Polymorphs of Titanium Dioxide: An Assessment of the Variants of Projector Augmented Wave Potential of Titanium on Their Geometric and Dielectric Properties

Abstract

Titanium dioxide (TiO₂) is one of the important functional materials owing to its diverse applications in many fields of chemistry, physics, nanoscience, and technology. Hundreds of studies on its physicochemical properties, including its various phases, have been reported experimentally and theoretically, but the controversial nature of relative dielectric permittivity of TiO₂ is yet to be understood. Toward this end, this study was undertaken to rationalize the effects of three commonly used projector augmented wave (PAW) potentials on the lattice geometries, phonon vibrations, and dielectric constants of rutile (R-)TiO₂ and four of its other phases (anatase, brookite, pyrite, and fluorite). Density functional theory calculations within the PBE and PBEsol levels, as well as their reinforced versions PBE+U and PBEsol+U (U = 3.0 eV), were performed. It was found that PBEsol in combination with the standard PAW potential centered on Ti is adequate to reproduce the experimental lattice parameters, optical phonon modes, and the ionic and electronic contributions of the relative dielectric permittivity of R-TiO₂ and four other phases. The origin of failure of the two soft potentials, namely, Ti_pv and Ti_sv, in predicting the correct nature of low-frequency optical phonon modes and ion-clamped dielectric constant of R-TiO₂ is discussed. It is shown that the hybrid functionals (HSEsol and HSE06) slightly improve the accuracy of the above characteristics at the cost of a significant increase in computation time. Finally, we have highlighted the influence of external hydrostatic pressure on the R-TiO₂ lattice, leading to the manifestation of ferroelectric modes that play a role in the determination of large and strongly pressure-dependent dielectric constant.

1. Introduction

Semiconducting crystalline TiO₂ has been experimentally observed in three different temperature phases.¹⁻⁴ These include the rutile phase, the anatase phase, and the brookite phase. The rutile phase, with a band gap of 3.0 eV,⁵ exists at temperatures around 1097 and 1172 K and is stable at ambient conditions.^1,2 Allen et al.⁶ have found that a mixture of the brookite and anatase phases of TiO₂ can be observable up to 772 K, whereas that of the anatase, brookite, and rutile phases can be observable around 872 K, and the rutile-only phase is observable at and above 1097 K. The anatase and brookite phases of TiO₂ are semiconducting systems with band gaps of 3.4 eV⁷ and 3.3 eV,⁸ respectively.

One of the fundamental physical properties of rutile TiO₂ (hereafter, R-TiO₂) is the relative dielectric permittivity (expressed as ε_r)⁹⁻¹⁶ that provides insight into the applicability of Ti-based materials in high energy density storage devices,^17,18 supercapacitors,¹⁹ among other technological areas.²⁰ A number of theoretical²¹⁻²³ and experimental^11−16,24 investigations have been centered on the determination of ε_r. Specifically, Schöche et al.²⁵ have discussed the low- and high-frequency contributions to ε_r reported by various authors. From these studies, it is apparent that different research groups have reported different values of ε_r (ε_r values between 100 and 10,000). The variability in the observed or proposed values of ε_r has caused many research attentions owing to the underlying controversy. Bonkerud and co-workers¹⁶ have recently uncovered that the giant ε_r reported in early studies^11,12 was due to an incorrectly designed experiment or an incorrect interpretation. Their argument was based on the early capacitance measurements that have considered the full thickness of the crystal, where metallic contacts were deposited on opposite sides. In any case, R-TiO₂ ceramics when co-doped with Nb + Tr (Tr = Al, In, Ga, Tl, Dy, Gd, La, Y, Sm, V, Li) feature colossal dielectric permittivity.²⁶⁻²⁸ The experimental ε_r values for these systems vary between 10,000 and 100,000. They are yet to be confirmed by first principles and/or density functional theory (DFT) calculations.

Lee et al.²⁹ have studied the dielectric properties of R-TiO₂ using a variety of exchange correlation energy DFT functionals, including the local density approximation (LDA), the LDA+U, the generalized gradient approximation (GGA) method, and the PBE0 and HSE06 hybrid functionals. The essence of the study was that the dielectric properties of R-TiO₂ vary upon changing the theoretical methods applied, and the frequencies of phonon vibrations may or may not be softened at high-symmetry points in k-space. The ε_r values calculated using LDA were in close agreement with experiment, and those calculated with the HSE06 functional and LDA+U method were overestimated and severely underestimated, respectively. For instance, the ionic contribution to ε_r was reported to be 63.7, 40.0, 176.5, 282.1, and 376.4 with LDA+U (U = 3.0 eV), LDA+U (U = 4.36 eV), LDA, PBE0, and HSE06, respectively (see the actual data in Table 1 of ref (29)). Our inspection of their data shows that ε_r is not meaningful with LDA+U and is partially (or significantly) overestimated with the LDA (or PBE0 and HSE06) methods compared to the corresponding experimental mean value of 159.7.¹⁴ Furthermore, the LDA result outlined above was abnormally different from that of 133.5 reported by Shojaee et al.,²³ even though the HSE functional gave a value of 129.3 (exact values 159 and 70 along the x–y plane and z-direction, respectively).³⁰ Interestingly, however, the electronic contribution to ε_r did not show any strong dependence on the quality of theoretical methods applied (PBE, PBE0, etc.).^21,31−33 The incipient ferroelectric behavior and the low-frequency dielectric constant are directly linked to lattice dynamics, with a strong dependence on the low-frequency transverse optical phonon mode A_2u.^34,35

In this study, we have theoretically investigated the lattice properties, zone-center phonon vibrations, and the low- and high-frequency-dependent (ionic and electronic contributions, respectively) dielectric permittivity of rutile (R-)TiO₂ using DFT at PBEsol³⁶ and PBE^37,38 levels, with and without incorporating the Hubbard term U.^39,40 Density functional perturbation theory (DFPT) was employed for the evaluation of phonon vibrations and dielectric properties.⁴¹⁻⁴³ The same properties were also investigated for the anatase and brookite phases, as well as for the two high-pressure structures of TiO₂ (called pyrite⁴⁴⁻⁴⁶ and fluorite⁴⁵⁻⁴⁸) for comparison.

The main objective of our study is to demonstrate the dependency of the above properties on the two DFT functionals employed, as well as that on the three commonly used variants of the projector augmented wave (PAW) potential^49,49,50 available in the Vienna ab initio simulation package (VASP).⁵¹⁻⁵³ The study is necessary because it is imperative to clarify how and to what extent the promotion of the number of core electrons to valence electrons in the PAW potential(s) affects the accuracy of the physical properties of TiO₂ (e.g., lattice geometry, relative permittivity, and phonon frequencies). The two GGA functionals were chosen because PAW potentials are specifically designed for use in conjunction with PBE- and PBE-like functionals (e.g., PBEsol); they are computationally inexpensive and are at the forefront of large-scale, high-throughput calculations for in-silico design of new functional materials. To verify the conclusions arrived at using the two PBE-based functionals, similar calculations for R-TiO₂ were performed at a relatively high level using hybrid functionals HSE06⁵⁴ and HSEsol.⁵⁵ Finally, we have compared our computed results with the low-temperature experimental results.

2. Computational Details

The electronic structure properties of all the five phases of TiO₂ were calculated using PBE and PBEsol; the VASP code was used. Hereafter, we refer the anatase-, brookite-, pyrite-, and fluorite phases of TiO₂ to as A-, B-, P-, and F-TiO₂, respectively.

The PAW potentials were used, in which the core electrons were frozen, and were replaced by pseudopotentials. There are four versions of PAW potential for Ti, distributed by VASP. Depending on the size of the cutoff radius used to define the potential around the nucleus, the PAW potentials have been considered hard or soft, meaning they differ from each other in the degree to which they treat inner-shell electrons as valence electrons. We used three commonly used versions of the PAW potential for Ti, keeping the same for the O site in TiO₂ unchanged. We label the standard PAW potential of Ti, which has four valence electrons 3d³4s¹, as Ti_std. The remaining two variants of the PAW potential for Ti are augmented versions, labeled as Ti_pv and Ti_sv, which include 2p⁶ and 2s²2p⁶ semi-core states as part of the basis set for valence bands, respectively. The outermost cutoff radii were 2.8, 2.5, and 2.3 Å for Ti_std, Ti_pv, and Ti_sv, respectively. The Wigner–Seitz radii of the corresponding PAW potentials were 1.323, 1.323, and 1.217 Å, respectively.

