# Fileset [apssamp.ver6.pdf](https://mdr.nims.go.jp/filesets/2fdcad7f-b06e-4a57-8d7c-a9baafd23a8f/download) ## Creator [Ryota Masuki](https://orcid.org/0000-0002-5407-844X), [Takuya Nomoto](https://orcid.org/0000-0002-4333-6773), [Ryotaro Arita](https://orcid.org/0000-0001-5725-072X), [Terumasa Tadano](https://orcid.org/0000-0002-8132-2161) ## Rights ## Other metadata [Continuous crossover between insulating ferroelectrics and polar metals: Ab initio calculation of structural phase transitions of

Li B O_{3}

(

B = Ta

, W, Re, Os)](https://mdr.nims.go.jp/datasets/5cbe71f2-e405-4772-b733-9b65bdfcbe09) ## Fulltext Continuous crossover between insulating ferroelectrics and the polar metals: Ab initio calculation ofstructural phase transitions of Li𝐵O3 (𝐵 = Ta, W, Re, Os)Ryota Masuki,1, ∗ Takuya Nomoto,2, † Ryotaro Arita,3, 4, ‡ and Terumasa Tadano5, §1Department of Applied Physics, The University of Tokyo,7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan2Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan3Research Center for Advanced Science and Technology,The University of Tokyo, 4-6-1 Komaba Meguro-ku, Tokyo 153-8904, Japan4RIKEN Center for Emergent Matter Science, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan5CMSM, National Institute for Materials Science (NIMS), 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan(Dated: September 8, 2024)Inspired by the recent discovery of a new polar metal LiReO3 by K. Murayama et al, we calculate thetemperature(𝑇)-dependent crystal structures of Li𝐵O3 with 𝐵 = Ta, W, Re, Os, using the self-consistent phonon(SCPH) theory. We have reproduced the experimentally observed polar-nonpolar structural phase transitions andthe transition temperatures (𝑇𝑐) of LiTaO3, LiReO3, and LiOsO3. From the calculation, we predict that LiWO3is a polar metal, which is yet to be tested experimentally. Upon doping electrons to the insulating LiTaO3,the predicted 𝑇𝑐 is quickly suppressed and approaches those of the polar metals. Thus, there is a continuouscrossover between ferroelectric insulators and polar metals if we dope electrons to the ferroelectric insulators.Investigating the detailed material dependence of the interatomic force constants (IFCs), we explicitly show thatthe suppression of 𝑇𝑐 in polar metals can be ascribed to the screening of the long-range Li-O interaction, whichis caused by the presence of the itinerant electrons. The quantitative finite-temperature calculations do not showsigns of unscreened long-range interactions by the weak electron-phonon coupling or enhancement of polarinstabilities by carrier doping, as expected in some previous works.I. INTRODUCTIONPolar metals, metals that show ferroelectric-like phase tran-sition in terms of crystal symmetry, have attracted considerableinterest since LiOsO3 was experimentally identified as a polarmetal for the first time [1]. With the coexistence of broken in-version symmetry and metallic conductivity, polar metals canhost emergent phenomena [2] such as superconductivity [3, 4],enhanced thermoelectricity [5, 6], nontrivial topology [7, 8],and multiferroicity [9].However, only a few materials have been discovered [2,10, 11], although half a century has passed since its first pro-posal [12]. This is because the long-range Coulomb interactionbetween local dipoles, which causes the ferroelectric instabil-ities in insulators [13, 14], is screened out by the itinerantelectrons. Thus, understanding the mechanism to overcomethis incompatibility between polar instability and metalicity isessential to designing and discovering new polar metals withdesired properties.As the first discovered polar metal, LiOsO3 has been exten-sively investigated experimentally and theoretically. LiOsO3shows second-order order-disorder structural phase transitionbetween 𝑅3𝑐 and 𝑅3̄𝑐 structures at 140 K [1, 15–18]. In thehigh-temperature 𝑅3̄𝑐 phase, the ferroelectric-like 𝐴2𝑢 phononis unstable, which is dominated by the displacements of the Liions. The origin of this instability has been discussed from dif-ferent