www.nature.com/scientificreports

OPEN

Permittivity boosting by induced
strain from local doping in titanatesfrom first principles
Alex Kutana , Yuho Shimano  & Ryoji Asahi *We examine the effect of isovalent substitutions and co-doping on the ionic dielectric constant ofparaelectric titanates (perovskite, Ruddlesden-Popper phases, and rutile) using density functional
perturbation theory. Substitutions increase the ionic dielectric constant of the prototype structures,and new dynamically stable structures with εion ~ ­102–104 are reported and analyzed. The boostingof ionic permittivity is attributed to local defect-induced strain, and maximum Ti–O bond length is
proposed as a descriptor. The Ti–O phonon mode that is responsible for the large dielectric constantcan be tuned by a local strain and symmetry lowering from substitutions. Our findings help explain
the recently observed colossal permittivity in co-doped rutile, attributing its intrinsic permittivity
boosting solely to the lattice polarization mechanism, without the need to invoke other mechanisms.
Finally, we identify new perovskite- and rutile-based systems that can potentially display colossal
permittivity.
Achieving materials with a very large (colossal) dielectric permittivity (CP) is crucial for various technologicalapplications, including miniaturization of on-chip capacitive elements in transistors, high density memory,
sensors, as well as high power density energy s­ torage1. Due to their temperature and electric field stability, paraelectric materials are preferred for CP over ferroelectrics whose dielectric constant is high but also unstable
near the phase transition temperature. With recent experimental reports on colossal permittivity in paraelectric­TiO2 rutile via co-doping2,3, microscopic mechanisms for permittivity boosting need to be well understood.
The apparent CP due to polarization of charge depletion layers at contacts, grain boundaries, interfaces, andother spatial h­ eterogeneities4,5 (the so-called Maxwell–Wagner effect), is classified as “extrinsic”. Other extrinsic
mechanisms include thermally activated conductivity by carrier hopping. A notable example is ­CaCu3Ti4O12,
having a colossal permittivity close to ­105 by the Maxwell–Wagner ­effect4,6–8. On the other hand, possible bulk
mechanisms including ionic polarization, metal–insulator transition, and charge density wave f­ ormation8 can beclassified into “intrinsic”, i.e. bulk mechanisms that are active at all temperatures. Of these the ionic polarization
is of greatest practical interest, as other mechanisms incur high dielectric losses, limiting applications. Since theearly reports of large, temperature-dependent permittivity in insulating tetragonal ­YBa2Cu3O6+x9,10, the overlapof intrinsic and extrinsic processes has complicated pinpointing the microscopic origins of the apparent CP. In
co-doped rutile, a thermally activated process, most likely the transport of hopping carriers, was found to be
responsible for the largest part of the apparent colossal p­ ermittivity3. At the same time, a nonactivated, intrinsic
mechanism of permittivity boosting coexists in Nb + In co-doped rutile. The mechanism does not switch off at
low temperatures; in fact, the enhanced permittivity increases with decreasing ­temperature3,11. The temperature
trend follows that of pure rutile, suggesting a common microscopic origin. While theory accounts well forthe observed dielectric constant of pure rutile, and attributes it largely to the ionic mechanism, no theoreticalexplanation exists for permittivity boosting in co-doped rutile. Originally, the “electron-pinned defect-dipoles”
mechanism has been proposed in In + Nb co-doped ­TiO2 ­rutile2, suggesting the electronic nature of the large
response. Further studies of In + Ta and Al, Ga, In + Nb co-doped ­TiO212,13 also attributed the observed colossal
permittivity to the electronic mechanism. On the other hand, the studies of Nb + In co-doped T­ iO2 single crystals
at sufficiently low temperature where thermally excited carriers are frozen suggested that the enhancement of
the permittivity should stem from intrinsic e­ ffects3,11.
Here we show that the ionic mechanism can indeed account for boosting the permittivity in co-doped rutile
and other substituted metal oxides. To this end, we perform a computational study of the ionic part of thedielectric permittivity of co-doped and substituted rutile and other Ti-containing dielectric oxides. We focus on
substitutions in stable paraelectric prototypes, such as rutile ­TiO2, perovskite ­CaTiO3, and Ruddlesden–Popper­Sr2TiO4 and and ­Sr3Ti2O7, modified via co-doping or isovalent replacement. From the analysis of the results,
Nagoya University, Nagoya, Aichi 464‑8603, Japan. *email: ryoji.asahi@chem.material.nagoya-u.ac.jp

