Triggered Ferroelectricity in HfO2 From Hybrid Phonons and Higher‐Order Dynamical Charges
Seongjoo Jung, Turan Birol
TL;DR
This paper reveals a new mechanism of ferroelectricity in HfO2, showing how polarization arises from complex interactions between structural modes without traditional instabilities.
Contribution
The paper introduces a novel 'hybrid-triggered' ferroelectric mechanism in HfO2, enabled by trilinear and quadlinear couplings between stable polar and nonpolar modes.
Findings
Hybrid-triggered ferroelectricity in HfO2 is enabled by trilinear coupling without structural instabilities.
High-order couplings of nonpolar phonons significantly contribute to polarization in HfO2.
The findings clarify the origin of ferroelectricity in fluorite-related structures.
Abstract
Ferroelectric HfO2 has emerged as a highly promising material for high‐density nonvolatile memory and nanoscale transistor applications. However, the uncertain origin of polarization in HfO2 limits our ability to fully understand and control its ferroelectricity. Ferroelectricity, the emergence of a spontaneous and switchable polarization in solids, is conventionally understood to be governed by unstable structural modes (phonons), arising either directly from an unstable polar phonon or indirectly through coupling of unstable nonpolar phonons with a polar mode. While these “proper” and “improper” mechanisms successfully explain ferroelectricity for most systems, they do not encompass all possible phenomena. Here, we present a novel mechanism of “hybrid‐triggered” ferroelectricity, where a polar order emerges through trilinear coupling without any structural instabilities. Our group…
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FIGURE 6- —Office of Naval Research10.13039/100000006
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Taxonomy
TopicsFerroelectric and Negative Capacitance Devices · Ferroelectric and Piezoelectric Materials · Semiconductor materials and devices
Introduction
1
The origin of ferroelectricity is typically classified as either proper, where polarization arises from an unstable polar structural mode [1] (phonon), or improper, where polarization is induced by one or more unstable nonpolar modes through their coupling with a polar mode [2, 3]. Both mechanisms rely on the existence of a phonon instability. Order parameters associated with unstable modes are termed primary, while those emerging through coupling with these modes are referred to as secondary. Phenomenological theories of ferroelectrics that focused on mechanisms involving unstable phonons have been widely successful in explaining and predicting materials' properties [4, 5, 6].
However, a system where such an approach has not been as successful is the ferroelectric HfO2‐based thin‐films [7]. HfO2 is often dubbed the “ferroelectric of the future,” and has many properties that make it suitable for applications, including compatibility with existing silicon technology [8, 9, 10, 11, 12, 13]. While the ferroelectric phase of HfO2 has been most commonly identified with the orthorhombic space group Pca2 (#29), the origin of ferroelectricity in this phase remains elusive. This lack of understanding limits the ability to effectively control unfavorable properties, such as the high coercive field, slow domain wall propagation and formation of non‐ferroelectric phases [14, 15, 16, 17].
HfO2 exhibits characteristics that both align with and challenge aspects of both proper and improper mechanisms. The absence of a polar structural instability, specifically the Γ4− mode, suggests that the ferroelectricity in fluorite HfO2 (Fm3¯m, #225) is improper in nature [9, 18]. Several other factors also support this theory. While the Born effective charges are anomalous (+38%) they are not as enhanced compared to typical proper ferroelectric oxides [4, 19, 20, 21], and depolarization field does not effectively suppress the polarization [8, 18, 22]. Also, out‐of‐plane polarization is induced when an in‐plane tensile strain is applied [23]. Together with the negative piezoelectric response, they suggest strong interactions between polar and nonpolar modes. However, the couplings necessary for conventional improper ferroelectricity are forbidden in fluorite structure [24].