viewpoints, such as instability of Li and O ions [19, 20],∗ masuki-ryota774@g.ecc.u-tokyo.ac.jp† tnomoto@tmu.ac.jp‡ arita@riken.jp§ TADANO.Terumasa@nims.go.jpshort-range geometric and bonding properties [21, 22], the de-coupling electron mechanism [23, 24], incomplete screeningof the dipole-dipole interaction [25], and hyperferroelectric-ity [26, 27]. Some other materials have been investigated aswell [24, 28, 29], but these works focus on individual materialsand systematic investigations have been lacking because of theminimal number of known materials.In addition, doping carriers to ferroelectric insulators hasbeen investigated as an effective way to design a polarmetal [30–33]. Some works predict that the polar atomic dis-placements are not suppressed and even enhanced by dopingin many materials, which are attributed to the so-called meta-screening effect [31, 32]. However, they do not consider thefinite-temperature effects. Furthermore, they focus on the re-lations of the ferroelectric insulators and doped ferroelectrics,but the relations between the intrinsic polar metals and thesetwo remain unclear.Recently, LiReO3 have been experimentally revealed to be apolar metal [34]. LiReO3 is isostructural to LiOsO3 and showsa polar-nonpolar structural phase transition between the high-temperature 𝑅3̄𝑐 and the low-temperature 𝑅3𝑐 phases at 170K, which are shown in Fig. 1. In addition, LiTaO3, anothermaterial in the Li𝐵O3 group with a 5𝑑 transition metal atthe 𝐵 site, has been reported to be a ferroelectric insulator.Therefore, we consider that Li𝐵O3 (𝐵 = 5𝑑 transition metals)is an ideal platform to perform systematic analysis on the polarmetals and to investigate the relations between ferroelectrics,doped ferroelectrics, and polar metals.In this work, we focus on Li𝐵O3 with 𝐵 = Ta, W, Re, andOs. From the basic electronic-structure calculations, we haveconfirmed that LiTaO3 is an insulator while the other threeare all metals in both the polar and nonpolar phases. Theharmonic phonon calculations show that the 𝐴2𝑢 mode at Γpoint has the largest instability in the nonpolar 𝑅3̄𝑐 phase,mailto:masuki-ryota774@g.ecc.u-tokyo.ac.jpmailto:tnomoto@tmu.ac.jpmailto:arita@riken.jpmailto:TADANO.Terumasa@nims.go.jp 2which is consistent with the structural phase transitions ofLiTaO3, LiReO3, LiOsO3.Furthermore, we calculate the 𝑇-dependence of the crys-tal structures of these materials based on the self-consistentphonon theory [35]. The calculated transition temperaturesaccurately reproduce the chemical trend. Based on the elec-tronic, phononic, and the structural calculations, we predictthat LiWO3 is another polar metal whose 𝑇𝑐 is slightly lowerthan LiReO3 and LiOsO3. The synthesis of LiWO3 has beenreported [36] but its detailed properties has not been mea-sured yet. Upon doping electrons to the insulating LiTaO3, thehigh 𝑇𝑐 is swiftly but continuously suppressed and approachesthose of polar metals. Thus, there is a continuous crossoverbetween the ferroelectric insulator and the polar metals, whichare connected by the doped ferroelectrics. The result showsthat the polar instability is suppressed with electron-doping ifwe consider the finite temperatures, although the magnitudeof the polar displacement remains intact (meta-screening ef-fect) [32]. Analyzing the interatomic force constants (IFCs) indetail, we explicitly show that the suppression of 𝑇𝑐 in polarmetals is caused by the screening of the long-range Li-O in-teractions. The calculation results suggest that the decouplingelectron scenario that the long-range Coulomb interactions areonly weakly coupled if the coupling between the electrons andpolar mode is small [24] is not the case for these materials.II. THEORYWe use the structural optimization method based on self-consistent phonon (SCPH) theory [35] to calculate the tem-perature dependence of the crystal structures. SCPH theory isa mean-field theory of the phonon anharmonicity, which hasbeen demonstrated to accurately reproduce finite-temperatureproperties of strongly anharmonic materials [37–42].SCPH theory is based on the variational principle of the freeenergyF = −𝑘𝐵𝑇 log Tr 𝑒−𝛽Ĥ0 + ⟨𝐻̂ − Ĥ0⟩Ĥ0≥ 𝐹, (1)where 𝐹 is the true free energy, and F is the variationalfree energy. 