Scientific Reports |(2023) 13:3761| https://doi.org/10.1038/s41598-023-30965-6

1
Vol.:(0123456789)

www.nature.com/scientificreports/

Figure 1.  (a) Ionic part of the dielectric constant, εion, as a function of max(dTi–O), the maximum length of theTi–O bond in substituted/co-doped titanates. Different colors correspond to different prototype structures;(b) <εion> vs. max(dTi–O) in rutile under uniform strain, showing a pole at the ferroelectric transition when thefrequency of the A­ 2u mode becomes imaginary. The shaded area corresponds to the range of the maximum
dielectric constants in (a); the vertical dashed line shows max(dTi–O) at equilibrium lattice constant.
the permittivity increase in these materials is attributed to impurity-induced local strain, which enhances thedielectric response by softening the Ti–O phonon mode. We find that the frequency of the mode dominating thedielectric response can be changed over a wide range by substitutions, offering a possibility to precisely engineer
the ionic dielectric constant.

Results and discussion

We examine the effects of isovalent substitutions and co-doping for titanates on phonons, Born effective charges,
and ionic part of the dielectric constant. The electronic part of the dielectric constant is typically ~ 1–10 acrossmost insulators and was not considered here in detail. The prototypes for substitution include mineral perovskite ­CaTiO3 with Pnma structure, rutile T­ iO2, as well as Ruddlesden-Popper phases S­ r2TiO4 and S­ r3Ti2O7. Ca,
Sr → Ba, Pb, Sr, Ca substitutions in perovskites, as well as III–V (Al, Ga, In, Sc, Y, La–V, Nb, Ta) and II-VI (Mg,
Ca, Sr, Ba–Cr, Mo, W) co-doping in rutile are performed. A more detailed description of the parent compounds
and substitutions is given in the Supplementary Information. A total of 150 rutile and 1178 perovskite andRuddlesden-Popper structures were calculated. The goal is to analyze the structures where substitutions increase
εion and identify relevant phonon modes. Among dynamically stable titanates (ν2 > 0), we found the maximum
Ti–O bond length, max(dTi-O), reflecting maximum local strain, to be a good descriptor of the highest achievableionic dielectric constant. The correlation between εion and max(dTi-O) is rationalized in terms of the softening of
the phonon mode that corresponds to the Ti–O bond stretching. The descriptor was considered by analogy with
Goldschmidt ­factor14, which is used as a descriptor for structural stability of perovskites. As high ionic dielectric
constant is dominated by the Ti–O mode, lowering its frequency significantly increases the dielectric response.
The ionic part of the dielectric tensor εion αβ is calculated as a sum over the gamma point phonon modes s
according ­to15,16∗ Z̄e2  Z̄sαsβs (2πν s )2
ε0 M 0 V∗