The crystal lattices of the five phases of TiO₂ were fully relaxed without any constraints, starting from their respective experimental structures wherever feasible (see discussion below). The cutoff energy for the plane-wave basis set was set at 520 eV, and the break condition for the electronic relaxation loop was set at 10^–8 eV. The force on each ion was less than 0.002 eV Å^–1. The k-meshes of 9 × 9 × 15, 8 × 8 × 10, 6 × 4 × 6, 10 × 10 × 10, and 8 × 8 × 8 were utilized for R-, A-, B-, F-, and P-TiO₂, respectively. High precision, together with a blocked Davidson iteration scheme, was invoked.

The ionic^41,56−58 and electronic (optic)⁵⁷⁻⁵⁹ contributions (ε_ij⁽⁰⁾ and ε_ij^(∞), respectively) to dimensionless total relative dielectric permittivity ε_r (ε_r = ε_ij⁽⁰⁾ + ε_ij^(∞)) were calculated at the same levels of theory (PBE and PBEsol) for all the three PAW potentials mentioned above. DFPT was employed since it provides better estimation of the relative dielectric permittivity at the expense of computational time, especially when compared with experiment. The finite difference method (FDM)⁶⁰⁻⁶² is computationally expensive, which was also considered in conjunction with the hybrid functionals and GGA to examine whether ε_r obtained with this method are comparable with those emerged from DFPT/GGA. The same k-meshes used for the convergence of crystal lattices and evaluation of other properties were used for the calculation of ε_ij⁽⁰⁾. The cutoff energy was set at 520, 600, and 700 eV that enabled us to test the dependence of the cutoff energy for plane-wave on ε_ij⁽⁰⁾ and ε_ij^(∞) and ε_r.

For reasons discussed in the following section, the hybrid functionals, HSE06 and HSEsol, were employed to obtain the relaxed lattice geometry, phonon modes, and dielectric permittivity of R-TiO₂. A 4 × 4 × 6 k-mesh was chosen both for geometry relaxation and phonon mode analysis. The cutoff energy for the plane-wave basis set, the total energy convergence for electronic relaxation, and the residual forces used for ionic relaxation were 520 eV, 10^–6 eV, and 0.002 eV Å^–1, respectively. The FDM was applied for the calculation of the phonon vibrational modes and relative dielectric permittivity. The cutoff energy for the plane-wave basis set and the ion relaxation loop were 600 and 10^–6 eV, respectively.

Phonon dispersion and Raman spectra were calculated using FDM.^63,64 The PBEsol relaxation geometries of R-TiO₂, A-TiO₂, and B-TiO₂ were used to construct supercells consisting of 72, 108, and 96 atoms, respectively. A 3 × 3 × 3 k-mesh was used to calculate phonon dispersion for all three systems above. For the calculation of the Raman spectra, the size of the k-mesh was chosen to be 9 × 9 × 15, 8 × 8 × 4, and 6 × 6 × 4 for the corresponding systems, respectively. The cutoff energy was set to 520 eV, and the energy for electronic relaxation loop and the residual force were converged to within 10^–7 eV and 0.02 eV Å^–1, respectively. For cubic P- and F-TiO₂, supercells with cell sizes close to 10 Å were used, consisting of 96 and 192 atoms, respectively. A k-mesh of 2 × 2 × 2 was used, together with the electronic relaxation loop and the residual force that were set converged to within 10^–8 eV and 0.02 eV Å^–1, respectively. The DFPT formalism was utilized, together with the Phonopy code.⁶⁵

3. Results and Discussion

3.1. Lattice Properties

The fully relaxed lattice geometries of all the five phases of TiO₂ are shown in Figure 1. Tables 123 list the calculated lattice properties of R-, A-, and B-phases of TiO₂ and compare them with experiment. The properties include the lattice constants (a, b, c, α, β, and γ), cell volume V, and mass density ρ.

Figure 1.

(a)–(e) Polyhedral models of the PBEsol relaxed unit-cell geometries of the five phases of TiO₂, obtained with Ti_std. The orientation of the crystallographic axes a, b, and c is shown for each case.

Table 1. Comparison of PBE, PBEsol, PBE+U, and PBEsol+U Computed Lattice Constants (a = b ≠ c and α = β = γ), Cell Volume (V), and Mass Density (ρ) of R-TiO₂ with Experiment^a,^b.

PAW	a = b/Å	c/Å	V/Å3	ρ/gcm–3	a = b/Å	c/Å	V/Å3	ρ/gcm–3
	PBE				PBE+U
Ti_std	4.661	2.970	64.52	4.11	4.687	3.023	66.41	3.99
Ti_pv	4.652	2.968	64.23	4.13	4.673	3.014	65.82	4.03
Ti_sv	4.645	2.969	64.05	4.14	4.665	3.007	65.42	4.05
	PBEsol				PBEsol+U
Ti_std	4.615	2.949	62.79	4.22	4.640	3.000	64.59	4.11
Ti_pv	4.600	2.943	62.28	4.26	4.623	2.989	63.89	4.15
Ti_sv	4.596	2.941	62.12	4.27	4.614	2.982	63.49	4.18
	HSEsol
Ti_std	4.572	2.934	61.33	4.32
Ti_pv	4.556	2.934	60.91	4.36
Ti_sv	4.551	2.931	60.70	4.37
	HSE06
Ti_std	4.603	2.947	62.43	4.25
Ti_pv	4.590	2.949	62.13	4.27
Ti_sv	4.584	2.947	61.92	4.28
expt.66	4.601	2.964	62.745	4.23	4.601	2.964	62.745	4.23

^aα = β = γ = 90° for the theoretical methods applied.

^bThree different PAW potentials used.

Table 2. Comparison of PBEsol and PBEsol+U Level Lattice Constants (a = b ≠ c and α = β = γ), Cell Volume (V), and Mass Density (ρ) of A-TiO₂ with Experiment^a,^b.

PAW	PBEsol					PBEsol+U
	a/Å	b/Å	c/Å	V/Å3	ρ/gcm–3	a/Å	b/Å	c/Å	V/Å3	ρ/gcm–3
Ti_std	3.800	3.800	9.547	137.85	3.85	3.858	3.858	9.553	142.19	3.73
Ti_pv	3.777	3.777	9.575	136.59	3.88	3.827	3.827	9.591	140.47	3.78
Ti_sv	3.773	3.773	9.564	136.14	3.90	3.817	3.817	9.575	139.49	3.80
expt.68	3.784	3.784	9.518	136.28	3.89	3.784	3.784	9.518	136.28	3.89

^aα = β = γ = 90° for the theoretical methods applied.

^bThree different PAW potentials used.

Table 3. Comparison of PBEsol and PBEsol+U Level Lattice Constants (a = b ≠ c and α = β = γ), Cell Volume (V), and Mass Density (ρ) of B-TiO₂ with Experiment^a,^b.

PAW	PBEsol					PBEsol+U
	a/Å	b/Å	c/Å	V/Å3	ρ/gcm–3	a/Å	b/Å	c/Å	V/Å3	ρ/gcm–3
Ti_std	5.149	9.203	5.463	258.88	4.10	5.228	9.262	5.519	267.22	3.97
Ti_pv	5.127	9.186	5.449	256.64	4.13	5.203	9.243	5.489	263.94	4.02
Ti_sv	5.124	9.177	5.444	255.95	4.15	5.190	9.227	5.476	262.23	4.05
expt.68	5.144	9.293	5.412	258.71	4.10	5.144	9.293	5.412	258.71	4.10
expt.69	5.137	9.714	5.452	256.94	4.13	5.137	9.714	5.452	256.94	4.13
expt.70	4.981	9.082	5.53	250.14	4.24	4.981	9.082	5.53	250.14	4.24

^aα = β = γ = 90° for the theoretical methods applied.

