Review on ferroelectric/polar metals

As a prototypical ferroelectric with a simple ABO₃ formula, BTO has played a significant role in understanding the origins of ferroelectricity as well as polar metals. In its bulk form, BTO is an insulator with a bandgap of 3.2 eV.⁵⁹⁾ It undergoes a series of phase transitions from high-temperature paraelectric cubic to ferroelectric tetragonal, orthorhombic and rhombohedral phases at 403, 287 and 197 K, respectively.²⁷⁾ The paraelectric-to-ferroelectric phase transition at the 403 K Curie temperature (T_c) is characterized by the evolution of the TO F_1u soft phonon mode.⁶⁰⁾ Although the electrical transport properties have been studied for decades in semiconducting n-type BTO,⁶¹⁾ it was not until 2008 that Kolodiazhnyi et al. reported the metallic behavior in n-type BTO when the carrier density exceeds 10²⁰ cm⁻³ by oxygen vacancy doping.²⁸⁾ Based on the observation that the low-symmetry phases of n-type BTO are preserved up to a critical carrier concentration of n_c ≈ 1.9 × 10²¹ cm⁻³, Kolodiazhnyi et al. proposed n-type BTO as a "ferroelectric" metal [Fig. 1(a)].²⁷⁾ This claim is supported by the theoretical work by Wang et al., who found that the polar displacements and the soft phonon mode in n-type BTO can be sustained up to a critical carrier density of 0.11 electron per unit cell (e/uc),¹²⁾ which is around 2 × 10²¹ cm⁻³. The persistence of polar distortions is attributed to a short-range portion of the Coulomb force (∼5 Å) with an interaction range of the order of the lattice constant, i.e. the Coulomb force is incompletely screened in this range.¹²⁾ However, Zhao et al. recently pointed out that this short-range interaction is not a portion of the Coulomb force but a force arising from the so-called "meta-screening effect", which is triggered by local lattice response accommodating the screening electrons.³⁹⁾ Furthermore, this "meta-screening effect" is proposed to be largely universal in other ferroelectric oxides, such as PbTiO₃ (PTO) and BiFeO₃ (BFO).³⁹⁾ As noted above, the polarity and conductivity in bulk n-type BTO have been found to exist in different phases by neutron scattering measurements.²⁹⁾ However, recent scanning transmission electron microscopy (STEM) measurements reveal that polarity and conductivity can coexist in a single phase in "ferroelectric" Ba_0.2Sr_0.8TiO₃ thin films³⁴⁾ [Figs. 1(b)–1(d)] and proximity-induced "ferroelectric" SrTiO₃ thin films.³³⁾ In addition, the critical electron density in BTO can be increased by compressive strain, improving the functionality of BTO-based polar metals.⁶²⁾

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While the ferroelectricity in BTO is stabilized by the long-range Coulomb forces and the hybridization between Ti 3d and O 2p orbitals, there is another driving force for the ferroelectricity in another class of ferroelectrics, which involves the displacement of the A-site atoms in addition to/instead of that of the B-site atoms as in BTO.^13,63,64) For example, in PTO, in addition to the Ti displacements, the Pb atoms are also displaced and contribute a significant portion to the total polarization.^13,63,64) The Pb displacement is due to hybridization between the Pb (6s, 6p) bands and O 2p bands, which reduces the short-range repulsions and enhances polarization.^13,63,64) This hybridization is called the lone-pair mechanism. It has been found that although itinerant electrons can effectively screen the long-range Coulomb forces, the lone-pair-driven A-site polar distortion is not strongly affected or even enhanced by electron doping.³⁵⁾ This is because the electronic states corresponding to the lone-pair mechanism are away from the Fermi energy if electrons are doped, and the bottom of conduction bands in these materials are often the B-site states.³⁵⁾ Thus the doping of electrons can be seen as a selective enlargement of the B-site ion radius, which stretches the A–O bonds.³⁵⁾ This lone-pair-driven persistence of polarity has also been found in BFO.³⁷⁾ It may also be applicable to other lone-pair-driven ferroelectrics, such as PbVO₃,⁶⁵⁾ SnTiO₃⁶⁶⁾ and BiMnO₃.⁶⁷⁾

