Pyroelectric Effect in Tetragonal Ferroelectrics BaTiO 3 and KNbO 3 Studied with Density Functional Theory

: We present a computational methodology for investigating the pyroelectric properties of anharmonic crystalline ferroelectrics. The description of phonon properties and pyro-electricity in perovskite ferroelectrics such as BaTiO 3 and KNbO 3 requires phonon anharmonicity to be taken into account. Our computational approach is based on density functional theory calculations. We use self-consistent phonon theory to describe phonon anharmonicity and to obtain finite-temperature atomic displacements. The Berry phase approach is used to obtain spontaneous polarization needed for the prediction of the primary pyroelectric coefficient. We calculate the fixed-volume primary pyroelectric coefficient and piezoelectric strain secondary pyro-electric coefficient for the tetragonal polymorphs of BaTiO 3 and KNbO 3 . We also report the anharmonic phonon dispersions for the studied systems. The computationally relatively straightforward and cost-efficient approach offers a way of screening the pyroelectric coefficients of novel pyroelectric materials for energy harvesting and other microelectronic applications.


■ INTRODUCTION
Pyroelectricity is a property intrinsic to polar materials, which can be observed as an ability to generate a temporary voltage when undergoing temperature fluctuations.Pyroelectricity can be quantified with the pyroelectric coefficient p that represents the ratio of change in spontaneous polarization P and change of temperature T, p = ΔP/ΔT. 1,2Pyroelectricity consists of primary p (1) and secondary p (2) effects.The primary pyroelectric effect results from the change of spontaneous polarization caused by a temperature change with the volume and shape of the crystal being fixed.The secondary contribution to pyroelectricity arises from the thermal deformation of the unit cell. 2 It is in fact a piezoelectric contribution and does not involve spontaneous polarization. 3yroelectricity is a property exhibited by all ferroelectric materials, and typically, ferroelectric pyroelectrics such as perovskite-type BaTiO 3 also exhibit the largest values of the pyroelectric coefficient.Pyroelectric materials find use in a number of devices such as sensors and detectors 3,4 and in potential novel applications, such as disinfection, 5 catalysis, 6 and energy harvesting. 7−12 Experimental approaches for the measurement of pyroelectric coefficients in bulk materials and thin films are readily available, 13,14 but considering the large number of polar materials reported in the crystallographic databases, pyro-electricity has been measured only for a limited number of them.A computationally efficient and robust scheme for the prediction of pyroelectric coefficients could be utilized in the high-throughput screening of the pyroelectric properties of crystalline materials.Many pyroelectrics are perovskite-type solid solutions, 7 and a computational scheme that can be used for such materials would be particularly useful.−18 However, due to the challenging temperature dependence of the pyroelectric effect, first-principles modeling of pyroelectricity itself has so far received relatively little attention.To the best of our knowledge, the first atomic-level modeling study on pyroelectricity was done by Peng and Cohen, 19 who used a first-principles-based shell model potential along with molecular dynamics (MD) for LiNbO 3 .A more recent example of employing ab initio MD for the study of pyroelectricity has been reported by Ponomareva et al. for pyroelectric response and phase transitions in PbTiO 3 . 20achine-learned force fields combined with MD simulations have likewise been used to study piezoelectricity and pyroelectricity in ZrO 2 by Kersch et al. 21The converse effect of the pyroelectric effect and electrocaloric effect has also been studied with effective Hamiltonians combined with MD simulations in lead zirconate titanate Pb(Zr 1−x Ti x )O 3 (PZT), 22,23 barium zirconate titanate Ba(Zr 1−x Ti x )O 3 (BZT), 24 and barium titanate BaTiO 3 , 25−27 for example.