^bThree different PAW potentials used.

Experimental lattice constants even reported for the three major phases of TiO₂ vary widely from one research group to another.^56,66−72 The variation may be due to the differences in the experimental conditions and the fitting methods adopted in measuring and refining the structural data. A similar attribute is notable of the data obtained using two DFT functionals and the three variants of the PAW potential, as well as when the Hubbard U (U = 3.0 eV) term was combined with GGA.

From Table 1, it may be seen that the lattice constants of R-TiO₂ follow the trend: a = b ≠ c and α = β = γ = 90°. PBEsol reproduces lattice constants and cell volumes, which are in fair agreement with the experimental range of reported values (not shown). In contrast, the PBE functional systematically overestimates the same properties (Table 1). The same is also observed when both of them were compared with to the high-precision experimental data reported so far with an R-factor (discrepancy index) as low as 1.47%.⁶⁶ For a particular GGA method, PBE or PBEsol, when passed from Ti_std via Ti_pv to Ti_sv, the lattice constants (a = b and c) and cell volume were systematically decreased; this is opposite to the trend in the mass density that was increased. The result suggests that the lattice constant or cell volume tends to decrease and mass density tends to increase as the PAW potential softens. Conversely, the HSE06 method gave improved lattice properties over HSEsol. The main difference between PBEsol and HSE lies in the lattice volume, in which the predictability of the former is relatively accurate compared to that feasible in the crystal.

The DFT-U method did not show any dramatic improvement in the lattice properties. Rather, it significantly overestimates all the lattice constants, with a consequent increase in the cell volume. The reducing behavior of the lattice constants observed with DFT was observed with DFT+U when passing from Ti_std via Ti_pv to Ti_sv.

A-TiO₂ is metastable. It is stable only up to a temperature of 600 °C. Beyond this temperature, a transformation to rutile phase occurs, even though the reported thermal analysis suggests that the transformation from anatase to rutile may start at a temperature of around 400 °C.⁷³ The geometry of A-TiO₂ is tetragonal, with space group I4₁/amd (141), and the unit and primitive cells contain four and two formula units of TiO₂, respectively. Table 2 summarizes the conventional unit-cell lattice properties of A-TiO₂, obtained from this work and reported experimentally. Since PBEsol-calculated lattice properties agree well with experiment for R-TiO₂ (cf. Table 1), we did not consider the PBE and DFT+U methods to compute the lattice properties for A-TiO₂. Table 2 summarizes the PBEsol results that demonstrate that Ti_std slightly overestimates the lattice constants and cell volume and underestimates the mass density compared to experiment.⁶⁸ There is no appreciable improvement in these properties upon increasing the size of the PAW potential from Ti_std through Ti_pv to Ti_sv. PBEsol+U has overestimated the lattice constants and cell volume and underestimated the mass density, all compared to experiment.

The crystal structure of B-TiO₂ is orthorhombic. It has eight formula units of TiO₂ (space group Pbca (61)). Several experimental studies have reported the crystal structure of B-TiO₂, showing a dispersion in the values of the lattice properties. For instance, Murugesan et al.⁶⁸ (Silva Junior et al.⁷⁴) [Nishio-Hamane et al.⁷⁵] {Rezaee et al.⁷⁰} have reported a = 5.144 Å (5.138; 5.259) [4.596] {4.492}, b = 9.293 (9.194; 9.202) [8.962] {9.101} Å, and c = 5.412 (5.449; 5.47) [4.823] {5.455} Å. Because of the variation in the lattice constants, the nature of both V and ρ is changed appreciably. The V values extracted from the unit cells of the corresponding crystals were 258.711 (256.844; 264.711) [198.656] {245.341} ų. Our PBEsol results are very close to those reported by Murugesan et al.⁶⁸ (Table 3). There is a monotonic decrease in the lattice constants with respect to the increase in the number of core electrons as valence electrons in the PAW potentials, an observation which is very similar to that found for R- and A-TiO₂ (vide supra).

Table 4 summarizes the calculated lattice properties of P- and F-phases of TiO₂. These are compared with the properties of corresponding crystals deposited to the Materials Project database⁷⁶ (IDs: mp-1102591⁷⁷ and mp-1008677,⁷⁸ respectively), as a result of the lack of experimentally known data. In any case, the PBEsol lattice constants are all slightly underestimated relative to the PBE, with a concomitant decrease in the cell volume and an increase of mass density. The trend in the cell properties for a given correlated method with respect to the three PAW potentials is consistent with that observed for R-, A-, and B-TiO₂ (see above). We have observed that F- and P-TiO₂ lattices are relatively denser than that of the brookite, anatase, and rutile lattices of TiO₂, in agreement with the computational data of Dharmale et al.⁴⁴

Table 4. Comparison of Lattice Constants (a = b = c and α = β = γ), Cell Volume (V), and Mass Density (ρ) of P- and F-TiO₂, Obtained Using PBEsol and PBE, with Those Catalogued in the Materials Project Database (Calculated with [PBE/Ti_pv]⁷⁶)^a.

PAW	PBEsol			PBE
	a = b = c	V/Å3	ρ/gcm–3	a = b = c	V/Å3	ρ/gcm–3
Pyrite (Cubic, P-a3̅)
Ti_std	4.861	114.84	4.62	4.905	118.03	4.49
Ti_pv	4.852	114.22	4.64	4.902	117.78	4.50
Ti_sv	4.848	113.97	4.65	4.899	117.55	4.51
mp-110259177	4.902	117.79	4.50	4.902	117.79	4.50
Fluorite (Cubic, Fm-3̅m)
Ti_std	4.789	109.87	4.83	4.836	113.12	4.69
Ti_pv	4.786	109.63	4.84	4.838	113.25	4.68
Ti_sv	4.783	109.40	4.85	4.835	113.05	4.69
mp-100867778	4.84	113.38	4.68	4.840	113.38	4.68

^aThree different PAW potentials used.

3.2. Phonon Vibrations and Dynamical Stability

Born’s stability criterion^79,80 suggests that a crystal lattice is said to be dynamically stable if the normal mode vibration frequencies of all the optical phonons are positive, that is, if the eigenvalues of the dynamical matrix are all positive. To this end, we investigated the nature of phonon vibrational frequencies for all the five phases of TiO₂, in combination with three PAW potentials. The frequencies of IR- and Raman-active modes of R- and A-TiO₂ are summarized in Tables 5 and 6, respectively, whereas those of B-TiO₂ are summarized in Table S1; the PBE and experimental phonon frequencies are included in these tables wherever applicable.

Table 5. Comparison of Zone-Center Fundamental IR and Raman Phonon Frequencies ω (cm^–1), as Well as IR Intensities I (km/mol) of Rutile-TiO₂, Obtained Using PBEsol and PBE Functionals, in Conjunction with the PAW Potentials Ti_std, Ti_pv, and Ti_sv Centered on Ti^abcdef.