εion αβ =(1)Here, e is an elementary
M0 is a convenient mass reference, typically taken to be 1 a­ mu16, V a unit charge,
1/2
∗
∗ =
Z
ξs,κβ is the αth Cartesian component of the unnormalized effeccell volume, Z̄sα(Mκ,αβ0 /Mκ )
κβtive charge vector for phonon mode s, and νs is mode frequency. ξs,κβ are the eigenvectors of the dynamical
matrix, normalized according to Σκβξs,κβ ξs’,κβ = δss’, and Z­ *κ,αβ are the atomic Born effective charges, describing the
15,17­proportionality
between forces on atoms the and macroscopic electric field as eZ ∗ κ,αβ = ∂Fκ,β /∂ Eα. Here,Fκ,β is the βth component of the force acting on the κth atom, and Eα is the αth component of the macroscopic
electric field. As the mixed second derivatives of the total energy, Born effective charges can also be expressed as
the derivatives of the components of the polarization vector Pα with respect to the atomic displacements τκ,β as:eZ ∗ κ,αβ = V ∂ Pα /∂τκ,β. By analyzing the contributions to the ionic dielectric constant from the sum-over-modes
Eq. (1), one can gain insights into the mechanisms of the dielectric constant boosting. Note that only modes
with nonzero effective charges contribute to the static tensor Eq. (1). In order for the component of the mode
effective charge Z̄ ∗ to be large in any given direction, both the atomic Born effective charges Z* and dynamical
matrix eigenvector ξ need to have large components in that direction for the same atom.
Figure 1a shows the dielectric constants in substituted/co-doped oxides as a function of max(dTi-O), with a visible peak around max(dTi-O) ≃ 2.02 Å. The average ionic dielectric constants of co-doped rutile ­TiO2 as a functionof max(dTi-O) is also shown in Fig. S1. Other geometric descriptors, such as the cationic radius of co-dopants, do
Scientific Reports |Vol:.(1234567890)(2023) 13:3761 |https://doi.org/10.1038/s41598-023-30965-6

2

www.nature.com/scientificreports/

Figure 2.  Phonon mode frequencies and their contributions to the sum over modes expression Eq. (1) for
the ionic dielectric constant in substituted and co-doped titanates with a high εion zz as well as the reference
perovskite and rutile structures. Positions of Ca, Sr, Ba atoms and Ti, In, Nb in the unit cell are shown forperovskites and rutile, respectively, and O atoms are omitted. Dark blue—Ca, orange—Sr, green—Ba, blue—Ti,
dark green—Nb, purple—In.

not provide a good correlation with <εion>, as seen from Figs. S5 and S6. The value of max(dTi–O) near the peakof εion is higher than the typical bond lengths in titanates with moderate dielectric constants (e.g., in rutile, withthe calculated <εion>  = 140, max(dTi–O) = 1.98 Å), raising the possibility of mode softening by local strain fromsubstitutions. The soft ferroelectric Ti–O mode has been studied extensively in t­ itanates18,19, and its sensitivity
to external perturbations such as volume changes or degree of Ti 3d–O 2p hybridization has been well recognized. In rutile, volume increase drives the Ti–O A­ 2u mode s­ oftening20, with frequency becoming imaginary andstructure developing instability and possibly becoming ferroelectric at a certain extensive strain. The instability
is manifested across different t­ itanates21, and stabilized by anharmonic e­ ffects22. In Fig. 1b, we show the increaseof <εion> with max(dTi–O) in Ti rutile under uniform strain.The dielectric constant has a pole where the ­A2u mode
becomes unstable with its frequency becoming zero. The shaded area near the pole corresponds to the range ofthe maximum dielectric constants in Fig. 1a. It is seen that the maximum εion in substituted/co-doped oxides isreached in the same range of max(dTi–O) where the softening of Ti–O mode occurs in rutile. By examining the
microscopic origin of the large response, it is indeed confirmed that the same soft Ti–O mode is responsible forthe large dielectric constant in these oxides. Note that we do not find similar mode softening in Si- or Sn-based
rutile structures, as seen from Figs. S2 and S3, where the dependencies of the dielectric constant on strain in­TiO2, ­SiO2, and ­SnO2 rutile structures are compared.
As an illustration, consider ­Ba2Sr2Ti4O12, obtained from the orthorhombic ­CaTiO3 by Ca → Ba, Sr substitutions, Fig. 2a and b. The structure has alternating planes of Ba and Sr normal to the z axis and is dynamicallystable. It has an anisotropic ionic dielectric tensor with <εion>  = 3373, εion zz = 9015, dominated by the lowest optical mode at ν4 = 0.521 THz, contributing 99.9% of εion zz. This is unlike C­ aTiO3, where many modes contributeto εion, as seen from the summary given in Table S1. In B
­ a2Sr2Ti4O12, max(dTi–O) = 2.01 Å, similar to that of the
dynamically unstable cubic B­ aTiO3, but unlike that structure, shorter Ti–O bonds are also present. Apparently,
the ferroelectric Ti–O mode of ­BaTiO3 is stabilized by insertion of Sr planes, preventing the formation of unstable
Ti–O chains. This effect can also be viewed as the softening of the infrared-active modes of ­CaTiO3 by Ca → Ba
and Ca → Sr substitutions. Just like the rutile, the infrared-active modes of ­CaTiO3 are softened by ­tension23. Projecting this low energy mode onto the modes of cubic ­BaTiO3 shows that it is mostly made up of the ferroelectric
Ti–O mode of ­BaTiO3, with a small contribution from the displacement of Ba against the Ti–O cage. Similarly, alarge εion zz = 1228 is achieved in a ­Ba2Ca2Ti4O12(α) structure with alternating Ba and Ca planes, Fig. 2c. Note thatthe frequencies of the dominant modes are similar in the B­ a2Sr2Ti4O12 and ­Ba2Ca2Ti4O12(α) structures, and the∗ = 3.72, compared to Z̄ ∗ = 1.33 in B
larger εion zz in ­Ba2Sr2Ti4O12 is due to its larger Z̄4z
­ a2Ca2Ti4O12(α) in Fig. 2c. The
4z
dominant mode frequency is higher in another, ­Ba2Ca2Ti4O12(β), structure shown in Fig. 2d, ν5 = 2.02 THz, with∗
∗ = 3.52, giving εZ̄5z
ion zz = 568 for that structure, as compared to ν4 = 0.53 THz, Z̄4z = 1.33 for the ­Ba2Ca2Ti4O12(α)
in Fig. 2c. Thus, the frequency and effective charge of the dominant mode is sensitive not only to composition,
but also to the atomic arrangement on the lattice at a fixed composition. By comparing the eigen displacements
Scientific Reports |(2023) 13:3761 |https://doi.org/10.1038/s41598-023-30965-6