Although the polar distortions in both A-site and B-site perovskites discussed above can be sustained upon itinerant carriers doping, they originate from electrostatic forces, which are susceptible to electrostatic screening by itinerant charge carriers. So one may ask if there is a non-electrostatic mechanism to sustain the polar distortions. Next, we review progress in another type of perovskite polar metal due to a (non-electrostatic) geometric-lattice mechanism, which is responsible for the polar metallic state found in layered perovskite (Sr, Ca)Ru₂O₆³⁰⁾ and Ca₃Ru₂O₇.⁴⁴⁾ This geometric-lattice mechanism is similar to hybrid improper or trilinear coupling ferroelectricity.

Here, we briefly introduce the difference between proper and improper ferroelectricity, interested readers can find comprehensive discussions in Refs. 68–71. Based on the Landau theory of phase transitions, a proper ferroelectric, such as BTO and PTO, has its electrical polarization as the primary order parameter.^70,71) Furthermore, the paraelectric-ferroelectric phase transition is typically described by zone-center soft phonons.⁶⁹⁾ In contrast, an improper ferroelectric is one which the primary order parameter is not electrical polarization, but a quantity having another physical meaning and possessing other transformation properties, i.e. the polarization does not drive the paraelectric-ferroelectric transition but instead is a "slave" to some other primary order parameter.^70,71) In addition, the paraelectric-ferroelectric phase transition is typically described by coupled zone-boundary phonons instead of zone-center phonons.⁶⁹⁾

In materials that display a proper ferroelectric transition such as BTO, the free energy can be expanded in terms of the polarization:⁷¹⁾

$$\begin{eqnarray*}&&F\left(P\right)=1/2\alpha {P}^{2}+1/4\beta {P}^{4},\end{eqnarray*}$$

where F (P) is free energy, P is polarization and paraelectric-ferroelectric transition occurs when α = 0.

In contrast, for the improper ferroelectric YMnO₃, where the polarization arises due to a nontrivial coupling to a zone-boundary lattice instability, the K mode, a simplified free energy is given by:⁷¹⁾

$$\begin{eqnarray*}&&F\left(P,K\right)={\alpha }_{02}{P}^{2}+{\alpha }_{20}{K}^{2}+{\beta }_{40}{K}^{4}+{\beta }_{31}{K}^{3}P+\ldots ,\end{eqnarray*}$$

where K corresponds to the primary order parameter associated with the zone-boundary K₃ phonon mode and P is the polarization. Here the quadratic coefficient of the polarization, α₀₂, does not soften to zero at the ferroelectric transition,^71,72) i.e. α₀₂ > 0 at any temperature. Instead, the spontaneous polarization arises because of the β₃₁ coupling, where for small K, P ∼ K³, and for large K, P ∼ K.⁷²⁾

Another prominent improper ferroelectric is the SrTiO₃/PbTiO₃ superlattice,⁷³⁾ where the polarization arises due to the combination of two rotational modes R₁ and R₂. And the two rotational modes couple linearly to the polarization, that is, there is a trilinear term in the free energy of the form:^69,74)

$$\begin{eqnarray*}&&{F}_{111}=\gamma P{R}_{1}{R}_{2},\end{eqnarray*}$$

where P is polarization. This type of improper ferroelectrics is thus called trilinear coupling ferroelectricity. The term "hybrid improper ferroelectricity" is also used to describe this ferroelectric mechanism in order to generalize the idea to include cases where the two distortion patterns do not necessarily condense at the same temperature.⁶⁸⁾