This observation led to proposal of proper ferroelectricity in HfO2. Notably, HfO2‐based thin‐films are not reported to encounter the same challenges in electric switching associated with improper ferroelectrics [25], and a drastic dielectric enhancement is observed near the ferroelectric transition temperature [26]. Alternative pathways to ferroelectricity have been suggested, involving lower symmetry parent structures such as Pcam (#57) [27, 28], Pcnb (#60) [29], Cmme (#67) [30] or P4/nmc [31] phases. However, the explanations of proper ferroelectricity in HfO2 requires one or more of the following assumptions, each with certain limitations [32]: a higher energy parent structure, a substantial strain, or polar phases that differ from the experimentally observed Pca2 phase. Importantly, the proposed mechanisms for proper ferroelectricity do not fundamentally align with the antiferroelectric‐ferroelectric continuity [7, 18, 33] in fluorite systems. Instead, they predict paraelectricity when ferroelectricity is suppressed, and does not account for change in energy curvature at a polarized state [34].
There are other explanations for ferroelectricity in HfO2 that do not fit into the conventional categories of proper or improper ferroelectrics. One notable mechanism is by the negative biquadratic coupling between polar (Γ4−) and nonpolar modes (X5,y+) [35]. A large tensile strain along one of the in‐plane axes is reported to induce the nonpolar mode instability, which in turn, leads to a polar instability through the biquadratic coupling [24]. A trilinear coupling of modes derived from the parent phase P4/nmc was also identified as a possible origin of ferroelectricity [36]. The polar phase of ZrO2, which shares the same symmetry, has been also investigated and attributed to non‐linear interactions between the X5+ modes, represented by a spline function [37].
While these studies offer valuable insight, there is no consensus on a model that explains the ferroelectricity in Pca2 HfO2. In this study, we employ symmetry‐guided expansions of energy and polarization by Landau‐Ginzburg‐Devonshire (LGD) theory [38] with first‐principles density functional theory (DFT) involving all modes in Pca2 HfO2, starting from the highest‐symmetry structure. Our results reveal two key, previously unreported findings. First, ferroelectricity in HfO2 is driven by what we dub “hybrid‐triggered” mechanism involving multiple trilinear and quadlinear couplings of stable modes. Second, the hybrid zone‐boundary modes—combinations of multiple nonpolar modes—exhibit exceptionally high polarization, contributing over −40% of the bulk value and offering novel properties from previously unexplored interplay between dipoles.
Results and Discussion
2
Mode Decomposition and Symmetry‐Allowed Couplings
2.1
We begin by analyzing the crystal structure of the polar phase of HfO2 (orthorhombic Pca2) by decomposing the difference between it and the high‐symmetry reference structure (fluorite Fm3¯m) into the symmetry‐modes. There are eight relevant modes, which are labeled by the irreducible representations (irreps) of the parent space group as shown in Figure 1 [37, 39]. Note that the biaxial strain imposes a tetragonal I4/mmm (#139) parent phase, but we reference irreducible representations from the cubic phase for direct connection with existing literature [37]. Irreducible representations from I4/mmm parent phase and corresponding displacement patterns can be found in Table S1 (Supporting Information).
Modes and unique couplings in fluorite HfO2. (a) Eight modes observed in the ferroelectric Pca2 phase of HfO2. Blue arrows and signs represent ionic displacements in the front half of the unit cell (x=0.75 in direct coordinates for oxygen), and the pink arrows represent those in the rear half (x=0.25). (b) Unique 2nd, 3rd, and 4th‐order couplings among modes. Modes that can condense in the dielectric phase are highlighted in green, whereas modes that condense in the triggered phase are shown in red.
P0 denotes the order parameter of the polar zone‐center mode (Γ4−), while the remaining parameters, Qi, correspond to nonpolar zone‐boundary modes. Q4 represents the only unstable mode (X2,x−) which has non‐zero amplitude in the centrosymmetric Aeaa (#68) phase. The Aeaa phase is a more likely candidate for the nonpolar structure of HfO2 under tensile strain than the frequently assigned P4/nmc (#137) phase according to the energies from DFT (Figure S1, Supporting Information), consistent with the experimental observations that the long axis of the nonpolar phase has to point toward in‐plane direction for the formation of out‐of‐plane polarization [23] and changes in interplanar spacing between polar and nonpolar phases [8, 18]. The Aeaa phase is also more stable compared to Pcnb (#60) phase derived from P4/nmc across the entire range of biaxial strain.