3
Vol.:(0123456789)

www.nature.com/scientificreports/

Figure 3.  Eigen displacements ξν,κα/Mκ1/2 of phonon modes relevant to the dielectric constant in substituted
titanates (a) mode 4 in ­Ba2Sr2Ti4O12; (b) mode 5 in ­Ba2Ca2Ti4O12(β); (c) mode 4 in Nb-In co-doped ­TiO2 rutile.
Blue—Ti, red—O, orange—Sr, dark blue—Ca, green—Ba, dark green—Nb, purple—In. For clarity, atoms
outside of Ti–O cages are not shown in (a) and (b).
of the dominant modes ν4 and ν5 for the structures in Fig. 2c and d, respectively, the former has a significant
motion of Ca and Ba planes against each other in the z direction, which is prevented in the latter by staggering
Ca and Ba as in Fig. 2d. This staggered arrangement leads to dominant mode hardening from 0.53 to 2.02 THz.
We find this mechanism of permittivity enhancement through softening/mixing Ti–O modes to act in othersubstituted titanates. We obtain εion zz = 252 in ­BaCa3Ti4O12 (not shown), a modest enhancement over the prototype ­CaTiO3 with εion zz = 108. The dominant mode 5 in ­BaCa3Ti4O12 at 2.51 THz contributes ~ 184 to its εion zz.
Projection onto the modes of Pnma ­BaTiO3 shows it to be a linear combination of the modes at 4.17i THz, 3.26
THz, and 5.26 THz of ­BaTiO3. The former is the ferroelectric Ti–O mode, whereas the latter two are Ba-onlymodes. Here, the stable A-site modes are mixed into the Ti–O mode, leading to its stabilization. Thus, lowering
the symmetry of the structure via substitution is important for the stabilization of the Ti–O mode. This is to be
expected, as the Ti–O chain structure instability is nonlocal, i.e., Ti and O atoms are coherently displaced along
an infinite Ti–O chain, and there exists a minimum correlation length for the mode to become u­ nstable24. The
instability has been extensively ­analyzed18,24, with Ti 3d–O 2p hybridization and a partially covalent nature of
the Ti–O bond being determined essential to the f­ erroelectricity18,19. The 1D chain-like nature of the instability
was noted from the mode’s nearly flat zero-frequency isosurfaces in the Brillouin ­zone25. Real-space analysis has­shown24 that the minimum length of the Ti–O chain for the instability to appear is ~ 10 atoms or 5 unit cells; thus
any stabilizing perturbation must act on a shorter scale. In the Nb-In co-doped rutile, a soft mode with a large
dielectric constant contribution appears at ν = 0.594 THz, as seen in Fig. 2e. In pure rutile, Fig. 2f, the active A­ 2umode has a frequency of 5.28 THz.
The visualizations of phonon modes responsible for the large dielectric response are shown in Fig. 3. In­Ba2Sr2Ti4O12 and B­ a2Ca2Ti4O12(α), shown in Fig. 3a and b, the atomic displacements are very similar to those of
the ferroelectric ­BaTiO3 mode, which is also confirmed by mode projections. In contrast, in the Nb-In co-doped
rutile, the mode is localized on a single Ti atom in the xy plane, and delocalized along the z direction, as seen in
Fig. 3c, giving it a chain-like character, possibly due to symmetry breaking by co-doping.
We also note that the value εion zz = 9015 obtained for one of B
­ a2Sr2Ti4O12 structures may be close to a practically achievable maximum value for the ionic dielectric constant. From Eq. (1), one has εion ≃205 × (Z*/ν)2,
assuming a cubic unit cell with a side of 6 Å, and mode frequency ν in THz. Thus, in order to have εion ~ ­104,
one needs Z*≃7 for ν = 1 THz. As larger Z* values to achieve even higher εion seem unrealistic, further increasing
εion then requires decreasing the mode frequency below 1 THz, which may eventually result in instability andswitching to a ferroelectric state.
In order to assess the feasibility of synthesis of doped titanates, we calculate their formation energies bycomputing the energies of the reactions

Ax Bx Ti2(1−x) O2 → (x/2)A2 O3 + (x/2)B2 O5 + 2(1 − x)TiO2
Ax Bx Ti2(1−x) O2 → xAO + xBO3 + 2(1 − x)TiO2
for III–V and II–VI substitutions in co-doped rutile, respectively, andAx A′1−x TiO3 → xAO + (1 − x)A′ O + TiO2
A(n+1)x A′(n+1)(1−x) Tin O3n+1 → (n + 1)xAO + (n + 1)(1 − x)A′ O + nTiO2 , n = 1, 2

Scientific Reports |Vol:.(1234567890)(2023) 13:3761 |https://doi.org/10.1038/s41598-023-30965-6