Density functional theory (DFT) has been employed in the modeling of pyroelectricity by Liu et al. for different systems, such as nonferroelectric GaN and ZnO, 28 2D materials such as GeS, 29 and ferroelectric HfO 2 . 30This approach is based on the Born−Szigeti theory of pyroelectricity and combines phonon thermal expansion and a thermal shift with the investigation of spontaneous polarization.Symmetry considerations reduce the number of phonon modes to be considered.The primary coefficient is then obtained as a sum of the active phonon modes.Recently, the pyroelectricity in GaN and ZnO has also been investigated with full finite-temperature optimization of the unit cell and atomic coordinates by Tadano et al. 31 This approach within the quasi-harmonic approximation (QHA) makes use of updating interatomic force constants with the change of the crystal structure.Pyroelectricity itself is calculated from the temperature derivative of spontaneous polarization obtained from Born effective charges.The secondary pyroelectric contribution arising from the piezoelectric effect is also included.
Calculation of finite-temperature properties of materials requires the understanding of their phonon properties. 32As ferroelectric materials are often anharmonic, the calculations become more complex, as the second-order force constants obtained through a typical finite-displacement supercell approach are not sufficient to describe the phonon anharmonicity.Calculation of higher order force constants becomes computationally increasingly expensive with this approach due to the much larger number of supercells needed.Instead, other sampling methods such as fitting forces from randomly displaced supercells or molecular dynamics simulations can be used to obtain higher order force constants.Methods to study the phonon anharmonicity and finitetemperature thermodynamics include the temperature-dependent effective potential (TDEP) method, 33 phonon gas model (PGM) with the quasiparticle frequency approach, 34,35 iterative self-consistent phonon calculations (SCPH), 36−38 and related methods such as self-consistent ab initio lattice dynamics (SCAILD) 39,40 and stochastic self-consistent harmonic approximation (SSCHA). 41,42−46 The anharmonic lattice dynamics of niobate perovskites KNbO 3 and NaNbO 3 have also been investigated using SCPH methods. 47erovskite-structured oxides are a large group of compounds exhibiting many functionalities along with diverse polymorphism of crystal structures and associated phase transitions.Discovery and development of more effective lead-free ferroelectric and pyroelectric alternatives to PZT or lead magnesium niobate Pb(Mg 1/3 Nb 2/3 )O 3 (PMN) have been one of the focus areas in perovskite-related research for past decades. 48Many of these proposed alternatives make use of derivatives of potassium sodium niobate (K x Na 1−x )NbO 3 (KNN).
To the best of our knowledge, no DFT-based lattice dynamics approach for modeling pyroelectricity in ferroelectric perovskites has been presented previously.Here, we present a computational methodology to calculate the primary pyroelectric coefficients for ferroelectric perovskites.As model systems, we use two well-known tetragonal ferroelectrics, BaTiO 3 and KNbO 3 (space group P4mm, Figure 1).The phase diagrams of the two compounds are relatively similar, with ferroelectric Curie temperatures of 396 49 and 708 K 50 for BaTiO 3 and KNbO 3 , respectively.At room temperature, BaTiO 3 is tetragonal and KNbO 3 is orthorhombic.−55 We report the anharmonic finite-temperature phonon dispersions for tetragonal BaTiO 3 and KNbO 3 obtained with DFT and use the finite-temperature phonon properties to derive atomic displacements and pyroelectric properties at finite temperatures.