symmetryf	exptg	expth,i	PBEsol						PBE
			Ti_std		Ti_pv		Ti_sv		Ti_std		Ti_pv		Ti_sv
			ω	I	Ω	I	ω	I	ω	I	ω	I	ω	I
B2g(R)	825.5	826 (826.6)	810.2	0.0	795.3	0.0	793.1	0.0	783.6	0.0	770.3	0.0	769.3	0.0
A1g(R)	609.8	612 (610.4)	605.8	0.0	591.0	0.0	589.8	0.0	580.9	0.0	568.4	0.0	568.0	0.0
Eu(IR)		500 (494.0)	495.3	20.7	476.9	13.4	478.4	13.2	481.8	15.3	467.8	10.5	468.9	10.3
Eu(IR)		500 (494.0)	495.3	20.7	476.9	13.4	478.3	13.2	481.8	15.3	467.8	10.5	468.9	10.3
Eg(R)	445.8	447 (444.9)	459.6	0.0	448.0	0.0	447.1	0.0	440.6	0.0	430.0	0.0	429.6	0.0
Eg(R)	445.8	447 (444.9)	459.6	0.0	448.0	0.0	447.1	0.0	440.6	0.0	430.0	0.0	429.6	0.0
A2g (S)			405.5	0.0	413.6	0.0	415.1	0.0	407.7	0.0	414.8	0.0	416.4	0.0
B1u(S)		(406.3)	395.8	0.0	372.5	0.0	373.9	0.0	369.7	0.0	354.3	0.0	357.1	0.0
Eu(IR)		388 (374.4)	375.0	9.7	366.4	4.0	366.9	3.7	356.9	3.2	353.9	1.0	355.1	0.9
Eu(IR)		388 (374.7)	375.0	9.7	366.4	4.0	366.9	3.7	356.9	3.2	353.9	1.0	355.0	0.9
A2u(IR)		167 (188.8)	180.5	179.2	61.2	168.0	71.2	165.3	99.6	184.2	106.1i	171.5	98.8i	168.3
Eu(IR)	235.5	183 (172.7)	158.7	105.2	93.0	110.1	98.4	108.5	95.4	115.5	62.8 i	114.1	54.3i	111.5
Eu(IR)	235.5	183 (172.7)	158.7	105.2	93.0	110.1	98.2	108.4	95.4	115.5	62.8 i	114.1	54.5i	111.7
B1g(R)	140.2	143 (141.6)	134.2	0.0	139.7	0.0	140.5	0.0	143.9	0.0	148.5	0.0	149.0	0.0
B1u(S)		(113.0)	118.3	0.0	74.9	0.0	78.3	0.0	84.8	0.0	30.1i	0.0	16.7i	0.0
A2u(IR)			0.0	13.7	0.1	8.5	0.4	8.2	0.0	8.9	0.1	8.3	0.6	9.0
Eu(IR)			0.0	9.1	0.1	8.5	0.3	8.3	0.0	8.9	0.1	8.3	0.4	11.6
Eu(IR)			0.0	9.1	0.1	13.0	0.2	12.6	0.1	14.2	0.1	13.2	0.4	10.1

^aExperimental frequencies are included wherever available, and an energy cutoff of 700 eV was used.

^bNegative value in columns 12 and 14 represent imaginary frequency.

^cSchöche et al.²⁵ reported the four room-temperature IR-active phonon modes (three degenerate E_u and one non-degenerate A_2u) to be 188.6 ± 1.2, 379.3 ± 0.2, 500.5 ± 0.3, and 172.3 ± 1.9 cm^–1, respectively.

^dGervais and Piriou⁸⁵ reported the four IR-active phonon modes (three degenerate E_u and one non-degenerate A_2u) at 189, 381.5, 508, and 172 cm^–1, respectively.

^eEagles reported the four IR-active phonon modes (three degenerate E_u and one non-degenerate A_2u) at 183, 388, 500, and 167 cm^–1, respectively.⁸⁶

^fR, IR, and S correspond to Raman- and IR-active and silent modes, respectively.

^gRef (83).

^hRef (82).

ⁱValues in parentheses represent coherent inelastic neutron determination of phonon frequencies at high-symmetry points of Brillouin zone.⁸¹

Table 6. Comparison of Fundamental IR and Raman Frequencies ω (cm^–1), as Well as IR Intensities I (km/mol), Obtained Using PBEsol and PBE Functionals, in Conjunction with PAW Potentials Ti_std, Ti_pv, and Ti_sv Centered on Ti of A-TiO₂^a,^b.

	exptc,d	PBEsol						PBE
		Ti_std		Ti_pv		Ti_sv		Ti_std		Ti_pv		Ti_sv
		ω	I	ω	I	ω	I	ω	I	ω	I	ω	I
Eg(R)	640 (632.7)	640.4	0.0	619.8	0.0	620.6	0.0	616.5	0.0	599.7	0.0	599.7	0.0
Eg(R)	640 (632.7)	640.4	0.0	619.8	0.0	620.6	0.0	616.5	0.0	599.7	0.0	599.7	0.0
B2u (S)		548.6	0.0	535.9	0.0	534.4	0.0	531.0	0.0	519.2	0.0	518.6	0.0
A1g(R)	519 (515.4)	516.7	0.0	510.8	0.0	510.7	0.0	499.6	0.0	493.9	0.0	494.4	0.0
B1g(R)	515 (515.4)	497.5	0.0	487.3	0.0	489.0	0.0	478.5	0.0	475.1	0.0	477.6	0.0
Eu(IR)	435 (TO)	446.0	76.7	419.0	51.1	420.7	50.1	412.0	57.9	395.3	34.6	396.5	32.8
Eu(IR)	435 (TO)	446.0	76.7	419.0	51.1	420.7	50.1	412.0	57.9	395.3	34.6	396.5	32.8
B1g(R)	400 (396.0)	385.4	0.0	374.2	0.0	374.1	0.0	374.9	0.0	361.7	0.0	361.8	0.0
A2u(IR)	366 (LO)	355.9	100.3	326.6	93.3	326.9	92.0	323.3	101.5	300.0	94.0	301.2	92.8
Eu(IR)	262 (TO)	240.8	75.4	227.6	90.5	231.5	89.2	226.3	95.8	209.0	107.9	211.0	106.9
Eu(IR)	262 (TO)	240.8	75.4	227.6	90.5	231.3	89.2	226.3	95.8	209.0	107.9	211.0	107.0
Eg(R)	197 (196.3)	164.7	0.0	167.1	0.0	169.2	0.0	168.4	0.0	173.4	0.0	175.9	0.0
Eg(R)	197 (196.3)	164.7	0.0	167.1	0.0	169.2	0.0	168.3	0.0	173.4	0.0	175.9	0.0
Eg(R)	144 (140.9)	151.4	0.0	135.2	0.0	137.9	0.0	130.7	0.0	113.9	0.0	115.2	0.0
Eg(R)	144 (140.9)	151.3	0.0	135.2	0.0	137.6	0.0	130.7	0.0	113.9	0.0	115.1	0.0
A2u(IR)		0.0	8.2	0.0	9.4	0.0	7.1	0.0	10.1	0.0	7.2	0.0	9.1
Eu(IR)		0.0	9.9	0.0	9.3	0.1	9.2	0.0	9.8	0.0	9.4	0.0	7.2
Eu(IR)		0.0	9.6	0.0	7.2	0.2	9.2	0.0	8.1	0.0	9.4	0.1	9.3

^aIR, R, and S represent IR-active, Raman-active, and silent modes, respectively.

^bRaman-active modes reported by Arsov et al. were 144 (E_g), 197 (E_g), 399 (B_1g), 516 (A_1g), 516 (B_1g), and 639 (E_g) cm^–1.⁸²

^cIR-active and Raman-active modes are from refs (92) and (93), respectively.

^dRef (83).

Figures 234 show the phonon dispersion curves of R-, A-, and B-TiO₂, respectively. Figure 5a,b illustrates the same curves for F- and P-TiO₂, respectively. For the first three cases, the phonon modes are all stable at the center of the zone, Γ(k (0,0,0)), and at other points of high symmetry along the direction of the wave vector of the first Brillouin zone. That is, the optical phonon modes do not involve any imaginary frequencies, confirming that the R-, A-, and B-TiO₂ lattices are dynamically stable irrespective of the three PAW potentials utilized.

Figure 2.

Phonon dispersion of R-TiO₂, obtained using DFPT in combination with PBEsol and the three PAW potentials centered at the Ti site of the crystal lattice: (a) Ti_std; (b) Ti_pv; and (c) Ti_sv. The phonon frequencies in the vertical axis are in THz (1 THz = 33.3564 cm^–1).

Figure 3.

Phonon band structure of A-TiO₂, obtained using DFPT in combination with PBEsol and the three variants of the PAW potential centered at the Ti site of the crystal lattice: (a) Ti_std; (b) Ti_pv; and (c) Ti_sv. The phonon frequencies in the vertical axis are in THz (1 THz = 33.3564 cm^–1).

Figure 4.