4

www.nature.com/scientificreports/

Figure 4.  (a) Formation energies and (b) average ionic dielectric constants as a function of band gaps for
Ti-containing oxides. Red—co-doped rutile, blue—substituted perovskite and Ruddlesden–Popper phases.
for the perovskites and Ruddlesden–Popper phases, respectively. The formation energy is defined as the energy
difference per atom between the compounds on the left and right sides of the reaction. The obtained formationenergies of titanates are shown in Fig. 4a, along with their band gaps. As can be seen from the figure, most of thesubstituted titanates have negative formation energies, favoring mixing, and moreover, have large band gaps.
On the other hand, achieving co-doping in rutile could be more difficult, and some of the resulting compounds
could be metallic. At the same time, there are a number of thermodynamically stable rutile compounds withmoderate band gaps. Configurational energies, defined as the maximum formation energy difference among
the structures of the same stoichiometry, are shown in Fig. S7. These can be rather small, e.g., ~ 3 meV/atom for­Ba2Sr2Ti4O12, compared with its formation energy of ~ − 145 meV/atom. The closeness in energy among different configurations indicates that their mixing should occur at finite temperatures, and thermodynamic averaging, together with consideration of anharmonic effects, should be performed in order to estimate the dielectric
constant at nonzero temperature.
CP materials must possess sufficiently large band gaps in order to prevent dielectric breakdown, which isbelieved to be electronic in n
­ ature26. Our calculated average ionic dielectric constants as a function of the band
gap are shown in Fig. 4b, and the electronic contribution to the dielectric constant is shown in Fig. S4. Note thatthe electronic contribution does not exceed 10 for materials with band gaps > 0.6 eV. It is seen that generally
insulators with larger dielectric constants tend to have smaller band gaps. This undesirable trend is consistent
with the empirically known inverse relation between the dielectric constant and dielectric breakdown s­ trength27.
Discovering CP materials with both high dielectric constant and wide band gap remains a challenge that must
be addressed in the future work.
Based on the results of first-principles calculations, we show that co-doping and substitutions offer a versatile and efficient pathway to increase the ionic dielectric constant in titanate paraelectrics. Most of the obtained
compounds are found to be both dynamically and thermodynamically stable, offering a good prospect of experimental synthesis. Large ionic dielectric constants εion ~ ­104 can be achieved in some compounds, establishingthem as new candidates for colossal permittivity materials. The microscopic origin of the permittivity increase,namely the softening of the optically active phonon mode by a local defect-induced strain, may point to a new
direction to search for colossal permittivity in other materials.

Methods

The ionic dielectric response was calculated using density functional perturbation theory (DFPT) approach,as implemented in the VASP package. PBEsol f­ unctional28 with Hubbard U correction was used. U = 3 eV wasapplied to d electrons in transition metals, except for Ti and Sc, where U = 0 was used. The full list of the U values
is given in the Supplementary Information. The k point grid density of 3000 k points·atom (approximately corresponding to a distance of ~ 0.2 Å−1 between the k points) was used for the integration over the Brillouin zone.
Full cell shape and atomic position relaxations were carried out prior to DFPT calculations.

Data availability

The datasets generated during the current study will be available at https://​doi.​org/​10.​17632/​n9j8h​4h9g3.1 following a 6 month embargo from the date of publication to allow for patenting of research findings.
Received: 30 November 2022; Accepted: 3 March 2023