■ METHODS Self-Consistent Phonon Calculations.Quantum chemical calculations have been carried out using the CRYSTAL23 program package 57,58 and hybrid PBE0 density functional method (DFT-PBE0, 25% exact exchange). 59,60The following localized Gaussian-type basis sets, derived from Karlsruhe def2 basis sets, were used: triple-ζ-valence + polarization (TZVP) for Ti, Nb, and O and split-valence + polarization for Ba and K. 61−63 Tight tolerance factors (TOLINTEG) of 8, 8, 8, 8, and 16 were used for the evaluation of the Coulomb and exchange integrals.Default CRYSTAL23 structural optimization convergence criteria and DFT integration grids were used.Spontaneous polarization was calculated with the Berry phase approach implemented in CRYSTAL (SPOLB keyword). 64,65The following Monkhorst−Pack-type k-meshes 66 were used in the DFT calculations for both BaTiO 3 and KNbO 3 : 12 × 12 × 12 for structure optimization, 3 × 3 × 3 for force calculations of 3 × 3 × 3 supercells, 3 × 3 × 3 for Berry phase calculations with 2 × 2 × 2 supercells, and 8 × 8 × 8 for The Journal of Physical Chemistry C the quasi-harmonic approximation calculations with 2 × 2 × 2 supercells.In the case of the Berry phase calculations, we tested the convergence of the spontaneous polarization with two different approaches: using a denser 4 × 4 × 4 k-mesh or a larger 3 × 3 × 3 supercell led to negligible differences compared to the 3 × 3 × 3 k-mesh and a 2 × 2 × 2 supercell.SCPH calculations 32 were carried out with the ALAMODE code (version 1.5.0). 37,38In order to perform calculations at the DFT-PBE0/TZVP level of theory, we have developed a CRYSTAL23 interface for ALAMODE.In the approach implemented in ALAMODE for the calculation of finitetemperature phonon properties, a number of randomly displaced or MD-based structures are first used for fitting force constants up to the quartic or higher order.These force constants are then used in the SCPH procedure, which calculates the temperature-dependent phonon frequencies and eigenvectors.The force calculations for the SCPH procedure were carried out with 3 × 3 × 3 supercells.40 supercells with randomly displaced atoms (displacement magnitude of 0.06 Å, 135 atoms in the cell) were used.Choice of this number of displacement force datasets was based on previous ALAMODE SCPH study on the SrTiO 3 perovskite. 38We also tested the impact of using a larger number of supercells ( 60), but we observed that the choice of the supercell size and displacement magnitude contributed more to the convergence of the SCPH procedure than the larger displacement force dataset size.Observed fitting errors of the quartic force constants (1.67% for BaTiO 3 and 0.62% for KNbO 3 ) are comparable to previously reported values (for example, 0.77 and 2.19% for GaN and ZnO, respectively 31 ).When calculating the forces on atoms for supercells, a tight SCF convergence criterion (TOLDEE keyword) of 10 −10 au was applied.The SCPH calculations were carried out on an 8 × 8 × 8 reciprocal space k-mesh, using a 4 × 4 × 4 interpolating mesh.Default ordinary least-squares model was used for fitting the force constants from displacement force datasets.
The results from the SCPH calculations employing force constants up to quartic order were used to plot the finitetemperature phonon dispersions and to obtain the temperature correction to second-order force constants for the calculation of the primary pyroelectric coefficient with the Berry phase approach (see below).LO-TO splitting of the phonon modes when approaching the Γ point has not been included in the phonon dispersion relations as nonanalytic correction.Inclusion of nonanalytic correction in the SCPH procedure has not been found to affect finite-temperature properties in a study of phase transitions in BaTiO 3 . 46In addition, the inclusion of nonanalytic correction in SCPH led to unphysical behavior of lowest-frequency phonon modes.LO-TO splitting has similarly been omitted in a previous study of pyroelectricity in GaN and ZnO. 31 We have however included LO-TO splitting with nonanalytic correction for the harmonic frequencies obtained at the Γ point with the approach implemented in CRYSTAL 67−69 and report them in the Supporting Information (Tables S5 and S6) along with experimental and previously reported theoretical values.Example ALM (for fitting cubic and quartic force constants) and ANPHON (for SCPH calculations) inputs are given in the Supporting Information.The reciprocal space path used in plotting the phonon dispersion relations is also included in the ANPHON input example given in the Supporting Information.