The wavevectors of the zone‐boundary modes are essential for understanding the couplings (Figure S2, Supporting Information). These wavevectors are not necessarily parallel to the direction of ionic displacements in real space; for instance, the ionic displacements of Q1 are along the z direction, while the wavevector points to the y direction. Unlike previous studies that label the modes by ionic displacement directions [9, 24, 37], we label them by their wavevectors.
Combinations of order parameter components that remain invariant under all symmetry operations of the space group (and hence transform as the Γ1+ irrep) constitute the most general invariant polynomial. Figure 1b lists the unique mode couplings that are present in this polynomial. Only terms derived from these couplings contribute to the LGD energy in its Taylor expansion around the high‐symmetry phase. Consequently, the energy landscape of ferroelectric Pca2 HfO2 can be represented using these terms up to a sufficient order, and insight about the mechanism behind ferroelectricity can be extracted from it. The second‐order terms are limited to simple quadratic or bilinear terms of the same irreps. In the third order, there are only trilinear terms that can be divided into two forms: ΓXαXα couplings (involving one Γ‐point mode and two X‐point modes of the same wavevector) and XxXyXz couplings (involving three X‐point modes of different wavevectors). Similarly, quadlinear terms include ΓXxXyXz and XαXαXα′Xα′ categories. While there are other fourth order terms, they do not play an important role in reducing the symmetry, and hence we do not list them. All structural order parameters in Pca2 HfO2 participate in odd‐powered couplings with order parameters of other modes and must vanish in the reference structure [32].
Minimal LGD Framework for Hybrid‐Triggered Ferroelectricity
2.2
With these insights, we now turn to investigate the microscopic origin of ferroelectricity in HfO2. To demonstrate the mechanism by which trilinear and quadlinear couplings give rise to ferroelectricity, we first introduce a simplified, hypothetical model describing the electric enthalpy H [34, 40] in terms of a polar (along c) and two nonpolar modes p0 and qi, all of which are stable. These quantities are distinct from the actual structural modes P0 and Qi used to characterize HfO2, but we consider the case where there is a trilinear coupling between them, similar to the case in HfO2. This illustrative example captures the fundamental mechanism through which the trilinear coupling between between stable modes can generate a polar response. Building on this conceptual framework, we will extend the analysis to the real case of HfO2, demonstrating that the same mechanism operates among its actual structural modes.
The quantities E, Ω, E, and P denote the energy, unit‐cell volume, electric field, and polarization along the crystallographic direction c, respectively. We expand the energy in terms of order parameters and adopt the reduced field variables [40]. v represents the voltage across a unit cell along c, and λ corresponds to the mode effective charge [41] associated with the polar mode, normalized by the lattice constant c. We consider the case where all coefficients are positive, thus there is no unstable mode that can give rise to ferroelectricity. The sign of γ does not matter, since it is determined by the origin choice and the definition of the zone boundary order parameters.