Scientific Reports |(2023) 13:3761 |https://doi.org/10.1038/s41598-023-30965-6

5
Vol.:(0123456789)

www.nature.com/scientificreports/

References

1. Yang, Z., Du, H., Jin, L. & Poelman, D. High-performance lead-free bulk ceramics for electrical energy storage applications: Design
strategies and challenges. J. Mater. Chem. A 9(34), 18026–18085. https://​doi.​org/​10.​1039/​D1TA0​4504K (2021).
2. Hu, W. et al. Electron-pinned defect-dipoles for high-performance colossal permittivity materials. Nat. Mater. 12(9), 821–826.
https://​doi.​org/​10.​1038/​nmat3​691 (2013).
3. Kawarasaki, M., Tanabe, K., Terasaki, I., Fujii, Y. & Taniguchi, H. Intrinsic enhancement of dielectric permittivity in (Nb + In)
Co-doped ­TiO2 single crystals. Sci. Rep. 7(1), 5351. https://​doi.​org/​10.​1038/​s41598-​017-​05651-z (2017).
4. Lunkenheimer, P. et al. Origin of apparent colossal dielectric constants. Phys. Rev. B 66(5), 052105. https://​doi.​org/​10.​1103/​PhysR​
evB.​66.​052105 (2002).
5. Cohen, M. H., Neaton, J. B., He, L. & Vanderbilt, D. Extrinsic models for the dielectric response of ­CaCu3Ti4O12. J. Appl. Phys.
94(5), 3299–3306. https://​doi.​org/​10.​1063/1.​15957​08 (2003).
6. Subramanian, M. A., Li, D., Duan, N., Reisner, B. A. & Sleight, A. W. High dielectric constant in A
­ Cu3Ti4O12 and A­ Cu3Ti3FeO12
Phases. J. Solid State Chem. 151(2), 323–325. https://​doi.​org/​10.​1006/​jssc.​2000.​8703 (2000).
7. Homes, C. C., Vogt, T., Shapiro, S. M., Wakimoto, S. & Ramirez, A. P. Optical response of high-dielectric-constant perovskiterelated oxide. Science 293(5530), 673–676. https://​doi.​org/​10.​1126/​scien​ce.​10616​55 (2001).
8. Lunkenheimer, P. et al. Colossal dielectric constants in transition-metal oxides. Eur. Phys. J. Spec. Top. 180(1), 61–89. https://​doi.​
org/​10.​1140/​epjst/​e2010-​01212-5 (2009).
9. Samara, G. A., Hammetter, W. F. & Venturini, E. L. Temperature and frequency dependences of the dielectric properties of­YBa2Cu3O6+x (X≊0). Phys. Rev. B 41(13), 8974–8980. https://​doi.​org/​10.​1103/​PhysR​evB.​41.​8974 (1990).
10. Rey, C. M., Mathias, H., Testardi, L. R. & Skirius, S. High dielectric constant and nonlinear electric response in nonmetallic­YBa2Cu3O6+δ. Phys. Rev. B 45(18), 10639–10646. https://​doi.​org/​10.​1103/​PhysR​evB.​45.​10639 (1992).
11. Taniguchi, H., Sato, D., Nakano, A. & Terasaki, I. Permittivity boosting in “yellow” (Nb + In) Co-doped T­ iO2. J. Mater. Chem. C
8(39), 13627–13631. https://​doi.​org/​10.​1039/​D0TC0​3539D (2020).
12. Dong, W. et al. Colossal permittivity with ultralow dielectric loss in In + Ta Co-doped rutile ­TiO2. J. Mater. Chem. A 5(11),5436–5441. https://​doi.​org/​10.​1039/​C6TA0​8337D (2017).
13. Tuichai, W., Danwittayakul, S., Srepusharawoot, P., Thongbai, P. & Maensiri, S. Giant dielectric permittivity and electronic structurein (A3+, Nb5+) Co-doped ­TiO2 (A = Al, Ga and In). In 10th Asian Meet. Electroceramics AMEC-10 2017, 43, S265–S269. https://​doi.​org/​10.​1016/j.​ceram​int.​2017.​05.​255.
14. Goldschmidt, V. M. Die gesetze der krystallochemie. Naturwissenschaften 14(21), 477–485. https://​doi.​org/​10.​1007/​BF015​07527(1926).