Phonon band structure of B-TiO₂, obtained using DFPT in combination with PBEsol and the three variants of the PAW potential centered at the Ti site of the crystal lattice: (a) Ti_std; (b) Ti_pv; and (c) Ti_sv. The phonon frequencies in the vertical axis are in THz (1 THz = 33.3564 cm^–1).

Figure 5.

(a,b) Illustration of phonon dispersion of F- and P-TiO₂, respectively, obtained using DFPT at the [PBEsol/Ti_std] level. The 4 × 4 × 4 (192 atoms) and 2 × 2 × 2 (96 atoms) supercells were utilized for the phonon calculations for F- and P-TiO₂, respectively. Phonon frequencies in the vertical axis are in THz (1 THz = 33.3564 cm^–1).

For F-TiO₂, the optical phonons are unstable along the path Γ–L–W–X, even though they are stable at the Γ-point (Figure 5a). For P-TiO₂, several low-frequency phonon modes are unstable throughout the Brillouin zone including zone center. The instability of the phonon modes is understood owing to their (negative) imaginary frequencies (Figure 5b). The result is not perplexed for pressure-driven crystal lattices, as are commonly observed to be dynamically unstable.

The rutile phase of TiO₂ has 15 optical phonon modes. Of these, three are acoustic phonon modes. At Γ(k (0,0,0)), the optical phonons are representation given by 1A_1g + 1A_2g + 1A_2u + 1B_1g + 2B_1u + 1B_2g + 1E_g + 3 E_u, in which E_g (non-polar), B_2g, B_1g, and A_1g are Raman-active and the A_2g (non-polar) and B_1u modes are infrared (IR) and Raman inactive silent modes.⁸¹ Each phonon mode E is twofold degenerate, and the phonon modes A_2u and E_u are polar.

The four experimentally determined Raman-active modes of R-TiO₂ appear at frequencies ω of 143 (B_1g), 447 (E_g), 612 (A_1g), and 826 cm^–1 (B_2g).^81,82 The corresponding frequencies measured by Ma et al.⁸³ center at 140.2, 445.8, 609.8, and 825.5 cm^–1, respectively. They have also measured a broad and moderately strong band that peaks at 235 cm^–1. This mode is composed of high-level anharmonicity and arises from two-phonon scattering. The same feature was observed by Tompsett et al.⁸⁴

The frequencies of the four vibrational bands observed by Ma et al.⁸³ and others^81,82 (see above) are in close match with our results (all within 15 cm^–1). For instance, they are centered at frequencies of 134.2 (B_1g), 459.6 (E_g), 605.8 (A_1g), and 810.2 cm^–1 (B_2g) with [PBEsol/Ti_std], respectively. When Ti_pv and Ti_sv were used, the two high-frequency modes, A_1g and B_2g, could be appreciably underestimated, except for the low-frequency phonon modes, B_1g and E_g. This comparison leads to a conclusion that the overall nature of the phonon modes reported using coherent inelastic neutron scattering measurements⁸¹ matches well with those computed with Ti_std. The PBEsol frequency of the lowest (silent) phonon mode calculated using Ti_std (ω = 118.3 cm^–1) is largely overestimated relative to that obtained using Ti_pv and Ti_sv (ω = 74.9 and 78.3 cm^–1, respectively). The latter two are unquestionably underestimated by 38.1 and 34.7 cm^–1 compared to experiment (ω = 113.0 cm^–1, see Table 5).

R-TiO₂ has four IR-active optical phonon modes. Schöche et al.²⁵ have found that the IR-active modes (three E_u and one A_2u) possess phonon frequencies of 188.6 ± 1.2, 379.3 ± 0.2, 500.5 ± 0.3, and 172.3 ± 1.9 cm^–1, respectively. The first three are doubly degenerate modes and the latter one is a non-degenerate mode. Gervais and Piriou⁸⁵ [Eagles⁸⁶] have reported that the corresponding phonon mode frequencies should show at frequencies of 189 [183], 381.5 [388], 508 [500], and 172 [167] cm^–1, respectively. The IR-active modes reported in different studies are not in disagreement with each other. However, the inelastic neutron scattering study did not observe the degenerate vibrational mode at 381.5 cm^–1, assigned by Gervais and Piriou.⁸⁵ Schöche et al.²⁵ have assigned the two E_u modes to have frequencies of 188.61 and 365.74 cm^–1, which are the first transverse optical (TO) and longitudinal optical (LO) modes, respectively.

Indeed, the experimental IR phonon modes are consistent with our PBEsol-based frequencies summarized in Table 5. This functional, together with Ti_std, has underestimated the lowest IR-active mode at 158.7 cm^–1, compared to the experimental value of 183 (172.7) cm^–1. Ti_pv and Ti_sv underestimated them further to appear at frequencies of 93.0 and 98.3 cm^–1, respectively. Interestingly, Ti_sv has predicted the frequencies of the degenerate phonon mode at 98.4 and 98.2 cm^–1, thus assigning them as nearly degenerate (which should not be!).

A discrepancy between experiment and calculation can be seen with the non-degenerate IR-active mode A_2u. ω for this phonon mode was 180.5, 61.2, and 71.2 cm^–1 with Ti_std, Ti_pv, and Ti_sv, respectively. The large difference in the wavenumbers with respect to the PAW potentials can be realized from the phonon dispersion curves shown in Figure 2a–c. From the shape of these curves, it is clear that the phonon modes below 2.5 THz are shallow-like (flattish) around high-symmetry M-point along the path Γ–M–X when the potentials Ti_pv and Ti_sv were used (Figure 2b,c, respectively). This may also mean that a slight distortion of the crystal lattice could cause softening of the phonon mode around the Γ–M–X path. This is not the case with the phonon modes calculated with Ti_std, in which the curves associated with the low-frequency optical modes are relatively steeper (Figure 2a), which match reasonably well with experiment.⁸¹

When PBE was used, the discrepancies between the phonon modes at low frequencies were very severe. From the data summarized in Table 5, one can see that the frequencies of all the phonon modes are underestimated for all the three PAW potentials utilized. Furthermore, Ti_pv and Ti_sv have predicted soft modes around the Γ-point that are linked with the three IR-active phonon modes and a silent mode; that is, they link with a doubly degenerate mode E_u, a non-degenerate mode A_2u, and a non-degenerate B_1u mode. The Ti_pv potential has predicted these phonons at frequencies of 62.8i, 106.1i, and 30.1i, respectively. Similarly, Ti_sv has predicted the corresponding frequencies at 54.4i, 98.8i, and 16.7i, respectively. These soft modes explain why the PBE-relaxed crystal lattice of R-TiO₂ is dynamically unstable at the center of the zone, Γ.

Montanari and Harrison^87,88 have demonstrated that the A_2u (TO) mode becomes soft at the Γ-point if the crystal of R-TiO₂ is expanded via LDA or GGA. This may be consistent with our PBE-level observation. The data suggest that the expansion of the rutile lattice by the two soft potentials softens the phonon modes associated with both the IR-active modes E_u and A_2u (see columns 12 and 14 of Table 5). The lattice expansion is evidenced in Table 1, especially when comparing the lattice parameters evaluated with Ti_std to those with Ti_pv and Ti_sv.

Mitev et al. have also reported an imaginary frequency of 86i corresponding to the A_2u (TO) phonon mode with the PBE functional using the CASTEP code.⁸⁹ We could not reproduce their result because of differing cutoff criteria used in our calculation. The authors of the study have interpreted that the softening of the A_2u (TO) mode at the Γ-point could be responsible for the incipient ferroelectric behavior of R-TiO₂ and hence should play some role in the determination of large and strongly temperature-dependent dielectric constant.

We now clarify the origin of the spurious (soft) nature of the phonon modes predicted by the two soft potentials Ti_pv and Ti_sv. For this, we investigated the nature of radial dependence of the charge density in one dimension for each of the three PAW potentials. We have used the PBEsol-, PBEsol+U-, and HSEsol-relaxed lattices of R-TiO₂. As shown in Figure 6a–d, the nature of the radial dependence of charge density is very similar regardless of the degree of exchange correlation incorporated in each of the DFT methods chosen. That is, the charge density in the region of space between bonded Ti and O atomic basins in R-TiO₂, in the close vicinity of Ti, is steep (attractive) when Ti_std was applied and shallow (somehow repulsive!) when Ti_pv and Ti_sv were employed. These results suggest that the softness of the PAW potential (but certainly not the correlation method) is the sole cause of softening of the low-frequency optical phonon modes as a result of increased repulsion between core electrons in the Ti–O bonding region (cf. Figure 2a,b).