Primary Pyroelectric Coefficient.Pyroelectricity is a tensor property, as the spontaneous polarization can occur along the three crystallographic axes.In the point group 4mm, only the coefficient in the polar z direction is nonzero, 2 and the primary pyroelectric coefficient p z (1) is labeled here as p (1) .In terms of irreducible representations (irreps), the 12 optical phonon modes at the Γ point are decomposed to three A 1 modes, one B 1 mode, and four doubly degenerate E modes. 70he A 1 irrep includes the linear function z, and A 1 modes correspond to atomic displacements in the direction of the polar c axis (Cartesian z direction).This was also confirmed by a visual inspection of the phonon modes belonging to this irrep.Illustrations of the three optical A 1 phonon modes are included in the Supporting Information (Figure S5).The primary pyroelectric coefficient was calculated for the three A 1 modes (mode indices 6, 12, and 15) of tetragonal BaTiO 3 and KNbO 3 .Analogously, the contributions of A 1 phonon modes to spontaneous polarization and pyroelectricy have been previously considered for GaN and ZnO (point group 6mm). 28The evaluation of spontaneous polarization contributions of all phonon modes (Figure S3) and contributions at various phonon wavevectors q (Figure S4) further illustrates that only the A 1 modes at the Γ point (q = 0) are necessary for evaluating primary pyroelectric contributions in BaTiO 3 .For this evaluation, Quantum Espresso 71 and plane wave basis (DFT-PBE/USPP) 72 were used, the general mode-specific behavior being consistent with results obtained with CRYSTAL (DFT-PBE0).
A schematic of the SCPH-based calculation of the primary pyroelectric coefficient p (1) is presented in Figure 2. The SCPH calculations were carried out using 40 random structures with 0.06 Å maximum displacement generated with tools implemented in ALAMODE.Temperature-corrected interatomic force constants from the SCPH calculation were used to calculate finite-temperature phonon eigenvectors.From them, finite-temperature displacements were obtained by modespecific displacement of atoms in a 2 × 2 × 2 supercell.The displacements were created with a modified version of the

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approach producing displacements in a normal coordinate Q basis implemented in the ALAMODE package.In our modification, the displacements are not drawn randomly from a normal distribution.Instead of randomized values over all phonon modes and phonon wavevectors q, modespecific displacements are carried out at the Γ point only.The magnitude of the displacement is Q 2 < > , calculated from the phonon frequency ω and the input temperature.When creating displaced structures from the phonon eigenvectors, the direction of the displacement was kept the same for each A 1 mode throughout the temperature range.Displacements are given in the Supporting Information (Tables S1 and S2 and Figures S1 and S2).
The primary pyroelectric coefficient p (1) at temperature T was obtained by calculating the numerical derivative of the spontaneous polarization P with respect to temperature using the central difference method p (1) = (P Td 2 − P Td 1 )/(T 2 − T 1 ), where T 2 = T + 20 K and T 1 = T − 20 K.The SPOLB approach in CRYSTAL allows us to obtain spontaneous polarization by calculating the polarization difference between two structures with the Berry phase approach.Typically, the spontaneous polarization of a ferroelectric material is obtained by calculating the polarization difference between the polar noncentrosymmetric structure and nonpolar centrosymmetric reference state.Here, we calculate P Td 2 − P Td 1 directly as the polarization difference between two polar noncentrosymmetric structures (supercells with finite-temperature displacements).We have confirmed that the resulting values are identical to values obtained as the difference of spontaneous polarization of structures at T 1 and T 2 .The mode-specific contributions to the primary pyroelectric coefficient were summed together to obtain the total primary pyroelectric coefficient p (1) .
