. For oIV-like phase, these three features decrease with α angle increases. At 90° of α angle, the E_b and E_c values are not able to be estimated since the oIV-like phase degenerates into t-phase [Figs. 4(b) and 4(c)]. Figure 4(a) shows that even at large α angle of 88°, P_r of oIV-like phase still maintains as high as 43.1 and 32.2 μC/cm² for HfO₂ and ZrO₂; however, the E_b, and hence, E_c values become much smaller than that of optimized HfO₂ and ZrO₂ oIV-phase with α angle of 83.9° and 85.9°. The E_c values are 3.44 and 1.04 MV/cm for HfO₂ and ZrO₂ oIV-like phase at α angle of 88°, which are very close to experimental observations.

On the other hand, the P_r values of HfO₂ and ZrO₂ oIII-like phase along [001] direction always keep in high values with some variations among different α angles [Fig. 4(d)]. To be noted, the P_r and E_b values of ZrO₂ oIII-like phase at 84° are missing because the structure could not be obtained in optimization for its high instability. The E_b and E_c values decrease with decreasing α angle for both HfO₂ and ZrO₂ oIII-like phase [Figs. 4(e) and 4(f)]. The E_b values for HfO₂ oIII-like phase are higher more than twice of ZrO₂ and correspondingly E_c values for HfO₂ oIII-like phase are much higher than ZrO₂. Unfortunately, the L–G fittings for HfO₂/ZrO₂ oIII-like phase with α angle smaller than 87° are not proper for getting positive $γ$ values in Eq. (1) by MATLAB⁶³ (see the supplementary material); only E_c values for α angles of 88° and 90° are shown in Fig. 4(f). The smallest E_c of oIII-phase is 6.5 MV/cm for ZrO₂ at α angle of 88°. Although shear strain does reduce the E_c value of HfO₂/ZrO₂ oIII-phase, E_c is still larger than most of the experimental observations. There could be other causes like oIV-like phase mediated or thermal energy assisted polarization switching as Qi et al. suggested.³⁸ If the oIII-like phase transforms into oIV-like phase by thermal energy, E_c will be much reduced at α angle of 88°. Other factors, such as HZO thickness, doping, defects, and substrate/HZO interface, should also affect the P_r, E_b, and E_c values for oIII-like and oIV-like phases since their effects on ferroelectricity and oIII-phase contents in the HZO film have been extensively investigated in the past decades.^{4–16,20–26,28,30,33–37,39,42} However, more systematic studies with extended models beyond the work by Wei et al.⁶⁴ need to be done to clarify how P_r, E_b, and E_c values are affected by the above-mentioned factors. We will continue to work on these subjects.

Most experimental reports recognized oIII-phase to be responsible for the ferroelectricity in the HZO film.^1,2 Few reports supported the rhombohedral phases contributing the ferroelectricity.^43,44 Huan et al. first proposed ferroelectric oIV-phase at 2014 via theoretical modeling,²⁷ but no experiment observed its existence until recently. Perevalov et al. reported the existence of oIV-phase in the La-doped hafnium oxide (La:HfO₂) film by comparing HRTEM image with simulation model.⁴⁵ Via theoretical modeling, Qi et al. found that with Zr substitution in HfO₂ to form Hf_0.5Zr_0.5O₂, the energy barrier for oIII-phase transforming to t-phase is lowered with respected to pure HfO₂, and oIV-phase becomes the intermediate state for polarization switching of oIII-phase via t-phase.³⁸ Antunes's theoretical work suggested the existence of oIV-phase by comparing the calculated XRD of HfO₂ phases with experimental GIXRD measurements of the HZO film under different Zr contents.⁴¹ Since the (121) and (12  $1 ¯$⁠) XRD peaks of HfO₂ oIV-phase are around 30° with 2.3° separation which are similar to m-phase (111) and (11  $1 ¯$⁠) XRD peaks with 3.4° separation, the experimental observed two shoulders around 30° [main peak is recognized as the (111) peak of oIII-phase or (011) peak of t-phase] in XRD could be resulted from oIV-phase. Our results in Figs. 3 and 4 further suggest oIV-phase serves as a buffer state to maintain the ferroelectricity of the HZO film under different structural distortions and, hence, oIV-phase could exist as a small portion in the HZO film.

According to Fig. 3, oIII-like phase is more stable than oIV-like phase when α angle larger than 87° for both HfO₂ and ZrO₂. Meanwhile, the transition from oIII-like phase to m-like phase is hindered by a barrier larger than 21.2 meV/atom for α angle larger than 87°. Although oIII-like phase is further destabilized for α angle smaller than 86°, the structure will very possibly transform into oIV-like phase instead of m-like phase with thermal agitation since the barrier for transforming into oIV-like phase is much smaller than that for transforming into m-like phase. At the same time, the ferroelectricity of the HZO film could be maintained according to the P_r estimation of oIV-like phase [Fig. 4(a)] but with some changes in overall magnitude and direction of P_r. The transition from oIII-like to m-like phase most likely happens at α angle around 86° by thermal agitation at high temperature but it has lower probability than transforming into oIV-like phase. Transition to m-like phase becomes even more suppressed for further smaller α angle. Hence, oIV-phase is like a buffer state preventing oIII-phase from transforming into m-phase when simultaneously destabilized by shear strain and thermal energy.

In some experimental reports, there were shoulders shown on the two sides of the oIII-phase (111) peak in XRD of the HZO film which also had good remanent polarizations and clear P–E loops.^{8,16,23,46,47} The shoulders could be the evidence of oIV-phase,⁴¹ but it needs more careful analyses to clarify. On the other hand, the structural distortion caused by shear strain is rarely analyzed in experiments until a recent work by Liao et al.²⁶ The FFT analysis of one HRTEM image was attributed to oIII-phase but having 88° between (011) and (⁠ $1 ¯$ 00) directions,²⁶ which clearly indicates the HZO lattice is distorted with shear stain. According to Fig. 3, oIV-like phase is more stable than oIII-like phase when α angle is smaller than 86°; however, m-like phase also becomes more stable than both oIV-like and oIII-like phase under the same condition. Hence, to distinguish oIV-phase from m-phase via HRTEM or HAADF-STEM would need careful analysis on the regions with lattice tilt angle smaller than 86°.

Based on the calculations in this work, HZO oIV-phase (Pmn2₁) serves as a buffer state preventing oIII-phase (Pca2₁) transforming into m-phase (P2₁/c) when the lattice suffered distortion of shear strain. The remanent polarization of HZO could be maintained to some certain degree when oIII-phase transforms into oIV-phase with α angle deviates from 90°. The coercive field of the HZO film would be reduced due to the shear strain on oIII-phase or transformation into oIV-phase. Our study would ignite more careful analyses to verify the existence of oIV-phase.