15. Gonze, X. & Lee, C. Dynamical matrices, born effective charges, dielectric permittivity tensors, and interatomic force constantsfrom density-functional perturbation theory. Phys. Rev. B 55(16), 10355–10368. https://d​ oi.o​ rg/1​ 0.1​ 103/P​ hysRe​ vB.5​ 5.1​ 0355 (1997).
16. Zhao, X. & Vanderbilt, D. First-principles study of structural, vibrational, and lattice dielectric properties of hafnium oxide. Phys.
Rev. B 65(23), 233106. https://​doi.​org/​10.​1103/​PhysR​evB.​65.​233106 (2002).
17. Souza, I., Íñiguez, J. & Vanderbilt, D. First-principles approach to insulators in finite electric fields. Phys. Rev. Lett. 89(11), 117602.
https://​doi.​org/​10.​1103/​PhysR​evLett.​89.​117602 (2002).
18. Cohen, R. E. & Krakauer, H. Lattice dynamics and origin of ferroelectricity in B
­ aTiO3: Linearized-augmented-plane-wave totalenergy calculations. Phys. Rev. B 42(10), 6416–6423. https://​doi.​org/​10.​1103/​PhysR​evB.​42.​6416 (1990).
19. Cohen, R. E. Origin of ferroelectricity in perovskite oxides. Nature 358(6382), 136–138. https://​doi.​org/​10.​1038/​35813​6a0 (1992).
20. Montanari, B. & Harrison, N. M. Pressure-induced instabilities in bulk T­ iO2 rutile. J. Phys. Condens. Matter 16(3), 273–292. https://​doi.​org/​10.​1088/​0953-​8984/​16/3/​008 (2004).
21. Sophia, G., Baranek, P., Sarrazin, C., Rérat, M. & Dovesi, R. First-principles study of the mechanisms of the pressure-induced
dielectric anomalies in ferroelectric perovskites. Phase Transit. 86(11), 1069–1084. https://​doi.​org/​10.​1080/​01411​594.​2012.​754442(2013).
22. Choi, W. I., An, J. S., Jang, I. & Kim, D. S. Strain and temperature-dependent dielectric permittivity of cubic S­ rTiO3: Self-consistentphonon theory calculations. Curr. Appl. Phys. 29, 78–82. https://​doi.​org/​10.​1016/j.​cap.​2021.​06.​004 (2021).
23. Cockayne, E. & Burton, B. P. Phonons and static dielectric constant in C
­ aTiO3 from first principles. Phys. Rev. B 62(6), 3735–3743.
https://​doi.​org/​10.​1103/​PhysR​evB.​62.​3735 (2000).
24. Ghosez, P. S. H., Gonze, X. & Michenaud, J. P. Ab initio phonon dispersion curves and interatomic force constants of barium
titanate. Ferroelectrics 206(1), 205–217. https://​doi.​org/​10.​1080/​00150​19980​80091​59 (1998).
25. Yu, R. & Krakauer, H. First-principles determination of chain-structure instability in K
­ NbO3. Phys. Rev. Lett. 74(20), 4067–4070.
https://​doi.​org/​10.​1103/​PhysR​evLett.​74.​4067 (1995).
26. Ridley, B. K. Quantum Processes in Semiconductors (Oxford University Press, 2013). https://​doi.​org/​10.​1093/​acprof:​oso/​97801​
99677​214.​001.​0001.
27. McPherson, J. W., Kim, J., Shanware, A., Mogul, H. & Rodriguez, J. Trends in the ultimate breakdown strength of high dielectricconstant materials. IEEE Trans. Electron Devices 50(8), 1771–1778. https://​doi.​org/​10.​1109/​TED.​2003.​815141 (2003).
28. Perdew, J. P. et al. Restoring the density-gradient expansion for exchange in solids and surfaces. Phys. Rev. Lett. 100(13), 136406.
https://​doi.​org/​10.​1103/​PhysR​evLett.​100.​136406 (2008).