It should be stressed that the methodology outlined here does not take into account thermal expansion of the unit cell, unlike, for example, in the work of Tadano et al. for GaN and ZnO. 31 In addition, a secondary piezoelectric contribution to the pyroelectric effect has to be taken into account separately (see below).A tertiary pyroelectric effect based on temperature gradients is also known, but obtaining it from atomic-level DFT calculations would be rather complicated, and the tertiary effect is neglected here.Finally, in ferroelectrics, the ferroelectric domain structure plays a key role, and the temperature dependence of the domain structure affects the pyroelectric response. 14econdary Pyroelectric Coefficient.The secondary pyroelectric coefficient can be calculated from the piezoelectric coefficients d (unit: CN), second-order elastic constants c (N/ m 2 ), and thermal expansion coefficients α (1/K), all of which are tensor quantities.The equation for a crystal in the ∞mm point group is given as p (2) = 2d 31 (c 11 α 1 + c 12 α 1 + c 13 α 3 ) + d 33 (2c 13 α 1 + c 33 α 3 ). 73−77 The piezoelectric and second-order elastic tensors at 0 K can be obtained with the ELAPIEZO keyword implemented in CRYSTAL. 65,78While the approach does not take the temperature dependence of d and c into account, the obtained values are nevertheless in reasonable agreement with the values derived from the experimental measurements (see below).In QHA calculations, we excluded the imaginary phonon modes of the ferroelectric crystal structures even though the effect of their inclusion would be negligible (ca.1% for BaTiO 3 at 300 K).
Experimental measurement of the primary and secondary pyroelectric effects separately is not typically carried out, but state-of-the-art experimental approaches allow the quantification of different pyroelectric contributions. 14RESULTS Barium Titanate.The optimized lattice parameters for BaTiO 3 (P4mm) are a = 3.969 Å and c = 4.092 at 0 K, giving a c/a ratio of 1.031.The experimental c/a ratio is 1.009 at 300 K, 79 and a value of 1.022 has been previously reported from DFT-PBEsol calculations. 45The value indicates slight supertetragonality in the geometry, which has been previously shown to occur for BaTiO 3 and PbTiO 3 with DFT-GGA. 80he anharmonic, temperature-dependent phonon dispersion relations obtained with SCPH at 300 and 600 K, together with the uncorrected, harmonic dispersion relations, are presented in Figure 3.The phonon dispersion relations illustrate how the imaginary phonon modes disappear, and the ferroelectric tetragonal structure becomes dynamically stable when anharmonic corrections are taken into account with SCPH.The energy ordering of the A 1 modes at the Γ point changes at 500 K, with the A 1 mode 6 becoming the mode 4, just above the acoustic modes.This illustrates the necessity of checking displacements from the phonon eigenvectors to correctly identify each mode in the finite-temperature displacements.Small numerical inaccuracies with the 3 × 3 × 3 supercells in CRYSTAL persist, resulting in small imaginary acoustic mode Figure 3. Top: harmonic (gray, dotted) and 300 K anharmonic (red, solid) phonon dispersion relations for tetragonal BaTiO 3 .Bottom: anharmonic 300 K (red, solid) and 600 K (blue, dashed) phonon dispersion relations for tetragonal BaTiO 3 .

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between Γ and Z points (ca.10i cm −1 ).However, this does not affect the optical A 1 modes that are responsible for pyroelectricity.We have included a fully converged finitetemperature phonon dispersion using 4 × 4 × 4 supercells in Quantum Espresso (DFT-PBE/USPP) and a harmonic dispersion with CRYSTAL using 2 × 2 × 2 in the Supporting Information (Figures S6 and S7).
Within the temperature range where the tetragonal polymorph exists, the total pyroelectric coefficient of BaTiO 3 calculated with DFT-PBE0 is −270 μCm −2 K −1 at 300 K, − 249 μCm −2 K −1 at 350 K, and −238 μCm −2 K −1 at 400 K (Figure 4).Measurements of the pyroelectric effect in a 95 weight-% BaTiO 3 -5 weight-% CaTiO 3 ceramic have been carried out by Lang et al. 81 in a temperature range from 4.9 to 400 K, through the orthorhombic−tetragonal and tetragonal− cubic phase transitions.More recently, the pyroelectric effect in a tetragonal thin-film sample of BaTiO 3 has been investigated by Adkins et al. 82 The experimental values of the pyroelectric coefficient are included in Mode contributions to the primary pyroelectric coefficient from the three A 1 modes (mode indices 6, 12, and 15 at 300 K) are presented in Figure 5.The relative contribution of the modes 6 and 12 changes at the orthorhombic−tetragonal phase transition: the contribution from mode 12 increases as .Calculated primary pyroelectric coefficient p (1) , total pyroelectric coefficient p (1) + p (2) , and experimentally determined 81,82 pyroelectric coefficients for BaTiO 3 .The temperature range between the vertical lines represents the range inside which the tetragonal polymorph exists.