𝐻̂ is the true Hamiltonian, and Ĥ0 is the trialHamiltonian, which we restrict to be a harmonic HamiltonianĤ0 =∑k𝜆 ℏΩk𝜆(𝑎̂†k𝜆𝑎̂k𝜆 + 12)in SCPH theory. The SCPHfrequencies Ωk𝜆 are considered as the variational parameters,which are adjusted to minimize F [43]. The minimization isperformed by solving a self-consistent equation of Ωk𝜆 [44],which we call the SCPH equation.In structural optimization, we consider the minimized vari-ational free energy (minΩk𝜆F ) as the approximate free energyand minimize it with respect to the crystal structures. We startfrom the Taylor expansion of the potential energy surface𝑈̂ =∞∑︁𝑛=0𝑈̂𝑛, (2)𝑈̂𝑛=1𝑛!∑︁{R𝛼𝜇}Φ𝜇1 · · ·𝜇𝑛 (R1𝛼1, · · · ,R𝑛𝛼𝑛)𝑢̂R1𝛼1𝜇1 · · · 𝑢̂R𝑛𝛼𝑛𝜇𝑛=1𝑛!1𝑁𝑛/2−1∑︁{k𝜆}𝛿k1+···+k𝑛Φ̃(k1𝜆1, · · · ,k𝑛𝜆𝑛)𝑞k1𝜆1 · · · 𝑞kn𝜆𝑛,(3)where 𝑢̂R𝛼𝜇 is the 𝜇(= 𝑥, 𝑦, 𝑧) component of atomic displace-ment of atom 𝛼 in the primitive cell at R, and𝑞k𝜆 =1√𝑁∑︁R𝛼𝜇𝑒−𝑖k·R𝜖∗k𝜆,𝛼𝜇√︁𝑀𝛼𝑢̂R𝛼𝜇 . (4)are those in the normal coordinate representation. 𝜖k𝜆,𝛼𝜇 isthe polarization vector of the phonon with mode 𝜆 and crystalmomentum k. 𝑀𝛼 is the mass of the atom 𝛼. We call theTaylor expansion coefficients Φ𝜇1 · · ·𝜇𝑛 (R1𝛼1, · · · ,R𝑛𝛼𝑛) andΦ̃(k1𝜆1, · · · ,k𝑛𝜆𝑛) as interatomic force constants (IFCs).The IFCs are the functions of atomic positions in the unitcell, which we denote as 𝑋𝛼𝜇 for the 𝜇(=𝑥, 𝑦, 𝑧) componentof the atomic position of atom 𝛼 [45]. As the solution of theSCPH equation is determined by the set of IFCs Φ̃, we canwrite down the SCPH free energy as𝐹SCPH (𝑋𝛼𝜇) = F (Φ̃(𝑋𝛼𝜇),Ωk𝜆 (Φ̃(𝑋𝛼𝜇))). (5)The crystal structure (𝑋𝛼𝜇)-dependence of the IFCs Φ̃ canbe calculated using the IFC renormalization [35, 46, 47].Thus, we can calculate the gradient of the SCPH free energy𝜕𝐹SCPH (𝑋𝛼𝜇 )𝜕𝑋𝛼𝜇and perform structural optimization with finite-temperature effects. Please see Refs. [35, 46] for more detailson the IFC renormalization and the structural optimization atfinite temperatures.III. SIMULATION DETAILSA. Phonon calculations and structural optimizations at finitetemperaturesWe use the ALAMODE implementation of the SCPH cal-culation and SCPH-based structural optimization [35, 44, 46,48, 49]. We use the 2 × 2 × 2 supercell that contains 80 atomsin the phonon calculations. Note that supercells used in thecalculations are based on the primitive cell, which is given bythe parallelepiped shown in Fig. 1. The reference structure isdetermined using the structural optimization of VASP, whoselattice constants are summarized in Table I.The harmonic IFCs are calculated using the small displace-ment method with atomic displacements of 0.01 Å. Theanharmonic IFCs are obtained using the compressive sens-ing method [44, 50], which enables efficient extraction ofIFCs from a small number of displacement-force data. Thedisplacement-force data is obtained with high-accuracy DFTcalculations on a set of randomly displaced configurations. Weuse the ab initio molecular dynamics (AIMD) simulation to3TABLE I. The lattice constants that are used in the phonon calcula-tions. The lattice constants are defined so that the lattice vectors ofthe conventional cell are (𝑎, 0, 0), (−𝑎/2,√3𝑎/2, 0), (0, 0, 𝑐). Theconventional cells of Li𝐵O3 contain 30 atoms.Materials 𝑎 [Å] 𝑐 [Å]LiNbO3 [34] 5.1818 13.6313LiTaO3 5.1885 13.6659LiWO3 5.1744 13.5222LiReO3 [34] 5.1267 13.3700LiOsO3 5.1116 13.0105generate the randomly displaced configurations. We performthe AIMD calculation at 300 K for 16000 steps with the stepof 1 fs for LiWO3, LiReO3, and LiOsO3. The first 1000 stepsare discarded as thermalization steps, and 300 snapshots aresampled uniformly from the rest 15000 steps. The configura-tions are generated by adding random atomic displacements of0.04 Å to the 300 AIMD snapshots. The procedure is similarfor LiNbO3 and LiTaO3. For these ferroelectric insulators,however, we perform AIMD calculations at 500 K and 750K for 8000 steps, respectively. In each AIMD