Acknowledgements

We thank Drs. Pradeep R. Varadwaj, Hiroki Taniguchi, Van An Dinh, Yoshitada Morikawa, and Koichi Hayashi
for their fruitful discussions. The work was also supported by the JSPS Grant-in-Aid for Transformative Research
Areas (A) (21H05560).

Author contributions

All authors contributed equally to the manuscript.

Competing interests

The authors declare no competing interests.

Additional information

Supplementary Information The online version contains supplementary material available at https://​doi.​org/​
10.​1038/​s41598-​023-​30965-6.
Correspondence and requests for materials should be addressed to R.A.
Reprints and permissions information is available at www.nature.com/reprints.

Scientific Reports |Vol:.(1234567890)(2023) 13:3761 |https://doi.org/10.1038/s41598-023-30965-6

6

www.nature.com/scientificreports/
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps andinstitutional affiliations.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International
License, which permits use, sharing, adaptation, distribution and reproduction in any medium or
format, as long as you give appropriate credit to the original author(s) and the source, provide a link to theCreative Commons licence, and indicate if changes were made. The images or other third party material in thisarticle are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to thematerial. If material is not included in the article’s Creative Commons licence and your intended use is not
permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly fromthe copyright holder. To view a copy of this licence, visit http://​creat​iveco​mmons.​org/​licen​ses/​by/4.​0/.
© The Author(s) 2023

Scientific Reports |(2023) 13:3761 |https://doi.org/10.1038/s41598-023-30965-6

7
Vol.:(0123456789)