Figure 6.

1D plot of the radial dependence of charge density for R-TiO₂, evaluated using several methods in combination with the three PAW potentials Ti_std, Ti_sv, and Ti_pv. Charge density calculations were performed on the (a) PBEsol lattice with PBEsol; (b) PBEsol+U lattice with PBEsol+U; (c) HSE06 lattice with PBEsol, and (d) HSEsol lattice with HSEsol. Values are in logarithmic scale.

A total of 18 phonon modes exist in case of the A-TiO₂ lattice.^84,90,91 Of these, 3 are acoustic modes and 15 are optical modes. The irreducible representation of the latter 15 modes is given by 1A_1g + 1A_2u + 2B_1g + 1B_2u + 3E_g + 2E_u. The modes with a subscript “u” are IR-active phonon modes except B_2u (a silent mode), and those with a subscript “g” are Raman-active modes. The PBEsol- and PBE-calculated vibrational frequencies, Table 6, show that the high- and low-frequency phonon modes that are both IR and Raman active are either slightly overestimated or slightly underestimated but were well reproduced by PBEsol in combination with the standard potential Ti_std. In particular, the eight high-frequency Raman and IR modes are all within 15 cm^–1 of the experimental values. The three doubly degenerate IR and Raman modes (E_u and E_g) predicted at frequencies of 240.8 (E_u), 164.7 (E_g), and 151.4 (E_g) cm^–1 are underestimated or overestimated but all within 35 cm^–1 of the experimental wavenumbers (Table 6).^84,90 The degree of underestimation of high-frequency phonon vibrations was severe especially with the Ti-centered potentials Ti_pv and Ti_sv; the same behavior was also observed for R-TiO₂ (Table 5). As mentioned above, the underestimation should not be due to the short Ti–O bond distance in the crystal lattice calculated using the two soft potentials. We confirmed this by calculating the frequencies of phonon modes with Ti_std, Ti_pv, and Ti_sv using bulk geometries of R- and A-TiO₂ obtained with Ti_pv or Ti_sv. As shown in Tables S1 and S2, the phonon frequencies with Ti_std are always higher than those calculated with Ti_pv and Ti_sv, a feature that was emanated irrespective of the type of the crystal lattice used. From this, it is clear that the softer the potential, the shallower the charge density profile in the close vicinity of Ti along the Ti–O bond of TO₂, thus softening (lowering) the frequency of phonon vibrations around the zone center. However, the phonon mode softening is less pronounced at the Γ-point for A-TiO₂ compared to that found for R-TiO₂. The performance of the PBE functional is homologous with that of PBEsol for A-TiO₂, but the phonon frequencies are largely underestimated compared to experiment (Table 6).

For B-TiO₂, the number of IR- and Raman-active modes is greater than that found in R- and A-TiO₂; this is attributed to the greater number of atoms (24) responsible for the unit cell of the brookite lattice. Because of this, and because B-TiO₂ has a lower orthorhombic Pbca symmetry, there are 36, 24, 9, and 3 Raman-active (A_1g, B_1g, B_2g, and B_3g), IR-active (B_1u, B_2u, B_3u), silent (A_1u), and acoustic vibrational phonon modes, respectively. They are represented by the irreducible representations at Γ: 9A_1g + 9B_1g + 9B_2g + 9B_3g, 8B_1u + 8B_2u + 8B_3u, 9A_1u, and B_1u + B_2u + B_3u. The frequencies of the PBEsol- and PBE-calculated phonon modes compared in Table S3 indicate that the phonon modes evaluated with PBEsol with potential Ti_stv are in better agreement with experiment.^84,94

3.3. Dependence of the Magnitude of Relative Dielectric Permittivity on PAW Potentials and DFT Methods

Since rutile and anatase TiO₂ are a uniaxial crystal and brookite TiO₂ is a biaxial crystal,⁹⁵ there are two and three independent non-zero components both for the ionic and electronic parts of the dielectric permittivity, respectively. The average of the former property, ε_r⁽⁰⁾, was added to that of the latter property, ε_r^(∞), so as to calculate the relative static dielectric permittivity, ε_r, for all the three phases of TiO₂. They are summarized in Table 7, together with the experimentally reported values wherever available.

Table 7. Comparison of Ionic (Low-Frequency) and Electronic (High-Frequency) Components (ε_ij⁽⁰⁾ and ε_ij^(∞), Respectively) of the Relative Dielectric Permittivity Tensor ε_r of R-TiO₂, Obtained Using Three Cutoff Energies in Combination with the PBEsol and Three PAW Potentials Ti_std, Ti_pv, and Ti_sv^a.

PAW potential	cutoff energy/eV	εij(0)				εij(∞)				εr
		εxx(0)	εyy(0)	εzz(0)	εr(0)	εxx(∞)	εyy(∞)	εzz(∞)	εr(∞)	εr(0) + εr(∞)
R-TiO2
Ti_std	520.0	124.0	124.0	163.0	137.0	8.0	8.0	9.6	8.5	145.5
	600.0	123.4	123.4	162.3	136.4	8.0	8.0	9.6	8.5	144.9
	700.0	122.9	122.9	161.3	135.7	8.0	8.0	9.6	8.5	144.2
Ti_pv	520.0	354.1	354.1	1135.2	614.5	7.6	7.6	9.1	8.1	622.6
	600.0	366.0	366.0	1257.8	663.3	7.6	7.6	9.1	8.1	671.3
	700.0	370.3	370.3	1327.0	689.2	7.6	7.6	9.1	8.1	697.3
Ti_sv	520.0	296.2	296.2	732.9	441.8	7.5	7.5	8.9	8.0	449.9
	600.0	309.1	309.1	799.3	472.5	7.5	7.5	8.9	8.0	480.5
	700.0	327.1	327.1	966.8	540.3	7.5	7.5	8.9	7.9	548.2
expt.14c		111	111	257	159.7
expt.25		84.69	84.69	152.97	107.45	5.96	5.96	7.16	6.36	113.81
expt.85		84.02	84.02	153.83	107.29	5.91	5.91	7.19	6.34	113.63
expt.89		81.8	81.8	167.1	110.23	6.00	6.00	7.80	6.60	116.83
expt.25		87.0	87.0	163.0	112.33	6.12	6.12	7.63	6.62	118.96
expt.25		89.8	89.8	166.7	115.43	6.32	6.32	7.80	6.81	122.25
expt.25		86	86	170	114.00	6.84	6.84	8.43	7.37	121.37
A-TiO2
Ti_std	700	44.1	44.1	21.2	36.4	7.4	7.4	6.7	7.1	43.6
Ti_pv	700	54.6	54.6	23.6	44.3	7.0	7.0	6.4	6.8	51.1
Ti_sv	700	52.4	52.4	23.3	42.7	6.9	6.9	6.3	6.7	49.4
expt.92		45.1	45.1	22.7	37.6	5.8	5.8	5.4	5.7	43.3
B-TiO2
Ti_std	700	48.2	50.7	45.0	49.4	7.6	8.0	7.4	7.7	57.2
Ti_pv	700	52.8	70.1	47.3	56.7	7.0	7.7	7.0	7.2	64.0
Ti_sv	700	52.4	68.6	47.9	56.3	6.9	7.6	6.9	7.1	63.4
Ref (23)b		64.1	86.7	55.2	68.7	6.8	7.5	6.8	7.0	75.7
Ref (23)b		50.2	58.2	50.0	52.8	7.0	7.5	6.9	7.1	59.9
expt.97,98					93					78

^aBoth low- and room-temperature experimental values of ε_r reported by various authors are included.

^bReported at the LDA and GGA levels, using Quantum Expresso software.

^cLow-temperature data refer T = 1.6 K.

As outlined in the Introduction section, the major controversy of TiO₂ lies in the experimental determination and proposed relative dielectric permittivity values of R-TiO₂. Some have reported the colossal nature of the ionic and total dielectric permittivities, and others reported appreciably large numbers for the same property for the same system. For instance, Nicolini, in 1952, has proposed a colossal dielectric permittivity of R-TiO₂, with ε_r approximately 10,000 for ceramic R-TiO₂. This value is close to the range 100–10,000 reported by Parker and Wasilik,¹² as well as by Chu.¹³ Parker and Wasilik¹² have reported a colossal dielectric permittivity in hydrogenated and reduced oxygen-rich rutile single crystalline TiO₂ [e.g., Nicolini (ε_r = 10,000) and Perker et al. (ε_r = (10,000–30,000)]. Later, in 1961, Parker reported values of 111 and 257 for ε_⊥⁽⁰⁾ = ε_xx⁽⁰⁾ = ε_yy⁽⁰⁾ and ε_||⁽⁰⁾ = ε_zz⁽⁰⁾ along the crystallographic a–/b- and c-axes of R-TiO₂ at low temperature, 1.6 K, respectively; these were 86 (58) and 170 (97) at 300 (1000) K, respectively.¹⁴ Indeed, these results were in agreement with Samara and Peercy [ε_r values along the a- and c-axes were 89.8 (114.9) and 166.7 (251) at 296 K (4 K), respectively].²⁴ Bonkerud et al.¹⁶ have recently argued that they fail to confirm the results of Nicolini¹¹ and Parker and Wasilik,¹² even though they could reconfirm the results individually published by Parker¹⁴ and Samara and Peercy.²⁴ Therefore, the anomalous dielectric permittivity reported by various authors has continuously been debated since some of the reported results above are the repercussion of any incorrectly designed experiment or an incorrect interpretation.

The trend in our PBEsol-based values of ε_⊥⁽⁰⁾ (ε_⊥⁽⁰⁾ = [ε_xx⁽⁰⁾ + ε_yy⁽⁰⁾]/2) and ε_||⁽⁰⁾ (ε_||⁽⁰⁾ = ε_zz⁽⁰⁾) (Table 7) calculated with Ti_std is in agreement with those of Samara and Peercy,²⁴ and Parker,¹⁴ as well as of Bonkerud et al.¹⁶ Because Ti_pv and Ti_sv underestimate wavenumbers of the low-frequency phonons (Table 5) compared to Ti_std, they are the cause of spuriously large ε_r⁽⁰⁾ for R-TiO₂. The feature is persistent regardless of the three energy cutoff values invoked during our calculation (Table 7). The discrepancy is understandable since the calculation of the dielectric permittivity due to ions strongly relies on the nature of the PAW potential responsible for the determination of the positive eigenvalues of the dynamical matrix, and the low-frequency phonon mode is the main factor in determining the net magnitude of ε_r⁽⁰⁾. The latter is not surprising given that the ε_r⁽⁰⁾ is inversely proportional to the square of the phonon frequency (ε_r⁽⁰⁾ α 1/ω²),⁹⁶ so that the smaller the phonon frequency, the larger the value of ε_r⁽⁰⁾. The notion is consistent with the rule of thumb that the combination of a higher frequency and a lower polarization factor should result in a lower dielectric constant in any specific direction. By contrast, ε_r^(∞) is nearly invariant with respect to the cutoff energies used.

Table S4 summarizes ε_r⁽⁰⁾ and ε_r^(∞) of R-TiO₂, calculated using PBE. The data reveal that neither of the three PAW potentials, in combination with the three energy cutoffs utilized, are suitable in reproducing experimental ε_r⁽⁰⁾. The ε_r⁽⁰⁾ was predicted to be spuriously large (ε_r⁽⁰ in the range 415–421) with Ti_std and unphysical with Ti_sv and Ti_pv. The ε_r^(∞) values (e.g., 7.9, 8.1, and 8.6 with Ti_sv, Ti_pv, and Ti, respectively) are slightly overestimated relative to the experimentally suggested range (ε_r^(∞) between 6.3 and 7.4).

We have calculated ε_r⁽⁰⁾ and ε_r^(∞) using PBE+U and PBEsol+U (U = 3.0 eV). This was instigated with the perception that the inclusion of the Hubbard on-site effective potential U centered on the d-orbital of the Ti atom provides the expected chemistry of localized electrons and hole polarons and their energetics for rutile, anatase, and brookite TiO₂ and that the GGA alone fails to do so.⁹⁹⁻¹⁰¹ The ε_r⁽⁰⁾ and ε_r^(∞) values in Table S5, show, first, that ε_ij⁽⁰⁾ and ε_ij^(∞) computed with a specific functional are largely unaffected changing the cutoff energy for a particular PAW potential. Second, when moving from Ti_std through Ti_pv to Ti_sv, there is a systematic increase in the magnitude of both ε_ij⁽⁰⁾ and ε_ij^(∞) for a specific functional and cutoff energy. Third, the GGA-based mean value of ε_ij⁽⁰⁾ is roughly two or three times smaller when computed with DFT+U (ε_ij⁽⁰⁾ in the range 44–69 (35–54) with PBE+U (PBEsol+U)), so there is a sharp disagreement between the calculated and experimental values of ε_r⁽⁰⁾. Similarly, the DFT+U-calculated difference between the parallel and perpendicular components of ε_ij⁽⁰⁾ is also unphysically small compared to the notable difference deduced from the corresponding experimental values. Fourth, the ε_ij^(∞) are almost oblivious with respect to the two DFT+U methods applied for any particular potential and cutoff energy and are close to the experimental range of reported values (6.3–7.4, see Table 7). The only feature that manifests with the combination between the Hubbard term and DFT is the disappearance of all soft (low-frequency IR-active) phonon modes, meaning that the combination does not predict the dynamic instability of the R-TiO₂ lattice predicted by Ti_pv and Ti_sv in combination with PBE (Table 6).

Tables 6 and S4 list the PBEsol- and PBE-calculated ε_r⁽⁰⁾, ε_r^(∞), and ε_r values for A- and B-TiO₂, respectively. For A-TiO₂, PBEsol/Ti_std shows the best prediction of ε_r (ε_r = 43.6 with PBEsol/Ti_std and 43.3 with experiment). This is compared to those predicted byPBEsol/Ti_pv and PBEsol/Ti_sv that have overestimated ε_r by 7.8 and 6.1, respectively. The corresponding overestimations by PBE were 10.5, 20.0, and 18.5 for Ti_std, Ti_pv, and Ti_sv, respectively. These values are indeed large, appearing due to numerically small frequencies associated with the low-frequency phonon modes. The lack of detailed experimental data for B-TiO₂ did not allow us to estimate the performance of PAW potentials and correlation methods in predicting the accuracy of ε_r values.

Our result suggests that the ε_r value for the R-TiO₂ lattice is larger than that of the A- and B-TiO₂ lattices (ε_r(R) < ε_r(B) < ε_r(A)). For mass density, ρ(A) < ρ(B) < ρ(R).

The ionic and optical dielectric constants for the two cubic phases of TiO₂ are listed in Table S6. The former could not be determined correctly and are unphysical, even though the large values of ε_r⁽⁰⁾ indicate the presence of any ferroelectric transition. The obvious reason for this is that the frequencies of the three low-frequency phonon modes were too small for F-TiO₂, and many of them were negative for P-TiO₂. The optical dielectric constant was close to 10.0–11.0 for both the systems. They are larger than those calculated for R-, A-, and B-TiO₂.

We found that ε_⊥⁽⁰⁾ < ε_||⁽⁰⁾ for R-TiO₂ with PBEsol; it is reversed (ε_⊥⁽⁰⁾ > ε_||⁽⁰⁾) for A-TiO₂. The anisotropic nature of ε_r⁽⁰⁾ is similar to that found for the electronic part of ε_r (i.e., ε_⊥^(∞) < ε_||^(∞) for R-TiO₂; ε_⊥^(∞) < ε_||^(∞) for A-TiO₂), showing that the polarization along the c-direction is strongest for the former than for the latter system. This may indicate that the excitonic wavefunctions are extended in one direction than the remaining two, which are expected of low-dimensional excitons.²¹ As mentioned by Mikami et al.,⁹⁰ Wemple¹⁰² has observed that the difference between the dielectric permittivity of the two TiO₂ phases was due to the smaller anion density of anatase TiO₂.