calculation,the first 1000 steps are discarded as thermalization steps, and140 configurations are similarly extracted. Thus, we get 280displacement-force data for LiNbO3 and LiTaO3, respectively.We choose the calculation settings so that the generated config-urations effectively sample the low-energy region of the poten-tial energy surface. Note that high accuracy is not necessaryin the AIMD calculations because they are just for generatingrandom structures.We use 8 × 8 × 8 𝑞-mesh in SCPH calculations. We fix theshape of the unit cell in SCPH-based structural optimizationsbecause the cell volumes of the considering polar metals donot drastically change on structural phase transitions [1, 34].B. DFT calculationsWe employ the Vienna Ab initio Simulation Package(VASP) [51] for DFT calculations. We use the PBEsolexchange-correlation functional [52] and the PAW pseudopo-tentials [53, 54]. In the high-accuracy calculations, we setthe convergence criteria of the SCF loop as 10−8 eV and thebasis cutoff as 600 eV. We use the 4 × 4 × 4 Monkhorst-Pack𝑘-mesh and accurate precision mode, which suppresses egg-box effects and errors. In the AIMD calculations, we set theconvergence criteria of the SCF loop as 10−6 eV and the basiscutoff as 400 eV. We use the 2×2×2 Monkhorst-Pack 𝑘-meshto reduce the computational cost. In both calculations, we useGaussian smearing with a width of 0.05 eV. The spin-orbitcoupling (SOC) is not included in the anharmonic phonon cal-culations because it does not affect the low-energy landscapeof the potential energy surface [55].In addition, we would like to add a short discussion on elec-tronic correlations. The target materials Li𝐵O3 can be stronglycorrelated electron systems due to the 5𝑑 transition metals inthe 𝐵 site. In particular, electronic correlations of LiOsO3 havebeen shown to be essential to precisely describe the electronicproperties [56–58]. However, we do not explicitly considerthe electronic correlations in this work. This is because theeffect of electronic correlation will be the largest in LiOsO3,which have half-filled 𝑡2𝑔 bands, but it is not strong enough tocause metal-insulator transition. Thus, the conventional DFTcalculations should be accurate enough to discuss distinctionsbetween metals and insulators and basic electronic structures.Furthermore, previous works show that the electronic corre-lation little affects the structural properties [19], which is themain topic of this work.IV. RESULTS AND DISCUSSIONNonpolar 𝑅"3𝑐 Polar 𝑅3𝑐FIG. 1. The crystal structure of LiReO3 in the nonpolar 𝑅3̄𝑐 phaseand in the polar 𝑅3𝑐 phase. The figure of the crystal structures isgenerated by VESTA [59].A. Electronic Structures of Li𝐵O3We calculate the electronic structures of Li𝐵O3 with 𝐵=Ta,W, Re, Os, which are shown in Fig. 2. All the materials showsimilar band structures with an almost isolated set of 12 bands(including spin degeneracy) near the Fermi level, which consistof hybridized 𝐵-site 𝑑-orbitals and O 𝑝-orbitals. LiTaO3 is aninsulator as the Fermi level lies in the middle of a band gap.As the number of 5𝑑 electrons increases, the Fermi level shiftsupward, and the bands become half-filled in LiOsO3. LiWO3,LiReO3, and LiOsO3 are all metals. In the supplementarymaterials [55], we summarize all the calculation results bothon the high-𝑇 𝑅3̄𝑐 and low-𝑇 𝑅3𝑐 phase, with and withoutSOC. According to Fig. S1 to S4, LiTaO3 is an insulator,while LiWO3, LiReO3, and LiOsO3 are metals in the low-𝑇 𝑅3𝑐 phase as well. Thus, the metal-insulator transitionsdo not occur with the structural phase transitions in the targetmaterials. This is consistent with the experimental observationthat LiReO3 is a polar metal [34], and supports our predictionthat LiWO3 is another polar metal, which we discuss later.4Γ L B1|B Z Γ X|QP1 Z|L P−5.0−2.50.02.55.0Energyfromε F[eV](a) LiTaO3, high-T R3̄c−4 −2 0 2 4Energy from εF [eV]01020DensityofStatestotalTa(d)Li(s)O(p)Γ L B1|B Z Γ X|QP1 Z|L P−5.0−2.50.02.55.0Energyfromε F[eV](b) LiWO3, high-T R3̄c−4 −2 0 2 4Energy from εF [eV]01020DensityofStatestotalW(d)Li(s)O(p)Γ L B1|B Z Γ X|QP1 Z|L P−5.0−2.50.02.55.0Energyfromε F[eV](c) LiReO3, high-T R3̄c−4 −2 0 2 4Energy from εF [eV]01020DensityofStatestotalRe(d)Li(s)O(p)Γ L B1|B Z Γ X|QP1 Z|L P−5.0−2.50.02.55.0Energyfromε F[eV](d) LiOsO3, high-T R3̄c−4 −2 0 2 4Energy from εF [eV]01020DensityofStatestotalOs(d)Li(s)O(p)FIG. 2. The electronic band structures and density of states (DOS) of(a) LiTaO3, (b) LiWO3, (c) LiReO3, (d) LiOsO3. The calculations areperformed on the high-temperature 𝑅3̄𝑐 phase, taking into accountSOC.B. Harmonic phonons of Li𝐵O3We calculate the harmonic phonon dispersions and atom-projected phonon density of states of Li𝐵O3. Note that weneglect SOC from this section because SOC hardly affects thelow-energy region of the potential energy surface, which weshow in Sec. II in the supplementary materials [55]. Fig. 3shows the calculation results of LiReO3. LiReO3 has a pair ofunstable modes in the high-𝑇 𝑅3̄𝑐 phase, which is dominatedby Li ions. The most unstable mode is the ferroelectric-like𝐴2𝑢 mode at Γ point that causes the transition to the low-𝑇 𝑅3𝑐phase, which is common to all 𝐵 = Ta, W, and Os cases [55].We also show the 𝑇 dependence of the SCPH dispersion ofthe high-temperature phases of Li𝐵O3 in section III of thesupplementary materials [55]. The ferroelectric-like mode atΓ point softens the most drastically as the temperature getslower, which is also consistent with the polar-nonpolar struc-tural phase transitions. The non-dispersive nature of the softmodes in the harmonic dispersions suggests that the transi-tion is caused by the on-site instability of the loosely bondedsmall Li ions, as suggested for LiOsO3 [18, 20]. The imag-inary phonon is lifted in the low-temperature phase, and theinstability disappears.In Section IV A, we saw that the electronic bands of Li𝐵O3near the Fermi level are dominated by the 𝐵-site 𝑑 orbitals andO 𝑝 orbitals, while the contribution of Li ions, which domi-nates the instability of the high-symmetry phase, is negligible.Thus, LiReO3 seems consistent with the decoupling electronmechanism [24], that the electronic state near the Fermi leveland the polar atomic displacements are decoupled, as discussedon LiOsO3 [20, 23].Γ L B1|B Z Γ X|QP1 Z|L P05001000Frequency[cm−1](a) LiReO3, high-T R3̄c0 500 1000Frequency [cm−1]0.000.050.100.150.20phDOS[cm](b) LiReO3, high-T R3̄ctotalLiReOΓ L B1|B Z Γ X|QP1 Z|L P05001000Frequency[cm−1](c) LiReO3, low-T R3c0 500 1000Frequency [cm−1]0.000.050.100.150.20phDOS[cm](d) LiReO3, low-T R3ctotalLiReOFIG. 3. Calculation results of the harmonic phonon dispersion andatom-projected phonon density of states of LiReO3 in the high-𝑇 𝑅3̄𝑐phase and in the low-𝑇 𝑅3𝑐 phase.C. Structural phase transitions of Li𝐵O3Based on the above discussions, we apply the SCPH-basedstructural optimization at finite temperatures to Li𝐵O3 with 𝐵= Ta, W, Re, Os. The temperature-dependence of the atomicdisplacements are shown in Fig. 4. The atomic displacementsare defined with respect to the reference structure, which is ob-tained by the structural optimization at zero temprature basedon DFT. The displacements of Li and 𝐵-site ions are zero athigh temperatures, while they are finite at low temperatures.The atomic displacements of O atoms can be finite in high-𝑇 phases because their internal positions are not completelyfixed by symmetry. Hence, the polar-nonpolar structural phasetransitions of these materials are reproduced by theoretical cal-culations. The transition temperatures, which are estimatedfrom the crossing points of the SCPH free energies, are sum-marized in Table II. We can see that the calculated 𝑇𝑐 of eachmaterial is compatible with the experimental values. In ad-dition, the calculation results reproduce the chemical trend,i.e., 𝑇𝑐 of the ferroelectric insulators (𝐵=Nb, Ta) are muchhigher than those of polar metals (𝐵=Re, Os). The chemicaltrends within each class (the ferroelectrics and the polar met-als) are also accurately reproduced. The SCPH theory cannot5TABLE II. The calculated and experimental transition temperatures(𝑇𝑐) of Li𝐵O3 