From the above discussion, we can see that improper ferroelectricity arises not because of zone-center soft phonon instability, as in proper ferroelectrics, but because of zone-boundary rotational modes. Since these rotational modes are due to the combinations of the different radii of the oxygen, A-site cation and B-site cation in ABO₃ perovskites, improper ferroelectricity is said to have a (non-electrostatic) geometric-lattice mechanism and hence it is believed to be resistant to electrostatic screening by itinerant charge carriers.⁶⁹⁾ In addition, because of the geometric-lattice mechanism, the polarization of improper ferroelectrics is also found to be resistant to the electrostatic forces provided by the depolarization field, i.e. the spontaneous polarization of ultrathin improper ferroelectrics is stable in the absence of metallic electrodes.⁷¹⁾ By selecting the suitable chemical species, one can properly design the geometric factor and realize rotation induced polarization in perovskite oxides.⁶⁹⁾

After explaining hybrid improper ferroelectricity, let us now discuss the mechanism of the polar metal Ca₃Ru₂O₇.⁴⁴⁾ Ca₃Ru₂O₇ belongs to the Ruddlesden−Popper oxides with a general formula of A_n+1B_nO_3n+1. Ca₃Ru₂O₇ consists of CaRuO₃ perovskite blocks stacked along the [001] direction with an extra CaO sheet inserted every 2 perovskite unit cells, taking a group symmetry Bb2₁m.^44,68) The structure exhibits RuO₆ octahedral rotations and tilts about [001] and [110], respectively.⁴⁴⁾ These main lattice modes can be identified as a polar zone-center mode ${{\rm{\Gamma }}}_{5}^{-},$ and two-zone boundary modes at the X (1/2, 1/2, 0) point—an oxygen octahedron rotation mode ${{\rm{X}}}_{2}^{+}$ and an oxygen octahedron tilt mode ${{\rm{X}}}_{3}^{-},$ as shown in Fig. 2.^44,68) It was found that the polar distortion in Ca₃Ru₂O₇ is mainly due to Ca and O displacement, which arises from a trilinear anharmonic interaction of the form αQ(${{\rm{\Gamma }}}_{5}^{-}$)Q(${{\rm{X}}}_{2}^{+}$)Q(${{\rm{X}}}_{3}^{-}$).^44,68) Here Q stands for the amplitude of the mode and α is a constant. One can see this mechanism is similar to the trilinear coupling or hybrid improper ferroelectricity. The Ca–O displacement is not screened because the Fermi surface originates from Ru⁴⁺ orbitals and Ru contributes little to the ${{\rm{\Gamma }}}_{5}^{-}$ mode, while the rotational mode ${{\rm{X}}}_{2}^{+}$ and the tilt mode ${{\rm{X}}}_{3}^{-}$ have geometric origins and hence are negligibly affected by the itinerant charge carriers.^44,68)

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One can clearly see that both the lone-pair mechanism and the "hybrid improper ferroelectricity" mechanism fit neatly into the DEM model in that the inversion symmetry is broken by the A-site atom while the conductivity is mainly contributed by the B-site atom.

As the first unambiguous polar metal, the discovery of LiOsO₃ in 2013 has attracted significant research interests.^{15,17,41,42,75–80)} At room temperature, LiOsO₃ takes a centrosymmetric structure (R$\overline{3}$c), which can be viewed as a result of the a⁻a⁻a⁻ octahedral tilting from the higher-symmetry cubic phase (Pm$\overline{3}$m). A phase transition from the centrosymmetric R$\overline{3}$c structure to the polar R3c structure takes place around T_s ≈ 140 K accompanied by a shift of about 0.5 Å in the mean positions of the Li atoms along the cubic perovskite [111] axis.¹⁵⁾ The loss of inversion symmetry below T_s is confirmed by neutron diffraction and convergent-beam electron diffraction (CBED) measurements [Figs. 3(a)–3(d)]. In the meanwhile, anomalies in the temperature dependence of the heat capacity, susceptibility, and resistivity was also observed [Figs. 3(e)–3(g)].¹⁵⁾ This phase transition of LiOsO₃ is similar to that of LiNbO₃ despite the fact that LiNbO₃ is an insulator while LiOsO₃ is a conductor.^81–84)