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the temperature increases, while the contribution from mode 6 decreases with increasing temperature.
In the case of the secondary pyroelectric coefficient p (2) , our DFT-PBE0 values for BaTiO 3 are 72 μCm −2 K −1 at 300 K and 78 μCm −2 K −1 at 400 K, which compare fairly well with 80 μCm −2 K −1 derived from the room-temperature experimental values reported. 73otassium Niobate.The optimized lattice parameters for KNbO 3 (P4mm) are a = 3.989 Å and c = 4.114 Å, giving a c/a ratio of 1.031.The experimental c/a ratio is 1.017 at 550 K. 83 The anharmonic, temperature-dependent phonon dispersion relations obtained with SCPH at 300 and 600 K, together with the uncorrected, harmonic dispersion relations, are presented in Figure 6.Similar to BaTiO 3 , the imaginary phonon modes disappear, and the ferroelectric tetragonal structure becomes dynamically stable when the anharmonic corrections are taken into account with SCPH.
Within the temperature range where the tetragonal polymorph exists, the total pyroelectric coefficient of KNbO 3 calculated with DFT-PBE0 is −122 μCm −2 K −1 at 500 K, − 97 μCm −2 K −1 at 600 K, and −84 μCm −2 K −1 at 700 K (Figure 7).As pristine tetragonal KNbO 3 is not the thermodynamically stable phase at room temperature, experimental data on the pyroelectric effect are scarce.Qiu et al. have recently used phenomenological Landau−Devonshire theory to investigate the γ-phase of a (001)-oriented KNbO 3 film. 84Their model shows reasonable agreement with experimental data on remanent polarization and piezoelectric coefficient.For the pyroelectric coefficient, they predict a value of −140 μCm −2 K −1 at zero misfit strain, T = 298 K, and 100 kV/cm external electric field.The total pyroelectric coefficient predicted by Qiu et al. agrees well with our DFT-PBE0 prediction (Figure 7), but we note that our calculations have been carried out for the tetragonal phase, which is not the thermodynamically stable phase at room temperature.The absolute magnitude of our predicted total pyroelectric coefficient for tetragonal Figure 6.Top: harmonic (gray, dotted) and 300 K anharmonic (red, solid) phonon dispersion relations of tetragonal KNbO 3 .Bottom: anharmonic 300 K (red, solid) and 600 K (blue, dashed) phonon dispersion relations of tetragonal KNbO 3 .
Figure 7. Calculated primary pyroelectric coefficient p (1) , total pyroelectric coefficient p (1) + p (2) , and Landau−Devonshire theory-based theoretical pyroelectric coefficient 84 for KNbO 3 .The temperature range between the vertical lines represents the range inside which the tetragonal polymorph exists.

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KNbO 3 is larger than the 93 μCm −2 K −1 measured at 300 K for a ceramic KNbO 3 sample. 85ode contributions to the primary pyroelectric coefficient from the three A 1 modes (mode indices 6, 12, and 15) are presented in Figure 8. Unlike for BaTiO 3 , the relative contributions of the modes stay rather uniform with little variation with increasing temperature.Despite the similar crystal structures and phonon dispersion relations, tetragonal BaTiO 3 and KNbO 3 thus show different pyroelectric responses at the level of individual phonon modes.