Minimizing H at fixed v is equivalent to imposing a boundary condition of a fixed voltage applied along the c axis. The solutions of the following system of equations determine the equilibrium state of the material under the applied voltage, expressed in terms of the order parameters p0, q1, and q2:
We are interested in the existence of nontrivial solutions of this system, beyond the trivial case q1=q2=0. To this end, one may either solve ∂H/∂q1=0 for q1 as a function of p0 and q2, or alternatively solve ∂H/∂q2=0 for q2 as a function of p0 and q1. Adopting the latter approach yields
Substituting this expression for q2 into ∂H/∂q1=0, we obtain an effective condition involving only p0 and q1:
The stability of the q1=0 solution is determined by the sign of ∂2H/∂q12. If this quantity is negative at q1=0, condensation of q1 lowers the free energy, indicating the spontaneous condensation of the hybrid mode q1q2. Conversely, a positive curvature implies that the mode remains inactive. Accordingly, the onset of the hybrid‐mode instability is identified by the condition under which ∂2H/∂q12 evaluated at q1=0 becomes negative:
This criterion defines the critical value of the order parameter p0, denoted p0,c, at which the instability occurs:
An illustrative solution of Equation (1) is presented in Figure 2a. Below |p0,c|, the system behaves as a dielectric with no contribution to energy from the trilinear coupling. This region is shown with blue lines. However, once |p0| exceeds |p0,c| by application of a voltage, the hybrid mode q1q2 becomes unstable inducing a phase transition to the polar phase where both q1 and q2 condense simultaneously (shown in red lines)—similar to the “avalanche” transition in Aurivillius compounds [42], and the predicted transitions by quadratic‐linear couplings [43]. The simplest explanation to this phenomenon is that γp0 acts as a tunable coefficient for second‐order hybrid mode term q1q2. We refer to the pre‐trigger phase as the dielectric phase and the post‐trigger phase as the “triggered” phase.
LGD model and DFT validation of hybrid‐triggered ferroelectricity. (a) Solution of the theoretical LGD model for hybrid‐triggered ferroelectricity described by Equation (1), formulated in terms of hypothetical order parameters p0, q1, and q2. (b) Same as part (a), but for Equation (7). The inclusion of an additional trilinear coupling term, q1q3q4, significantly reduces the critical polarization required to induce the triggered ferroelectric transition. (c) DFT‐calculated energies of HfO2 as functions of the structural order parameters P0, Q1, and Q5, providing first‐principles validation of hybrid‐triggered ferroelectricity. The amplitudes are constrained such that Q1/Q5 is 4.4 (Figure S4, Supporting Information). Results are shown for cases where all other order parameters are constrained to zero (left) and where they are fully relaxed (right), revealing a substantial reduction of the critical polarization P0,c from 0.49 Å to 0.17 Å.
Triggered Ferroelectricity versus Improper Ferroelectricity
2.3
A key distinction exists between improper and triggered ferroelectricity, though both arise from couplings between polar and nonpolar modes. Improper ferroelectricity involves unstable nonpolar modes (demonstrated by imaginary phonon frequency or negative quadratic coefficient of the order parameter)—either single or hybrid [4, 6, 44]—coupled to the polar mode. As a result, polarization switching in improper ferroelectrics requires changing the direction of a nonpolar mode present in the structure, which introduces an energy barrier that does not directly couple with applied voltage (Figure 3). Triggered ferroelectricity, by contrast, does not rely on unstable modes [35]. It arises from couplings of stable nonpolar modes with the polar mode (Figure S3, Supporting Information). When polarization surpasses a critical threshold due to applied voltage, hybrid modes condense and the system moves to a different global minimum of energy. Unlike either proper and improper ferroelectricity, the mechanism underlying triggered ferroelectricity requires no additional instabilities.
Coherent switching barrier of hybrid‐triggered and hybrid improper ferroelectrics. Example of an energy contour diagram from LGD model of (a) Hybrid‐triggered ferroelectrics from Equation (1) and (b) Hybrid improper ferroelectrics. The former switch though a barrier nonparallel to p0 axis, while the latter encounter an intrinsic energy barrier parallel to the p0 axis associated with the switching of either one of the nonpolar modes, which does not directly couple with voltage.
This fundamental difference renders triggered ferroelectrics more useful for applications than improper ferroelectrics. For example, the energy barrier associated with switching of a nonpolar order parameter is absent in triggered ferroelectricity. The triggered mechanism also aligns with the observed dielectric increase upon heating, akin to a first‐order transition between ferroelectric and dielectric phases [43]. Notably, this is the only mechanism proposed so far that is compatible with the continuous evolution of antiferroelectricity to ferroelectricity in ZrO2‐based thin‐films. P‐v plot of Figure 2a shows existence of two distinct critical voltages, vd derived from the dielectric phase, and vt from the triggered phase. In the case where transition to the dielectric phase during polarization switching is merely transient (vd<vt), it results in a ferroelectric hysteresis. Conversely, if the dielectric phase persists and manifests as a non‐transient state during polarization switching (vd>vt), it leads to the formation of an antiferroelectric‐like double hysteresis.