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To form a macroscopic metallic polar phase, the following questions have to be answered: What is the electronic structure and origin of the metallicity of LiOsO₃? What is the origin of the polar instability: is it displacive or order-disorder type? What are the driving forces of the long-range ordering of the local dipoles? It is generally accepted that the Fermi level is mainly contributed by the Os and O d–p hybridization with minimum contribution from Li.^75–77) The three electrons of Os⁵⁺ populates the t_2g orbitals while e_g orbitals are empty.⁷⁵⁾

Over the years, there have been some debates over the origin of the polar instability. Some groups argue it is displacive,^17,41,42,75) while others believe it belongs to the order-disorder type.^16,76,77,80) For the displacive type, the atoms remain associated with their average positions and the phase transition occurs as the Li atoms move along the polar axis and change its symmetry [Figs. 4(a), 4(b)]. For the order-disorder type, the structural model involves partially occupied sites, and the transition occurs as the symmetry of the occupational distribution is broken [Figs. 4(c), 4(d)]. If the corresponding phonon mode frequency decreases to zero near T_s, it is a signature of a displacive case (soft mode behavior). In the order-disorder case, the relevant phonon frequency stays temperature independent.⁸⁵⁾

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We first discuss the views of the pro-displacive group. Through phonon mode analysis, Xiang found that the polar instability of LiOsO₃ can be explained by the softening of the zone center A_2u (or Γ₂₋) phonon mode, which is mainly dominated by the displacement of the Li atom with a small contribution from the O atoms [Figs. 4(a), 4(b)].⁴¹⁾ This picture is well supported by other theoretic works from Refs. 17 and 75.^17,75) Although it is generally accepted that Li displacements play a major role in the polar instability of LiOsO₃, there is some dispute over the contribution from Os atoms. Xiang argues that the 5d t_2g states of the Os⁵⁺ ion are partially occupied, and hence the second-order Jahn–Teller effect (or the d–p hybridization) could not take place.⁴¹⁾ Whereas Giovannetti and Capone contend that although the 5d t_2g orbitals are half-filled, the e_g orbitals are empty and open for d–p hybridization, which enhances polar distortions, as in common ferroelectric insulators, such as LiNbO₃.⁷⁵⁾

We next discuss the polar instability from the perspective of the order-disorder mechanism. Based on the fact that the 140 K (∼12 meV in energy term) phase transition temperature is much smaller than the depth of the double potential wells, which are around 44 meV, Liu et al. argued that the phase transition is of order-disorder type [Figs. 4(c), 4(d)].^77,86) This assertion is supported by the experimentally observed incoherent charge transport in LiOsO₃ above the transition temperature [Fig. 3(g)], which could be attributed to the scattering induced by disorder of Li off-center displacement.^15,77) In addition, it is further supported by the absence of the A_2u soft phonon mode in the Raman spectroscopy measurements by Jin et al.⁸⁰⁾ However, the A_2u soft phonon is recently detected by Laurita by ultrafast optical pump–probe experiments.⁴²⁾

These contradicting arguments over the displacive or order-disorder nature of the polar instability may suggest that LiOsO₃ has characters of both, as suggested by Sim et al.⁷⁶⁾ and Laurita et al.⁴²⁾ This fact should be understandable as even the prototype "displacive" ferroelectric BaTiO₃ was found to have both displacive and order-disorder components.^13,85,87,88) In addition, LiNbO₃ has also been suggested to have both displacive and order-disorder characters.⁸⁹⁾ It may be worthwhile to carry out some nuclear magnetic resonance experiments to clarify the origin of polarity in LiOsO₃ as done in BTO.⁸⁷⁾