We have calculated the dielectric permittivity and phonon frequencies for R-TiO₂ using the relaxed lattices of HSEsol and HSE06, in which the frozen phonon approximation was invoked. The results are summarized in Tables S7 and S8. As observed with GGA (see above), the HSE results show that the magnitude of ε⁽⁰⁾ depends significantly on the type of the PAW potential employed. Overall, the ε⁽⁰⁾ value of [HSE06/Ti_std] agrees well with low-temperature experimental results discussed above.¹⁴ The ε^(∞) values are also in better agreement with experiment due to a systematic improvement of the band gap with the HSE methods.

3.4. Dependence of Low-Frequency Phonon Modes and Static Dielectric Constant on Hydrostatic Pressure

We have examined the isotropic pressure dependence of the geometry, zone-center phonon modes, and relative dielectric constant of R-TiO₂. Figure 7a–c shows the dependence of the harmonic frequencies of the lowest five Γ-center phonon modes on the applied pressure of R-TiO₂. Except for the degenerate IR-active mode E_u (ω = 375.0 cm^–1 with PBEsol/Ti_std) and the non-degenerate Raman-active mode B_1g (ω = 134.2 cm^–1 with PBEsol/Ti_std), the phonon frequencies of the other three IR-active bands with mode symmetry A_2u, B_1u, and E_u are affected by applied pressure. Figure 7 That is, ferroelectricity can be induced to the rutile lattice of TiO₂ via negative hydrostatic pressure, and the most sensitive ferroelectric mode is the TO mode A_2u. The other two modes (B_1u and E_u) soften as the pressure increases, which causes further expansion of the rutile lattice. The softness of the E_u mode, which is the analogue of the A_2u mode in the a–b plane, was previously observed.⁸⁷

Figure 7.

(Left) PBEsol level external (hydrostatic) pressure dependence of the lowest five Γ-centered phonon modes of R-TiO₂, obtained with PAW potentials: (a) Ti_std; (b) Ti_pv; (c) Ti_sv. (Right) The dependence of dielectric permittivity (ionic) on applied pressure, obtained PAW potentials: (d) Ti_std; (e) Ti_pv; and (f) Ti_sv. The labeling of the phonon modes is consistent with the data shown in Table 5.

The three PAW potentials give qualitatively similar insight into the development of the three ferroelectric modes, thus leading to the creativity of lattice instability. However, the application of the two soft potentials Ti_pv and Ti_sv confirms that the phonon frequency of the A_2u mode vanishes faster than that of the B_1u and E_u modes and becomes more negative when the lattice is expanded. That is, for the PAW potentials Ti_pv and Ti_sv, the frequency of the A_2u mode vanishes at hydrostatic pressure at about −1.0 GPa, while it is about −9.3 GPa with Ti_std. These results explain why the static dielectric constant increases dramatically and more rapidly with Ti_pv and Ti_sv compared to Ti_std, which may be captured from Figure 7d–f. Also, the anisotropy in the in- and out-of-plane dielectric constants predicted with the two soft potentials are quite significant than that predicted using Ti_std. The component of dielectric constant parallel to the c-axis is increased prominently compared to the in-plane dielectric constants for all the three occasions exploited, meaning that an enhancement of the dielectric constant can be made possible via the expansion of the rutile lattice of TiO₂.

4. Conclusions

This study has utilized three most commonly used variants of the PAW potential for Ti to examine the physical properties of TiO₂. In particular, the combined application of these pseudopotentials with PBE, PBEsol, and PBEsol+U has enabled us to assess the accuracy of the ground-state lattice properties, phonon vibrations, and relative static dielectric permittivity. We have shown that the change in the size of the potential can have a non-negligible effect on the temperature-dependent lattice geometry of all the five phases of TiO₂. Their effect is particularly pronounced when the low frequencies of optical phonon modes, as well as the ionic contribution to the relative dielectric permittivity, were compared. The origin of the underlying difference was uncovered when radial dependence of the charge density was analyzed around the close vicinity of Ti. We have argued that the radial dependence of charge density is relatively steep in the close vicinity of Ti when the standard potential Ti_std was invoked; it was shallow when the two soft potentials Ti_pv and Ti_sv were invoked. Because of this nature of the charge density profile, Ti_std has produced higher phonon frequencies (and relatively accurate ε_r⁽⁰⁾) than that predicted with Ti_pv and Ti_sv. Higher-level calculations using HSEsol and HSE06 have shown to have some improvement on the accuracy of the dielectric properties, despite being computationally intensive. In addition, we have shown that ferroelectric instability can be induced in the R-TiO₂ lattice by applying negative hydrostatic pressure. At least three low-lying phonon modes (A_2u, B_1u, and E_u) of R-TiO₂ are observed to be sensitive not only to the onset of ferroelectric phase transitions but also to the hard and soft nature of the three PAW potentials. Finally, we recommend using the standard PAW potential, Ti_std, for modeling the ground-state lattice properties and phonon-related features of TiO₂-based materials, rather than choosing a softer PAW potential.

Data Availability Statement

This research did not report any data.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.3c02038.

• Comparison of the IR and Raman frequencies ω (cm^–1) and IR intensities I (km/mol) obtained using PBEsol in combination with PAW potentials Ti_std, Ti_pv, and Ti_sv centered on Ti of R-TiO₂; comparison of the IR and Raman frequencies ω (cm^–1) and IR intensities I (km/mol) obtained using PBEsol in combination with PAW potentials Ti_std, Ti_pv, and Ti_sv centered on Ti of R-TiO₂; comparison of the IR and Raman frequencies and IR intensities obtained using PBEsol and PBE functionals, in conjunction with PAW potentials Ti_std, Ti_pv, and Ti_sv centered on Ti of B-TiO₂; comparison of the PBE level electronic and ionic components (ε_ij⁽⁰⁾ and ε_ij^(∞), respectively) of the relative dielectric permittivity tensor ε_ij for R- and A-TiO₂, computed in conjunction with three variants of the PAW potential, viz., Ti_std, Ti_pv, and Ti_sv; comparison of the PBE+U and PBEsol+U (U = 3.0 eV) level electronic and ionic components (ε_ij⁽⁰⁾ and ε_ij^(∞), respectively) of the relative dielectric permittivity tensor ε_r for R-TiO₂, computed in conjunction with three variants of the PAW potential, viz., Ti_std, Ti_pv, and Ti_sv; comparison of the PBEsol level electronic and ionic components (ε_ij⁽⁰⁾ and ε_ij^(∞), respectively) of the relative dielectric permittivity tensor ε_ij for F- and P-TiO₂, computed in conjunction with three variants of the PAW potential, viz., Ti_std, Ti_pv, and Ti_sv; ionic (low-frequency) and electronic (high-frequency) components (ε_ij⁽⁰⁾ and ε_ij^(∞), respectively) of the dielectric permittivity tensor ε_r of R-TiO₂, obtained using HSEsol and HSE06 in combination with the three PAW potentials Ti_std; and comparison of the IR and Raman frequencies ω (cm^–1) obtained using HSEsol and HSE06 in combination with PAW potentials Ti_std centered on Ti of R-TiO₂ (PDF)

Author Contributions

Conceptualization, P.R.V.; computation, data acquisition, formal analysis, and investigation, P.R.V.; computation of phonon modes of R-, A-, and B-TiO₂: R.A.; computational verifications and critical discussions, V.A.D. and Y.M.; supervision, R.A.; drawing of figures, preparation of tables, and writing—original draft, P.R.V.; writing—review and editing, P.R.V. and R.A.; All authors have read and agreed to the published version of the manuscript.

This research was funded by JSPS Grant-in-Aid for Transformative Research Areas (A); Grant numbers: 21H05560 and 20H05883.

The authors declare no competing financial interest.