First‐Principles Verification and Additional Couplings
2.4
To connect the minimal LGD framework introduced above with the case of HfO2, we now explicitly map the abstract order parameters of Equation (1) onto the symmetry‐adapted modes from section 2.1. In this mapping, the polar variable p0 corresponds to the zone‐center polar mode P0, while pairs of nonpolar variables (q1,q2) represent symmetry‐allowed combinations of zone‐boundary modes (Qi,Qj) that participate in trilinear couplings of the form P0QiQj. This correspondence enables a direct test of the hybrid‐triggered mechanism using first‐principles calculations.
The modes P0, Q1, and Q5 in HfO2 exhibit symmetry properties consistent with those predicted by Equation (1). (Analogous symmetry relationships are also found for P0Q3Q6 and P0Q3Q7; the P0Q1Q5 trilinear coupling is selected here merely as a representative example. As will be shown, each of these coupling terms contributes to a single, unified ferroelectric mechanism.) We set all other displacements to zero and compute the crystal's energy as a function of the hybrid mode Q1Q5, maintaining a fixed ratio between Q1 and Q5, while P0 is held at progressively increasing values. The resulting energy plots are displayed in the left panel of Figure 2c, where the critical value for triggered ferroelectricity is |P0,c| is 0.49 Å. When P0 is below this threshold, the hybrid mode increases the energy of HfO2. For P0 exceeding 0.49 Å, the energy of HfO2 begins to decrease as a function of increasing hybrid mode order, revealing the hybrid mode instability.
Nevertheless, attributing the emergence of ferroelectricity to a single trilinear coupling term, P0Q1Q5, proves insufficient, as the calculated critical value |P0,c| exceeds the spontaneous polar distortion in the ferroelectric Pca2 phase, |P0,s|=0.40 Å. This discrepancy arises because the trilinear and quadlinear couplings that participate in the triggering mechanism of HfO2 are interconnected in a manner that does not produce multiple, independent triggering events. Instead, these couplings act cooperatively to reduce the critical threshold of a single, unified trigger.
This behavior can also be illustrated within the LGD framework. As an example, consider an extension of Equation (1) in which an additional trilinear coupling term q1q3q4 is introduced, together with higher‐order terms to ensure stability:
Here, q4 corresponds to the order parameter of an unstable mode, implying the condition β4<0. The couplings p0q1q2 and q1q3q4 each involve one mode that condenses in the dielectric phase (p0, which couples directly to the applied voltage, and q4, which is intrinsically unstable) and two modes that condense only in the triggered phase (q1q2 and q1q3). Crucially, both couplings share a common component that remains inactive prior to triggering, namely q1. A representative solution of Equation (7) is shown in Figure 2b. Compared to Equation (1), this extended model exhibits a unified triggering event in which p0, q1, and q2 condense simultaneously at a lower critical polarization than that predicted by Equation (6). The detailed derivation is provided in the Supporting Information.
An analogous situation occurs in HfO2, where all relevant couplings collectively drive a single transition in which every mode that is inactive in the dielectric phase condenses at a unified and reduced critical value |P0,c|. Figure 1b illustrates how nearly all trilinear and quadlinear couplings contribute to the triggering mechanism in HfO2, with the exceptions of P0Q2Q4, Q1Q3Q5Q6, and Q1Q3Q5Q7. In the right panel of Figure 2c, we present DFT energies computed for the same fixed values of P0, Q1, and Q5 as in the left panel, but now allowing all other order parameters to relax to nonzero values. This relaxation leads to a substantial reduction of the critical polarization, lowering P0,c from 0.49 Å to 0.17 Å.