■ CONCLUSIONS
We have demonstrated a DFT-based lattice dynamics approach that enables, for the first time, the determination of primary pyroelectric coefficients for anharmonic crystalline ferroelectrics such as BaTiO 3 and KNbO 3 .We take phonon anharmonicity into account by combining the density functional theory and self-consistent phonon theory.Finitetemperature atomic displacements obtained from the selfconsistent phonon calculations and spontaneous polarizations obtained from the Berry phase approach are used to calculate the primary pyroelectric coefficient with numerical differentiation.We have used the DFT-PBE0 hybrid density functional method and the outlined computational approach to calculate the primary pyroelectric coefficient of two model systems, tetragonal ferroelectrics BaTiO 3 and KNbO 3 .Secondary pyroelectric coefficients have also been evaluated within the quasi-harmonic approximation.Overall, the absolute magnitude of the predicted total pyroelectric coefficient of tetragonal KNbO 3 is smaller than that obtained for tetragonal BaTiO 3 .For tetragonal BaTiO 3 , the calculated values are in line with experimental data, while for tetragonal KNbO 3 , there are not yet enough data on the pyroelectric properties.The computational methodology demonstrated here for primitive tetragonal crystal structures offers a relatively straightforward and cost-efficient approach to predict the primary pyroelectric coefficient of ferroelectric materials.Extension of the methodology to other crystal systems besides the tetragonal crystal system is also computationally feasible.Most importantly, the supercell-based approach presented here can be extended to perovskite-type solid solution materials such as lead-free potassium sodium niobate (KNN) perovskites.Atomic-level understanding of primary pyroelectricity in such compounds is particularly interesting because the most efficient known pyroelectrics are perovskite-type solid solutions.
Finite-temperature atomic displacements of BaTiO 3 and KNbO 3 ; mode-specific contributions to spontaneous polarization of BaTiO 3 ; mode-specific of contributions to spontaneous polarization of BaTiO 3 at different phonon wavevectors; computational details of supporting Quantum Espresso calculations; example inputs of the ALAMODE ALM and ANPHON SCPH calculations; tabulated values of the primary, secondary, and total pyroelectric coefficients; visualizations of optical A 1 phonon modes; additional phonon dispersion relations for BaTiO 3 ; and comparison of phonon frequencies of BaTiO 3 and KNbO 3 with the previous literature; modified ALAMODE displace.pyand GenDisplacement.pyscripts are available at https://github.com/aalto-imm/pyro-displacements (PDF) ■ AUTHOR INFORMATION Corresponding Author Antti J. Karttunen − Department of Chemistry and Materials Science, Aalto University, FI-00076 Aalto, Finland; orcid.org/0000-0003-4187-5447;Email: antti.karttunen@aalto.fi

Figure 1 .
Figure 1.Crystal structure of tetragonal BaTiO 3 (P4mm).Ba atoms in green, Ti atom and its coordination polyhedron in blue and O atoms in red.The polar c axis is responsible for spontaneous polarization and the pyroelectricity of BaTiO 3 .Tetragonal KNbO 3 is isostructural, with K replacing Ba and Nb replacing Ti.Illustration made with VESTA software.56

Figure 2 .
Figure 2. Schematic illustration of the procedure used here for calculating the primary pyroelectric coefficient from the supercells with mode-specific atomic displacements obtained with finitetemperature phonon dispersion relations.

Figure 4 .
Close to room temperature, the values from Lang et al. vary between −180 and −230 μCm −2 K −1 , 81 while the value from Adkins et al. is 270 ± 90 μCm −2 K −1 . 82The calculated total pyroelectric coefficient is closer to the value reported by Adkins et al., but the error bar in the experimental value is relatively large.The sharp decrease in the experimental data of Lang et al. around 380 K is due to the pyroelectric coefficient sharply changing near the tetragonal−cubic phase transition.

Figure 5 .
Figure 5. Mode contributions to the primary pyroelectric coefficient (p (1) ) of BaTiO 3 .Mode indices are based on the energy ordering at 300 K.The temperature range between the vertical lines represents the range inside which the tetragonal polymorph exists.