Next, we review the theories proposed to explain the driving forces of the long-range ordering of the local dipoles. The key here is that if there is no interaction between these local dipoles, then the local dipoles may be disordered and the system will be non-polar macroscopically. By employing an effective Hamiltonian to study the interaction between local polar modes, Xiang found that the short-range interactions between some local dipole modes survive the electronic screening and stabilizes the ferroelectric-like ordering [Figs. 5(a), 5(b)].⁴¹⁾ On the other hand, by approximating the dipole interactions with Li–Li pairs, Liu et al. found that these interactions are only slightly screened along certain directions [for example, there is almost no conduction charge distribution between pair 1 in Fig. 5(c)] and hence long-range ordering is possible along these directions.⁷³⁾ This is because that the electronic density mainly concentrates between Os and O ions due to the strong hybridization between Os 5d and O 2p, while there is almost no conduction charge in a relatively large space around Li-ion [Figs. 5(c)–5(e)].⁷⁷⁾ This incomplete screening scenario is consistent with the claim by Vecchio et al.,⁷⁹⁾ which states that LiOsO₃ is a strongly correlated bad metal close to Mott localization due to the half-filling configuration of the Os 5d t_2g orbitals. It is also supported by the fact that the resistivity of LiOsO₃ exceeds that of a normal metal by two orders of magnitude²⁴⁾ since higher resistivity means less effective electronic screening.

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These theoretical works strongly support the DEM model proposed by Puggioni and Rondinelli.³⁰⁾ It is recently proved in the ultrafast spectroscopy experiments by Laurita et al., in which they demonstrated that the intra-band photo-carriers relax by selectively coupling with a subset of the phonon spectrum (¹E_g and ²E_g), while they couple extremely weakly to the A_2u soft polar mode.⁴²⁾ In addition, this DEM model is also supported by the dual nature of the d orbitals of Os atoms, in which the electrons in t_2g orbitals are responsible for the conductivity while the empty e_g orbitals hybrid with O 2p orbital to enhance the polar distortions as in common ferroelectric insulators.⁷⁵⁾

While the origin of the polar instability and the mechanism of the long-range ordering of local dipoles in LiOsO₃ have been well explained, its switchability by external electric field has not been demonstrated experimentally so far, despite theoretical predictions.^41,90) In the next section, we review the progress in the first electric-field-switchable ferroelectric metal WTe_2.

As a member of the transition metal dichalcogenides family, WTe₂ has been extensively studied over the past years due to its exotic topological properties.^91–94) It takes a layered orthorhombic structure (1 T'), which belongs to the polar space group Pmn2₁.^95,96) While two-dimensional ferroelectricity has been widely reported in a variety of two-dimensional materials, such as SnTe,⁹⁷⁾ CuInP₂S₆⁹⁸⁾ and IV–VI group compound,⁹⁹⁾ most of them are semiconductors. Interestingly, WTe₂ is a Weyl semimetal.⁹¹⁾ In 2018, through detailed electrical transport measurements under external electric field by sandwiching 2–3 WTe₂ layers between two hexagonal boron nitride dielectric sheets, Fei et al. presented WTe₂ as the first external-electric-field-switchable ferroelectric metal with an out-of-plane polarization [Figs. 6(a), 6(b)].¹⁹⁾ Later, Sharma et al. demonstrated that WTe₂ is a ferroelectric metal even in its bulk form through extensive piezoresponse force microscopy (PFM) measurements [Figs. 6(c)–6(g)].²⁰⁾

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The natural questions following the demonstration of the first ferroelectric metal are: what is the origin of the ferroelectricity in WTe₂ and what is its switching mechanism? Following the suggestion by Fei et al. that "the polarization could principally involve a relative motion of the electron cloud relative to the ion cores, rather than a lattice distortion", Yang et al.¹⁰⁰⁾ and Liu et al.¹⁰¹⁾ demonstrated that the polarization stems from uncompensated interlayer charge transfer. The calculated charge transfer of ∼3.2 ×10¹¹ e cm⁻² is consistent with the experimentally measured polarization of 2 × 10¹¹ e cm⁻² at 20 K.¹⁹⁾ This charge transfer can be clearly seen in the differential charge diagram shown in Fig. 7 between the upper and lower layers of WTe₂. These authors continue to suggest that the out-of-plane polarization can be switched by an in-plane interlayer sliding,^100,101) which has been used to explain the switchability of other two-dimensional van der Waals materials, such as In₂Se₃^102,103) and MoS₂.¹⁰⁴⁾ From Fig. 7(a), one can see that the polarization can be switched by moving the upper layer along the −y axis by a distance of Δ₁ + Δ₂, as evidenced by the inversion of the charge distribution between the upper and lower layers in Figs. 7(b)–7(c). Yang et al. also showed that the switchability of multilayer WTe₂ can also be described by this model by selectively sliding some specific layers.¹⁰⁰⁾