Higher‐Order Dynamical Charges
2.5
Even after addressing the roles of multiple trilinear and quadlinear couplings, an important aspect of this transition remains unexplained. As demonstrated in previous studies [24, 27], when the DFT energy of HfO2 is accurately plotted as a function of the polar order parameter, a cusp emerges in the curve. This cusp is does not appear in our demonstrations of hybrid‐triggered ferroelectricity so far, as it is not a feature necessarily derived from the mechanism itself. A major contribution to this energy cusp is from an underappreciated factor which also connects to the origin of strong trilinear couplings in HfO2: the polarization arising from hybrid modes which are nonpolar by themselves. The hybrid modes Q1Q5, Q2Q4, Q3Q6, and Q3Q7, which couple with the polar mode in HfO2, break all the inversion symmetries present in the crystal and transforms exactly as the polarization itself (Γ4−) (Table S2, Supporting Information).
The broken inversion symmetry from nonpolar modes is a well‐established characteristic of hybrid improper ferroelectrics [44, 45]. In such systems, the coupling between two nonpolar modes typically induces a nonzero amplitude of the polar structural order parameter, which serves as the dominant source of polarization. However, HfO2 deviates from this conventional picture. In Figure 4, the ground state structures of the proper ferroelectric LiNbO3 (R3c, #161), the hybrid improper ferroelectric Ca3 Ti2 O7 (Cmc2, #36), and HfO2 (Pca2, #29) are each decomposed into two components: one arising from distortions from polar modes and the other from nonpolar modes. Each component is interpolated with its respective high‐symmetry parent phase (Pm3¯m, I4/mmm, and Fm3¯m), and the resulting polarization is plotted. As expected, the nonpolar distortion of LiNbO3 produces no polarization, while the hybrid improper Ca3 Ti2 O7 exhibits only a minute polarization induced through the trilinear coupling that enables its ferroelectricity.
Higher‐order dynamical charge arising from nonpolar modes in HfO2. Interpolation between the high‐symmetry reference structure and the polar ground state configuration, decomposed into polar and nonpolar mode contributions. HfO2 exhibits a substantial higher‐order polarization component originating from nonpolar modes (accounting for approximately –41% of the bulk response), in sharp contrast to the proper ferroelectric LiNbO3, where such contributions are symmetrically forbidden (0.0%), and to hybrid improper ferroelectric Ca3 Ti2 O7, where they are allowed but negligible (0.4%).
In contrast, the interpolated structures of HfO2 exhibit a distinctly different behavior, highlighting the critical influence of both trilinear and quadlinear couplings on the emergence of its ferroelectricity. The nonpolar distortions produce a nonlinear polarization response, contributing up to approximately –41% of the total bulk polarization. These results indicate that, unlike in conventional improper or hybrid improper ferroelectrics, the direct coupling of hybrid modes–or, equivalently, higher‐order dynamical charge interactions with the electric field—together with local electrostatic effects, plays a decisive role in the ferroelectric behavior of HfO2. This interpretation is further supported by recent reports of ion motion that reverses the direction of the simulated electric field [46, 47, 48]. a phenomenon that cannot be simply explained within the traditional framework of linear dynamical charges (Born effective charges).
Accordingly, a new term representing the electronic polarization induced by the hybrid mode must be added to Equation (1), such that λp0→λp0+μq1q2.
Here, μ is derived from second‐order mode effective charge [49], and corresponds to the second derivative of polarization with respect to nonpolar atomic displacements. Such a variable is not commonly introduced before in the context of ferroelectricity. Solution of the resulting equation with the same coefficients as Equation (1) is shown in Figure 5a. The polarization of the hybrid mode is against the direction of the polar mode and increases the value of p0,c. This addition generates the cusp in the energy versus polarization plot and reveals more aspects of the phase transition. Not only are there overlapping regions of polarization in the transition pathway, but also the voltage and the order parameters in these overlapping regions are discontinuous, resulting in a disruptive phase transition where all the order parameters abruptly shift as polarization increases.