with 𝐵 = Nb, Ta, W, Re, Os.Materials calculation [K] experiment [K]LiNbO3 1350 [34] 1480 [60]LiTaO3 723 910 [60]LiWO3 193 -LiReO3 267 [34] 170 [34]LiOsO3 207 140 [1]treat the deviation of the atomic distribution function from theGaussian distribution of the effective harmonic Hamiltonian.Thus, it is less suited for the order-disorder structural phasetransition than for the displacive structural phase transitions.However, because the SCPH theory calculates the effectiveharmonic Hamiltonian based on the variational principle, wecan expect that it approximates the spread of the atomic dis-tributions around the average positions. In fact, the agreementof 𝑇𝑐 with the experimental results suggests that the effectivemean field theory accurately captures the overall behaviour ofthe materials. Such limitations of the SCPH theory can be onereason for the errors of calculated 𝑇𝑐.As shown in Fig. S8 in the supplementary materials [55],the ferroelectric-like 𝐴2𝑢 mode that drives the polar-nonpolarstructural phase transition has the largest instability in the high-temperature 𝑅3̄𝑐 phase of LiWO3. In addition, LiWO3 ismetallic in both 𝑅3̄𝑐 and 𝑅3𝑐 phases, which we can see fromFig. S2. Thus, we predict that LiWO3 is another isostructuralpolar metal whose transition temperature 𝑇𝑐 is slightly lowerthan LiReO3 and LiOsO3.As depicted in Fig. 5, the calculated 𝑇𝑐 of the polar met-als Li𝐵O3 (𝐵 = W, Re, Os) do not change drastically withthe number of 𝑑 electrons per 𝐵 site, while LiTaO3 signifi-cantly deviates from this trend. Here, we focus on the regionbetween the ferroelectric insulator and the polar metals. Wedope electrons to the insulating LiTaO3 by changing the num-ber of electrons in DFT calculations and investigate the changeof 𝑇𝑐. From Fig. 5, the 𝑇𝑐 of LiTaO3 is quickly suppressedand approaches those of polar metals upon electron doping.Note that such calculations to change the number of electronsare not necessarily accurate because of the possible artifactfrom the uniform positive background. In fact, we fix theshape of the unit cell when we change the number of electronsbecause the lattice constants optimized by DFT calculationslargely deviate from the chemical trend. However, we considerthe quick suppression of 𝑇𝑐 in doped LiTaO3 is qualitativelycorrect because 𝑇𝑐 changes much more slowly and follows thetrend when we reduce the number of electrons in LiWO3, asshown in Fig. 5.Because the change of 𝑇𝑐 is continuous, we consider thatthere is a continuous crossover between the ferroelectric in-sulators and the polar metals, which are connected by dopedferroelectrics. Upon doping electrons to ferroelectric insula-tors, the polar instabilities is suppressed at finite temperatures,even when the polar displacements at zero temperature remainunchanged.250 500 750 1000 1250temperature [K]0.00.20.4atomicdisplacement[Å](a) LiTaO3LiTaO100 200 300 400temperature [K]0.00.20.4atomicdisplacement[Å](b) LiWO3LiWO100 200 300 400temperature [K]0.00.20.4atomicdisplacement[Å](c) LiReO3LiReO100 200 300 400temperature [K]0.00.20.4atomicdisplacement[Å](d) LiOsO3LiOsOFIG. 4. Calculation results of the temperature dependence of theatomic displacements of Li𝐵O3 with 𝐵 = (a) Ta, (b) W, (c) Re, (d) Os.The solid lines show the cooling calculations, in which we start fromthe high-temperature and add a slight atomic displacement along theunstable 𝐴2𝑢 mode until the transition to the low-temperature phaseis induced. The dashed lines show the heating calculations, in whichwe start from the low-temperature 𝑅3𝑐 phase.0.0(B=Ta) 1.0(B=W) 2.0(B=Re) 3.0(B=Os)number of d-electrons per B site0200400600800transitiontemperatureTs[K] dope electrons to LiTaO3reduce electrons of LiWO3LiBO3, (B = Ta, W, Re, Os)FIG. 5. The calculated transition temperatures of Li𝐵O3 withdifferent numbers of electrons per 𝐵 site. The dotted lines are guideto the eye.D. Chemical trend of 𝑇𝑐 in the polar metals Li𝐵O3In this section, We discuss the origin of the trend of 𝑇𝑐 ofpolar metals in Fig. 5. Up to the quartic order, the double wellpotential along the soft mode can be written as𝑈 (𝑞𝜆) =12Φ̃(0𝜆, 0𝜆)𝑞2𝜆 +14!