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This interlayer sliding model explains some important aspects in WTe₂ but also left some critical questions unanswered. For example, how does the out-of-plane electric field induce an in-plane interlayer sliding? In addition, the magnitude of the required interlayer sliding (Δ₁ + Δ₂ ∼ 50 pm) is much larger than the atomic distortions (∼10 pm) in conventional ferroelectric switching.¹⁰³⁾ It can be experimentally tested with microscopic structural characterization tools such as STEM. This model also imposes some limitations. For example, it needs to selectively slide the layers to switch the polarization of multilayers,¹⁰⁰⁾ which is unrealistic to explain the switchability of bulk WTe₂.

The sliding mechanism of polarization switching in WTe₂ is reminiscent of the DEM model in that the orientation of polarization and the direction of the polarization switching are spatially separately, which we would like to call "decoupled space mechanism (DSM)." We also note that another paper has proposed a similar switching mechanism of ferroelectric metals through interfacial coupling with a ferroelectric insulator, where the polarization and its switching are located in different regions in the system. Fang et al. proposed BiPbTi₂O₆ to be a polar metal, where the conductivity comes from the Ti 3d states and the polar instability originates from the 6s lone-pair electrons in Bi and Pb similar to BiFeO₃ and PbTiO₃.⁴⁵⁾ By growing BiPbTi₂O₆ on PbTiO₃ with in-plane ferroelectricity, which can be achieved with tensile strain,^105,106) the in-plane polarization of BiPbTi₂O₆ can be switched if that of PbTiO₃ is switched.⁴⁵⁾ This approach circumvents the need to directly apply an external voltage on the polar metal, which will induce an electric current instead of an electric field. We hope this DSM mechanism may give researchers some hints to study the large number of polar metals whose switchability is yet to be demonstrated. Although some important progress has been made in understanding the physics of the first ferroelectric metal WTe₂, some important questions need to be investigated. For example, is the ferroelectricity a type of prototypical proper ferroelectricity, hyperferroelectricity (discussed below) or improper ferroelectricity? Can the ferroelectric metallic state be explained by the DEM model?

In the above section discussing improper ferroelectrics, we mentioned that because of its geometric-lattice origin of polarization, it is resistant to the electrostatic forces provided by itinerant charge carriers and depolarization field. In 2014, Garrity, Rabe, and Vanderbilt proposed another type of ferroelectricity, coined hyperferroelectricity, which is also resistant to the depolarization field.¹⁰⁷⁾ Since the polarization of hyperferroelectric is resistant to the electrostatic forces provided by depolarization field, one natural question we can ask is whether it can persist under the electrostatic forces provided by itinerant charge carriers. Recently, several hyperferroelectric metals have been proposed, such as LaPtSb,⁵³⁾ LaAuGe,⁵³⁾ SrHgPb,⁴⁹⁾ KMgSb_0.2Bi_0.8,⁵⁰⁾ CaAgBi,⁵¹⁾ CrN⁵⁵⁾ and CrB₂.⁵⁵⁾

Let us briefly introduce hyperferroelectricity and discuss its differences with prototypical proper ferroelectricity and improper ferroelectricity before summarizing the recent progress in hyperferroelectric metals. Hyperferroelectrics is first proposed by Garrity, Rabe, and Vanderbilt in hexagonal ABC compounds (LiGaGe type).¹⁰⁷⁾ Figures 8(a) and 8(b) show the non-polar high-symmetry P6₃/mmc structure and the polar P6₃mc structure, respectively.¹⁰⁸⁾ The structure is a hexagonal variant of the half-Heusler structure and can be described as a wurtzite structure "stuffed" with a third cation (the A atom in ABC notation).¹⁰⁸⁾ The polar phase is reached primarily by a buckling in the honeycomb layers as the atoms move from sp² environment towards sp³ bonding, resulting in polarization in the z direction.^107,108)