The effect of polar hybrid modes on triggered ferroelectricity. (a) Plots for a hybrid‐triggered ferroelectric material with polar hybrid mode, including energy‐polarization (top left), polarization‐voltage (bottom left), order parameters‐polarization (top right), and 2D energy‐p0,P (bottom right) modeled with additional hybrid mode contribution to polarization to Equation (1). The triggered ferroelectric phase is separated from the dielectric phase in the 2D energy plot, introducing an additional layer of hysteresis and a disruptive phase transition. (b) Coherent switching pathway of HfO2 via the hybrid‐triggered mechanism. Five order parameters that remain zero in the dielectric phase condense together at Pc=0.30 C m−2 disruptively.
The energy plotted as a function of p0 and P provides further insight into this transition. (Figure 5a, bottom right). When μ is zero, the data in this plot is restricted to a 1D line, as P cannot assume a value different from λp0, and vice versa. However, when μ is non‐zero, P and p0 are independent. Two distinct local minima emerge in this plot: the dielectric phase where the hybrid mode is inactive, and the triggered phase where they condense. As μ increases, the triggered phase's local minimum separates more from the P=λp0 line, eventually disconnecting entirely from the dielectric phase's local minimum. Thus, a slight increase in polarization above Pc causes a sudden jump in all order parameters, including p0. The pathway from the triggered phase to the dielectric phase differ from the pathway from the dielectric phase to the triggered phase as the local minima are completely separated, leading to overlapping polarization regions and additional hysteresis in the switching pathway.
Full Energy Landscape and Coherent Switching Pathways
2.6
We apply these insights to HfO2 under 1% epitaxial tensile strain to map the complete coherent ferroelectric phase transition pathway. Note that switching in the real material does not necessarily happen in a coherent, homogeneous way, and what we discuss here is not an exact reproduction of factors such as domain wall motion which likely affect the switching energy barriers [50]. However, what we present is a complete energy landscape of most commonly observed Pca2 HfO2 that involves all possible parent structures of Pca2 which are also a subgroup of I4/mmm (strained Fm3¯m) within the order parameter space. This understanding is crucial for interpreting the complex nature of real switching behavior and domain boundaries, since even materials with intricate domain structures exhibit coherent switching in the local limit.
We derive the symmetry‐adapted expansion of the energy up to fourth order and of the polarization up to second order (Equation (S3), Supporting Information) for HfO2, and fit these expressions to DFT‐computed energy and polarization data sampled over 8D grid in order parameter space around the I4/mmm and Pca2 phases of HfO2 (Figure S4). The resulting solutions obtained from the fitted coefficients are presented in Figure 5b. The critical polarization is 0.30 C m−2. In the dielectric phase without applied voltage, only Q4 is non‐zero; upon applying voltage, P0 and Q2 condense. At 0.47 V/unit cell (u.c.), ferroelectricity is triggered, causing a simultaneous jump in five other order parameters. The voltage needed to switch polarization is higher than the triggering voltage, reaching about 1.4 V/u.c.. The separation of two local minima is clearly visible in the 2D energy landscape. Note that the proper ferroelectric switching pathway via the Pcam (#57) phase is also observed from this model, but requires higher voltage of 2.3 V/u.c., and is geometrically restricted in non‐periodic boundary conditions. Thus, it is reasonable to conclude that HfO2 will switch through the Aeaa phase via the hybrid‐triggered mechanism under homogeneous conditions.
Origin of Polar Hybrid Modes from Nonpolar Modes
2.7
Finally, one critical question remains: what makes the hybrid modes so polar? The answer lies more in the fluorite structure itself rather than in a unique property of the Hf‐O bonds. In fact, the Hf‐O bond is relatively simple electronically due to the high electropositivity of Hf, compared to bonds such as Ti‐O which are known to enable ferroelectricity in perovskites through hybridization [51]. Although the Hf–O bond is mostly ionic, it gains more covalent character as the Hf–O bond length decreases. This results in bond‐to‐bond charge transfer associated with ionic displacements in each mode [49]. Figure 6a shows a (010) plane of oxygens (y=0.75) and surrounding Hf atoms in the conventional cell. When ions displace according to −Q4 or Q3, charge transfers toward +x between the Hf–O bonds in the upper half, and toward −x in the lower half. Figure 6b shows this charge transfer from DFT, in the form of cuts of charge density change on Hf–O planes. The opposing direction of charge transfer between bonds results in a overall nonpolar structure.