Φ̃(0𝜆, 0𝜆, 0𝜆, 0𝜆)𝑞4𝜆 (6)=12Φ̃2𝑞2𝜆 +14!Φ̃4𝑞4𝜆, (7)6TABLE III. The interatomic force constants along the ferroelectric-like soft mode and the estimated transition temperatures. Φ̃2 and Φ̃4 arethe IFCs along the soft mode (0𝜆), which are defined as Φ̃2 = Φ̃(0𝜆, 0𝜆) and Φ̃4 = Φ̃(0𝜆, 0𝜆, 0𝜆, 0𝜆). 𝑇𝑐,est are the transition temperaturesestimated from the assumption that𝑇𝑐 ∝ 3Φ̃222Φ̃4and𝑇𝑐 of LiReO3 is 267 K.𝑇𝑐,calc is the𝑇𝑐 calculated by the SCPH-based structural optimization,which are also summarized in Table II.Φ̃2 [Ry/(𝑎2𝐵amu)] Φ̃4 [Ry/(𝑎2𝐵amu)2] 𝑇𝑐,est [K] 𝑇𝑐,calc [K]LiTaO3 -1.963 ×10−3 0.706 ×10−3 586 723LiWO3 -1.091 ×10−3 0.741 ×10−3 173 193LiReO3 -1.541 ×10−3 0.955 ×10−3 267 267LiOsO3 -1.591 ×10−3 1.326 ×10−3 205 207where we defined Φ̃2 = Φ̃(0𝜆, 0𝜆) and Φ̃4 =Φ̃(0𝜆, 0𝜆, 0𝜆, 0𝜆) for notational simplicity. 𝑞𝜆 is the atomicdisplacement along the ferroelectric-like soft mode in normalcoordinate representation. The depth of this double well po-tential is 3Φ̃222Φ̃4, which is roughly proportional to 𝑇𝑐 as shownin Table III. In Table III, 𝑇𝑐,est are the transition temperaturesestimated from the assumption that𝑇𝑐 ∝ 3Φ̃222Φ̃4and𝑇𝑐 of LiReO3is 267 K. Since 𝑇𝑐,est reproduces the trend of the calculatedtransition temperatures 𝑇𝑐,calc, we can conclude that the as-sumption 𝑇𝑐 ∝ 3Φ̃222Φ̃4holds approximately. Thus, the chemicaltrend of 𝑇𝑐 of Li𝐵O3 can be explained by the change of Φ̃2and Φ̃4 among these materials.According to Table III, Φ̃4 monotonically increases from𝐵=Ta to 𝐵=Os. This is presumably because the quartic in-teraction arises from the short-range repulsive forces betweenions, which generally get larger as the lattice constants getsmaller. |Φ̃2 | takes the largest value in LiTaO3, which leads tothe highest 𝑇𝑐 among the target materials. In the polar metals,|Φ̃2 | also monotonically increases from 𝐵=W to 𝐵=Os. If theinstability originates from the competition of the short-rangerepulsion and the long-range dipole-dipole interaction, whichis usually the case in ferroelectric insulators, the chemicaltrend of Φ̃2 would be opposite because the short-range repul-sion quickly gets prominent when the lattice constant shrinks.Thus, this 𝐵 site-dependence of |Φ̃2 | supports that the insta-bility of the polar metals has a short-range origin, which hasbeen suggested for LiOsO3 in previous research [21].E. Difference between ferroelectric insulators and polar metalsLastly, we discuss the origin of the difference in the transi-tion temperatures between the ferroelectric insulator LiTaO3and the polar metals Li𝐵O3 (𝐵 = W, Re, Os). In Fig. 6, we plotthe 𝑧-𝑧 components of the interatomic force constants (IFCs)of the 𝑛-th nearest neighbor (n.n.) shells from a Li ion. 𝑧direction is defined along the 𝑐 axis of the conventional cellof the rhombohedral structure. 𝑥 and 𝑦 axis are defined sothat the 𝑥𝑦𝑧 defines a rectangular coordinate system. Notethat similar discussions can be done on other components ofthe IFCs, which is shown in Section V in the supplementarymaterials [55]. The IFCs of different element pairs are plottedseparately. The 𝑛-th nearest neighbor shell of the element pairLi-𝐴 is defined as the 𝑛-th nearest group of 𝐴 atoms when wefix a Li atom and classify the 𝐴 atoms around it in terms ofthe distance from the fixed Li atom. The atomic distances ofthese 𝑛-th nearest neighbor shells of LiReO3 are summarizedin Table IV. The cases of 𝐵 = Ta, W, Os are shown in Tables S2to S4 in the supplementary materials [55], which are almostthe same as the LiReO3 case. Since the soft modes of Li𝐵O3are dominated by Li and O displacements, let us consider theresults of the element pairs Li-Li and Li-O [Fig. 6 (a), (c)].In fact, the contributions of the Li-𝐵 IFCs to the potential en-ergy surface along the soft modes are small as we later discussin this section. The short-range IFCs with atomic distancessmaller than 3.0 Å [0-th n.n. shell of Li-Li (Li onsite) andfirst and second n.n. shell of Li-O] are finite in both the fer-roelectric insulator and the polar metals. These