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Let us first discuss the difference between hyperferroelectricity and prototypical proper ferroelectricity. Hyperferroelectricity is a type of proper ferroelectricity, in this article we use the term "prototypical proper ferroelectrics" to describe those proper ferroelectrics other than hyperferroelectrics. The ferroelectric phase transitions can be understood by the lattice dynamics of their high-symmetry phase. For proper ferroelectrics, the high-symmetry phase has at least one unstable TO mode, specifically, a Γ mode that is unstable under zero macroscopic electric field (ε = 0) boundary conditions.¹⁰⁷⁾ The frequency of this mode can be obtained from the first-principles computation of the force-constant matrix with the usual periodic boundary conditions.¹⁰⁷⁾ If the depolarization field is unscreened, corresponding to the case of electric displacement D = 0, the structural instability is determined by the LO modes, which can be obtained by adding to the dynamic matrix a non-analytic long-range Coulomb term that schematically takes the form (Z*)²/${\varepsilon }^{\infty },$ where Z* is the Born effective charge and ${\varepsilon }^{\infty }$ is the electronic contribution to the dielectric constant, generating the well-known LO–TO splitting.^107–110) In prototypical proper ferroelectrics, such as BTO and PTO, due to their large Born effective charges and relatively small ${\varepsilon }^{\infty },$ the LO–TO splittings are huge, such that all LO modes are stable.^107–110) Therefore they lose ferroelectricity if the depolarization field is not well screened. In contrast, in the hexagonal ABC hyperferroelectrics, the LO–TO splittings are small, such that even the LO modes can become unstable.^107–110) The unusual electric properties of hyperferroelectrics mean that the ferroelectric phase transition temperature, which decreases with decreasing screening of the depolarization field, will still be nonzero even under D = 0 boundary conditions.^107–110) At this temperature T_D, the LO mode becomes unstable and the material becomes a hyperferroelectric.^107–110) Consequently, the polarization in these materials is very robust against the depolarization field. Although hyperferroelectrics share the same depolarization-field-resistant property as improper ferroelectrics, it also differs significantly with improper ferroelectrics in the primary order parameters and the phonon modes. As hyperferroelectric is a type of proper ferroelectrics, the differences between proper and improper ferroelectrics discussed in Sect. 4.1 also apply here, interested readers can find more comprehensive discussions in Refs. 107–110.

After introducing hyperferroelectricity, let us now discuss hyperferroelectric metals introduced at the beginning of this section. Strictly speaking, the term "hyperferroelectric metal" or even "hyperferroelectricity" is not scientifically rigid at this moment because their polarization has not yet been experimentally shown to be switchable, to the best of our knowledge. But as discussed in Sect. 2, the misuse of terms has happened a lot in this confusing field that studies the coexistence of contra-intuitive properties. A more proper term for these materials would be "hyperpolar metals" before its switchability is shown. Anyhow, we decide to follow the convention and continue to call them hyperferroelectric metals in this work.

Let us take LaPtSb as an example of hyperferroelectric metal.⁵³⁾ It takes the typical hexagonal ABC structure as shown in Fig. 8. Recently, Du et al. found that LaPtSb is a hyperferroelectric metal with a resistivity that is nearly an order of magnitude lower than the well-studied oxide polar metals.⁵³⁾ By utilizing a density functional theory chemical pressure (CP) analysis to analyze the local bonding interactions between La, Pt and Sb, they found that the buckling of the Pt–Sb (or B–C in ABC notation) plane is a result of the relief of intraplanar positive CPs created by local bonding preferences.⁵³⁾ Because these bonding preferences are local, the resulting polar bucking is reasonably expected to be resistant to electrostatic screening by itinerant charge carriers. Despite some progress has been made in hyperferroelectric metals, there have been some major issues left unsolved. Such as the experimental observation of the LO soft phonon and of course the switchability of polarization.