The origin of polar hybrid modes in fluorite structure. (a) A (010) plane of oxygens (y=0.75) and surrounding Hf atoms in fluorite HfO2 unit cell with gradient‐colored arrows indicating charge transfer from the elongated Hf–O bonds (blue) to the shortened bonds (red). (b) Charge transfer resulting from the −Q4 or Q3 mode around highlighted oxygen ion. The plane containing the two bonds directed toward +x shows charge loss (left), whereas the plane containing the two bonds directed toward −x shows charge gain. (c) Charge transfer resulting from the Q2 or −Q7 mode starting from condensed −Q4 or Q3 mode. There is minimal charge transfer between the two extended bonds in the +x direction (left), while there is significant charge transfer between the two shortened bonds in the −x direction resulting in net polar structure.
However, if ions then displace according to Q2 or −Q7 starting from the distorted structure by −Q4 or Q3, charge is transferred upward around each oxygen ion in the crystal, as the charge transfer from the shortened bonds surpasses that from the elongated bonds (Figure 6c). This results in a significant downward polarization purely from electronic dipole effects, without any ionic contribution [49]. This aligns with the observation that the Born effective charges are dramatically different in different phases [21]. The nontrivial polarization in this system originate less from the first derivatives of polarization with respect to displacements, but more from the second derivatives. Also this purely electronic dipole, coupled with an oppositely oriented ionic dipole, can enhance the stability of ferroelectricity in ultra‐thin HfO2, as the depolarization field from one polarization source stabilizes the other, further strengthening the trilinear interaction.
Conclusion
3
Our findings transform the traditional interpretation of polarization in oxides as a linear function of charged ions' displacements, especially in HfO2. HfO2 was thought to consist of nonpolar spacer layers and polar layers, based on the displacement of oxygen ions. We instead show that HfO2 can exhibit substantial polarization Pz even in the absence of ionic displacements along z. We term this newly identified mechanism of ferroelectricity, which arises from strong multilinear couplings among stable modes, hybrid‐triggered ferroelectricity. This phenomenon is conceptually analogous to triggered ferroelectricity [35], in the same way that hybrid improper ferroelectricity [44] relates to improper ferroelectricity [52]. These insights resolve longstanding debate regarding the ferroelectric behavior of thin‐film HfO2, and also establish a conceptual framework for designing next‐generation ferroelectric materials and devices.
Computational Methods
4
Periodic density functional theory (DFT) calculations were performed with the Vienna ab initio simulation package (VASP) 6.4.1 [53]. Spin‐orbit coupling as well as spin‐polarization was tested to contribute no qualitative difference to the system. A generalized gradient approximation (GGA) exchange‐correlation functional PEBsol was used [54]. The projector augmented wave (PAW) method was used to describe atom cores and the plane wave basis set was expanded to a kinetic energy maximum of 520 eV for Kohn–Sham orbitals, using valence configurations 5p65d36s1 for Hf and 2s22p4 for O. Tetrahedron smearing with Blöchl corrections imposed on electrons at the Fermi level. VESTA software [55] was used for building and visualizing crystal structures. More details on the computational method can be found in Supporting Information.
Author Contributions
S.J. conceived the idea, designed, performed the calculations and analyzed data. T.B. supervised the research. S.J. prepared the manuscript. S.J. and T.B. edited the manuscript.
Conflicts of Interest
There authors declare no conflicts of interest.
Supporting information
Supporting File: adma72722‐sup‐0001‐SuppMat.pdf.
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