IFCs seem tochange almost linearly when the 𝐵-site ion is changed. On theother hand, the long-range IFCs, which we define as IFCs withatomic distances larger than 3.0 Å in this paper [e.g. 3rd n.n.shell of Li-Li, 3rd, 5th, and 6th n.n. shells of Li-O], are finitein LiTaO3 but are close to zero in the polar metals.This distinction between the short-range and long-rangeIFCs can be seen more clearly in Fig. 7, in which we plotthe 𝐵-dependence of the IFCs. As we see from Fig. 7 (a), theshort-range IFCs show systematic dependence on 𝐵 site ions,and there is no apparent difference between the ferroelectricinsulator and the polar metals. Fig. 7 (b) shows representativeexamples of long-range IFCs that have relatively large contri-butions. These long-range IFCs have a significant finite valuein LiTaO3, but they are small and almost constant in polarmetals, which can ascribed to the screening of the Coulombicinteraction by the itinerant electrons. Although we discuss theimportance of O-O interactions in the supplementary materi-als [55], we do not consider them here because it is difficult tochoose important components such as 𝑧-𝑧 components. As theO-sites have lower symmetry, their displacements in the softmode are not along the 𝑧 direction, and atoms that belong tothe same nearest neighbor shell can be displaced in differentdirections.70 1 2 3 4 5shell number n−0.004−0.0020.0000.002harmonicIFC[Ry/a2 B](a) Li-Li IFC (z-z component)LiTaO3LiWO3LiReO3LiOsO31 2 3 4shell number n−0.03−0.02−0.010.000.01harmonicIFC[Ry/a2 B](b) Li-B IFC (z-z component)LiTaO3LiWO3LiReO3LiOsO31 2 3 4 5 6 7 8 9 10shell number n−0.0050.0000.005harmonicIFC[Ry/a2 B](c) Li-O IFC (z-z component)LiTaO3LiWO3LiReO3LiOsO3FIG. 6. The 𝑧𝑧 components of the interatomic force constants ofthe 𝑛-th nearest neighbor shells from a Li ion of Li𝐵O3 (𝐵 = Ta, W,Re, Os). The 𝑛-th nearest neighbor shells of the element pair Li-𝐴is defined as the 𝑛-th nearest group of 𝐴 atoms when we fix a Liatom as the center and classify the 𝐴 atoms around it in terms of thedistance from the fixed Li atom. The plotted IFCs of the 𝑛-th nearestneighbor shell of the element pair Li-𝐴 is Φ𝑧𝑧 (center Li atom,R𝛼),where 𝑛-th R𝛼 is included in the nearest neighbor shell. Note thatwe consider the onsite IFC as the IFC of the zero-th nearest neighbor.TABLE IV. 𝑛-th nearest neighbor shells from a Li ion of LiReO3.The shell number 𝑛 with the number of atoms in the 𝑛-th n.n. shell,and the corresponding atomic distances. The results are calculated onthe crystal structure in the high-temperature 𝑅3̄𝑐 phase without SOC.Note that the number of atoms in the shells are calculated within the2×2×2 supercell considering the periodic boundary condition.Li-Lishell number 𝑛 num. of atoms distance [Å]0 1 0.00001 6 3.70492 3 5.12673 3 5.35004 2 6.32535 1 7.4098Li-𝐵shell number 𝑛 num. of atoms distance [Å]1 6 3.16262 2 3.34253 2 6.02374 6 6.1200Li-Oshell number 𝑛 num. of atoms distance [Å]1 3 1.98522 6 2.73683 3 3.14144 6 4.20335 6 4.47736 6 4.73147 6 4.85748 6 5.27699 3 5.706510 3 6.204180.0(B=Ta) 1.0(B=W) 2.0(B=Re) 3.0(B=Os)number of d-electrons per B site−0.0050.0000.005harmonicIFC[Ry/a2 B](a) short-range IFCs, z-z componentsLi-Li onsiteLi-O, 1st n.n.Li-O, 2nd n.n.0.0(B=Ta) 1.0(B=W) 2.0(B=Re) 3.0(B=Os)number of d-electrons per B site−0.0020.0000.0020.004harmonicIFC[Ry/a2 B](b) long-range IFCs, z-z componentsLi-Li, 3rd n.n.Li-O, 3rd n.n.Li-O, 5th n.n.Li-O, 6th n.n.Li-O, 7th n.n.FIG. 7. 𝐵-site dependence of the 𝑧-𝑧 components of the interatomicforce constants (IFCs) of 𝑛-th nearest neighbor (n.n.) shells of dif-ferent element pairs. (a) The short-range IFCs with atomic distancesmaller than 3 Å. (b) The long-range IFCs with atomic distance largerthan 3 Å.Furthermore, we apply cutoffs to the harmonic IFCs andinvestigate the change of instabilities along the ferroelectric-like soft modes. Fig. 8 shows the cutoff dependence of thecurvature of the potential energy surface. Here, we set theharmonic IFCs with atomic distance larger than the cutoff aszero and calculate𝑑2𝑈𝑑𝑞2𝜆=∑︁dist(0𝛼,R1𝛼1 )